Verbalized Machine Learning: Revisiting Machine Learning with Language Models
Authors: Tim Z. Xiao
Robert Bamler
Bernhard Schölkopf
Weiyang Liu
Venue: Accepted by TMLR ()

Abstract: Motivated by the progress of large language models (LLMs), we introduce the framework of verbalized machine learning (VML). In contrast to conventional machine learning (ML) models that are typically optimized over a continuous parameter space, VML constrains the parameter space to be human-interpretable natural language. Such a constraint leads to a new perspective of function approximation, where an LLM with a text prompt can be viewed as a function parameterized by the text prompt. Guided by this perspective, we revisit classical ML problems, such as regression and classification, and find that these problems can be solved by an LLM-parameterized learner and optimizer. The major advantages of VML include (1) easy encoding of inductive bias: prior knowledge about the problem and hypothesis class can be encoded in natural language and fed into the LLM-parameterized learner; (2) automatic model class selection: the optimizer can automatically select a model class based on data and verbalized prior knowledge, and it can update the model class during training; and (3) interpretable learner updates: the LLM-parameterized optimizer can provide explanations for why an update is performed. We empirically verify the effectiveness of VML, and hope that VML can serve as a stepping stone to stronger interpretability.

Published in Transactions on Machine Learning Research (01/2025)
Verbalized Machine Learning:
Revisiting Machine Learning with Language Models
Tim Z. Xiao
shenchong..zooo@@uiii- www.toonouuuuuuuuueeeeeed.dddd
Max Planck Institute for Intelligent Systems, Tübingen 8 University of Tübingen
Robert Bamler
robert. Snterttbemmeerr@uuudttueennnn.ee
University of Tübingen
Bernhard Schölkopf
Maz Planck Institute for Intelligent Systems, Tübingen
Weiyang Liu'
Mas. Planck Institute for Intelligent Systems, Tübingen University of Cambridge
Reviewed on OpenReview: https: // openreview. net/ forum? id= K3A96RuJE9
Abstract
Motivated by the progress of large language models (LLMs), we introduce the framework of verbalized machine learning (VML). In contrast to conventional machine learning (ML) models that are typically optimized over a contimuous parameter space, VMI. constrains the natural language. Such a constraint leads to a
parameter space to be human-interpretalle new perspective of function approximation, where an LLM with a text prompt can be viewed as function parameterized by the text prompt. Guided by this perspective, classical ML problems, such regression and classification, and find that these problems can be solved by an LLM-parametterized learner and optimizer. The major ad advantages of VML include (1) easy encoding of inductive bias: prior knowledge about the problem and hypothesis class can be encoded natural language and fed into the L.M-paaameteerized learner; (2) automatic model class selection: the optimizer can automatically select a model class based on data and verbalized prior knowledge, and can update the model class during training; and (3) interpretable learner updates: the puM-paramtterized optimizer can provide explanations for why an update is performed. We empirically verify the effectiveness of VML, and hope that VML can serve slepping stone to stronger interpretability.
revisit
"The limits of my language mean the limits of my world."
Ludwig Wittgenstein
1 Introduction
The unprecedented success of large language models (LLMs) has changed the way people solve new problems in machine learning. Compared to conventional end-to-end training where neural network is trained from scratch some curated dataset, it has become increasingly more popular to leverage a pretrained LLM and design good prompis that contain in-conlext examples and elfective instructions. These two ways of problem-solving lead to an intriguing comparison. Traditionally, we would optimize a neural network in a continuous numerical space using gradient descent, while in the new approach, we optimize the input prompt of an LLM in a discrele natural language space. Since neural network is effectively function parameterized by its numerical weights, can pretrained L.LM act as a function parameterized by its text prompt?
Driven by this question, conceptualize the framework of verbalized machine learning (VML), which uses natural language as the representation of the model parameter space. The core idea behind VML is that
*Corresponding author

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Published in Transactions on Machine Learning Research (01/2025)
can define machine learning model using natural language, and the training of such model is based on the ilerative update of natural language. This framework enables many new possibilities for interpretability, as the decision rules and patterns learned from data are stored and sunmarized in natural language. Specifically, we propose to view the input text prompt of LLMs as the model paramcters that are being learned. However, oplimization over such a natural language parameter space also introduces additional difficuluies. Inspired by previous work [5, 25] where the optimizer is viewed as a function parameterized by a neural network, we parameterize the optimizer function as another LLM, which produces the next-step model parameters by taking in the current model parameters, batch of training data points, and the loss function. Therefore, VML requires the optimizer LLM to update the learner LLM iteratively towards the training objective.
Compared to conventional numerical machine learning, the VML framework brings a few unique advantages. First, VML introduces an easy and unified way to encode inductive bias into the model. Because the model parameters are fully characterized by human-interpretable natural language, one can casily enter the inductive bias using language. This linguistic parametcrization makes machine learning modcls fully interpretable and adjustable. For example, if the input and output data are observed to be linearly correlated, then one can use this sentence part of text prompt. How to effectively encode inductive bias is actually longstanding problem in machine learning, and VML provides unified way to inject the inductive bias through natural language-just like teaching a human learner. Second, VML performs automatic model selection during the learning process. The optimizer LLM can automatically select a suitable model class based on the training data and verbalized prior knowledge. Third, each update of the model is fully interpretable in the sense that the optimizer LLM can give an explanation of why chooses such an update. One can even interact with the optimizer LLM in order to inject new prior knowledge or obtain detailed reasoning.
VML can be viewed as natural generalization of in-context learning (ICL). Specifically, ICL is single-step implicit learning process, while VML is a multi-step iterative learning process where the in-context examples are summarized into verbal pattern and knowledge. Moreover, VML offers a sequential (or conditional) way for scaling inferenco-time compute [7, 48]. Compared to the best-of-N re-sampling, VML itcratively updates its model parameter prompt by laking into account the learner's past predictions.
An important concept of VML is its unified token-level representation of both data and model. Unlike numerical machine learning, language models in VML do not differentiate data and model, and treat both of them as part of the text prompt. This shares striking connection to stored-program computers, also known as the von Neumann architecture, where the key idea is to represent programs as data rather than wiring setups. The link between language models and stored-program computers underscores the importance of text prompts, which play a similar role to computer programs, and, along with LLMs, can become a powerful zero-shot problem solver. Our contributions are as follows:
We formulate the framework of verbalized machine learning, where pretrained language models are viewed function approximators parameterized by their text prompts. Then, we revisit few simple machine learning problems and show that VML is able to solve them.
We design a concrete VML algorithm with text prompt template. This algorithm parameterizes both the learner model and the optimizer as LLMs, and enables the iterative verbalized training.
We conduct empirical studies for the injection of verbalized inductive bias and show that it is promising to use natural language unified way to encode prior knowledge. Moreover, we validate the elfectiveness of VML in different applications (Section 4, Appendix A, E,F.G.H).
2 Related Work
LLMs for planning and optimization. Language models are used to perform planning for embodied agents [49, 60, 26, 28]. such that they can follow natural language instruction to complete complex tasks. More recently, LLMs have been used to solve optimization problems [63]. Specifically, the LLM generates a new solution to an optimization problem from a prompt that contains previously generated solutions and their loss values. The LLM optimizer [63] shares a high-level similarity to our work, as we also aim to solve an optimization problem with LLMs. The key difference to [63] is our function approximation view of LLMs, which enables us to revisit classical machine learning problems and solve them in the VML framework.
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Natural language to facilitate learning. [45, 23, 24, 37, 71] show that natural language captions serve effective supervision to learn transferable visual representation. [35, 38, 40, 34, 66, 61] find that natural language descriptions can easily be turned into zero-shot classification criteria for images. [4] proposes to use natural language as latent parameters to characterize different tasks in few-shot learning. In contrast, VML uses the text prompt of LLMs to parameterize functions and learns this prompt in a data-driven fashion.
Prompt engineering and optimization. There are many prompting methods [56, 72, 73, 54, 67, 68, 58) designed to clicit the reasoning ability of LLMs. To reduce the efforts in designing good prompts, prompt optimization [72, 73, 63, 41, 57, 11, 27, 32, 50, 70] has been proposed. VML can be viewed as a special instance of prompt optimization, but unlike many current generic prompt optimization methods that search the optimal text prompts through best-of-N sampling without reasoning, VML updates its text-based parameters by explicitly reasoning about the incorrect predictions and learning the underlying data pattern based on the reasoning outcome, which ensures that the learner VML remains fully interpretable. To summarize, the difference between VML and current generic prompt optimization (e.g., [73]) is similar to the difference between grudienl-based and gradienl-free optimization. Another subtle difference that separates the two is that the goal of VML (like classical machine learning) is to learn model to recognize generalizable pattern in given training set, while the goal of current prompt optimization (like classical optimization) is more general, which is to simply optimize an objective function (without the need to learn the underlying pattern, e.g., [41]). We can phrase the goal of a machine learning problem into an optimization objective, and use the tools from optimization to solve it, but the focus of the two areas are fundamentally different. We compare the difference of the two in the experiments (see Section 4.8 and Appendix D). We observe that current prompt optimization often results in a generic instruction rather than a description of the data pattern.
LLMs for multi-agent systems. Due to the strong instruction ability, LLMs are capable of playing different roles in multi-agent systems. [42, 59, 15, 22] study multi-agent collaboration system for solving complex tasks like software development. VML can also be viewed as a two-agent system where one LLM plays the learner role and the other LLM plays the optimizer role.
3
Verbalized Machine Learning
Data x
Model
Output y
Numerical space
3.1 From Numerical to Verbalized Machine Learning
(a) Numerical Machine Learning
Classical machine learning models (e.g., neural networks) are typically trained within a continuous numerical parameter space. Once trained, these models are stored as collection of numerical values that are not interpretable and remain a black box. Motivated by the strong universal problem-solving capability of LLMs, we find it appealing to view an L.L as a function approximator that is parameterized by its own text prompt. This perspective leads to our VML framework. Similar to general-purpose modern computer whose functionality is defined by its running program, function that is defined by an LLM is characterized by its text prompt.
Text prompt
Data x
Text output
LLM
Output
Model 0
Natural langinge
(b) Verbulized Machine Learning
Figure 1: A comparison between nuner- ical machine learning and VML.
Due to the fully human-inkerpretable text prompt, the VML framework provides strong interprelability for its learned function and is also easy to trace the cause of model failure. Figure 1 gives a comparison between numerical machine learning and VML. In the proposed VML framework, both data and model are represented in a unified token-based format, while numerical machine learning treats data and model differently.
3.2 Natural Language as the Model Parameter Space
VML parameterizes machine-learning model with natural language. More formally, VML places strong constraint the model parameters 0 (01,02. ... 0.) is a text token sequence, 6e E A, Vt is some text token from large token set A, and Olanguage denotes the set of all ural language sequences that humans can understand. The model parameter space in VML has the following properties: (1) discrete: the natural language space is discrete; (2) sequential: the natural language space is soquential, and the next word is dependent on its previous words. In contrast, the parameter space numerical machine learning is not sequentially dependent; and (3) human- iiman-interpretable: the natural language that characterizes the model is human-interpretable. More discussions are given in Appendix J.
Olanguage to exchange for interpretability, where 0
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Text prompt template for the learner
Text prompt template for the optimizer
You the model. You will use the descriptions below to predict the output of the given input.
You are the optimizer for model, your goal to learn the best descriptions for the model. The model used the Current Pattem Descriptions below produced the oulputs the given inputs. You are given the target outputs, please optimize the Pattern Descriptions for better prodiction.
Patiern Descriptions:
You designed do regression, ie., to prediet the output of any given input. Both input and output real numbers.
w* Iaputs (a batch of Lid. data):
**
[[0.59] [1.55] [0.64] [1.43] [0.28] [0.02] [0.84] [0.39] [0.02] [1.28]]
A batch of data (xt, ..op Xce]
Model parameters !
** Current Pattern Description: **
x* Input: [0.59]
You are designed to do regression, L.e., to predict the output of any given imput. Both input and output real numbers.
Data x
Model parameters 0
Please give your ourput strictly the following format:
x* model oulputs: **
[|0.59] [3.88] [1.28] [1.43] [0.53] [0.02] [1. [0.39] [1. 1 [0. II
Prediction ( .... {o]
Explanations: (Your step-by-slep analyses and resulis]
The target ourputs:
[|5.84] [8.51] [5.92] [8.09] [4.98] [3.91] [6.46] [5.23] [3.88] [7.88]]
Ourput:
Ground Truth (y. J4]
(Your ourput MUST be in REAL NUMBER ROUNDED TWO DECIMAL POINTS; make nocessary assumptions necded, it MUST be in the same format the Input]
If the model is doing well, you can keep using the current descriptions. However, if the model is not performing well, please optimize the model by improving the 'New Pattern Descriptions', The model uses the 'New Pattern Deseriptions' should/better predict the target outputsiof the given inputs, as well as the next batch of kid. Input data fromithe same distribution. If previous 'ptimization Step' are provided. you if it's helpful. Please think step by step and give your outputsstriclly in the following formut:
the information from your last optimization step
Please ONLY reply according to this format, don't give me any other words.
Optimizer parameter yi
Verbalized loss function
update model parameters
Reasoning:
[be explici and verbose, improve the Current Patiern Descriptions by yoursell]]
Learner LLM fassi
New Paltern Descriplions:
Iterative optimization
[put your new descriptions here; MUST be specific and concrete;]
provide inference resalts
Optimizer LLM for
Please ONLY reply according this format, don't give any other words.
Figure 2: An overview of iterative optimization and text prompt templates of the learner and the optimizer in the regression example.
One of the most significant advantages to use natural language as the model parameters is the casy incorporation of our prior knowledge about the problem and the desired inductive bias into the model training. When the model parameters get updated during training, the model is fully interpretable, and one can observe and understand what gets added and what gets modified. Our empirical evidences also supports our interpretability claim, we find that the model parameters O are typically a language description of the underlying pattern that the model discovers from the training data.
3.3 Language Models Function Approximators
The core idea behind VML is using pretrained language model to act as a function approximator, param- eterized by its natural language prompt. Specifically, we denote the language model as f(x; 0) where æ is the input data and 0 is the function parameter. can be represented such as images if the LLM supports vision input), and the model parameters 0 is also represented with text tokens. In VML, f()) is typically a frozen large language model that is pretrained on a large corpus of text (e.g., DeepSeek- V3 [29], Llama-3 [51], GPT-4 [I]). If we consider a static function, we can set the temperature parameter of the LLM as zero, which theoretically makes the output deterministic. If we set the temperature high (see Appendix for more discussion), fax; 0) can be viewed as performing sampling from some distribution. We revisit how a classical machine learning problem can be formulated in the VML framework. Suppose have in total N training data points {z,, Vn W. where In is the input fcature vector and yn is the target output value. As an illustrative example, consider the following least square regression problem using the function fmodel (x; 0) that is parameterized by natural language description A:
text tokens (or other format
min Eregression
1 2N
Σ (yn (2r ())) S.L. 0 θ Oanguagee
(1)
where minimizing the objective function with respect to the discrete token-based model parameters 0 is actually quite difficult. Back-propagating gradients through discrete variables (e.g., policy gradients, Gumbel-softmax [16]) is typically known be sample-inefficient and sub-optimal.
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3.4 Iterative Training by Prompt Optimization
Because the model parameters 0 in VML are text prompts, optimizing 0 is effectively a prompt optimization problem. Different from current prompt optimization [73], where the goal is to produce generic prompt without adding new information, the training in VML focuses on updating the model's language characteriza- tion, which involves both the addition of new prior information and the modification of existing information. To optimize the model parameters, we start with the gradient of the regression objective in Equation 1:
V Volrggrsssinn N Σ (Un (1x; 0)) 0 madeel (Fn; 0)
s.t. 0 17 V Vo regresssio E O languaag
17-
(2)
00
where oj is the learning rate, and the constraint is to ensure that the updated model parameters are still in the natural language space. It seems to be infeasible to compute this gradient. To address this, we view the gradient as a function of the data (æ. y) and the current model paramcters 0. Then we directly approximate the nexi-step model parameters using another pretrained language model denoted by (2t 0, y. 0 1b) where i is the model prediction from the learner fmodel. +/ denotes the optimizer parameters that characterizes the optimizer scttings, and we can use language to specify the update speed, the momemtum, etc. The largest possible batch size of the optimizer LLM is determined by its context window. The optimizer LLM can already output natural language that satisfies the constraint, so we simply ask the LLM to play the optimizer role, which has been shown effective in [63]. More importantly, our VML framework gets better as LLM's instruction-followii ability gets stronger. An overview of the iterative optimization and the prompt templates in the regression example are given in Figure 2. The training procedure is given in Algorithm 1.
Using an LLM as the optimizer offers several unique advan- tages. First, the optimizer can perform automatic model selection. When the learner model can not make correct predictions for the training data, the optimizer will auto- matically update the learner to more complex and capable model (see the polynomial regression experiments in Sec- tion 4.2 as an example). Second, the optimizer can provide detailed explanations of why a particular update should be performed, which helps us to understand the iuner working end mechanism of the verbalized optimization process. Third, the LaMamaaaacteriieed optimizer allows users to interact with it. This not only helps us to casily trace model failures, but more importantly, it also allows us to inject prior knowledge to improve optimization.
A Algorithm 1 Training in VML
Initialize model parameters Ou, iteration number T, batch size M and optimizer parameters v;;
for 1, ,T do Sample M training examples "1,""","M;
for = 1,,,... M do 9m fmodel (mm; 0 1);
end
0,- (f3m Don Vnn M O 0i-1;));
Different optimizer parameterizations. In this paper, we use direct parameterization, i.e., param- eterizing the optimizer as a single function fopt, which couples the gradient and the update functions together. Alternatively, we can use an indirect parameterization where the gradient and the update are two separate HLM-parameterizeer functions. The gradients are known as "textual gradients" in prompt optimization [41, 70j. The update of learner's model parameter is given by 0, = fupdate (0, 30 de 0,-1) and similarly, äy parameterized by LLMs. Compared to direct parameterivation that only takes one LLM call, this process has to use several LLM calls, After empirically comparing both methods in Section 4.7 and Appendix B.3, we find that in most sccnarios, the direct parameterization yiclds better performance.
0t where
de is computed by fgrad( (3) 80
øt is computed by fgraa(e, 9). Both fupdate and fgrad are
3.5 Discussions and Insights
VML as framework to encode inductive bias. A unified framework to encode arbitrary inductive bias has been pursued for decades. For different types of data, we need to design different models to encode the inductive bias (e.g., graphical models [18] for random variables, recurrent networks [14] for sequences, graph networks [17] for graphs, and convolution networks [21] for images). VML uses a unified natural language portal to take in inductive biases, making it very flexible for encoding complex inductive bias. To incorporate an inductive bias about the hypothesis class or prior knowledge about the problem, we can simply concatenate a system prompt pprior (i.e., some constant prefixed text that describes the inductive bias) with the model parameters 0. The final model parameters are (Oprior, 0) where 0 is learnable and 0 Qprior is given by users.
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VML enables interpretable knowledge discovery. Because the model parameters 0 are already in natural language, it is easy to understand the underlying pattern that leads to the prediction and the decision rules that the model uses. Unlike numerical machine learning where the knowledge is learned within a black box, this property enables VML to discover novel knowledge that humans can also learn from.
VML as "the von Neumann architecture" in machine lcarning. Machine learning usually treats the model parameters and the data differently, similar to the Harvard architecture that stores instruction and data separately. VML stores both data and model parameters in the text prompt as tokens, which resembles the von Neumann architecture that stores instruction and data in the same memory.
4 Applications and Case Studies
We demonstrate the features and advantages of VML by revisiting some classical machine learning tasks followed by realistic medical image classification task. In these tasks, we are given data Derain (æ.,, Vn I Inmi and we want to find 0* such that fmodel( 0*) best describes the mapping x + y. Our experiments below show in detail how VML is able to solve these tasks and find 0-
Experiment sctups. We use the instruction-tunce Llama-3 70B [51] for the LLM unless specified otherwise. The training set for each task consists of 100 data points. For all tasks, we use batch size of 10 for each optimization step (see Figure 2 (right) an example), which corresponds to 10 steps per training epoch. To
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Figure 4: Training dynamic for VML based polynomial regression. The model is trained for 2 epochs, each with 10 steps.
evaluate regression performance, we look at the training loss, and the model predictions in both interpolation and extrapolation settings. For classifications, we use additional test sets (20 data points), and evaluate both training and testing accuracies. Inspired by the momentum from classical optimization, we provide the last step (i.c., one step only) of the optimization history to the optimizer LLM for training stability.
Training logs. The results of our experiments are showed using: (a) training loss, which is computed by parsing the model output (string) and converting it in to the same data type as the target value (y), then we use mcan squared error for regression, and zero-one loss mcan (f.e., average accuracy) for classification. The computed training loss is for logging purpose only, it is not required for training in VML (see Algorithm 1).; (b) visualization of the learned model, which is also done through parsing and converting the model output; (c) the model parameter at each training step i before optimization (i.e., 0;,1), and the optimizer output for the updated 8;. For > 1, the full model parameter before optimization is 0;-1 = (fu, 0-11}, but in our figures below we only show the 0-1 to save space.
Compute. The LLM is ran on a node of 8 x A100 using the inference engine provided by vLLM [20]. During each step (1) of training, query the LLM 10 times for evaluating the model fmode)(;; 65-1) over batch, and 1 time for requesting the newly optimized 0, We also evaluate the entire test set at each step, which, depending on the size of the evaluation set, requires between 20 to 100 LLM queries. Overall, for the regression lasks, they lake around 10 minutes for each epoch of training. The classification tasks, take around 16 minutes for each epoch of training. The visualization of the decision boundary takes around 6-minute.
4.1 Linear Regression
We generate Derain from a lincar function with Gaussian noise, i.e., y = 3z + + G, where ~ NO and x ~ (0. We initialize the model parameter 0, by only specifying that the task is a regression task from IR. to R (see Figure 3(c) Step 1). Figure ((a) shows that training improves the model, and that it converges.
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Thamiin Tina
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(c) Model parameters Llama-3 GPTT44 at Step 28
Figure 5: Demonstration of prior injection, and comparison of Llama-3, GPT-4o and a neural net in the sinusoldal regression setting.
The subplots (b) and (c) show details of the model and optimization at steps 1, 3 and 15. At step 1, since Oo only contain the definition of 1-D regression task, the model, is randomly guessing (see the dashdot line). The optimizer, says that it notices a linear relationship between the input and the target outputs, hence introducing lincar regression model to capture such relationship, which results in model, being a straight line. From step 2 onward, the optimization focus switches to fitting the identified linear regression model to the data. For example, at step 3, we can see that optimizer, says it notices that the outputs of modelg are generally smaller than the target, suggesting the scaling factor is too small, hence increases it. Similarly, at step 15, optimizer1s also says it notices the model14 overestimates the target; hence, it reduces the scaling factor. We can see from (b) that the resulting model1s closely approximates the ground truth.
4.2 Polynomial Regression
We generate Dirain from polynomial function with Gaussian noise, i.e., y = 3x + +2+ where ~ (O 1) and x U(-3, 1). Similarly, O is initialized by only specifying that the task is regression task from R to I (see Figure 4(c) Step 1). Figure 4(a) shows that training is effective and converges. Subplots (b) and (c) show details of the model and optimization at steps 1, 2 and 3. At step 1, modelo randomly guesses the outputs. The optimizer, says that it notices y has a larger range than x, and that they seem to have positive correlation; therefore, it updates model, to be a simple linear model. This lincar model assumption leads to a jump in the training loss (see subplot (a)), as it is far from the ground truth. Consecutively, al slep 2, optimizer, soys the poor performance makes realize that the linear model oversimplifies the relationship between x and y. It notices non-linearity between and y, and to capture this, it uses a quadratic model. This leads to better model and large decrease in the training loss. At step 3, optimizer, switches from model class selection to fitting the quadratic model. The resulting modelz closely fits the ground truth.
4.3 Sinusoidal Regression
We generate Dersin from sine function with Gaussian noise, i.e., y == sin(x) + 2 + 0.01e, where -~ N 1) and æ ~ (1-3, 3). Fitting sine function is known to be difficult for neural nets in terms of extrapolation. Here, we lry GPT-4o, a more powerful model than Llama-3. Figure 5(b; right) shows that when Op contains only the definition of 1-D regression, it results in a linear model after training (see (c; right)). We can add a prior to by simply saying that the data looks like samples generated from a periodic function, which results in a very good approximation and it extrapolates much better than neural net (see (b,c; left)). We also find that adding the same prior to Llama-3 is not as effective (see (b,c; mid)), indicating the capability of VMT.
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is highly dependent on the capability of the LLM. However, this suggests VML grows with LLM's scaling law the effectiveness of VML can improve along with the capability (size) of the LLM.
4.4 Two Blobs Classification
We generate a linearly separable Dtrain from two blobs a 2-D plane. Oo is initialized by only specifying that the task is binary classification on a 2-D plane (see Figure 6(c) Step 1). Subplot (a) shows that training is effective and it converges. At step 1, optimizer, says the current batch of data has the pattern that data points with x > 0 belong to class 2, and data points with x < 0 belong to class 1; hence it updates modelı to have a linear decision boundary at x = 0, which happens to be perfect. However, Figure 6(a) shows that. the training loss does not immediately converge. We can investigate the cause and "debug" the optimizer by looking at optimizer, From (c) Step 2, optimizer, says model, is already quite simple yet accurate, but it wants to further improve the model and utilize the new information from the current batch. Guided by this reasoning, modelso becomes a very deep decision tree, and the decision boundary has reasonable margin towards the data (see Figure 6(b, c; right)). The results also reveal that VML's interpretable may lead to complex pattern for easy problems, highlighting the importance of specifying a proper inductive bias.
4.5 Two Circles Classification
We generate a non-linearly separable Dirain by creating data points on two concentric circles as the two classes. Besides the description of binary classification on a 2-D plane, we add a sentence to encode our inductive bias that the decision boundary a circle into 0, (sec Figure 7(c) Step 1). At step 1, optimizor, utilizes the prior, and updates model, to have a circle decision boundary. For the rest of the training, the optimizer mainly tries to find good fit. for the radius and the center of the decision boundary. At step 41,
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optimizer says modelao scems to be a good fit for the data, and no changes are needed. Hence, it uses the same Ou for model - Without the prior, VML can also learn a good model, but the performance shows large variance at the beginning of training (see Figure 7(a; dashed)) due to the model selection process similar to Figure 3(a). Figure 7(c; bottom right) shows the resulting 040 without the prior, which is a decision tree.
4.6 Medical Image Classification
To demonstrate the capability of VML beyond simple machine learning problems, we evaluate the effectiveness of VMT. for image classification. We use GPT-4o, which supports visual inputs, to take into account both
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image and text data. The task is to classify whether a X-ray image has indications of pneumonia not, see Figure 8(b) for image examples. Due to the cost. of GPT-4o, we create subset of the dataset PneumoniaMNISST [64]. Our dataset consists of 100 training data and 100 test data (half pneumonia and half normal for both sets). Models are trained for 5 epochs. We try out two different model parameter initializations, one with prior and one without. We encode the inductive bias by simply adding a senlence as the prior, which states that the input is an X-ray image for identifying pneumonia, along with the definition of binary image classification (see Figure 8(c)). The test accuracy in (a) shows that both models are able to improve their performance on the task as the training epoch increases, and the model initialized with prior outperforms the model without (in terms of both testing accuracy and training convergence). Additionally, by inspecting the parameters of modelço (see (d)), we observe that the model parameters Os0 for the learner with prior has more medical domain knowledge associated to fealuress of pneumonia (such as "acute infection", ", "pneumatocele formation"), while the model parameters 4s0 for the learner without any prior mainly use generic visual knowledge associated to features of lung (such as "visible opacitics", "uniform texture"). This observation validates the effectiveness of using natural language to encode inductive bias. Our experiment also demonstrates the usefulness of learning in VML (i.e., the generalization performance can be improved over time), which is distinct from existing prompt engineering methods. Additionally, the interpretable nature of VML's model parameters is crucial for applications in medical domain. The learned models can be validated by medical professionals, and their predictions are grounded by their verbalized reasoning.
4.7 Ablation Study and Exploratory Experiments
Quantitative comparison to in-context learning. Since
Task Reg-L(() Pl)-P(L) Cla-TB(t) Cla-TC(1) CIs-MI(*)
VML can be viewed a generalization of ICL, we therefore compare VML to ICL in all previous applications. Results are given in Table 1. The ICL results are chosen from the best one across runs. The metrics used for regression (Reg) and classification (Cls) are mcan square error (MSE 4) and test accuracy (t), respectively. We abbreviate linear regression as Reg-L, polynomial regression as Reg-P, two blob classification as Cls-TB, two circle classification as Cls-TC and medical image classification CI.-MI. We can observe that VML consistently outperforms ICL.
TCL.
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(b) Direct vs. Indirect panmeterization
Direct vs. indirect parameterization. As discussed in Section 3.4, we can use either a direct or an indirect way to parameterize the optimizer. We compare both parameterizations using the linear regression setting (as described in Section 4.1). Figure 9(b) shows that the direct parameterization outperforms the indirect one. The direct parameterization leads to faster convergence while requiring less LLM calls. Detailed experimental settings and more discussions are given in Appendix B.3.
Figure 9: Training loss dynamics. For each configuration,
show 5 individual runs (thin) and their mean (thick).
4.8 Comparison Between Generic Prompt Optimization and VML
To better disferentiate VML from prompt optimization, we compare VML to a generic prompt. optimization method called Automatic Prompt Engineer (APE) [73] both conceptually, and qualitatively on two tasks, We use Llama-3-70B for both APE and VML. Even though both mothods aim to optimize a prompt towards a pre-defined objective function, there are fundamental differences between the two. Conceptually, APE first samples a set of candidate prompts directly given only the training data, and then chooses the one
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with the best objective value. The prompt generation process is best-of-N sampling, and each sampling is independent from another. In contrast, VMI. produces new prompt by asking an LLM to explicitly reflect on the old prompt and the corresponding predictions ÿ, then reasons on how to produce a prompt that can better predict the target y for the given input. This is conceptually similar to the distinction between gradient-free and gradient-based optimization. For more discussion and the implementation details of APE, please refer to Appendix D. Note that the qualitative comparisons in this section do not imply superiority between the two methods.
Automatic Prompt Engineer Tos los: 35:87 calculate a mathemmatical function. Can you guess what the fanction it?
Automatic Prompt Engineer
Linear regression in Sec- tion 4.1. Figure 10(a) shows that the result. from APE is vague and general. Such a de- scription can easily be derived by humans through visual in- spection of the data, and it does not learn deeper insights from the data, whereas VML is able to learn useful new in- formation that is difficult. to obtain by visual inspection. We can observe that VML au- tomatically performs effective pattern summarization from data, which differs from naive prompt optimization.
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Text classification. Adopted from the Google BIG-bench[6],, the task is to classify whether name is more likely to be associated to female or male. Figure 10(b) shows that APE does return correct description of the task, but it is, once again, very general. Conversely, VML is able to learn more detailed knowledge about the data pattern which cannot be done casily through visual inspection.
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Figure 11: VML is able to learn to reason and solve symbolically generated GSM8K [36] questions with Llama-3.11-8
4.9 VML Enables Robust Mathematical Reasoning
Recent work [36] shows that if we modify the original GSMSK [10] question by changing only the variable values (e.g., Figure (a), the accuracy of many LLMs on the modified dataset will decline, which might be due to data contamination during pretraining. Our experiments show that VML can reduce such a performance variation and enable robust mathematical reasoning without changing the internal weights of
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70-
Thiining
Model parameters 0i
300
10-
0.. [Before Ops.)
9. (Ater Opt.
The model should use the linear transformation
150
0" (G/T;
M = -4.00x + 5.00 i) to predict the output.
f(r.
Thirs
30-
1200
Model parameters 0,s
20-
0.00
The model should use the linear transformation My - -2.00x 7.00 to predict the output.
Step
500p-19
-10
Step
()) Training dynamics the failure case
(b) Visualization the learned function
(c) The corresponding learned model parameters
Figure
Some failure cases where the optimization was trapped in bad local minima the task polynomial regression.
LLMs. Specifically, we randomly generate training set and a test set, both of size 100 (without overlap), using the template in Figure 1i(a). If we directly evaluate the test set using Llama-3,1-8B, the average accuracy over 5 runs is around 80%. We use VML to Iearn a set of instructions for this task, the initial one is given in Figure I1(c). We use a batch size of 10 and train for 10 steps. Figure [1(b) shows that, on average over runs, the test. performance increases with the number of training step, and VML enables the model to achieve achieves 98% accuracy on the test set. Figure 11fd) shows the optimization outputs for step 1, 3, and 6 for selected run in Figure 11(b). We can see that after step 1, the new instructions already recover the correct mathematica reasoming for the lask, but the lest accuracy is only around 91%. At step 3, optimizer, realizes that the error is mostly due the inaccurate calculations rather than the correctness of the instructions for model2 Hence, the new instructions for modelz includes emphasis on calculation verification, which brings the test performance up to 96%. At step 6, optimizer, says it notices many of models 's mistakes are stiill related to incorrect and incomplete calculations, therefore, it breaks down the instructions into a more detailed lists allow easier reasoning and checking. Therefore, VML can enable LLMs to improve their reasoning ability at test-time by themselves without changing the internal weights.
4.10 Failure Cases
Figure 12 shows a failure run for the polynomial regression task. Our log for this run records that after the first optimization step, the model are updated to a linear regression model, and the rest of the optimization steps are simply trying to this linear model to the data. Therefore, we can see the training dynamic plot show a fluctuating line, and the step with the lowest training loss (i.e., Step 19) still has lincar regression model. Unlike the other successful runs, where the optimizer realizes a quadratic function can be better model class than linear function, in this failure case the optimization clearly trapped in a local minima of a linear model. One possible way to reduce such failure case is to use more powerful LLMs. See Appendix C for more detailed discussions and three other examples where VML failed to learn desirable model. Some of these failures can be avoided by changing the VML prompt template, while others could occur less frequently if we switch to a more powerful LLM.
5 Current Limitations
Our paper introduces a verbalized way to perform machine learning and conducts several case studies on regression and classification tasks. The experiments show that VML can effectively perform these classical machine learning tasks with low-dimensional input data, validating the potential of LLMs as function approximators. Despite the empirical effectiveness, there are still limitations that remain to be addressed.
Large variance in learning. Training in VML still suffers from a relatively large variance. This is partially due to the stochasticity from the LLM inference, the capability of the LLM, as well as the prompt design of the optimizer. See Appendix C for failure cases and analyses.
Scalability in terms of data dimension, model paramcter, and number of optimization steps. The input data dimensionality and batch size are largely limited by the size of context window in LLMs. In addition, when high-dimensional data are represented in raw text, current LLMS find it hard to grasp the
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information in the data, and therefore, it can lead to poor performance in VML. Refer to Appendix B.6 for detailed experiments and discussions. Similarly, since VML. is parameterizing model in natural language space, the dimension of the learned model parameter is correlated with the number of tokens used in the text-based parameter. Hence, the complexity of the model parameter is largely limited by the size of context window in LLMs. Specifically, we empirically observe that the VML model parameters start with a simple model class and gradually shift to more complex model class during training. Due to the limited budget of querying LLMs and the main focus of this work being showcasing the concept of VML and its effectiveness in diverse set of machine learning tasks, our tasks are all relatively small-scale, i.e., with less than 200 training data points for each task and the data points are usually low-dimensional (our experiments mostly use 2-dimensional data as the raw input to LLMs). Our experiments only have a small number of optimization steps (i.e., less than 100), which is sufficient for simple machine learning lasks. However, how to effectively scale VML with more training iterations, digeer-dimeniionll data and more complex downstream tasks remains open questions.
Error for numerical tasks. The output numerical error in LLMs results in inevitable fitting error (see Appendix F). Concretely, even if the LLM correctly understands the underlying symbolic expression, there is still an output numerical error when performing inference on specific input values. This also suggests the intrinsic difficulty within LLMs to properly understand numbers (see [43, 69)).
Hallucination of LLMs. Hallucination is a well known problem in LLMs. Even though in our experiments we did not witness any empirical evidence that hallucination is affecting the reliability of VML, hallucination potentially can lead to large variance in learning as well as causing error for numerical tasks, Hence, this points us to an opportunity for future improvement in VML, since methods that can mitigate hallucination in LLMs should also improve the performance of VML. See Appendix K.5 for detailed discussion.
6 Future Directions
One future direction is to study various aspects in VML using insights and concepts from classical machine learning. Some interesting questions include: Can we find a better design for the optimizer so that the training is more robust and efficient? How does the optimization landscape in VML differ from classical ML, what does it look like? Another interesting dircction is to investigate the learning dynamics of VML, and compare it with how human learns. Since human also has a language model in mind, the same experiments in the paper can be conducted on human through messaging software. Improving VML's ability for handling high-ddimeniional data is also an important direction. More concretely, we need to develop better LLMs to support more data modalities, and make sure they can reason in those modalities well. In addition, we need to improve their ability in tools calling so that they know when to use existing tools (such as Python) to preprocess the data if they find them too difficult to reason in the current high-dimensional representation.
Acknowledgment
The connection between LLMs and computers was discussed in our blog'. The VML framework is naturally motivated by the idea of LLMs acting as a modern computer, since we view the verbalized model parameters as way to "program" " the LLM. We then connect. VML to the von Neumann architecture in the sense that. both data, and program instruction are in the format of text prompt in VML.
WL was supported by the German Research Foundation (DFG): SFB 1233, Robust Vision: Inference Principles and Neural Mechanisms, TP XX,, project number: 276693517. This work was partially funded by the Deutsche Forschungsemminnchaaft (DFG, German Research Foundation) under Germany's Excellence Strategy EXC number 2064/1 Project nunber 390727645. This work was supported by the German Federal Ministry of Education and Research (BMBF): Tübingen AI Center, FKZ: 011S18039A. RB acknowledges funding by the German Research Foundation (DFG) for project 448588364 of the Fammy Noether Programme. The authors thank the International Max Planck Research School for Intelligent Systems (IMPRS-IS) for supporting Tim Z. Xiao. TX acknowledges support from G-Reearrch'' PhD Grant Programme.
1 See the blog entitled *Large Language Models Are Zero-Shot Problem Solvers Just Like Madern Computers" in https: //tpp::////ixeeeee wewtaxx.ee/bbeee /log 2023/ httt///omputeer-ee vaateerrrr-----eee 1lms/.
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Appendix
Table of Contents
A More Case Studies
22
A.1 Digit Pattern Discovery
22
A.2 MNIST Image Binary Classification
23
B Details for Ablation Study and Exploratory Experiments
24
B.1 Comparison Between In-context Learning and VML
24 ................................
B.2 Larger and More Powerful LLMs Learn Faster Better
................................
B.3 Direct and Indirect Optimization
21
B.4
Evaluations
diverse set of L.LMs
25
B.5
Using Language Other Than English
B.6
High-dimensional Two Blobs Classification
27
C Failure Cases
28
D Details the Comparison between VML and APE
30
APE Experiments Details...
30
D.2
Differences between APE and VML
30
E Effect of Accurate Loss Feedback
33
Numerical Error of LLMs in Representing Symbolic Functions
34
G Mitigating Numerical Error by Tool Calling
36
Connection between Prediction Variance and Model Parameters VML
H.1 From Vague to Concrete Model Parameters
37
H.2 Semantic Invariance of Model Parameters
39
1 Probabilistic View on VML
I.1 Posterior Predictive Distribution
1.2 From Functions to Stochastic Processes
41
J Discussions Natural Language Model Parameters
J.1
Different Mechanisms to Update Model Parameters Direct Optimization
42
J.2 Occam's Razor, Constrained-lengdla Model Parameters, and Kolmogorov Complexity Connection to Nonparametric Models and In-context Learning
43
J.3
43
J.4
Distinctions between the Data Dimension and the Parameter Dimension
43
K
Broader Discussions
45
K.1
How is 'the function should be = m x + more interpretable?
45
K.2
Is controlling the hyperparameters of LLM optimizers more difficult?
45
K.3
Are language models fundamentally restricted for machine learning tasks?
45
K.4
Is it meaningful use LLMs for applications such as machine learning?
46
K.5 What about Hallucination? Can we trust LLMs to handle complex tasks such as VML?
46
L Complete Training Template at Initialization
48
T.1 Linear Regression
48
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L.2
Polyuomial Regression..
49
L.3
Sinusoidal Regression
50
L.4
Two Blobs Classification
51
L.5
Two Circles Classification
................................
52
L.6
Text classification
53
M Detailed Training History
54
M.1
Linear Regression (Llama-3-70B without prior)
................................
51
M.2
Polynomial Regression (Llama-3-70B without prior)
61 
M.3
Sinusoidal Regression (GPT-4o with prior)
68
M.4
Two Blobs Regression (T.Lama-3-70B without prior)
75
M.5 Two Circles Regression (LLama-3-70B without prior)
84
M.6 Two Circles Regression (LLama-3-70B with prior)
91
M.7 Text Classification (LLama-3-70B without prior)
98
M.8 Medical Image Classification (GPT-4o with prior)
103
M.9 Medical Image Classification (GPT-4o without prior)
110
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A More Case Studies
A.1 Digit Pattern Discovery
Optimizeer output Step
Optimiizer output Step 2
Optimizer outpur Step
3. 31
Itawcing:
Ressoring
Reewein:
Stop
10
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Figure 13: Binary digit pattern discuvery of 4-integer vectors. No prior is injected to the model parameter in this experiment.
To further demonstrate the interpretability of VML, we create a binary classification task on vectors of 4 digits. Class 0 contains vectors that only have digit '0' in the first position, and Class 1 contains vectors that only have digit '0' in the last. position (see Figure 13(b)). Our dataset consists of 100 training data and 20 test data (half for both classes). Models are trained for 5 epochs (i.e., 50 steps with batch size 10). Figure 13(a) shows that both the training and test accuracy improves with the nunber of steps, hence learning is effective. The model is initialized with the definition of the task. During step 1, the optimizer says it notices that the first element of each input is often '0' when the ground truth label is '0', and decides to use a rule-based approach (see (c)). The resulting model description is half correct, which captures the pattern that 'if the first element. is 0, predicts O' After a few more steps, the optimizer able to learn the correct description: 'If the first element is O, predicts Otherwise, if the last element is 0, predict 1. Compared to the regression and 2D plane classification results, the learned model here is more interpretable than learning neural network. Also, without any prior information, one will normally choose universal approximator such as neural network to solve this task, which will perform equally well but certainly not as interpretable. We also evaluate the performance of in-context learning (ICL) for this task as a baseline. Our result shows that VML is able to achieve 100% test accuracy with an interpretable description of the pattern, while ICL can only achieve 87.5% and does not explicitly output pattern description.
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A.2
MNIST Image Binary Classification
4
4
Model parameters O Step
Model parameters Oos at Step 25 If the digit features single loop and no extension beyond its main body, output 'I''22 the digit has multiple, distinct, and separate loops noticeable curves extending outside its main structure, output '
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Test Accuracy: 61%
Test Accuracy: 62%
Test Accuracy: 97%
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(a) Examples
Model parameters Step 1, 25,
Figure MNIST binary classification with InternV1.2-.llmm3-7688 [9]. We group digit injected to the model parameter in this experiment.
class 0, and digit '9" 1. No prior
We create a binary classification task from the MNIST dataset. We assign digit '4' to class 0, and digit '9' to class 1. Following the same selup in Section 4.6, both of our training and test set has 100 MNIST images, half for each class. The model is trained for 5 epochs with batch size 10. We use InternVL22 L-L-mm2--- [9] for as the inference engine, which support image input. We also tried out other LLMs that support image imput such Claude, GPT-4o and Llama 3.2, but they have been finetuned to reject the digit recognition task, possibly due to some safety reason. Therefore, we do not have results for those LLMss Note that the task can be easily solved if we directly instruct the LLM to classify digit '4' to O and '9' to 1. However, this is a knowledge from human inspection over the training set. In this toy MNIST example, such classification knowledge can be relatively easy to obtain by inspecting the images, but this way of obtaining knowledge is not scalable and requires massive human efforts if the image classífication problem is complex. In contrast. directly instructing LLM to perform classification tasks, VML can automatically discover the pattern behind the training data and learn perform classification without human interference. In this MNIST classification task, show that VML can easily learn this classification instruction without any human prior knowledge.
Results in Figure 14 show that VML is indeed able to learn from the training data and achieve 97% test accuracy with the learned model. Figure 14(c) shows that the model first learns to use visual features such as 'single loop' to describe the pattern for each class. Then, it learns that the classification rule is related the digit appears in the image, which is a semantic feature. In the end, the model parameter has redundant information (e.g., classify "2" as '0'), but it does include the correct description of the decision rule, hence, it. has good test performance.
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B Details for Ablation Study and Exploratory Experiments
Here we provide additional details for the experiments in Scction 4.7 and additional ablations.
B.1 Comparison Between In-context Learning and VML
In-context learning (ICL) is a popular method for adapting LLMs to downstream tasks. Here, compare the performance of VML and ICL in various lasks from previous sections. For all tasks, provide the entire training set as in-context examples, and query the individual test data independently. The resulting predictions for regression and 2D classification are plotted in Figure 15. The full comparison between VML and ICL are shown in Table 2. We can see that VML outperforms ICL in regression and medical image classification, and has the same performance to ICL the simpler classification tasks, c.g., two blobs and two circles. Within our framework, ICL can be understood as nonporameterie method, while VML is a parameteric one (see Appendix J.3 for more discussion).
Taniin Dula.
Erisza, Duta
VML VML MSE 2.26 ICL MSE 02.00
30
VMI. 038
Tea
-15 -1.0
(5) Linear Regression
(b) Polynomial Regreasion
(c) Two Blobs Classifieation
(d) Two Circles Classfication
Figure 15: Predictions of in-context learning (ICL) for the same regression and classification tasks with Llama-3 70B.
Task
(4) Reg- Rig-Linear
(1) Reg-Poly.
(1) Cls-Two (f) Cle-Two Circles
(1) Cls-Medical Tmg
VML
0.12
2.38
100%
95%
74%
ICL
0.38
62.96
100%
95%
18%
Table 2: Test performance for in-context learning (ICL.) and verbalized machine learning (VML) on various tasks from previous section (without adding prior information). ICL results are chosen from best across runs. The metrics used regression (Reg) and classification (Cla) are mean square error (MSE and test accuracy (f) correspondingl.
B.2 Larger and More Powerful LLMs Learn Faster and Better
To verify whether the performance of VML scale with the capability of LLMs, we compare three Llama-3.1 models of different sizes, i.e., 8B, 70B, and 405B, in the linear regression setting. Figure 16 shows the training loss 5 individual runs (thin) and their mean (thick) for each LLM. Note that due to the high variance nature of using LLMs for optimization, we select the best runs out 10 runs for this comparison. We see that more powerful LLMs (e.g., 405B) learn faster and achieve lower training loss.
B.3 Direct and Indirect Optimization
There are different ways to implement the optimization step in VML. We choose to directly update the model parameters O in a single LLM call by providing all the necessary information, i.e., 0: = Sope ((#m: fn. Vm 1M 1 0,-1; in Algorithm If we choose a lower abstraction level, we can decom- pose the direct single step optimization into indirect multi-step optimization. Algorithm 2 illustrates how fopt be decomposed into four consecutive functions, which resemble the operations of computation graphs in most numerical machine learning frameworks. Specifically, we calculate the following step-by-step: (1) the quality of the predictions (i.e., evaluate the loss function finss); (2) the 'gradient' of the loss & w.r.t. the predictions å denoted as θE/0g: (3) the 'gradient" of the loss w.r.t. the parameters 0,-1 denoted as ó 0t/00.-1 (4) update the current 0:-1 to 0, using the 'gradient' de/00, 0- The 'gradients' here are known as 'textual gradients' in prompt optimization literature [41, 70], which are essentially text- text-based feedback from L.M.s.
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8B
70B
405B
10.
Training Lasss
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Figure Llama-3.1 LLMs scale VML training performance linear regression setting. individual (thin) and (thick) for euch LLM.
We compare the two approaches in the lincar regression sctting using Llama-3.1 70B. Figure 17 shows, for both the direct and indirect optimization, the training loss individual runs (thin) and their mean (thick). We can see that the indirect method performs slightly worse than the direct method. The reason can be there are 3 more prompt templates design, which is harder than designing just one, and has higher risk of losing information in the pipeline.
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Figure Training loss of direct and indirect optimization linear regression setting using Llama-3.1 70B. The lines show individual runs (thin) and mean (thick) for each approach.
B.4 Evaluations on a diverse set of LLMs
Most of our experiments Section are done with Llama-3-70B. In this section, we evaluate various LLMs other than Llama-3-70B on four tasks from Section 4, including linear regression (Reg-Linear), polynomial regression (Reg-Poly), two blobs classification (Cls-Two Blobs), and two circles classification (Cls-Two Circles).
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Model
Run
Run
Run
Best
(MSE 4) Lineer (English)
Claune-3.5-Sonnet
0.000
0,006
0,001
0.027
0.002
0.000
GPT-4o
0.002
0.015
1.394
0.192
4.082
0.002
DeepSeek-V3
0.109
0.378
0.009
0.005
0.005
Swen2.5-72BInnstuuee
0.854
0.537
0.022
0.769
0.267
0.022
(MSE 1) Reg-Polg.
(Engtish)
Claudn-3.5-Sonnet
7.687
3.789
1.015
643.629
3.095
1.015
GPT-4o
3.245
0.614
3.200
DeepSeek-V3
1.581
356.805
11.020
2.389
7.622
1.58
Oweu2.5-72B.. 5-22---tur- Instruct
1.786
1.958
416.148 140
395.6442
6.531
1.786
(Acc.
Cws-Two Blobs (Bnglish)
Clando-3.-SSomnc
100%
GPT-4g
100%
100%
100%
90%
100%
100%
DeepSeck-Vi V3
100%
100%
95%
100%
95%
1 100%
Qwen2.5-72BB -nsttuuutt
100%
100%
100%
45%
30%
100%
(Ace. f)
Cávvlles (Rnglish)
Claude-3.-Soomet
100%
100%
100%
100%
100%
100%
GPT-44
100%
45%
100%
100%
100%
100%
DeepSeek-V3
40%
35%
100%
50%
100%
100%
Qwen2.55-7B-Instruue
45%
35%
35%
85%
35%
85%
(MSE L) Reg-Lin Lan (Chinese)
Mauude-35-Sommae
0.017
0.024
0.121
0.014
0.013
0.013
GPT-4o
0.035
3,420
3.667
0.227
3.475
0.035
DeepSeck-V3
0.769
0.652
0.336
1.236
0.725
0.336
Qwen2.5- I-Sttueet
1.867
0.465
0.251
0.046
0.631
0.046
(MSE 4) Reg-Poly.
(Chinese)
Claude-3.5-Somn
9.401
6778.058
1.701
12.280
16.852
4.704
GPT-4o
5.722
9.033
345.066
1.537
30.282
1.537
DoepSeck-V3
558.362
484.519
96.766
70442.614
42.830
Qwen2.5- 702-5---2----nteee
3407.283
5.682
127.100
11.144
1090.363
5.682
Table 3: Test performance for VML using various LLMs on regression and classification tasks from previous section (without adding prior information). The last tasks done with prompts in Chineso, For each sotting, include results for runs, and we highlight. the runs with worse performance than the ICL English best@5 baselines (see Table 2) in red.
As for LLMs, we use two proprietary LLMs (Claude-3.5-Somee) and GPT-40), and two open-source LLMs (DeepSock- V3 and epochs of training for regression and epochs of training for classification.
The experiments are done the same sctting as in Table 2, í.e.,
Results in Table 3 show that all four LLMs are able to perform well in the same settings. In particular, when comparing with the best@5 results using Llama-3-70B in Table 2, all four LLMs here have better performance in Reg-Linear and Reg-Poly, which might due to the fact. that these four LLMs are new and more capable than 1lama-3-70BB As for the two classification tasks, all four LLMs matches the performance of Llama-3-70B in Cls-Two Blobs, and three out the four outperform Llama-3-70B in Cls-Two Circles.
Overall, other than the fact that more powerful LLMs can learn faster and achieve lower loss (similar finding in Section 4.7 and Appendix B.2), there is not too much difference between them (no matter being proprietary open-source). Therefore, if cost is not constraint, one should always choose the most powerful LLMs for doing VML.
B.5 Using Language Other Than English
Our experiments in Section 4 are all done using English. In this section, we provide experiments using Chinese only, de.,, the learner and optimizer templates, and the model initial parameters are all in Chinese. We construct these Chinese prompts by first asking ChatGPT for a translation the existing English version, then asking a native Chinese speaker to verify the translation. We replace the original English prompt with the Chinese version, and run the same VML algorithm for linear regression (ineaL) and polynomial regression (Reg-Poly) using the same setting in Table 2. We four different. L.MSs for the
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experiments, two propitiatory (Claudo-3.5-Sonnct and GPT-4o), and two open-source (DeepSeek- V3 and Qwen2.5-7BB-hnstruc)))
The bottom two sections in Table 3 show the results. We can see that the performance is slightly worse than the English version (the top two sections in the same table) across all LLMs. But the results are still better than the TCI. English best@5 baselines in Table 2. This is likely due to most the L.M developers putting their efforts into the English corpus and English benchmarks, which highlights weakness of the existing models.
B.6
Two Blobs Classification
The two blobs classification task in Section 4.4 has only two feature dimensions for each data point. In this section, extend the same task to higher numbers of data dimensions, from 2-D) to 10-D. Note that these high-dimensional data are represented in raw text and are processed by the text encoder of an LLM during VML. This is different from the data used in medical image classification (fe., images in Section 4.6), which are also nigh-dimensional data, but they are processed by the image encoder of a vision- lasiua-language model rather than the text encoder.
The experiments are done with the same setting as in Section 4.4 but with different number of feature dimension. Table shows the best test accuracy out of 5 runs, and the average test accuracy over the 5 runs. We can see from the results that with Llama-3-70B, VML straggles to perform well when the data dimensions are larger than 7-D. There are few possible explanations, including: (1) the current LLMs are not trained to understand the text representation of high-dimensional data; (2) the current LLMs do not handle long context well, they cannot grasp the information in high- high-dimennsional data.
Dimension
2-D
3-D
4-D
5-D
6.1
7-D
8.D
9-D
10-D
Best@5
100%
100%
100%
95%
100%
95%
80%
90%
70%
Avg.
87%
99%
86%
51%
74%
79%
46%
65%
56%
Table 4: Test performance for high-dimmesion blobs classification using Llama-3-700 The data points the corresponding n-D space on a 2-D hyperplane. The tuble shows the best results out of 5 runs, as well as the average performance over the 5 runs.
B.6.1 How should VML handle high-dimensional data?
The ability of VMI for handling high-dimensional data is mainly constrained by the inference backbone, i.c., LLMs. The experiments in this section together with the experiments in Section 4.6 demonstrate two different approaches to handle high- gich-dimensional data, i.e., either representing the data in raw text and processing them with the text encoder of an LLM, or representing the data in image and processing them with the image encoder of a vision-language model. We see that if there is a corresponding encoder for the high-dimensional data (e.g., image), VML can handle tasks that involve these high-dimensional data easily.
Of course, we can also finetune text-only LLM to handle high-dimensional data in raw text, which might improve the performance of the corresponding VML task. However, if we think about how humans handle high-dimensional data, we know that humans rarely do inference directly on the raw text representation of high-dimensional data, which is very difficult. For many high-dimensional data such as sound and images, humans have dodicated encoders to process them. For those that do not have dedicated encoders (c.g., radio wave), humans often use tools to preprocess the data into the representation that can be easily encoded, then do inference afterwards.
This points us lowards a possible path to improve VML's ability for handling high-dimensional data. First, need to develop better inference engines to support more data modalities, and make sure they reason in those modalities well. Second, we need to improve the inference engines' ability in tool calling so that they know when to use existing tools (such as Python) to preprocess the data if they find them too difficult to reason about in the current format.
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C Failure Cases
R.. (lkcoe Ugt)
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(Aber (CT)
The model should use the lincar transformation
291
y - 4.00x 1) to predict output.
Model parameters On
100
The model should use the linear transformation M(y -2,00x 7,00 predict the output.
Step
Training Dynamies
Vissalization of learned function
3. The lourned model parmeterrs
(a) Failure Cuse: Trupped in local minima
Model parameters
Model parameters
relationship between the input and ground output can be described by a function of the form =="x"3* ("log(x)* e, where a, b, parameters to be determined. This updated equation captures the more pronounced curvature the data, with the cubie term allowing for more flexible und nuanced representation of the relationship between the input and output. The inclusion of the logurithmic term still accounts for the consistent underestimation of the outputs lower input values, and the offset term (c) accounts for the overall shift in the output.
You are designed to do bisary classification for the presence of condition visible in medical fmages. If anomalies present in structural features such as lungs, you should output T,, otherwise ourput 'O. Normalize the images consistent of 224x224 pixels, use convolutional layers with ReI.U activation, max-pooling, batch normalization (with batch size of 32), followed by fully connected layers (2 layers: 1024 and 512 neurons), using learning rate of 0.001 and cress-cntropy loss for training. Train with a dataset containing at least 10,000 annotated images.
b*x*2
variability
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(d) Failure Case: Aggressive fitting
Figure 18: Four examples of failure runs in VML. (a) polynomial regression; (b) medical image classification; (c) linear regression; (d) polynomial regression. For (b) and (c), we only show the learued model, the training dynamic less relevant these two cases.
In this section, show case four different examples where VML failed to learn desirable model. Some of these failures can be avoided by changing the VML prompt template, while others could occur less frequently if we switch to a more powerful LLM.
Trapped in a local minima. Figure 18 (a) shows a failure run for the polynomial regression task. Specifically, this corresponds to the run in Table Reg-Poly (English) with Frre225-72B-Instaaae Our log for this rum records that after the first optimization step, the model are updated to a linear regression model, and the rest of the optimization steps are simply trying to fit this linear model to the data. Therefore, we can see the training dynamic plot show fluctuating line, and the step with the lowest training loss (i.e. Step 19) still has linear regression model. Unlike the other successful runs, where the optimizer realizes a quadratic function can be a better model class than a linear function, in this failure case the optimization clearly trapped in a local minima of a lincar model. One possible way to reduce such failure case to use more powerful LLMs. In the same Table 3 Reg- Poly (English), we can see that when we use more powerful GPT-4o, all five runs have much lower test loss (ie. 10) than this failure case (i.e., 400).
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Not describing the data pattern. Figure 18 (b) shows failure case for the medical image classification task in Section 4.6. In this run, instead of semantic pattern description for the two classes of images (e.g., Figure 8), the optimizer returns a description of a two layer neural networks (without exact values for the weights) and the training procedure. Using this description to do inference directly on the input image will undoubtedly lead to a useless answer. The model description afler the next update is not showed Figure 18, but our log shows that, due to the expected poor performance of the current description, the optimizer proposes to increase the number of layers in the neural networks to make it more capable. One way to avoid such failure is to add instructions in the prompt template of the optimizer specifying, for example, "the new model description should be decision rule which must base on the features in the input image". "
Missing crucial information when describing a parametric model. Figure 18 (c) shows a failure case for the linear regression task in Section 4.1. There are two issues with this learned model. One is that the function in proposed by the optimizer is too complex for a linear regression task, which indicates overfitting, i.e., trying to all the data perfectly. The other more significant issue, which directly leads to the failure of evaluating the model on any given data point, is that the function in the description consists parameters with unknown values. Such function is not fully defined, therefore, the inference error will be large. Similarly, can avoid such failure by adding instructions to the prompt template of the optimizer specilying, e.g., "must provide the exact value of the parameters if the description potentially involve unknown or learnable parameters".
Aggressive fitting. Figure 18 (d) shows a failure run for the polynomial regression task. Specifically, this run corresponds to the run 2 Table Reg-Poly (English) with DeepSock-V3, We can see from the figure that after the first step of optimization, the learned model is already quadratic function, and it is quite close to the ground truth. However, as the training progresses, the learned model deviates from the ground truth model class and becomes more complex function, which does have low training loss but cannot extrapolate outside of the training data distribution (i.e., for x < -3 and > 1). This is similar to cases where the learning rate is too high in classical machine learning, which causes the optimizer to escape from good local minima and end up in a worse solution. This failure happens less frequently for more powerful LLMs. as we can see from Table 3 Reg-Pot Poly (Euglish) where GPT-4o has 0 failure run out of 5.
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D Details on the Comparison between VML and APE
D.1 APE Experiments Details
For our APE experiments in Section 4.8, we use the code from the authors' GitHub repo?. Unlike in the APE paper [73] which uses GPT-3 as the LLM, here we use Llama-3-70B. Note that our VML experiments in Section 4.8 are also done with Llama-3-700 3-BB for a fair comparison.
Template for Candidate Prompt Generation
Template Evaluation
gave friend instruction. Based the instruction they produced the following Inpur-outpot pairs:
Instruction: <PROMPT>
Inpat:
X Sauuput < yj
Input: <x>>>
..
Output: <y>
Inpet:
x, Duttptt "3y
instruction was COMPLETED BY
Figure 19: Prompt templates used for our APE experiments.
The workflow of APE mainly has two steps that rely on LLM calls. The first step is to use the provided data in batches construct proposal queries to sample set of possible candidate prompts (see Figure 19(left) for the template we used). The second step is to evaluate each candidate prompt with the data (see Figure (9iiht) for the template we used), and choose the best candidate based on some metric. We use general metric for our experiments, which is the likelihood of candidate prompt.
There are a few hyperparameters for the APE algorithm. We tried out different batch size for the proposal queries, and we choose batch size the end. Another important hyperparameter we can set is mar lokens, the maximum number of tokens allowed in the completion. We tried both 50 and 500. The prompts we show in Section 4.8 Figure 10 are results for setting max tokens to 50. This gives the most concise and reasonable prompts, but due to the hard cutoff at length 50, the prompt can be incomplete. If we allow longer response by setting max _oc_tt to 500, it is still possible to have incomplete candidate prompts. At the same time, these longer prompts are often worse, as we know the ground truth prompt (or pattern description) is around one or two sentences. See Figure 20 for the result of the same text classification task in Section 4.8 but with mac tokens being 500.
D.2 Differences between APE and VML
Even though APE and VML are both trying to optimize a prompt towards a certain target, there are fundamental differences between the two. We provide pseudo code algorithms for each of them here to compare their differences. When applying APE's algorithm (see Algorithm 3) to a learning problem, it can be sunmarized with two steps. First, we generate a set of candidate prompts from the training data by letting LLMs compete the instruction needed to produce the æ, y pairs. Then, use score function to rank each candidate prompl, and choose the best one. In the case of VML (see Algorithm 4), we start with an initial prompt, and we infer the corresponding ÿ for each æ in the current batch of training data. Then, we ask an LLM to generate a better prompt that can explain the current batch of æ, y pairs taking into account the ý produced by the current prompt, and we iterate the same process on next batch of training data until convergence. We highlight two distinctions between APE and VMT. below.
'Gradient' v.s. (Gradient'-based the training data (see Figure 19 (left) for the template), then the one with the highest score selected within the set. In VML, the prompts are generated by asking LLM to explicitly reflect on the last prompt and the corresponding prediction ÿ, then propose a new prompt that can better predict the target y for the given æ. This process requires the optimizer LLM to explicitly reason the following: why does the last prompt produce the current タ; how to modify the last prompt to minimize the prediction error; what is a better description for the relation between æ and y. If we use the language from classical machine learning, we
In APE, the candidate prompts are sampled directly given only
Phttps://github. com con/ hwtp//eirp//ae httm:///trr/aataaaiccccomm htcpliis__ppppeeeeeeeee grompt_enngineer
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APE: (max: tokens-5000) Text Classiflcation
the number of vonels the input string.
The friend actually not doing this, they are implemenring a different rule.
you figure out they are implementing?
Are the outputs boolean values indicating true false? - Abhirath Mahipal Ang 12 '19 16:46 good question! you can assume outputs are values indicating true Aug
The rule your friend is implementing appears be: "Is the input femnale given name?"
Here's how the ourputs match the inputs:
y:
female
a e Marlene is female given name (1)
* es Ernest is male given name (0)
paa
"Paternal grandmaa" given name, "Grandma"
female given
*co rite s n: Courtesan not traditional given name, but it has been used as femaale given name in some contexts (1)
Henry male
rucc Bruce male given name (0)
R Rosee ez Rose is female given name (1)
Nathan male given
Courineys Courtney is female given name (I)
this rule 100% commonly associated with one gender in certain cultural contexts, However, based on the provided iuputs outputs, appeurs be the most likely explanation. Abbirath Mahipal Aug 12 '19 at 17:17
be for both males and females,
This
looking for. hoping it's the Stefam Aug '19 18:17
would figure the underlying pattern, rather than
just mathematical fanction, done? * Thank you! I'm glad I could help uncover the
Figure 20: Resulting prompt from APE text classification task with mose tokens beling 500,
say that the optimization in VML makes use of the 'gradient" information from the last prompt and the current batch of training data, while the optimization in APE is 'gradient-free.
Numerical score function v.s. Self-evaluation. Another important distinction that APE requires a predefined score function that can give numeric score to each candidate prompt. For example, one can use the log-likelihood of prompt the score, which normally requires access to the weights of LLM, hence it is only possible for the open-source models. In contrast, VML does not require such score function to evaluate the prompt. Evaluation of prompt in VML is done by the LLM itself purely in natural language. This is more flexible and agnostic to different LLMs (e.g., proprietary or open-source).
Algorithm 3 APE (Simplified)
Given: Derain (#n. )N Batch size M, Score function a(-);
// Step 1: Sample candidate prompts P P=[]
for i 1=,..,,/MM .N/M do
Get a batch of training examples #1.... Ats P.extend(.preppc (#1.... ..M,)) //
See Figure 19 (left) for the template for Sprop. (-)
end
// Step 2: Evaluate candidate prompts
S =1
for each p P do
Get batch of training examples #1.". MaMS Sp 0;
for m = 1, ...
Sp 5p s(p; mm Vml); end
// See Figure (right) for the template s(y; x, y);
Sexxtendss,/M]:
end
return pd C s.t. P.index(p") = arg max; S(i);
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Algorithm VML (Same as Algorithm but with more details)
Given: Daaii = (Z yu N, Initial prompt Ou, Iteration number T, Batch size M
for i = 1.... , .T'de do
1/ Stop 1: (Forward Pass) Use current prompt to do predictions Get a batch of M training examples #1,'!" mMi
for m = 1, .... M do
Gm = model [2m: 11
// See Figure 2 (left) for the template for smodat());
end
11 Step 2: (Bockword Pass) Update the prompt base on the predictions, current batch of data, and current prompt 0, Sopt (1 (Lm, Dms Mr m= 0z-1);
// See Figure 2 (right) for the template for fope(.);
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E
Effect of Accurate Loss Feedback
Inference result
Learner LLM LaMi fanid
Loss calculation
250
w/o Loss Feedback
Loss Feedback
Inference result
200
Update model parameters
Input text prompt
Train Loss
Optimizer fiM
150
(a) VML with accurate loss feedback
100
Learner LLM fanid
Update
Inference result
model parameters
Optimizer LLM fip
11
13
15
17
19
Step
(b) VML without accurate loss leedback
(c) Comparison of training dynarmies
Figure 21: Training dynamics for two different optimization settings in the polynomial regression setting. One has the accurate loss computation, and the other does not.
The VML algorithm at Algorithm specifies that the arguments for fopa(.) consist of the inputs the predictions D, the targets y, the current model parameter 0;-1 and the optimizer configurations sb Hence, there is no explicit definition of the loss function for the optimizer (see Figure 2(right) for an example of the verbalized loss function). is up to the optimizer itself to evaluate the difference between the prediction ý and the target y. We are interested in question that whether having access the real training loss (defined and computed for logging purpose); mean squared error in this case, can help the optimizer to better navigale the training trajectory.
The orange line in Figure (1(c) shows that having such accurate loss feedback might not help, and might even decrease the performance this scenario. One possible explanation is that the single loss value itself does not contain too much information. Moreover, as the exact form of the loss function can be fed to LLM easily, the LLM might spend additional efforts to estimate the exact form of the loss function, which makes the convergence even more difficult. It actually makes intuitive sense that verbalized loss function (i.e., using natural language to explain the target of the loss function) works better in the VMT. framework. For example, knowing how does each prediction contributes to the loss value can be more informative and single overall loss value, since the model might be doing well for some data but not the others, and we only want to improve the model for points with the bad predictions.
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F Numerical Error of LLMs in Representing Symbolic Functions
(x)
fxx 3x +x+2
f(x sin(x) +
Ground Truth
8-
12-
60
6-
40
8-
20
2
-20
-2.5
2.5
Model parameters in VMI.
Model parameters in VMI
Model parameters in VMI.
Output 4 Input
Output=3 3 Iapur Input 2
Output = sin(lopat) +
Figure 22: Functions evaluations and numerical Llama-3 70B
=
f = 3x +x
f = sin(xx +2
Ground Truth
70
12
Mesn over 10 nins
60
10
50
f(3)
20-
10
4
2.5
0.0
2.5
Model parameters in VML
Model parameters VML
Model parameters VML
Output Toput
Output Inpur"2 Input
Output sin(lopat)
Figure 23: Functions evaluations and numerical crror in GPT-4o.
LLMs designed do language modeling, rather than exact calculations. Hence, their performance evaluating functions be unreliable, and might result in error. Figure 22 shows that Llama-3 is very comfortable in evaluating the given linear and polynomial function, as the mean is quite accurate. The variance over 10 runs is also pretty small, except for one or two points. However, for a more complex function such as sin(r), Llama-3 is only able to return small crror approximately in the range of x E (-2, Both the error and the variance are large out side of this range. This explains the non-smoothness for the function Figure 5(b; right), which has sin(x 1.0) in the learned model parameters.
By switching to the more powerful model, GPT-4o, the variance decrease. In particular, for sin(x), GPT-4o returns smaller error in larger range, (i.e. x (-2.5, 5.0)). This implies that as the capability of LLMs improves, their performance in evaluating more complex functions also improves.
from Figure 23 that both the error and
Nevertheless, this is currently still a limitation for VML the optimizer chooses to use complex mathematical functions as the model parameter. If the evaluation of the function has an error, then during training, the optimizer will update the model parameters based on noisy signal. This can lead to large variance in training and slow convergence. Future work should look into methods for minimizing the numerical error in LLMs function evaluation.
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Model parameters
take sumber, number undergoes specific
fasction, muthematical
2.5
pattern repeats every SZipi Eapprosiastet: 6.28).
The values the sine function range from to and follow this persodic paitern: When Imput the value
1.5
(
3) the inpat increases to Sipt22 (opprosimately 1.57), the sine value increases smoothly
(oppresimately 314), the sine value continues
0.5
bock
4) As the Input renches 23/p (approsimately 4.71), the sine value dicreases farther to -I:
Flaully, iaput
S2ipi5 tapprosinaaell 6.28), cycle repeats.
0.5
After applying the sine funetion to the input, you always add to the resalting value. This shifts the sine wave vertically by unios
-5.0
-2.5
0.0
2.5
5.0
-5.0
-2.5
2.5
5.0
(a) Text description the target funciton
(b) Llama3-708
(c) GPT-4o
Figure 21: Function evaluations based the natural language description of the corresponding symbolic sine function.
Figure 24 shows that if we use natural language to describe the symbolic sine function (see sub- sub-figureaa)) GPT-4o is able to produce more accurate evaluations than using the symbolic function (scc (c)). The accuracy of Llama-3 70B also increases, even though it still under performs GPT-4o (see (b)). This is likely due to Llama-3 is less capable in instruction following than GPT-4o. This observation implies that in VML, we
might want to instruct the optimizer to avoid using complex symbolic functions in the update and to prefer the natural language description of the function.
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G Mitigating Numerical Error by Tool Calling
In this section, we supplement experiments of Llama-3 70B with a python interpreter. Despite the fact that LLMs are able to perform numerical data tasks, the incorporation of a python interpreter further improves LLMs ability to deal with numerical values. Specifically, we use the open-inlerpreder" library to add python interpreter to Llama-3 70B, such that the LLM has the ability to use python programs to evaluate symbolic functions or perform numerical operations. We follow the same experimental settings as in Section 4.3 (sinusoidal regression of y sin(x) +2). The training data only sampled from [-3,3] with additive Gaussian noise. The in-domain testing data is sampled from the same range, while the out-of-domain testing data is sampled from [-6, -3] and 6].
Test Domsin
Neural Nets
Llamad with peice Llamai with prior and code Interpeoter
In Domain
0.0073
0.7385
1.0e-05
Out Domain
0.58-0
11.1735
1.0c-05
Overall
0.2345
Table 5: Evaluation (using squared error 4) sinusoidal regression Figure 5(b) for three different models including (1) neural networks, (2) Llama3 with prior, and (3) Liame8 with prior and code interpreter.
From the table, we can observe that with the python interpreter, Llama-3 70B can effectively learn periodic functions, while in the original experiment (í.e., Figure 5(b)), the same LLM is unable to approximate periodic function even with prior. The results show that the tool-using ability can further improve the
learnability of VML. The example logs for inference with the learned model is showed below.
Inference History: Llama-3 70B with Code Interpreter Example Logs
Forward Pass Input
Pattern Descriptions: use the descriptions to predict output given input.
model.
You are designed to do regression, l.e., to predict output of any given input. Both lnput output are real numbers, looks like the data generated periodic function. Myy Nsin(1x 0)+ 2\)
Input:
[-0.09]
Please give your FINAL output "STRICTLY IN THE FOLLOWING PORMAT" at the end:
=*
Explanations: [Your step-by-step analyses and results; please use python for evaluating aymbolic function]
Output:
[Your output MUST be REAL NUMBER ROUNDED TO TWO DECIMAL POINTS; MUST be the same format the
Input]
Please ONLY reply uccording to this format, don't give me any other words.
Forward Pass Output
Step 1:
Define the function
The given function Wy 1 Veinn(x 0)+ 2W.. This is simple sine function with an amplitude of 1, a period of W/2VPpiV)) and phase shift of The function is then vertically shifted by units.
Step 2: Evaluate the function at the given input The input
(9. We need evaluate the function input.
import nath
def ovaluate_ (aaluato_fucctio(e))
math.
-0.09
result avalnate, Sttiiiattttuccoiinnoo function
(x)
print(resulty
1.910121450801989
Explanations: ** We defined the function sin(1x 2 and evaluated it the given imput -0.09. The result of
the function evaluation is approximately .991012111080199..
** Output: *
[1.91]
Forward Pass Output
** Explanations: **
We defined the function
sin(1x 2 and evaluated it the given imput -0.09. The result of the function evaluation
approximately 1.91012445801996
Output:
[1.91]
3GitHub Repo: http:///ss.t..tccceeaaeaeeeteeee
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H
Connection between Prediction Variance and Model Parameters in VML
H.1 From Vague to Concrete Model Parameters
Ground Truth
12
Mean over 10 runs
12
10
10
8
f(x)
4-
-2-
-2
parameters
parameters VML
parameters
There lincar relationship with positive slope between iaput the output.
Output -b a Input
(a)
(b)
(c)
20
12.5
10.0
15
f(x)
10
2.5
0.0
Model parameters VML
Model parameters VML
Model parameters VML
Output = + lapur, where is in the range of range []
Output = h++a Iupor, where a is the range of [2. the range [3,5]
Output b - Iaput, where is in the range range of
(d)
(f)
Figure 25: Evaluations model parameters using vague Llama-3-70B.
descriptions. Results
The base LLM is
The model parameters generated by VML optimizer can be vague concrete. We are curious for those with vague descriptions, how would the LLM evaluations look like, and whether they have large variance. Figure 25 shows the results on Llama-3 70B for six different model descriptions, including:
(a) None
(b) "There is linear relationship with positive slope between the input and the output."
(c) "Output = + Input."
(d) "Output = + Input, where a is in the range of [0, 6] and b is in the range of [0, 8]"
(e) "Otputt
Input, where is in the range of [2, and b is in the range of [3, 5]"
"Output
Input, where a is in the range of [3, and b is in the range of [4, 6]"
(a) shows that if we only provide the information that the task is a regression task and do not specify the model at all, the LLM tends to predict linear function (slope 1) with increasing variance æ moves away
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from O. (b) shows that if we specify there is linear relationship between inputs and outputs, the LLM will predict a linear function with a similar slope as (a) but with smaller variance. (c) shows that if we specify the explicit form of the linear function, the slope will still be around 1, but the variance are larger when I > 1. (d, c, f) show that by providing a range for the values the unknown variables, the LLM tends to use the mid-point of the range for the values, and a smaller range does correspond to a smaller variance prediction.
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H.2
Semantic Invariance of Model Parameters
14
Ground Truth
14
14
12
12
12
10
10
10
(x)
2-
0-
0-
Model parameters VML
Model parameters in VML
Model parameters in VML
The new pattern deseriptions will be hased Uncar regression model scaling factor and term of 3.28. The now pattern descriptions are:
The updated pattern definitions utilizze linear regression
characterized slope 3.34 and an intercept of 3.28. The resised pattern equations are expressed as:
3,34s 3,28
= 3,34x 3.28
3,34x 3.28
where y the predicted output and is the Input.
where denotes the estimated outpur, and represents input variable.
(a)
(b)
(c)
14
15.0
12.5
[(]
f(0
5.0
2.5
0.0
0-
Model parameters VMI
Model parameters YML
Model parameters in VMI.
The new pattern deseriptions will be derived from model that predicts outcomes based on iaput and This model the flnal prediction. Here. the predicted result is determined by this specifie adjustment and addition applied the Input.
Amaddi the danox cimberrs nee deias takes berrs
Where putta converus clezance
# x 2 #
imput factor
And pasdly mummus. Three pait distt'
A *
value
generate
With ispuls crolled saftty dae paliens deo reveal.
skotched sumbers, praphncy made
MALEASA. *
lincus harmuss, they find ther rigktfal piace.
So live it the seng of and cxtniaed.
autlonstical preesiss redefined.
(d)
Figure 26; Evaluation on the model parameters from Figure 3(c; Step 15) using six different rephrasing with the same semantic meaning. 10 The base LLM is Llama-3-70B.
Results
In natural language, there are different ways to express a concept with the same semantic meaning. Hence, the model parameters generated by VML optimizer might vary lot between runs, even though they are semantically invariant. We are curious whether such variance in descriptions will lead to variance in model evaluations. Figure 26 shows that results on Llama-3 70B for six different but semantically invariant descriptions of the model from Figure 3(c; Step 15), i.e.,
(a) "The new pattern descriptions will be based on a lincar regression model with scaling factor of 3.34 and bias term of 3.28. The new pattern descriptions are:
y 3.34x 3.28
where y the predicted output and x is the input."
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(b) "The updated pattern definitions utilize linear regression framework characterized by slope of 3.34 and an intercept of 3.28. The revised pattern equations are expressed as:
y = 3.34x + 3.28
where y denotes the estimated output, and x represents the input variable."
(c) "y = 3.34x + 3.28"
(d)
"The new pattern descriptions will be derived from model that predicts outcomes based on input values. This model adjusts the input by a factor of 3.34 and adds a constant value of 3.28 to generate the final prediction. Here, the predicted result is determined by this specific adjustment and addition applied to the input."
(e) "Amidst the dance of numbers, a new design takes form, Where linear paths converge with clegance and charm. scaling factor whispers, 'Three point three four,' And bias gently murmurs, 'Three point two eight,' no more.
With imputs cradled softly, the pattern docs reveal, A future sketched in numbers, prophecy made real.
Y, the destined output, unfolds from X's grace, In linear harmony, they find their rightful place.
So here it is, the song y and x entwined, mathematical ballet, precision redefined." "
(f)
K #HEREAE,, (BЕクーパ
# F:
C HO 1
tR L X, 01
These rewrites are generated by GPT-4o based on (a). The description (a) is the original 01s from Figure 3(c; Step 15). (b) rephrases the descriptions from (a) slightly. (c) only keeps the symbolic equation from (a). (d) is rewrite without using math expression. (e) uses the poetry style. (f) is translation of (a) into Literary Chinese. The results in Figure 26(a,b,c) are similar, and have small variance across the 10 runs. The results in Figure 26(d,e.f) are also very accurate average. However, the poetry rewrite (e) and the Chinese rewrite (f) do have slightly larger variance. Overall, we see that if the various descriptions preserve the same semantic, then their evalustions through Llama-3 70B are likely to be similar.
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A Probabilistic View on VML
The output of a language model usually comes with randomness. In the paper, we typically consider to set the lemperature in LLMs as zero to remove the randommess from sampling, which indicates that LLMs will always output the text with the largest probability (É.e., largest confidence logit). However, we want. to highlight that such a sampling process actually gives us another probabilistic perspective to study VML. We will briefty discuss this perspective here.
Posterior Predictive Distribution
Because we can easily sample multiple possible model parameters by setting a proper temperature for the optimizer LLM, we view as way sample multiple learner models. This is well connected to Bayesian neural networks, where Bayesian inference is applied learn probability distribution over possible neural networks. We start by writing the posterior predictive distribution (D is training data):
P( = / p(ù ()p = E {p(0)0)}
where we can easily sample multiple model parameters and compute its probability with logits. Specifically, ... .,. 1,- D). Using this idca, it actually quite casy obtain the ensembled output that is weighted by posterior distribution.
we have that p(0)D) II. P(0,
1.2 From Functions to Stochastic Processes
With non-zero lemperature, we can view the oulput of LLMs as a sampling process from a distribution over text tokens, which means each output token can be viewed a random variable. Then the output of LLMs is effectively sequence of random variables, and therefore easy verify that it is stochastic process. This view makes it possible for VML to perform probabilistic modeling.
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J Discussions on Natural Language Model Parameters
There are many interesting properties regarding the natural language model parameters. Many traditional machine learning models can be revisited in the scenario where model parameters text prompts in the LLM.
J.1 Different Mechanisms to Update Model Parameters for Direct Optimization
Naive re-writing. Given the model parameters Og the step i, the simplest way to update the model parameters at the step & + 1 is to whatever the optimizer generates. We denote the optimizer L.M generates the new model parameters of new This is essentially
0e+1 f- O O
(4)
An simple extension to naive re-writing is add text prompt to instruct the optimizer LLM to take the previous model parameters 0₂ into considcration at the step + 1. Thus we have the conditional ro-writing, namely 0c+1 (opi(0)) This is also what we use in the main paper.
Incremental updating. Alternatively, we can choose to update the model parameters in an incremental fashion without remove the previous model parameters completely. We denote the optimizer LLM generates the new model parameters Ieww Then the model parameters 0.+1 at the step t +
0+11 + -0,, 0 n
(5)
However, the incremental updating will make the model parameters an increasingly longer text prompt. This is not ideal since the context window of a LLM is typically quite limited. The incremental updating mechanism can interpreted as using a small learning rate to train the learner. This will easily lead to bad local minima (because the previous incorrect model parameters will be kept and may affect the future learning a prior knowledge in the text prompt), but it may improve the training convergence.
Incremental updating with summarization. To avoid the infinite increasing length of model parameters, we can instruct the optimizer LLM to summarize the previous model parameters into fixed length. This yields
oc+1 +
C(O.)
O'ow
(6)
fixed token length
where C(.) is some text summarization scheme.
Connection to standard optimizers. There are many interesting connections between the optimizer LLM and the standard numerical optimizer. Usually the behavior of the optimization is determined by the optimizer parameters vb which is also a text prompt. This is usually a text description of the target of the optimizer. For example, we can instruct the optimizer LLM to serve as the first-order optimizer (e.g.. momentum SGD) and feed all the necessary information into the text prompt. Then the optimizer L.L will essentially become an optimizer mapping function that maps all the necessary information (including the previous model parameters) to the model parameters of the next step. To implement the momentum in the optimizer LLM, one can simply instruct the optimizer LLM to maintain the previous model parameters as much as possible. This is to say, everything we want to implement in the optimizer are realized through text prompts. It will inevitably depend on the instruction-follwviii ability of the LLM, and it is possible that there will some unrealizable optimization functionalities (e.g., we instruct the optimizer LLM to be a socond-order optimizer and the optimizer LLM may be likely ignore this instruction). However, we want highlight that as LLMs become more powerful, this problem will be less and less significant. In general, implementing advanced optimizer in VML is still an important. open problem.
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J.2 Occam's Razor, Constrained-length Model Parameters, and Kolmogorov Complexity
We are interested in how Occam's 's razor can be applied in VML. One natural way of doing so is to constrain the model parameters to be a small and fixed length. This essentially is
O Oc+1
{
C(O,)
O
(7)
new
fixed tokeen length fixed token length
We can see that as long as we constrain the text token length of the model parameters to be small, the learner will perform an automatic model simplification, as it will try to discover the data pattern with concise and simple text. There are many more ways to implement the Occam's razor in VML. More interestingly, it is also possible to incorporate structural constraint to the model parameters. For example, it can be causal knowledge (e.g., text reprosentation of a causal graph), logic formula or decision trees. Our work opens up many more possibilities on Occam's razor in VML, and rethinking the form of Occam's razor in VML is very crucial in unlocking the strong interpretability and controllability of inductive biases.
Another perspective on the length of the model parameters in VML related to Kolmogorov complexity [19], which is defined as the shortest effective description length of an object. The principle of Occam's razor is basically saying that hypotheses with low Kolmogorov complexity more probable [53]. By constraining
the length of model parameters to be small, we are effectively trying to find the minimum description length (MDL) of model in natural language. The theoretic Kolmogorov complexity of a model is usually impossible to compute, however, VML might, provide an estimation for Kolmogorov complexity by using the shortest. effective length of the learned model parameters in natural language.
J.3 Connection to Nonparametric Models and In-context Learning
Nonparameteric methods get around the problem of model selection by fitting a single model that can adapt its complexity to the data [55, 39, 12). These methods allow the model complexity to grow with the number of observed data. This is different to panametric models which have fixed number of parameters. In VML, as showed in Section 4.2, the model complexity is also flexible and adapts to the data during training. Similarly, the concept of in-context learning (ICL) can also be understood nonparametric methods in the lens of LLMs as function approximators. ICL denoles the method of using LLMs to solve new lasks by only providing the task demonstrations or examples in the prompt with natural language. Given a new data point, an LLM predicts its output using information in the provided demonstrations. From the perspective of VML, ICL in an LLM essentially defines a nouparametric model implicitly using the demonstrated examples in the natural language space.
J.4 Distinctions between the Data Dimension and the Parameter Dimension
We would like to point out that there is a distinction between the input data dimension and the parameter dimension. For example in Appendix B.6, even though the input data dimension is 10-D, the parameter space (f.e., the description of the model in natural language) is actually much larger than 10-D. As we are optimizing the natural language parameters, rather than the input data. In our experiments, the natural language based parameters can have the dimension of 10 tokens (see Figure 4(c) model parameter 0₃) to 600 tokens (see Figure 6(c) 0s1).
The task we show in Figure a 2-D plane binary classification task. Fach data point on the plane only has 2 dimensions/ features. This, of course, docs not mean that the optimization dimension is only in 2-D space. Depending on the model class we choose for the (ask, the parameter space can have various sizes of dimensions, even infinite dimension. For example, if we use a parametric model such a three-layer neural network with weights of shapes [2 10) and [10 1], then the parameter space has 30 dimensions. However, if we use a non-parametric model such as a decision tree, then we do not have a fixed number of parameters. The number of parameters for a decision tree depends on the number of nodes in the tree, which grows as the tree adapts to the training data.
Contrary to the two classic numerical based models above (ie., neural nets and trees) where the parameters are numerical values, one innovative concept of VML. as have described in Section 3.1 to 3.4, is to use
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natural language space the parameter space. This means that the optimization of model in VML is done by changing the text, which might happen to be digits as digits is a type of text, but it does not have to be digits.
For example, assuming we have the space 100 tokens available for describing the model, then the optimizer is free to use these 100 tokens describe the model with a formula "y az b', where 'a' and 'b' will be the only two parameter here in the classical sense, but in VML the parameters are all the token in the string of "y ar + b'. This is why in Section 4.2 Polynomial Regression and Figure 4, the optimizer is able to update the VML model parameters from 'output 2.5 imput + 1.5' to 'output 2.2 input "2 + 1,8* imput + 0.6.. This optimization step is not possible in the classical way if we only view 'a' and 'b' as the paramcters.
Therefore, in each optimization step, from VML's perspective, the optimizer can add new lokens and optimize the existing tokens. These tokens might correspond to adding new parameters (e.g., Step in Figure 4: from 'y = ar + D' to 'y = ax? + bæ +') or optimizing the existing parameters (e.g., Step Figure 4) the classical definition of parameters. Since the real parameters in VML are the tokens, both of these two operations can happen at the same step. Hence, we should really use tokens to understand the parameter space in VML.
Another example is the medical image classification task in Section 4.6 and Figure 8. The optimizer updates the model to consist of only semantic descriptions features without any numbers. In this case, if not using number of tokens as the parameter dimension as VML, it is hard to define the corresponding classical parameters and the dimensions.
As for the experiment in Figure 6, like the other experiments, we do not add any prior information the model class, so the optimizer is free to choose any appropriate class of model to solve the task. The Figure 6 shows an example run which ends up with using decision trees as the model class. As we mentioned above, the learned model is also a result of optimization in the text space, which is in the dimension of the number of tokens. Even if we do use a classical decision tree algorithm (instead of optimizing in the text space with VML), assuming we get the model in Figure 6(c), the parameter dimension still much larger than 2 as this decision tree has many nodes (e.g., the number of if-else pairs).
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K Broader Discussions
In this section, we use the format of Q&A to discuss a list of interesting topics that are loosely related to VML, but are more broadly tight to the capability of LLMs. Some of the questions might seen philosophical or ideological, but were asked by fellow researchers before. Nevertheless, we still include them into this section in case some readers find them insightful.
K.1 How is the optimizer's statement 'the function should be y = m x + more interpretable than learning linear function?
The interpretability from VML is on the framework itself. Using natural language to characterize the model can reveal exactly what pattern the model learns from data, which is very different from training neural networks from scratch. As for the case of regression problems the paper, interpretability comes from (1) automatic model selection with explanations: this is different from common practice where we assume the data is linear and use linear regression model. In our experiment, we don't have such a prior and the optimizer will learn this linear pattern purely by exploring the data. The closest equivalent from classical ML methods would be to train an "universal approximator", e-g., neural networks, which might decide to fit a function that is roughly linear, but has a lot more parameters and less interpretable; (2) another source of interpretability comes from the property that the user can easily interact with the optimizer and follow up with more questions to sock explanations.
K.2
Is controlling the hyperparameters of LLM optimizers, such as learning rate and regularization, more difficult than controlling those of traditional ML optimizers?
Exploring the hyperparameters of LLM optimizers is important yet challenging. It is great rescarch question for VML. One of the reasons that VML is particularly interesting is that it brings lot of new research questions.
LLM optimizers have both advantages and disadvantages. The procise control of learning rate and momentum can be difficult. However, adding the qualitative effect of high /slow learning rate and momentum is in fact quite easy. One can simply use language to describe it. In our optimizer prompt, we the concept of momentum (e.g., "update the model parameter without changing it too much" and provide a constant amount of optimization historics). In terms of regularization, it is also casy to add regularization to control the complexity of the model in VML, e-g., the word length of the model parameters (i.e., a form of Occam's 's razor). qualitative hyperparameter control for LLM optimizers is simple, while this can be challenging for classic ML.
K.3 LLMs are optimized for natural language understanding and generation, not for numerical data tasks typically associated with machine learning. Are LLMs fundamentally restricted for machine learning tasks?
Numerical data tasks are heavily studied in LLMs, for example, mathematical problem-solving. The popular MATH dataset [13] requires strong numerical data processing from LLMs, and this dataset is used as a standard evaluation benchmark for LLMs. Moreover, there exists many LLMs (e.g., DeepsockMath [47], WizardMath [31]) that are capable of solving competition-level mathematics problems.
Moreover, LLMs have shown remarkable potential in numerical data tasks for machine learning, and our work is one of the first methods to reveal such a potential. Some concurrent works [46, 70] also gave empirical evidence that LLMs can be fundamentally suitable for machine learning tasks.
Verbalized machine learning aims to provide a framework for LLMs to deal with machine learning tasks, with the ability to fully interpret the learned knowledge with natural language. We believe this framework will be increasingly more powerful, as LLMs get more powerful. We have already observed the performance improvement. of VML. by switching from Llama-3 to GPT-4o.
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K.4 The fundamental nature of LLMs is to predict (the next token) based on probability distribution over the vocabulary. One might argue this process is based on statistical choice rather than on true understanding. Is it meaningful to use LLMs for applications such as machine learning tasks?
We believe VML does represent meaningful direction to explore, as there is current no evidence that LLMs can not perform ML tasks. On the other hand, we already have quite a few applications that demonstrate the effectiveness of VML (e.g. medical image classification). In fact, even in-context learning can already perform a few ML tasks (as introduced by GPT-2 and GPT-3 papers [44, 8)). We believe there are a lot of applications to be unlocked in the VML framework.
Whether one should use LLMs for tasks other than language modeling is indeed an important open question, which is currently under active research with a significant number of researchers in the field investigating the boundary of LLMs' capability, and trying to explain the 'seemingly' emergence of such abilities from the simple language modeling training objective.
The argument that LLMs can not elicit true understanding due to its statistical training is debatable. Firstly, it is unclear what it means to train a model based on true understanding. One can not perform such a training without an explicit form of loss function. On the other hand, there some analyses that show that next-token prediction induces a universal learner [33]. Secondly, we believe that there is a distinct difference between low-level statistical training and high-level knowledge understanding. Whether one can induce another is unknown and is also out of the scope of our paper.
Currently, there has already been substantial evidence that LLMs possess a form of understanding that is functionally relevant for many real-world tasks. The fact that they can consistently generate useful and accurate outputs across various domains, including numerical math [13, 2], theorem proving [65], biology [30] (just to name few), challenges the argument that LLMs lack real understanding.
Hence, we believe to argue against the use of LLMs for tasks other than language modeling, such as math related problems, will require an equally substantial amount of empirical evidence or theoretical proof, which is missing at the moment.
K.5 Hallucination remains a significant issue for LLMs. How can we trust them to handle complex tasks such as VML?
Even though hallucination an observation associated with LLMs, does not fundamentally limit the performance of VML. Note that by definition, hallucination an event with a low probability, if an LLM always hallucinates, we will not call it hallueination and it will not be able to outperform humans in many benchmarks. In the case of VML, there only two types of LLM calls, i.e., fmodel(-) and fapa(-). Let's go through what happen will if there is hallucination for each of these calls, and why hallucination is not a fundamental limitation for VML.
Hallucination in (-)dal()) is when the LLM does not follow the model parameter to infer the output of given imput. For example, if the model parameter is 'output 4 + 3 Input' and it returns incorrect answer, this will be an example of hallucination. This can happen when we apply it to numerical tasks (see Appendix F). However, based on the fact that hallucination is a low probability event, it will not hallucinate random outputs for most of the training data. Therefore, the set of predictions ÿ would still provide useful information for the optimizer, even though it is noisy. This is exactly what happened for many of our experiments. For instance, in Section 4.2 Figure 4(b; both mid and right) we can see a small outlier (a dent) in the plot of the quadratic function, which due to such hallucination. Nevertheless, VML is still able to learn very good model.
Hallucination in fopt(-) is when the optimizer does not produce a sensible model parameter for the current step of training. From our experiment scction, we can indeod see that many of our case studies have a non-monotonic training loss, some of the fluctuation can be explained by hallucination (see faihure cases in Appendix C for example). However, since hallucination is low probability event, unsatisfied model
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parameter will most likely be corrected in the next step of optimization. Therefore, in most runs, VML eventually learns very good model. For cases where it cannot correct itself from the hallucination, they will be identified as failure cases by simply checking the learn model, the training loss, or the test set performance. As discussed in Appendix C, by adding more detailed instruction to the optimizer prompt template and switching to a more powerful LLM, these failure cases (a superset of hallucination) will occur less frequently.
Across our experiments, we did not witness any empirical evidence that hallucination is affecting the reliability of VML. The core of VML is that a model is characterized by the text prompt of an LLM. Whether the model parameter is good depends model parameter works well in the downstream task, it is highly unlikely that the model parameter is based on hallucination, because the hallucinated text is unlikely to demonstrate consistently good performance for all the data points. If the learned well-performing model parameter seems unexpected, it is more likely that the LLM-based optimizer discovered new knowledge from the training data. than have hallucinated.
its downstream performance (e.g., the training accuracy). Therefore, if
More importantly, hallucination in LLMs does not limit the usefulness and significance of using LLMs to solve numerical tasks (such solving math problems). Therefore, we don't view hallucination for VML, but rather opportunity for future improvement. In reality, we observe that hallucination does not limit the development of utilizing LLMs to solve math problems, which requires the capability of LLMs in understanding numbers and mat hematical functions. The math task even requires more precise reasoning from LLMs. However, LLMs has still achieved tremendous progress in math reasoning, despite the potential of hallucination. For example, DeepMind's AlphaProof[3] (based on Gemini) recently reached IMO silver level performance. One can also observe the progress of LLMs on the MATH dataset from around 15% [52] to 84% [62] (The MATH dataset has competition-level mathematics problems with many numerical computations). The inherited problem of hallucination does not seem to invalidate the field of Al for math. Therefore, it is hard to be certain, in particular without empirical evidence, that hallucination will be critical limitation for VML.
problem
We agree that current LLMs have many problems (e.g., hallucination, finite context-window). All these shortcomings of LLMs are being actively studied today. We believe VMI. actually an orthogonal independent contribution to these existing LLM research topics. VML studies how to enable interpretable learning using natural language. As LLM gets more powerful and more faithful, VML will also become more useful.
We also want to highlight that the goal of this work is not to propose method to replace numerical machine learning, nor to claim superiority of VML over numerical machine learning. We are simply sharing this now possibility doing learning in the LLMs era, and showing evidence that it can indeed learn for many lasks. VML does have many features which the numerical machine learning does not have, such as interpretability and adding prior with natural language, but it is also not scalable at the moment.
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L Complete Training Template at Initialization
L.1 Linear Regression
Text prompt template for the learner
You are the model. You will use the descriptions below to predict output the given input.
Pattern Descriptions:
You designed to regression, i.., to predict the output of any given input. Both input and output real numbers.
** Input: ** [SData]
Please give your output strictly in the following format:
Explanations:
[Your step-by-step analyses and results]
Ouiput:
[Your output MUST be REAL NUMBER ROUNDED TO TWO DECIMAL POINTS; make necessary assumptions if needed; it MUST be the same format the Input]
Pleuse ONLY reply according this format, don't give any other words.
Text prompt template for the optimizer
You are the optimizer for a model, your goal to learn the best descriptions for the model. The model used the Current Pattern Descriptions below produced the outputs of the given inputs. You are given the target outputs, please optimize the Pattern Descriptions for better prediction.
Inputs (a batch of L..dd data):
[[SData] [SData] [SData] [SData] [SData] [SData] [SData] [SData] [SData] [SData]]
** Current Pattern Descriptions: **
You designed to do regression, i..., predict the output of any given input. Both input and output are real numbers.
The model outputs:
[[SPrediction] [SPrediction] [SPrediction] [SPrediction] [SPrediction] [SPrediction] [SPrediction] [SPrediction] [SPrediction] [SPrediction]
The target outputs:
[[SGroundTruth] [SuroundTruth) [ruthundTruth) [SGroundTruth (SGroundTruth) [SGroundTruth) [SGroundTruthi [SGroundTruth) [SrroundTruthh] [SGroundIruthi]
If the model is doing well, you can keep using the current descriptions. However, if the model is not performing well, please optimize the model by improving the 'New Pattern Deseriptions! The model uses the 'New Pattern Deseriptions' should better predict the target outputs of the given inputs, as well as the next batch of l..dd input data from the same distribution. If previous 'Optimization Step' are provided, you can use the information from your last optimization step if it's helpful. Please think step by step and give your outputs strictly in the following format:
Reasoning:
[be explicit and verbose, improve the Current Pattern Descriptions by yourself:]
New Pattem Descriptions:
[put your new descriptions here; MUST be specific and concrete;]
Please ONLY reply according this format, don't give any other words.
Figure 27: Prompt templates of VML for the learner and optimizer for the linear regression (Llama-3-70BB without prior).
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L.2 Polynomial Regression
Text prompt template for the learner
are the model. You will use descriptions below to predict oulput the given input.
** Pattern Descriptions: **
You are designed to do regression, ie., to predict the output of any given input. Both input and output are real numbers.
Input: **
[SData]
Please give your output strictly in the following format:
Explanations: [Your step-by-step analyses and results]
Output:
[Your output MUST be in REAL NUMBER ROUNDED TO TWO DECIMAL POINIS; make necessury assumptions if needed; it MUST be in the same format as the Input]
Please ONLY reply according to this format, don't give me any other words.
Text prompt template for the optimizer
You the optimizer for model, your goal Ieam the best descriptions for the model. The model used the Current Pattem Deseriptions below produced the outputs of the given inputs. You given the target outputs, please optimize the Pattern Descriptions for better prediction.
** Inputs (a batch i.i. data): **
[[SData] [SData] [SData] [SData] [SData] [SData] [SData] [SData] [SData] [SData]]
Current Pattern Descriptions:
You are designed to de regression, i.c., to predict the output any given input. Both input and output are real numbers.
The model outputs:
[[SPrediction] [SPrediction] [SPrediction] [SPrediction] [SPrediction] [SPrediction] [SPrediction] [SPrediction] [SPrediction] [SPrediction]]
The target outputs:
[[SGroundTruthh [SGroundTruth] [SGroundTruth) [SGroundTruth) [SGroundTruth) [SGround Truth]
(SGroundTruth) [SGroundTruth)] (SGroundTruth) [SGroundTruthi]
If the model is doing well, you can keep using the current deseriptions. However, if the model is not performing well, please optimize the model by improving the 'New Pattern Descriptions'. The model uses the 'New Pattern Deseriptions' should better predict the target outputs of the given inputs, as well as the next batch of i..dd input data from the same distribution. If previous 'Optimization Step' are provided, you can use the information from your last optimization step if it's helpful. NOTE: both the model and you can only operate on the numerical precision of one decimal points! Please think step by step and give your outputs strictly in the following format:
Reasoning:
[be explicit and verbose, improve the Current Pattern Descriptions by yourself, please show your work; note that you don't have access to computer]
New Pattern Descriptions:
[put your new descriptions here; MUST be specific and concrete; MUST provide the exact value of the parameters if the descriptions potentially involve unknown or learnable parameters!!!a4aay
Please ONLY reply according to this format, don't give any other words.
Figure Prompt templates VML the learner and optimizer for the polynomial regression (Ilama-3-70BB without prior).
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L.3 Sinusoidal Regression
Text prompt template for the learner
You are the model. You will use the deseriptions below to predict the output of the given input.
Pattern Descriptions: **
You are designed to do regression, I.e., to predict the output of any given input. Both input and output are real numbers. looks like the data is generated by a periodic function.
Input:
[SData]
Pleuse give your output strictly in the following format:
Explanations: (Your step-by-step analyses and results]
Output:
[Your output MUST be REAL NUMBER ROUNDED TO TWO DECIMAL POINTS; make necessary assumptions if needed; it MUST be in the same format the Input]
Please ONLY reply according this format, don't give me any other words.
Text prompt template for the optimizer
You are the optimizer for model, your goal is to learn the best descriptions for the model. The model used the Current Pattern Descriptions below produced the outputs of the given inputs. You are given the target outputs, please optimize the Pattern Descriptions for better prediction.
Inputs (a batch of i..dd. data):
[|SData] [SData] [SData] [SData] [SData] [SData] [SData] [SData] [SData] [SData]]
** Current Pattern Descriptions: **
You designed to do regression, i.e., predict the output of any given input. Both input and output real numbers. It looks like the data is generated by periodic function.
** The model outputs:
[[SPrediction] [SPrediction] [SPrediction] [SPrediction] [SPrediction] [SPrediction] [SPrediction] [SPrediction] [SPrediction] [SPrediction]]
Thee target outputs:
[rrrroundTruth TrrthundTruth) [SGroundTruth) [SGroundTruth] [SuroundTruthh] [SGroundTruthh [SGroundTruthh] [SGroundTruth] [ruthundTruth] [ruth]]
the model is doing well, you can keep using the current descriptions. However, if the model is performing well, please optimize the model by improving the 'New Pattern Descriptions'. The model uses the given inputs, as well the next batch of i.i.d. input data from same distribution. If previous 'Optimization Step' are provided, you can use the information from your last optimization step it's helpful. NOTE: both model and you can only operate on the numerical precision of one decimal points! Please think step by step and give your outputs strictly the following format:
'New Pattern Descriptions' should better predict the target outputs outputs of
Reasoning:
[be explicit verbose, improve Current Pattern Descriptions by yourself; please show your work; note that you don't have access to computer]
New Pattern Descriptions:
[put your new descriptions here; MUST be specific concrete; **** MUST provide the exact value of parameters if the descriptions potentially involve unknown or learnable parameters!!! s
Please ONLY reply according to this format, don't give me any other words.
Figure 29: Prompt templates of VML for the learner and optimizer for the sinusoidal regression (GPT-10 with prior).
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L.4 Two Blobs Classification
Text prompt template for the learner
You are the model.
Model Descriptions:
are designed binary classification. The input is point a plane [x yl; you need output a vector containing two probabilities such that each corresponds how likely the data belongs each class, iee. [class prob. class 2 prob.j. The sum of the vector MUST be 1.0.
** Input: *4
[SData]
Please give your output strictly in the following format:
Explanations:
[Your step-by-step analyses and results]
Output:
[ONLY A PURE probability vector, where cach value is between 0.0 and 1.0 WITH TWO DECIMAL POINTS;
make necessary assumptions if needed]
Please ONLY reply according this format, don't give any other words.
Text prompt template for the optimizer
You the optimizer for a model, your goal is to learn the best deseriptions for the model. The model used the Current Model Descriptions below predicted how likely the given inputs belong to class, You given the target values, please optimize the Model Descriptions for better prediction.
** Inputs batch i..dd. data 2-D plane: [x y]): **
[[SData] [SData] [SData] [SData] [SData] [SData] [SData] [SData] [SData] [SData]]
Current Model Descriptions:
You are designed to do binary classification. The input is point 2-D plane [x yl: you need to output vector containing two probabilities such that each corresponds to how likely the data belongs to each class, i.e. [class prob. class prob.j. The sum of the vector MUST be 1.0. If 0, output [0.0, 1.0). Ifx if 8.5, output [0.0, 1.0), otherwise output [1.0, 0.0].
** The model predictions ([class I prob. class prob.]): **
[|SPrediction] [SPrediction] [SPrediction] [SPrediction] [SPrediction] [SPrediction] [SPrediction] [SPrediction] [SPrediction] [SPrediction]]
The targets ([class 1 prob, class 2 prob.J):
[|SGroundTruth) [SGroundTruth) [SGroundTruth) Truth) [SGroundTruthh] [SGroundTruth)
[ruth [SGroundTruthh] [SGroundTruthi [SGroundTruthh]
Please update the model by improving the 'New Model Descriptions', which should have lower classification error both on current and the next batch of f.i.d. data. previous 'Optimization Step' are provided, you can use the information from your operate the numerical precision of THREE decimal points. You are bad with numerical calculations, so be extra careful! Please think step by step and give your outputs strictly in the following format:
optimization step if it't's helpful. Both model you MUST ONLY
Reasoning:
[be explicit and verbose, improve the Current Model Descriptions by yourself, please show your work; note that you don't have access to computers]
New Model Descriptions:
[put your new decision rules here; MUST be concise and concrete; MUST PROVIDE THE EXACT VALUE OF THE PARAMETERS if the descriptions potentially involve unknown or learnable parameters!!!***9!
Please ONLY reply according to this format, don't give me any other words.
Figure 30: Prompt templates of VML for the learner and optimizer for the two blobs classification (Llama-3 1-ammn---7000 70B without prior).
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L.5
Two Circles Classification
Text prompt template for the learner
You are the model.
Model Descriptions:
You designed binary classification. The input is point 2-D plane [x yl: you need output the class label, Le., an integer the {0, 13. The decision boundary circle.
** Input: **
[SData]
Please give your output strictly in the following format:
Explanations: [Your step-by-step analyses and results]
Ouiput:
[ONLY the integer class label; make necessury assumptions if needed]
Please ONLY reply according this format, don't give me any other words.
Text prompt template for the optimizer
You are the optimizer for a model, your goal is to Iearn the best deseriptions for the model. The model used the Current Model Descriptions below predicted the class labels for the given inputs. You are given the target labels, please optimize the Model Descriptions for better prediction.
Inputs (a batch of i.i.d. data on 2-D planc: yx))):
[[SData] [SData] [SData] [SData] [SData] [SData] [SData] [SData] [SData] [SData]]
**
Current Model Descriptions: **
You are designed to do binary classification. The input is a point on 2-D plane [x y); you need to output the class label, L.e, integer in the set (0, 13. The decision boundary is a circle.
The model predictions:
[|SPrediction] [SPrediction] [SPrediction] [SPrediction] [SPrediction] [SPrediction] [SPrediction] [SPrediction] [SPrediction] [SPrediction]]
** The targets: **
[[SGroundTruthh] [SGroundTruth] [SGroundTruth) [SGround Truth] [SGroundTruth) [SGroundTTruth) [SGroundTruuhh] [SGroundTruth) [SGroundTruth) [SGroundTruthh]
the model is doing well, you can keep using the current descriptions. However, if the model is not performing well, please update the model by improving the 'New Model Descriptions', which should have lower classification error both on the current and the next batch of Li.d. data. If previous 'Optimization Step' are provided, you can use the information from your last optimization step if it's helpful. DON'T use symbolic representation for the model! Please think step by step and give your outputs strictly in the following format:
Reasoning:
[be explicit and verbose, improve the Current Model Descriptions by yourself, please show your work; note that you don't have access computers] New Model Descriptions:
[put your decision rules here; MUST be concise and concrete; MUST PROVIDE THE FXACT VALUE OF THE PARAMETERS if the descriptions potentially involve unknown or learnable parametons!!!**aag
Please ONLY reply according to this format, don't give any other words.
Figure 31: Prompt templates of VML for the learner and optimiser for the two circles classification (Llama-3-70P with prior).
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L.6 Text classification
Text prompt template for learner
the model.
** Model Descriptions: **
You are designed to do binary classification. The input is term; need to output the class label, Lee,, an integer in the set (0, 13.
Input: **
[SData]
Please give your output strictly in following format:
Explanations: (Your step-by-step analyses and results]
Output:
[ONLY integer class label; make necessary assumptions if neodeed]
Please ONLY reply according to this format, don't give me any other words.
Text prompt template for the optimizer
You the optimizer for model, your goal learn the best deseriptions for the model. The model used the Current Model Descriptions below predicted class labels for the given inputs. You are given the target labels, please optimize the Model Descriptions for better prediction.
** Inputs batch of ii.d. text): **
[[SData] [SData] [SData] [SData] [SData] [SData] [SData] [SData] [SData] [SData]]
Current Model Descriptions:
You are designed to do binary classification. The input term; you need to output the class label, Le., integer the set {0, 1}.
** model predictions: **
[|SPrediction] [SPrediction] [SPrediction] [SPrediction] [SPrediction] [SPrediction] [SPrediction] [SPrediction] [SPrediction] [SPrediction|]
The targets:
[|SGroundIrruth [SGround'rruth] [SGround'Truth] [Srround'ruuth) [Srround[rruth) [SGroundTruth) [SGroundTruth) (ruth) [SGroundTruthh] [SGroundTrutth]]
model doing well, you can keep using the current descriptions. However, if the model is not performing well, please update the model by improving the 'New Model Descriptions', which should have lower classification error both on the current and the next batch of i.l.d. data. previous 'Optimization Step' are provided, you can use the information from your last optimization step it's helpful. Please think step by step and give your outputs strietly in the following format:
Reasoning:
[be explicit and verbose, improve the Current Model Descriptions by yourself; please show your work; note that you don't have access to computers] New Model Descriptions:
[put your new decision rules here]
Please ONLY reply according to this format, don't give me any other words.
Figure 32: Prompt templates of VML for the learner optimizer for the text classification (Llama-3-70B without prior).
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M Detailed Training History
M.1 Linear Regression (Llama-3-70B without prior)
Training History: Linear Regression - Step
Epoch 0 Stop
Forward Paas Prompt
model. You will use the descciptions below to prodict output of she given iaput.
Puttern Deseriptions:
designed regresston,
predies ouspus any given tnpost tuput output
Input:
Please your scrietly in the following formati
Explanation:: (Your step-by-step anulyaes and resulta]
Output:
Your oulput MUST be in REAL NUMBER ROUNDED TO TWO DECIMAL POINTS; make necessary nosupption if neoded; it MUST be in the pomi format as the Iaput]
Please ONLY reply according to this format, don't give me any other words.
Forward Paxs Output
Rephoations::
there is no spacific pattern finction provided, make an ascmpptton that the outpur came au inport.
Oulpat: [1.80]
Explanations: Forward Output
prediction. Without pattern relaticnship, will simple
model.
[8.70]
Many outpute for the rest data
Forward Pass Output
Explanations: Since there's 's specific pastern or formula provided, I'll make an sosumption based on she given input. The Inpas [0.26] seoms to be small positive value, so I'll predict the output to be slightly larger, but still small positive value.
Output: [o.31]
36.34138
Optimization Prompt
Deseriptione opsimiser given impute. are given target ourpiss, please opstmixe Pastern Descriptions prediation. produced ontyute
your gool learn she descriptions for the model,
**
L..d. data):
[1.6
(0.36) [0.97] (1.86) (0.26) [1.80]]
Currout Pattern Deseciptions: we
You designed to
to prodies the ouspus of
Both and output real numbers.
** The model outputa: **
[[3.27] [3.0 ] [0, 88] 86) (o (0.42) [0.5] [0.97] [3.7] J [o.33] [1,89]]
The target ontputt::
[[775] [9.43] [6.95] [9.55] [4,8]] [5.23] [6.9d] [9.02] [4.75] [9.02]
the model is doing well, you con keep wiig the current desctiptions However, if the model is not performing well, please optimise the model by improving the "New Patsern Deseripsions The moddl use the "Now Patteru Deseription' should better predies the targei oulputa of the givea inpula, as
as the next basch ta.d. ipput data from same distribation. previous Optimitattion Step' are provided, you can use the information from your helpfal. DON'T use symbolic reprasentation for modell Pleas think step by step and give your outpuse strictly in
optimization step following format:
explictt and
the
Deseriptions yourself show your work; that
have computes)
Now Pattern Deseriptious:
deseriptions here; MUST specitic and **SMUSST provide
of the parameters if the deseriptions potentially
involve unknown learnable parametura!lt**kkg
Please ONLY raply according to this format, don't give mh any other words.
Optimization Output
Reasening:
The current model outputs are nos close to the sarget outputs, indicating that the curront pattern deseriptions are not effective, Upon examiting ing the isputs and target. outputs, notice that the target outputs are generally larger than the inputs, suggesting possible linear relationship with positive stoge. However, the current model cusputs are mosely simllar to the ispute, indicsting that the carrent pastern descriptions not capture this relationahlp. To optimixe the pastern description, will introduce a acaling factor to capture the lincaz relatiomship between the imputa and target cutputo. will small seran account any comptant shift in the target cutpulz.
Deseriptiane:
descrsptions will
regresston input.
scaling
5.5
descriptions
Updated Theta
The pattern descriptions will be baced liner regresstion model with scaling factor of 5.5 and biae term of 2.5. TThe descriptione
5.6x 2.5
where is the prodicted output and is the inpus.
54

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Published in Transactions on Machine Learning Research (01/2025)
Training History: Linear Regression Step
Epooh Step
the discciption ** Pattern Deveriptions: es
model. You
predict output of iaput.
You designed to do regrasston, te,, to predict the outpat of given Input. Both and real numbers. The deceriptinn model with scaling factor 5.5 and biue of 2.5. The new pattern descriptions
will be based on ltnear
5.6x = 2.0
where is the prodicted ousput and is the Inpus.
Input:
(0.86)
Please give your output skrietly in the following format:
Bephoations:: [Your stpp-by-etop analyzes and resulte]
Output:
output MUST in NUMBER ROUNDED DECIMAL POINTS; make necessary assumptions needed: MUST tn Inputt]
Please ONLY according
don't other words.
Poss Output
Replanations: 0.28, plug into the equation 5.5x 2.5 the output Given
y 5.5 (50 + 2.5
1.54 2.5
4.D4 Oulput: [4.04]
Many outputs for the rest data points ...
Forward Puss Output
Replanations:
given regression First, let's calculase 5.5(1.50) 5.740, add get 8.745 2.5 11.246. Rounded two decimal points, output 11.25. [11 26)
find output given Ingust Pingging
-.5(1150) 2.5.
Overall Low
2.8077400000n0000
Optimization Prompt
Yon are the optimizer for a modal. your goal io to learn the best descripctons for the modal. The model used the Corrant Pattern Descriptions below produced the outputs of the given iapute. You are given the target ourpuce, please opstiiie the Pattern Deseriptions for batter prediction. Imputs (a. bateb of 13.d. data):
[[0.86] [0.02] [1:1 [1 59) [1.9] [0.25] [1.89] [1.20] [0.79] [o.01]]
Current Pattern Descriptions:
desixned
regression, ie., to prodics the output of nay siven isput. Both isput ouiput ure numbers. The new paticrn deseription biu tern 2.6. new putteen descriptions arc:
be based on lincer regressiou model with scaling fuctor
y
iapus.
*e The model outputs: ee
[1/2111 2.61] 55555 [11:25]
[4.04] [12.92] [9.50] (6.55] 2.50]]
** The target
e*
[[6,00] [8.95] [7.29] [8.6 [7.7] [4.77] [0.67] [8.08] [6.43] [4.01]] If the model doing well.
can keep minn the cuzrent desetiption. However, if the model is not persor mingg well, plense oplimiee the model by improving the "New Patsern Descripstions The modal uswe the 'Now Pattern Descriptions should better predict the target outpute of the given tnpists, as well as the next baseh of Lt.d, ipput data from the same distribution. If previous Opttiisation Step' are provided, you cas use the informacion from your last optimization step if it's selpful, DON'T use symbolic representation for the model! Please think step by stop and give your outputa strictly in the following format:
Reasoning:
explicit and varhose, improve the Current Pattern Deseriptious by younelff please show your work; note that you don't have access computes)
Pattern Descriptions:
[put
descriptions
specific concrete; UUMUST provide learnable parameturs!!!***"
paramaters descriptions potestially
Please ONLY caply according to this format, don't Optimization Output
other words.
Reasoning:
The current model ousputs are nos close to the target ousputs, indicating that she currens pattern descriptions are nos offective. Upon examining the Inpate, target ousputs, and model outputs, notion thas the model outputs are generally larger than the target outputs, suggeesing that the scaling factor will adjus the bise term to beiter capture the overall shift in the target outpuls. To optimise the patteru deseriptio,, I will caleulate the average difference between the target outpatz and madal ouspute to determine
of 5.5 is teo large. I will reduce the scaling factor to bring the model suiputs closer to the target outputa, Additionally, enggeest
adjuetment needed the bis term. The average difference is approcmately -1.55 will subtract this value tarm of 2.5 get new hias term of 1.05.
from current
New Pattern Deseraptione:
pastern descelptions
based lleerr regression
scaling factor
1.05. The pattern descriptione
4.2x 1.06
wherre prodicted
the inpui.
Updated Theta
The pastern descriptions will be bazed linear regresston model with scaling factor of 4.2 and hinx term of 1.05. The patters descriptions
y 4.2x - 1.05
where 1s the prodicted ousput and x le the input.
55

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Published in Transactions on Machine Learning Research (01/2025)
Training History: Linear Regression Step
Epooh Step
model. You ** Pattern Deveription:: es
the discciption below predict output of the iaput.
You designed to do regravston, te,, predict the outpat of given Input. Both and ousput real numbers. The deseriptinn model with scaling factor of 4.2 and bise 1.05. The new pattern descriptions
will be based on ltnear
4.2x 1.05
where is the prodicted ousput and is the inpus,
Input:
(0.49)
Please give your output strictly in the following format:
Bephoations:: [Your step-by-steng analyses remules
Ousput: (Your autput MUST be in REAL NUMBER ROUNDED TO TWO DECIMAL POINTS; make necemary assumptions needed; it MUST be some
taput]
Please ONLY according
don't
other words.
Puss Output
Explarations:
Using the ltnear regression modal, aubetitute with the taput 0.05.
4.2x 1.05
4:2(0 05) + 1.05
0:21 1.05
1.36
Oulput: [1:26]
Many outputa for the rest data points ...
Overall
Optimization taudel, your goal to learn the best deseciptions for the mndel. The model Curreat Paticrn Desetiption below outpuss, please opsimise the Pottern Descriptions
You ** Inputs batch of i...d. data): we
produced the
given
prediction.
[[0.49] [0.74] [0.15] [6.05] [0.84] [1.21] [1:26] [L.79] [0.00] [0.79]] ** Current Pattern Descriptions:
You are designed to do regresstio, Le,, to prodict the ourpor of any given imput. Both input and ousput are real numbers, The new pattern descrpptinns will be based on linear regresston model with sealing factor of 4.2 and bise term of 1.05. The new pattern deseriptions are:
y 4.22 1.05
where is the prodieted ousput and x in the Inpus.
The model outpute:
[[8.11] [4.19] [1.6] ]
(4.68) (0.13) (0.48) (8.52) (5.21) [4.26]]
Larget outputa:
[[8.00] [6.23] [4.31] [4.00] [6.30] [7.70] [7.87] [0.17] [6.97] [6.47]
parforming well, pleaee optimise the model by the use the Information tnpats, Deseripsional. madel distribution. provious "Optimization Step' provided,
uiing
descriptions. Deseriptione'
smproving 'New
the basch L.d. data from
lost optimization step helpful. DON'T symbolic representation for the modell Please think step by step and give outpuss strictly the following format:
Reasoning:
be explicct and varhose, improve the Current Pattern Descriptions by yourself; please show your works note that you don't have accers to computer]
New Pattern Descrsptions:
(put your new descriptions here; MUST be specific and concrete; UUSM provide the excact value of the parameters if the descriptions potentially involve un unknown leornable paramederatii-een
Please ONLY reply according to this format, don't give mh any other words. Optimisation Output
Reasoning:
model ousputs
outputs, the model ousputa
dnseriptios
Upon
inputa, target
model ousputa.
than Additionaly,
scaling better capture the overall zhift in the target outputs and model outpute datermine the adjuctment needed for the bisr The average difference approsimately 1.43. will add this value to She cuerent bias term of 1.05 to
small.
the
the bis diffarence between the
To optimixe the pattern desrriptinn, will caleulate the
now bins term of 2.45. I will also analyse the relationslip between the iupuis and target outputo to detecmine the new scaling factor, Upon examining the data, notice that the target ousputs increase appeoximately 5.5 units for every 1 unit Increase in the input, will use shis relationehip to eot the new scaling factor to 5.5.
New Pattern Descriptions:
The new pastern descriptions will be based on a lineat regression model wish scaling factor of 5.5 and a bis term of 2.48. The new putiern deseriptions are:
y 5.bx 2.48
where the predicted output
in the input.
Updated
pastern descetptions
regresston
sealing factor
pattern descriptione
6.6x. 2.45
wherre prodicted oulput in the inpus.
56

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DECIMAL POINTS; make necessary assumptions needed: MUST In same Published in Transactions on Machine Learning Research (01/2025)
Training History: Linear Regression Step
Epoch Step Paza Prompt
the discciption below predict output of the iaput. ** Pattern Deveriptions: es
model. You
You designed to do regravston, te,, to predict the outpat of given Input. Both input and ousput real numbers. The deseriptinn model with scaling factor of 5.5 and bise term of 2.48. The new pattern descriptions
will be based on ltnear
5.6x = 2:48
where is the prodicted ousput and is the inpus,
Input:
(0.08)
Please give your output strictly in the following format:
Bephoations::
[Your step-by-xtop analyzes and resulte]
Output:
output MUST in REAL NUMBER ROUNDED Inputt]
Please ONLY according this
don't
other words.
Poss Output
Rephanation::
Given the iaput 1. I will use the linear zegression model y 6.6x + 2.48 to predict the output Subasitating x 1 Into the equation, I 5.5(1) + 2.45 = 7.88.
Outpot: [7:96]
Many outputs for the rest data points
Overall Loss
2.0120009999999
Optimiaation Prompt
optiminer
descriptions your gool given target ourpuse, please optimite
model Dascriptions predsction.
Deceriptione
produced putpote i..dd data):
given tmpite.
** Inputs
[[0.08] [0.44] [0.11] [0.54] [1.48] [0.02] ** Curront Pattern Desoriptione: we
[1.39] [L.7] (1.38)]
You designed to do regrassion, i.c., to prodict the putput of aay given input. Both input and ousput real numbers. The new pattern descriptions
will be based on lincar regression model with sculing factor of 5.5 and bis term of 2.48. The new pattern descriptions
y = 5.5s 2:48
where le the predicted output and i te the inpus.
The model outputs:
[[2,00]] 4,90] | 3.00) (.1] [10.62] 2.000 7.098 [10.1] [11.93] [10.07])
The target outpute:
[[4.27] [5.19] [4.24] [6.65] [8.01] [8.80] [6.90] [8.15] [9:17] [7.96]
the model doing well, you cun keep uing the current desctiptions However, model not persormiux well, pleuse optimiae the model by improving lug 'New Patiern Deseripiion"
model use the "New Patteru Deseriptions should better predies targei outputa of the givea inpuls, previous 'Optimization Step' are you the symbolse representation
iaput
optimization step helpfal. following
model!
step by step give your autpuss strictly
Reasening:
be explicat and vathose. improve the Current Puttern. Descriptions by yoursilf; please show work: note that don't have access to computer)
New Pattern Deseriptious:
(put your new descriptions here; MUST be specific and concrete; UUMUUT provide the excact value of the paramaters If the descriptione potentially involve unknown or isarnable parmmetara!!t**kkg
Please ONLY raply according to this format, don't give mh any other words. Optimization Output
Reasoning:
cuereat model ouiputs are uas cluse largei outputs, indicating that the cureen palleru deseriptions are not effective. Upon examining Impate, target cusputo,
outpute generally the determine the scaling factor. Upon examining the data, notice that the in the taput. However, the curreat acaling factor of 5.5 is close to this value, I will only make small adjustmeat to the scaling factor. I will increase she scaling factor to 5.7 to better capturee the relationship between the Inpute and target ousputs.
outpute, notice
shan target outputs, suggesting that Additionally, will descriptions, difference upproximately 1 and
relasiomslip outputa this value to the current hise term of 2.48
outputs. To
differance
adjuitmest nceded
The
blar term of 3.71. will also analyse the relationship between the outpute increase by appresimately 5.8 units for avery unit incraaxe
New Pattern Descriptions:
The new pastern descriptions will be based on a linear regression model wish scaling factor of 5.7 and a biss term of 5,71, The new patiern descriptions are:
y 5.7x 3.71
where the predicted output and ie the input.
Updated Theta
new pastern descetptions
based linear regression model wish scaling factor 5.7 and term 3,71, The pattern descriptions
6.7x 9.71
wherre prodicted ouiput is the inpus.
57

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Published in Transactions on Machine Learning Research (01/2025)
Training History: Linear Regression Step
Epoch Step Paza Prompt
the discciption below predict output of the iaput. ** Pattern Devoription:: es
model. You
You designed to do regravston, te, to predict the outpat of given Input. Both input and ousput real numbers. The new deseriptinn model with scaling factor 5.7 and bise term of 3.71. The new pattern descriptions
will be based on ltnear
S.7s = 3.71
where is the prodicted ousput and is the inpus,
Input:
(1.50)
Please give your output skrietly in the following format:
Bephoations:: [Your stpp analyzes and resulte]
Output:
output MUST in NUMBER ROUNDED TWO DECIMAL POINTS; make necessary assumptions needed: MUST In same Inputt]
Please ONLY according this don't other words.
Poss Output
Replanations: Given the y 5-7(1 29) + 3:71
1.39, plug into the equation 5.7x + 3.71 the output
7.813 + 3.71
11.523 Oulput: [11.52)
Many outputa for the rest data points ...
Overall Low
8.08032
Optimization
taudel, your goal to learn she best deseciptions for the mndel. The model used Cutreat Paticrn Desetiption below target outpuss, please opsimise the Pottern Descriptions
You ** IIpput batch of id.d. data); we
produced the
given
prediction.
[[1,50] [0.25] [1.72] [0.98] [1.39] [L.7 [1.54] [0.40] (0.27] [0.93]] ** Current Pattern Deseriptions: **
You ane designed to do regresstion Le,, to predict the ourpor of any given input. Both input and cusput are real numbers, The new pattern descriptions will be based on linear regresston model with sealing factor of 5.7 and bise term of 3.71. The new pattern deseriptions are: y = b.78 3.71
where is the prodieted ousput and x in the Inpus.
The model outpute:
[122.79] [5002] (18.61] 9:8 [11.62] [18.4] [11.42] [ 6.49] [ 6.266 8.099]]
The Larget outputa: **
[10.02] [4.64] [9.2 [7.02]
(9.14) (8.68) (5.47) (4.77) (6.78) winng
model parforming well, pleaee optimise the model by the use the Information from Deseripsional. madel distribution. provious "Optimisation Step" provided,
descriptions. Deseriptione'
improving "New
the basch L.d.d data from
lost optimization step helpful. DON'T symbolic representation for the modell Please think step by step and give outpuss strictly the following format:
Reasoning:
be explicit and varhose, improve the Current Pattern Descriptions by yourself; please show your works note that you don't have accers to computer]
New Pattern Descrsptions:
(put your new descriptions here; MUST be specific and concrete; *USMUS provide the excac value of the parameters if the descriptions potentially involve un unknown or leornable parameceratii-een
Please ONLY raply according to this format, don't give mh any other words. Optimisation Output
Reasoning:
model ousputs
outputs, indicating the model
Enseriptions
Upon that the scaling the difference betwenn
impata, ousputa. model outpula,
claser the the Additienalyy, outpute. To optimiza the decermine the séjustment nasded for the hiss term. The average difference approcimately -1.4. I will sustract new bias term of 2.27. I will also analyse the relationship between the imputs and Sarget outpuis to
acoling factor and
better capture the relactonehip between the the
descriptions, I will calenlate the
and modal this value from the ourrent bins term of 3.71 to determine the new scaling factor, Upon examining the data, nosice that the sarget outputs increase by approximately 4.2 units for every I unit Incroase in the input. The currens scaling factor of 5.7 is larger than this value, so I will decrease the acaling factor to 4.4 to better capture the relationship between the isputs and targes outpata.
New Pultern Descriptions:
new pasteern descriptione will bazed on a linaar regression model with scaling factor 4.4 and bins term of 2.27. The new pattern descriptions are:
y 4.1x- 2.27
prodicted output
the Inpus.
Updated
pastern descriptioas will based lincat regression model wish scaling factor and 2.27. The deseciptions
4.4s 2.27
where the predicted output and to the ipput.
58

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Published in Transactions on Machine Learning Research (01/2025)
Training History: Linear Regression Step
Epooh Step
the discciption below predict output of the iaput. ** Pattern Deveription:: es
model. You
You designed to do regravston, te,, to predict the outpat of grven imput. Both inpur and ousput real numbers. The deseriptinn model with scaling factor 3.26 and biux sarm of 3.45. The pattern descriptione
will be baxed on ltnea
8.20x 3.45
where is the prodicted ousput and is the inpus,
Input:
[1:26]
Please give your output skrietly in the following format:
Bephoations::
[Your step-by-xtop analyzes and resulte]
Output:
output MUST in NUMBER ROUNDED DECIMAL POINTS; make necessary assumptions needed: MUST tn same Inputt]
Please ONLY according this don't
other words.
Poss Output
Replanations:
Given the liner regresston model 3.28x + 9.45, plug in the input 1.26. y 3.26(1.26) (.2)) + 3.45 = 4.08 + 3.45 7.53.
Outpot: [7.53]
Many outputs for the rest data points
0.1138909999992
Optimiaation Prompt
** opsiminer given tmpite. You grvwn target ourpuse, please optimiiee
your
descripcions
modal Dascriptions predsction.
Dectriptione
produced putpote i..dd. data):
[],:26] [1,00] [0.32] [0.53] [1.3 [0.24] [1.87] (1.16) [1.2 [1:26]]
e* Curront Pattern Desociptione: we
You designed to do regrassion, s.c., to prodics the putput of aay given imput. Both input and ousput real numbers. The new pattern descriptions
will be based on lincaz regression model with scaling factor of 3.26 and biss serm of 3.45. The new pattern descriptions y 3.266 + 3-45
where le the predicted output and it te the inpus.
The model outpute:
[[7.50] [5.55] [4.49] [6.17] [7/73] [4:25] [9.58] [7.10] [7:06] [7.54]
The target outpute:
[[7.87] [8.09] [4.93] [6.30] [8.18] [4.78] [9.37] [7.80] [7.73] [7.73]
model doing well, you cun keep uing the current desctiptions However, model not performinx well, pleuse optimiae model by improving lug 'New Pataern Deseripiion" model use the "New Patteru Deseriptions should better predies targei outputa the givea inpuls,
you the previous 'Optimization Step' step by step give your autpuss model!
optimization step helpfal. following
symbolic representation
Reasening:
[e explicat and vathose. improve the Current Puttern. Descriptions by yoursilf please show work: note that don't have access to computer)
New Pattern Deseriptious:
(put your new descriptions here; MUST be specific and concrete; UUMUUS provide the excact value of the paramaters If the deseriptione potentially involve unknown or isarnable parmmetara!!f**kk
Please ONLY raply according to this format, don't give mh any other words. Optimization Output
Shortening the staze from 5 to
Reasoning:
curreni model oulputo are gencrally close to target outputs, but there are some deviation, Upon cxmmining the iaputs, target oulputo, ond model the
outputs,
Inpata, and tveressiaaing othere, suppeste that model 3.54. still incresiing with the and the rate increaze elightly faster than the current sealing factor 3.26. This suggete that the scaling factor 3.25 might be slightly small. I will mncraae the sealing factor 0.01
the outputs be approximately
underssionainn targes outputs calculate
outputs. The eclationship
factor
needs
dats, notice that the
sealing factor 3.27.
Now Pattern Deseriptious:
The new pattern descriptions will be based on lnear regression model wish a scaling factor of 3.27 and bins term of 3.54, The new pattern descciptions arm:
y 3,27x J. 3.64
wheree is the predicted ouipul and a in the inpus.
Updated Theta
new pattern deseriptions will baxed on linear regrasston model with a scaltng factor 3.27 bine term 3.54. The new pattern descriptions
are:
y prodicted output
input.
59

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DECIMAL POINTS; make necessary assumptions needed: MUST to same Published in Transactions on Machine Learning Research (01/2025)
Training History: Linear Regression Step 20
Epooh Step
the discription below predict output of the iaput. ** Pattern Deveription:: es
model. You
You designed to do regravston, te,, to predict the outpar of given imput. Both inpur and ousput real numbers. The deseriptinn model with scaling factor 3.27 and biux sarm of 5.54. The pattern descriptione
will be baxed on ltnear
3.27x 3,54
where is the prodicted ousput and is the inpus,
Input:
(0.71)
Please give your output skrietly in the following format:
Bephoations::
[Your etep-by-xtup analyzes and resulte]
Output:
output MUST in NUMBER ROUNDED Inputt]
Please ONLY according this don't
other words.
Poss Output
Replanations:
Using the given linear rogression modal, 3.27x + 3.84, and the imprat calculate the output followa: y 3.27(1) + 3.54
3.27 3.51
6.81 Oulput: 6.91
Many outputs for the rest data points ...
Optimization taodel, your goal
she deseciptions for the mndel. The model used Cutreat Paticrn Desetiption below outpuss, please opsimise the Pottern Descriptions
You ** Inpput batch i.-dd data); we
produced the
given
prediction.
[[0.71] [0.6 [0.51] [1.34] [0.92] [0.16] [1.12] [0.14) (1.07] [L. ** Current Pattern Deseriptions:
You are designed to do regresstion Le, to predict the ourpore of any given impot. Both input and cusput are real numbers. The new pattern descriptions will be based on linear regresston model with sealing factor of 3:27 and bixs sarm of 3.54. The new pattern descriptions are:
y 3.271 + 3.54
where is the prodieted ousput and x in the Inpus.
The model outpute:
[[6.80] [5.6 [6.22] [T:91] (0.67] (4.07) (7:21) (3.92) (9. [0.81]]
Larget outputa:
[[6.00] [6.6 ] [6.48] (8.1] (0.50] (4.68] [7.37] [4.48] [8.85] [6.00]] naing
parforming well, please optimise model by the use the Information from Deseripsioni: madel distribution. provious "Optimization Step" provided,
descriptions. Deseriptione'
smproving "New
the basch L.d. data from
lost optimization step helpful. DON'T symbolic representation for modell Please think step by step and give outpuss strictly the following format:
Reasoniug:
be explicct and varhose, improve the Current Pattern Descriptions by yourself; please show your works note that you don't have accers to computer]
New Pattern Descrsptions:
(put your new descriptions here; MUST be specific and concrete; UUSMUST provide the excact value of the parameters if the descriptions potentially involve un unknown leornable paramederatiiseen
Please ONLY raply according to this format, don't give mh any other words. Optimisation Output
Reasoning:
outputs, model
close undersstimating the
and model that blas term of 3.89. will and octpote to determine if the sealing factor of 3.27 needs to be acgueted. Upon examining the
the
3.64 model be approximately 0.15. I will add this valne the currens blus term of 3.54 outpute. The average difference also analyse the relationchip between the data, notioe thot the targel outpats are stil incrensing with the ispuls, and the rate of jacrease slightly fnster than the eureent acaling factor of 3.27. This suggests that the scaling factor of 3.27 might be alightly too small. I will increase the scaling factor by 0.02 to a new scating facsor of 3.29. New Pattern Descriptione:
calculate
difference
model
The new pattera descriptions will be based on a lincar zegression medel with a acaling foctor of 8:29 and bina term of 3.69, The new pattern descriptions are:
y 3,29x 3.69
where the predicted output ie the input.
Updated Theta based liearr regression model scaling factor 3.29
new pattern desoriptions
3.69. pattern descriptions
3.29x 3.69
whecre prodieted ouipul the inpus.
60

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Published in Transactions on Machine Learning Research (01/2025)
M.2 Polynomial Regression 1-70a without prior)
Training History: Polynomial Regression Step 1
Bpoch Step
Forward Pass Prompt
You the model. You will use the descriptions below to predict the output of the given imput.
Pattern Descriptions:
designed do regression, i.c,, to predict output any given input. Both imput and output real mumbers.
Inppu:
[-2.4]
Please give your output strictly in the following format:
Explanations: [Your step-by-step analyses and results]
Output:
(Your output MUST he in REAL NUMBER ROUNDED
DECIMAL POINTS; make nocessry assumptions needed;
MUST be in the same format as the Input]
Please ONLY reply according to this format, don't give any other words.
Forward Pass Output
Explanations:
Since there no specific pattern relationship provided, will make assumption that the output constant value. In this case, I will assume the output is 0.00.
Output: [0.00]
Many outputs for the rest data points
Forward Pass Output
Explanations:
Since there is no specific pattern or function provided, I will make an aesumption that the iuput and output are linearly related with constant difference of 1.2. Based on this assumption, will add 1.2 the input to get the output.
Output: [0.50]
63.1 13.993990999999999 Overall Loss
Optimization Prompt
You the optimizer for a model, your goal to learn best descriptions for the model. The model used the Current Pattern Descriptious below produced the outputs of the given inputs. You are given the target outputs, please optimize the Fattern Descriptions for better prediction.
** Inputs (a batch of i.i.d. data):
[|-2.4] [ 0.6] 0.9] [-1.8] [-0.7] [-0.9] [-1.2] [-u. [-0.2] [03]]]
** Current Pattern Deseriptions: **
You are designed to do regression, j.e., to predict the output of any given input. Both input and output are real numbers.
** The model outputs: **
[[-2.4] [ 0. U.. [.0. (..] 0.5] [ [-1.2] [0. [-0.2] [.55]]
** The target outputs: **
[[17.8] [.7]] 6.6] [10. 3.6] [2.5] [.44] 1.1] 0.9] [2.6]
the model is doing well, you can keep using the current descriptions. However, if the model not performing well, please optimize the model by improving the 'New Pattern Descriptions: The model
the 'New Pattern Descriptions' should better predict the target outputs of the given inputs, as well as the next batch of L.d.d input data from the same distribution. If previous 'Optimization Step' are provided, you use information from your optimization step it's helpful. NOTE: both the model and you can only operate on the mumerical precision of one decimal points! Please think step by step and give your outputs strictly in the following format:
Rensoning:
[be explicit and verbose, improve the Current Pattern Descriptions yourself; please show your work; that you don't have access to computer] New Pattern Descriptions:
(put your new descriptions here; MUST be specific and concrete; MUST provide the exact value of the parameters if the descriptions poteutially involve unknown or learnable parameters!!!ewng
Please ONLY roply according to this format, don't give me any other words. Optimisation Output
Reasoning:
After examining the model outputs and target outputs, mostly around D. while the target outputs have a larger range of values. This suggests that the model is not rapturing the underlying pattern in the data. Upon closer inspection, notice that the imputs and target outputs seem to have a positive correlation. This means that the imput increases, the target output also tends increase. To improve the model, will intraduce description that captures this positive correlation. I will use simple linear relationship between the input and output, with a slope and intercept that can be learned from the data. New Pattern Descriptions:
that the model not performing well. The mudel outputs
pattern
The new pattern description output 2.5 imput
Updated Theta
The new pattern description is: output 2.5 Input 1.5.
61

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Published in Transactions on Machine Learning Research (01/2025)
Training History:
Polynomial Regression Step 2
Epoch Step
Forward Pass Prompt
You are the model. You will use the descriptions below to predict the output of the given imput.
** Pattern Descriptions: **
You are designed to do regression, ie., to predict the output of any given input. Both input and output are real mumbers. The new pattern description is: output 2.5 * input + 1.5.
** Input:
[-2.4]
Please give your output strictly in the following format:
Explanations: (Your step-by-step analyses and results]
Output:
Yourr output MUST be in REAL NUMBER ROUNDED TO TWO DECIMAL POINTS; make necessary sesumptions needed;
MUST be in the same format as the Input]
Please ONLY reply according to this format, don't give me any other words.
Forward Pass Output
Explanations: The output caleulated multiplying the input by 2.5 and adding this case, input is 1, so output
2.5 +1 1.5.5 =.5
Output: [1.00]
Many outputs for the reat data points
Forward Pass Output
Explanations:
To find the output, need to apply the given pattern description: output = 2.5 imput + 1.5. In this case, the input is -1.2, so we plug it into the equation: output 2.5 (-1.2) + 1.5.
First, nultiply 2.5 and -1.2: 2.5
(-1.2) -3.0.
Then, we add 1.5: -3.0 + 1.5 = -1.5.
Output: [-1.50]
Overall Loss
257.99
Optimization Prompt
You are the optimizer for a model, your goal is to learn the best descriptions for the model. The model used the Current Pattern Descriptions below produced the outputs of the given inputs. You given the target outputs, plense optimize the Pattern Descriptious for better prediction.
** Inputs batch i.i.d. data):
[[-2.4] [-2.9] [-2.1] (-1.8] [-1.2] [-0.6] [-2.2] [1. [1.9] [1.]
** Current Pattern Descriptions: **
You designed to do regression, i.e., to predict the output of any given imput. Both imput and output real numbers. The new
pattern description is: output 2.5 input + 1.5.
** The model outputs: **
[[-4.5] [-5.8] [-3.8] [-3. [-1.5] [0.]] [4. 44. [3:2] 44. Il
** target outputs: **
[[19.5] [26.1] [12.4] [10.8] [.5] [0.8] [13.8] [.4] [9.7] 5.44]
the model is doing well, you can keep using the current descriptions. However, if the model not performing well, please optimize the model by improving the "New Pattern Descriptions' The model uses the 'New Pattern Descriptions' should better predict the target outputs of the given inputs, as well as the next batch of i.i.d. imput data from the same distribution. If previous 'Optimization Step' provided, you use the information from your last optimization step it's helpful. NOTE: both the model and you can only operate on the numerical precision of one decimal points! Please think step by step and give your outputs strictly in the following format:
Reasoning:
[e explicit and verbose, improve the Current Pattern Descriptions by yourself; please show your work; note that you don't have acress to computer)
New Pattern Descriptions:
[put your new descriptions here; MUST be specific and concrete; MUST provide the exact value of the parameters if the descriptions potentially involve unknown learnable parametors!lisssn]
Please ONLY reply according to this format, don't give me any other words. Optimization Output
Reasoning:
After examining the model outpute and target outpute, it's clear that the model not performing well. The model outputs are close to the target outputs, and the pattern description output- = 2.5 imput + 1.5 seems to be oversimplifying the relationship between the input and output. Upon closer inspection, notice that the target outpute are generally larger than the model outputs. This suggests that the model is underestimating the target outputs. also notice that the target outputs seem to have non- linear relationship with the inputs, with larger inputs corresponding to larger outputs, but not in a linear way. To improve the model, pattern deseription that captures this non-linear relationship. I will input and output, with coefficients that can be learned from the data. This will allow the model to better fit the curved relationship between the inputs and target outputs.
will introdure
quadratic relationship between the
New Pattern Descriptions:
The new pattern description is: output = 2.2 impput ' + 1.8 imput + 0.6.
Updated Theta
The new pattern description is: output 2.2 imput"? 2 + 1.8 input + 0.6.
62

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Published in Transactions on Machine Learning Research (01/2025)
Training History:
Polynomial Regression Step
Epoch Step
Forward Pass Prompt
You are the model. You will use the descriptions below to predict the output of the given imput. ** Pattern Descriptions: **
You are designed to do regression, i.., to predict the output of any given input. Both input and output are real mumbers. The new pattern description is: output 2.2 input 22 + 1.8 input +
** Input:
[-1.]
Please give your output strictly in the following format:
Explanations: (Your step-by-step analyses and results]
Output:
Your output MUST be in REAL NUMBER ROUNDED TO TWO DECIMAL POINTS; make necessary aesumptions needed;
MUST be the same format as the Input]
Please ONLY reply according to this format, don't give me any other words.
Forward Pass Output
Explanations:
Given input = -1.9, we need find output using the formula: output 2.2 Inputt2 1.8 input 0.G. First, calculate imput"? = "2 3.61,
Now, calculate output 2.2 - 3.61 188 (-1.9) + 0.6 -2.42.
Output: (-244]
Many outputs for the rest data points...
Forward Pass Output
Explanations:
Given the input 0.8, nced to plug it into the equation output 2.2 * 'input"2 1.8 "imput 0.6.
First, we square the input: 2 = 0.64.
Then, multiply 0.64 by 2.2: 2.2 +
1.408.
Next, we multiply 0.8 by 1.8: 1.8 0.8 1.44.
Now, we add 1.408 and 1.44: 1.408 + 1.44 2.848.
Finally. add 0.6 to 2.818: 2.818 0.6 3.448. Output: [3.45]
Overall Loss
44.34
Optimization Prompt
You are the optimizer for a model, goal is to learn the best descriptions for the model. The model used the Current Pattern Descriptions below produced the outputs of the given lnputs. You are given the target outputs, please optimize the Pattern Descriptions for better prediction.
Inputs (a batch of i.i.d. data):
[[-1.] [-2.7] [066 [-1.9] [-2.6] [-1.8] [-2.1] 0.8] [-0.8] [-2.1]]
Current Pattern Deseriptions:
designed do regression, i.c., to prodict output any given input. Buth input output are numbers. The new
pattern description is: output = 2.2 input"2 "2 1.8 input 0.6.
** The model outputs: **
[ [12.8] 2.5] [-2.4] [10.8] [4.5] [6.5] 3.4] 0.6 6.5]
The target outputs:
[[4.2] [21.2] [2.4] [11.7] [18.1] [8.3] [12.2] [6.8] [13.7]]
If the model is doing well, you can keep using the current descriptions. However, if the model is not performing well, please optimize "New Pattern Descriptions.. The model uses 'New Pattern Descriptions' should better predict the well as the next batch of L.dd. input data from the same distribution. If previous 'Optimization Step' are provided, you can use the information from your last optimization step it's helpful, NOTE; both the model and you only operate the numerical precision of decimal points! Please think step by step and give your outputs strictly the following format:
model by improving
target outputs of the given inputs,
ne
Reasoning:
explicit and verbose, improve the Current Pattern Descriptions by yourself; please show your work; note that you don't have access computer]
New Pattern Descriptions:
[put your new deseriptions here; MUST be specific and concrete; MUS provide the exact value of the parameters if the descriptions potentially involve unknown or learnable parameters!!i
Please ONLY reply according to this format, don't give me any other words. Optimization Output
Reasoning:
After examining the model outputs and target outputs, it's clear that the model not performing well. The model outputs are not close to the target outputs, and the pattern description output 2.2 * input 22 + 1.8 input + 0.6 seems be undershooting the target outputs. Upon closer inspection, notice that the target outputs are generally larger than the model outputs, especially for larger inputs. This suggests that the model underestimating the target outputs, especially the target outputs seem to have stronger quadratic relationship with the inputs, with larger iuputs corresponding to even larger outputs. improve the model, will introduce a new pattern description that captures this stronger quadratic relationship. will inerease the coefficien of the quadratic term capture the stronger curvature of the relationship. will also adjust the linear term to better fit the data.
larger inputs. I also notice that
New Pattern Descriptions:
The new pattern description is: output 2.8 imput 2 + 1.2 Input + 0.4.
Updated Theta
The new pattern description is: output 2.8 imputt 22 + 1.2 imput + 0.4.
63

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Published in Transactions on Machine Learning Research (01/2025)
Training History:
Polynomial Regression Step
Epoch 0 Step
Forward Pass Prompt
You are the model. You will use the descriptions below to predict the output of the given imput. ** Pattern Descriptions: **
You are designed to do regression, ie., to predict the output of any given input. Both input and output are real mumbers. The new pattern description is: output = 2.8 input 2 + 1.2 input + 0.4.
** Input:
[0.2]
Please give your output strictly in the following format:
Explanations: (Your step-by-step analyses and results]
Output:
Yourr output MUST be in REAL NUMBER ROUNDED TO TWO DECIMAL POINTS; make necessary aesumptions needed;
MUST be in the same format as the Input]
Please ONLY reply according to this format, don't give me any other words. Forward Pass Output
Explanations:
Given the pattern description: output 2.8 * imput "2 + 1.2 input + 0.4.
laput 0.4. So, output = 2.8 (0.4)" + 1.2 0.4 0.4 0,448 + 0.48 + 0.4 1.328.
Output: [1.33]
Many outputs the data points
Forward Pass Output
Explanations:
Given the input -1.7, we need to find the output using the formula output = 2.8 'uputt '2 + 1.2 'mpput + 0.4.
First, calculate the square of the iuput: (-1.7) 2.89.
Then, we multiply it by 2.8: 2.89 2.8 = 8.092.
Next, we multiply Now, we add 8.092 and -2.04: 8.092 2.04 = 6.052.
input by 1.2: -1.7
-2.04.
Finally, we add 0.4: 6.052 + 0.4 6.452.
Output: [6.45]]
Overall Loss
8.585
Optimization Prompt
are the optimizer for a model, your goal learn descriptions the model. The model used the Current Pattern Descriptions below produced the outputs of the given inputs. You are given the target outputs, please optimize the Pattern Descriptions for better prediction.
** Inputs (a batch of i.i.d. data):
[| 0.2] [-1.7] [:2.8] [-1.1] 0.4] [0.8 [-0.1] [-0.1] [ [088 [-0.7]] ** Current Pattern Deseriptions: **
0.8)
You are designed to do regression, i..., to predict the output of any given imput. Buth iuput and output are real mumbers. The new pattern description
nutput
* input + input +
** The model outputs: **
0.8] [ 6.4] [19. ] [ [13] [3.2] [.3] [ [0.3] [3:2] [-0.5]]
** The target outputs:
3. [8.6] [25.6] [ 6.5] 1.2] 5.5] 0.7] [ 0.8] 5.1] [:2]]
descriptions. However, if the model not performing well, please optimize the model by improving the 'New Pattern Descriptions'. The model uses the 'New Pattern Descriptious' should better predict the target outputs the given inpats, as well Step' are provided, you can use the information from your last optimization step If it's helpful. NOTE: both the model and you can only operate on the numerical precision of one decimal points! Please think step by step and give your outputs strictly in the following format:
the model is doing well. you can keep using the
next batch i.i.d input data
same distribution. previous 'Optimization
Reasoning:
[be explicit and verbose, improve the Current Pattern Descriptions by yourself; please show your work; note that you don't have access computer)
New Pattern Descriptions:
(put your new descriptions here; MUST be specific and concrete: MUS provide the exact value the parameters if the descriptions potentially involve unknown learnable parameters!!
Please ONLY reply according to this format, don't give any other words. Optimization Output
Reasoning:
After examining the model outputs and target outputs, it's clear that the model is not performing well. The model outputs not close to the target outputs, and the pattern description output = 2.8 imput 2 + 1.2 input + 0.4 seems to be undershooting the target outputs, esperially for larger inputs. Upon closer inspection, notice that the target outputs the model outputs, especially for larger inputs. This suggests that the model is underestimating the target outputs, especially for an even stronger quadratic relationship with the inputs, with larger
generally larger than
larger inputs. also notice that the target outputs imputs corresponding to even larger outputs. To improve the model, I will introduce a new pattern description that captures this even stronger quadratic relationship. relationship. will also adjust the linear term better fit the data. New Pattern Descriptions:
will increase the coefficient of the quadratic term to capture the stronger curvature of the
The new pattern description is: output 3.2 imput + 0.8 * input + 0.6. Updated Theta
The new pattern description output 3.2 imput ' + imput + 0.6.
64

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Published in Transactions on Machine Learning Research (01/2025)
Training History:
Polynomial Regression Step
Epoch Step
Forward Pass Prompt
You are the model. You will use the descriptions below to predict the output of the given imput. ** Pattern Descriptions: **
You are designed to do regression, i.., to predict the output of any given input. Both imput and output are real mumbers. The new pattern description is: output 3.2 input 2 + 0.8 input +
** Input:
[-2.3]
Please give your output strictly in the following format:
Explanations: (Your step-by-step analyses and results]
Output:
Your output MUST be in REAL NUMBER ROUNDED TO TWO DECIMAL POINTS; make necessary aesumptions needed;
MUST be the same format as the Input]
Please ONLY reply according to this format, don't give me any other words. Forward Pass Output
Explanations:
Given the pattern description output 3.2 input + 0.8 Input + 0.6, and the input is -2.3, first, we calculate imput's 2 which la (-2.3)"2 5.29. Then, plug the values into the equation: output 3.2 * 5.29 + 0.8 -2.3 + 0.6 16.928 1.84 + 0.6 15.708. Output: [15.71]
Many outputs for the rest data points...
Forward Pass Output
Explanations:
Given the pattern description output 3.2 inpput22 2 + 0.8 'npput + 0.6, will this equation to find the output for the given imput -2.4.
First, calculate iuput (-2.4)"2
Next, I will calculate 3.2 imput 2: 3.2 5.76 = 18.43.
Then, will calculate 0.8 input: -2.4 -1.92
Now, will add the results of the previous stepe: 18.43 1.92 0.6 17.11.
Output: [17.11]
Overall Loss
2.3660000000000
Optimization Prompt
You are the optimizer for a model, goal is to learn the best descriptions for the model. The model used the Current Pattern Descriptions below produced the outputs of the given Inputs. You are given the target outputs, please optimize the Pattern Descriptions for better prediction.
Inputs (a batch of i.i.d. data):
[[22.3] 0.3] [-2.4] 0.5] [-2.3] [-1.4] [-1.7] [-3.] [-2.3] [-0.5]]
Current Pattern Deseriptions:
designed do regression, i.c., prodict output any given input. Buth input output are numbers. The new pattern description is: output = 3.2 input"2 0.8 imput 0.6.
** The model outputs: **
[[15. ] [111] [17.1] 1.8] [15.7] 5.8] 8.5] [27. [15.7] [1.
The target outputs:
[[14.8] [2.4] [17.1]
2.9] [17.3] [6.3] 8.7] [22.9] [16.6] [1.5]]
If the model is doing well, you can keep using the current descriptions. However, if the model is not performing well, please optimize
model by improving
"New Pattern Descriptions.. The model uses 'New Pattern Descriptions' should better predict the
well as the next batch of L.dd. input data from the same distribution. If previous 'Optimization Step' are provided, you can use the information from your last optimization step it's helpful, NOTE; both the model and you only operate the numerical precision of decimal points! Please think step by step and give your outputs strictly the following format:
target outputs of the given inputs,
ne
Reasoning:
explicit and verbose, improve the Current Pattern Descriptions by yourself; please show your work; note that you don't have access computer]
New Pattern Descriptions:
[put your new deseriptions here; MUST be specific and concrete; MUS provide the exact value of the parameters if the descriptions potentially involve unknown or learnable parameters!!i*xxy
Please ONLY reply according to this format, don't give me any other words. Optimization Output
Reasoning:
After examining the model outputs and target outputs, it's clear that the model is not performing well, especially for larger inputs. The model outputs generally higher than the target outputs, indienting that the model overshncting the target outputs. Upon closer inspection. I notice that the target outputs seem to have a stronger quadratic relationship with the inputs, especially for larger inputs. The curreut pattern description output To improve the model, I will adjust the coefficient of the quadratic term to reduce the steepness of the relatiouship. will also adjust the linear term better the data. Additionally, will adjust the constant term to better fit the data. New Pattern Descriptions:
imput"2
input + seems
steep, resulting in overshooting.
The new pattern description is: output 2.9 imput ' + 0.5 input + 0.4.
Updated Theta
The new pattern description is: output 2.9 Imput 2 + 0.5 Input + 0.4.
65

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Published in Transactions on Machine Learning Research (01/2025)
Training History:
Polynomial Regression Step 19
Epoch Step
Forward Pass Prompt
You are the model. You will use the descriptions below to predict the output of the given imput.
** Pattern Descriptions: **
You are designed to do regression, i.., to predict the output of any given input. Both input and output are real mumbers. The new pattern description is: output = 3.2 input 22 + 1.8 input + 1.7.
** Input:
[-1.]]
Please give your output strictly in the following format:
Explanations: (Your step-by-step analyses and results]
Output:
(Yorr output MUST be in REAL NUMBER ROUNDED TO TWO DECIMAL POINTS; make necessary aesumptions needed;
MUST be in the same format as the Input]
Please ONLY reply according to this format, don't give me any other words. Forward Pass Output
Explanations:
The given input is 0. Using the pattern description output = 3.2 imput"2 + 1.8 iuput + 1.7, we get output = 3.2 * 0 + 188 * 0 1.7 1.7. Output: [1.70]
Many outputs for the rest data points...
Forward Pass Output
Explanations:
The given pattern description is output 3.2 'iuput"2 1.8 * imput 1.7. We are given an input of 0.3.
First, we square the input to get 0.3
Then, we multiply 0.09 by 3.2 to get. 3.2 0.09 0.288.
Next, we multiply the input by 1.8 to get 1.8 * 0.3 0.54.
Finally. add 1.7 to the sum of the previous resulta to get 0.288 + 0.51 1.7 2.528.
Output: [2.53]
Overall Loss
1.1 1.14999999999999
Optimization Prompt
You are the optimizer for a model, your goal is to learn the best descriptions for the model. The model used the Current Pattern Descriptions below produced the outputs of the given lnputs. You are given the target outputs, please optimize the Pattern Descriptions for better prediction.
Inputs (a batch of i.i.d. data):
[[-155] [-1.8] [-0.8] [0.] [0.3] [-2.6] [0.9] [-0.1] [0.2] [-0.5]
Current Pattern Deseriptions:
designed do regression, i.c., to prodict output any given input. Buth input and output are numbers. The new pattern description is: output = 3.2 input 1.8 input 1.7.
** The model outputs: **
6.2) 8.8] [ 2.3] 1.7] 2.5] [18.6] 55.9 [16] 2.2] 16
The target outputs:
[[ [788 [10.9] [.6] [.5] 3.9] [19.5] [4.8] [.I] [18] [15]
If the model is doing well, you can keep using the current descriptions. However, if the model is not performing well, please optimize model by improving
"New Pattern Descriptions.. The model uses 'New Pattern Descriptions' should better predict the
well as the next batch of L.dd. input data from the same distribution. If previous 'Optimization Step' are provided, you can use the information from your last optimization step it's helpful, NOTE; both the model and you only operate the numerical precision of one decimal points! Please think step by step and give your outputs strictly the following format:
target outputs of the given inputs,
ne
Reasoning:
explicit and verbose, improve the Current Pattern Descriptions by yourself; please show your work; note that you don't have access computer]
New Pattern Descriptions:
[put your new deseriptions here; MUST be specific and concrete; MUS provide the exact value of the parameters if the descriptions potentially involve unknown or learnable parameters!!i*xxy
Please ONLY reply according to this format, don't give me any other words. Optimization Output
Reasoning:
After examining the model outputs and target outputs, notice that the model still not capturing the curvature of the data well, esperially for inputs with large absolute values. The current pattern description output 3.2 input 1.8 * imput + 1.7 seems to be underestimating the outputs for inputs with large absolute values. To improve the model, will increase the coefficient of the quadratic term to better fit the data, and adjust the linear step, where I decreased the coefficient of the quadratic term from 3.4 to 3.2. This time, will increase it to 3.3 to better capture the curvature. Upon examining the data, notice that the model is performing pootly for inputs with large absolute values. will increase the coefficient of the quadratic term to 3.3 to better capture the curvature the data for these inputs. I will also adjust the lincar term to 1.9 to better capture the trend. Finally, will adjust the constant term to 1.6 to better fit the data. New Pattern Descriptions:
better capture tread. I cousider the previous optimization
The new pattern description is: output = 3.3 Imput 2 + 1.9 Input + 1.6. *....0*..**. Updated Theta **0.0*.**0**
The new pattern description is: output 3.3 impput 22 + 1.9 imput + 1.6.
66

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If the model ls doing well. you can keep using the current descriptions. However, if the model is not performing well, please optimize the model by improving the 'New Patteru Descriptions: The model uses the 'New Pattern Descriptions' should better predict the target outputs of the given inputs, Stop' are provided, you can use the information from your last optimization step it's helpful. NOTE: hoth the model and you decimal points! Please think step by step and give your outputs strictly the Published in Transactions on Machine Learning Research (01/2025)
Training History: Polynomial Regression Step 20
Epoch Step 10
Forward Pass Prompt
You are the model. You will use the descriptions below to predict the output of the given imput.
** Pattern Descriptions: **
You are designed to do regression, i.., to predict the output of any given input. Both input and output are real mumbers. The new pattern description is: output = 3.3 * input + 1.9 input + 1.6.
** Input:
[-0.1]
Please give your output strictly in the following format:
Explanations: (Your step-by-step analyses and results]
Output:
(Yorr output MUST be in REAL NUMBER ROUNDED TO TWO DECIMAL POINTS; make necessary aesumptions needed;
MUST be in the same format as the Input]
Please ONLY reply according to this format, don't give me any other words. Forward Pass Output
Explanations:
Given the input -3, we need to calculate the output using the formula: output 3.3 inputt2 1.9 * input 1.6. First, calculate imput 2: (-3)"2 9. Now, substitute the values: output 3.3 9 1.9 (+3) 1.6.
output = 29.7 5.7 1.6.
output
Output: [25.60]
Many outputs for the rest data points
Forward Pass Output
Explanations:
Given the input we ueed apply the formula output 3.3 inpput" + 1.9 'iuput + 1.6.
First, we calculate the square of the input: 0.9"2 = 0.81.
Then, we multiply 0.81 by 3.3: 0.81
3.3 2.673.
Next, we multiply 0.9 by 1.9: 0.9
1.9 = 1.71.
Now, we add 2.673 and 1.71: 2.673
+ 1.71 = 4.383.
Finally, add 1.6 to 4.383: 4.383
1.6 5.983.
Output: [5.98]
1.508000000000002 Overall
Optimization Prompt
You the optimizer for model, your goal is to learn the best descriptions for the model. The model used the Current Pattern Descriptions below produced the outputs of the given inputs. You are given the target outputs, please optimnie the Pattern Descriptions for better predlctino.
Inputs (a batch of i.idd. data):
[[-O.]] [ 0.6) 0.7] [-3. ] [0.9] [=1.9] [-2.6] [0.8] [-0.3] [0.]]]
** Current Pattern Descriptions:
You are designed to do regreesion, iee, to predict the output of any given imput. Both input and output are real numbers. The new pattern description is:
= 3.3 input 2 1.9 input 1.6.
** The model outputs: **
[[ [.44 [39] 4.6] [25.6]
The target outputs: *4 [6.3] [18.9] [5.2 [ [13] [0][22.) [2.]]
[[ [[.I] [4..] [3.6] [26.7] 4.1] 8.6] [20.1] 5.7] [ 2.6 [28]]
well the next batch of i..d. imput data from the same distribution. If previous 'Optinization
only operate the numerical precision following format:
Reasoning:
[b explicit and verbose, improve the Current Pattern Descriptions by yourself; please show your work; note that you don't have access computer]
New Pattern Descriptions:
[put your descriptions here; MUST specific and concrete; MUS provide the exact value of the parameters if the descriptions poteutially involve unknown or learnable parameters!!
Please ONLY reply according to this formst, don't give any other words. Optimization Output
Reasoning:
After examining the model outputs and target outputs, I notice that the model still not capturing the curvature of the data well, input
especially inputa with large absolute values. The current pattern description output be underestinating the outputs for iuputs with large absolute negative values. To improve the model, will increase the coefficient of the guadratic term better fit the data, and adjust the linear term to better capture the trend. will also adjust the constant term to better fit the data. Upon examining the data, I notice that the model is performing poorly for inputs with large absolute negative values. will increase the coefficient of the quadratic term to 3.4 to better capture the curvature of the data for these inputs. I will also adjust the linear term 2.1 to better capture the trend. Pinally, I will adjust the constant term to 1.5 better fit the data. New Pattern Descriptions:
imput + seems to
The new pattern description is: output 3.4 imput + imput + 1.5.
Updated Theta
The new pattern description is: output 3.4 imput 2 + 2.1 imput + 1.5.
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M.3 Sinusoidal Regression (GPT-4o with prior)
Training History: Sinusoidal Regression - Step 1
Epoch O Step 1
Forward Pass Prompt are the model. You will use the dascriptions below to predict the output of the given taput- are designed to do regression, sue,, to predict the output of any given tupot. Both input and output are real numbers, looks like the data generated by a persodic function.
Inputi
Please your
strictly the following format:
Explemation:: siepaly-stty annlyses and resulta]
Output:
(Your output MUST be in REAL NUMBER ROUNDED TO TWO DECIMAL POINTS; make necessary nsuupptions if nooded: MUST be in the zame format as the Input]
Please ONLY raply according to this format, don't give me any other words.
Forward Pass Output
Explonation::
The data le generated by periodic function. Since the function la periodic, might resemble common pertodie functions such au sine costne. For the impus -0.3, we will assue function could be relited to the sine function due to ita common uange in periodic data. The sinc -0.8 radians approximately -0.72.
Output: [-0.72]
outputs for the data
Forward Poss Outpot
Explunaiion::
The data secma to be generated by periodic function, Amsming it follows common periotic pattern auch sinc cosine wave, I'll use the sine function for assimation. First, I normalise the inpuc to align with the expected partod of the sine function. Grven the input [-2-4], I'll compute the sine valus and then scale it appropriately.
Step-hy-step:
1. Assuiee a sypical eine function with amplitude and frequency adjussments: 'y a A din[B x () + D:
2. Without knowing exact values of A, B, G, and D, aceume A = 1, = 3, C O and o for simplicity "y stn(s))
3. Compute sine normalized tnpot: 'sin(-2.4): Using exlenlator:
sia(-2.4) -0.07546
Resull: Roundiug two decimal places: -0.68
Outpot: [-0.68]
3.961
Optimization Prompt
Van the optimiser for model, your goal to learn the best descriptions for the model. The model ueed the Current Pattern Deceription below produced the outpato of the given inputo. You are given the target outiuts, pleas optimine the Pottern Descriptions for better prodiction. Inputs (a batch of L..d. data):
[l [[:18]] 2.4] [.2.] 1.0] -1.5] [-0.7] (.0.4] [1.9] [+1.3] [.o.s]]
Current Pattern Descriptions:
You aze designed so do rogeeusion, 1.6., to prodict the output of any given inpet. Both input and output are roal numbern. It looks like the data is generated by partodic fimesion..
The model outputx:
[[a.. [ [0.7] [+0-7] [1 [-1-]] [-0.8] [-0.4] [1. ] [ [o.2]]
target outputa: [[). [2.7] [1.8] (2.9) [R: [1.4] [LG] [3. [L. 1 [1.3]]
model doing you can keep using current deseriptions. However, model not performing well, please optimiia
by
improving 'New
from Deseriptiona' Ssep' provided. you the Please think munerical precision of decimal
data
distribution.
aptimisation ontgote strictly in the following format: by step and
helpful. NOTE; both
and you
Reasoning:
(be explicit and verhose, Improve the Current Pattern Desoriptions by youseelf please show your work; note that you don't have access to computer]
New Pattern Desoriptione:
(put your new deseriptions here; MUST be specific and concrete; ***MUUTT provide the exaat value of the parameters if the descriptions poteatially involve unkaown or learnable paramcicass!!
Please ONLY caply according to this format, don't give me any other words. Optimization Output
Reasoning:
which suggests appears uxing pertodie
pattern cutpute.
date. ouspuse matching targes
properly different addisional componenis which ouggesis that fauetion generating these oulputs possible optimization could that the model should take skould ideally be achieve this could be shrough combination stmusoidal function and linear transfomattin encure valuns fall
periodic
outpuls positive to what
beiween
model hiss and scaling factor. One within the turget New Pattern Deseriptions:
Given
You are designed so do regrwesion, Le,, to predict the outpot of any given Inpet. Both Input and outpar are real nombers. It looks like the data le generazed by function that could be combimation of partodic funetion and a linee trandormation, Specifically, the model should use the traneformation: Oulput = 1.5 sin(0.5 ingut) |
This should eusuce that the cuipul values range from 1 to 3,
Updated Theta
You are designed to do rogcession i.., prodict the output any given input. Both isput and output are renl numbers, looks like the data generated function that could be combination periodic fuaction and lincur trandoranttion. Specifically, the model ahould wee the tramfforaaiio: Ingot)
Output
should
outpot range
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Training History: Sinusoidal Regression Step
Epoch Step
Forward Pasa Prompt
You are the model. You will use the descriptions below to prodics the output of she given iaput. Pattern Descriptions:
You are designed to do regression, se,, to predict the outpot of any given tnpot. Both Inpur and ousper are real numbers. It looke like the data ie generazed by a pertodie function. You are designed to do regresstan, te, to predict the outpot of any given tsput. Both taput and outpot are real sumbars, It lonke like the data is generated by a fuaction that could be a combination of a periodie Iunction and a llaeer tronsformation. Specifioally, the model should use the traneformasion
Output 1.5 sin(0.5 input) +
This should cusure that the cuiput values range from to
Iupal:
(-2-6)
give your output scrietly in the follawing
bo
Explanations: [Your step-by-step analyses and resulta]
Your output MUST be in REAL NUMBER ROUNDED TO TWO DECIMAL POINTS; make neccusary noumptions if nsoded; it MUST be in the sume fermat oz the laput]
Please ONLY raply according to this format, don't give me any other words.
Forward Puss Output
Explanation::
The given Inpet -1.4,
Compate inscr value of the siac function:
-1.12.
Compate
of -1.12 sin(-1.12) -0.501.
Apply tomooomaation:
-0.901 + -1.852 2 0.65,
ousput
places.
[0.65]
Many outpute for the rest data
Forward Pess Output
Explanations:
Step 1: Ideatify the input value which is 1.8.
Step 2: Use the traufarmation equation Output = 1.8 sinr0.5 input) + 2.
Step 3: Calculate the valus iaside the sinc function: 0.8 1.8 1.44.
Step 4: Caleniata the xine this value: (1.(1.44).
Step 5: Mulsiply the sine recult by 1.5.
Step Add to result from step
Step Round
resuts two dectmal points. Calculation:
1.5 + 0.99 1.485
1.485 2 3.485
Rounded value 3.49
Output: [3.49]
0. 0.2000000000000000 Overall Loss
Optimization Prompt
You are the oplimiser for a model. your goal is to learn the best deseciptions for the model. The model used the Curreal Pattern Desetiption below produced the outguts of the siven imputo. You are given the target outputo, pleas opsimine the Pottern Description for better prediction.
Inputs batch of L.dd. data):
[[225] [-0.3] [1.5] 0.8] 2.2] [2.b] 3.] [-0.6] [ [122] [-1.4]
Current Pattern Descriptions:
designed
regression, tee,, predict outpot any given ingut. input cuepot
like genarszed
partodie
You derigned
that regreszion, to prediet autpot of any given inpus. input and outpott sumbars. combination periodic fuaction
by
Specifiiully, ahould
the tramfoomation:
Outpot 5 sin(0.8 a Ingut)
This should unaure that the output values from to 3.
e* The model outputs: **
[[o,6] [1.0] [3:5] [2.0] [3.5] [3.4] [3. [ [1.3] [3:2] [0.6]]
The target outputt::
[[1.6] [1-7] [s.. [2.4] [2.5] [2.6] [2:2] (1.8] [2.9] [1. I]
If the model is doing well, you con keep using the current deseriptions. However, if the model is not performiig well, please optinise the model by
improving the "New Patsern Descripsione" The model usee the "Now Pattern Descriitiona' should better predies the targes outpots of the given inpots, ad
as the next batch of La.d, imput dastn from the same distribation. previous Optimization Step' are provided. you can use the information from your last aptimisation step if it's helpful. NOTE; both model and you can ealy operate on the namerical precision of one decimul points! Please think step by step and give your outputo strictly in the following Jormut:
(be explicit and verbose, improve the Current Pattern Desoriptions yourself) please show your work; that you don't have computer) New Pattern Desoriptions: [put
descriptions here: MUST be specific and concreta; + provide the exact value of the parameters if the deseriptione potentially involve unknown leurnable duramcccss!!*eeey
Please ONLY caply according to this format, don't give me any other words.
Optimization Output
Reasoning:
Upon examining the current model outputs and their correrponding target outpute, appeare the current pattern description Involving a combination sinmentdal function and a liear trandfrmation not ylelding easiefectrry reentte. The model outports are often aither too high too low compared target values. indicating and biss teru better oligo the ousputs with larget,. Observing the largel outputs suggesto biss term close 1.5 with a alightly dampened outputo closer keeping
nced more Eno-tuning
our model. new palteru deseriptions should coasider adjusting the amplitdde frequenty.
amplitude might aceded, This adjantment could ges the approach
desired rango aligament. Let's zevise current poriodic the frequency to
parameter
We will
valuea:
New Pattern Vou designed do regreasion, se,, predict the output of any given Both Input and by function that could be combination pertodic function and linear trandfomation. Spectfically, the model abould the traneformaiion: Outpul - 2 * sin(0.9 * input) + 1.6.
real numbers. It looks like the dats generated
This adjustment nims to improve the aligament of output values with the Sarget zange (1 to 3). Updated Theta
You are designed so do rogeeasion i.c., to prodict the output of any given input. Both input and output are real numbern. It looks like the data is generated by a funciion that could be combination of periodic fuaction and a lincur transfomaation Specifically, the model abould use the tramsformaiion Output = 1.2 (Dn(D. ingat) + 1.5.
This adjuctment aime to improve the aligament outpot values with the target range (1 to 3).
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Training History: Sinusoidal Regression Step3
Epoch Step Pass Prompt
the discciptiona below to predict oulput of the given iaput. ** Pattern Devoriptions: ee
model. You
You designed to do regresion, se., predict the output of grven Both Input and by periodie function. You are derigned to do regression, to predict the autpot of any given tepus. Both toput and autput are real sumbers. It looks like the data is gonerated by Tuaction that could be combination of a periodio lunction and a ltacez tronsfommation. Specilically, the model ahould use She trausformasion
real numbers. It looks like the dats generated
Outpot = 1.2 (D.9 Inpet)) + 1.5
This adjustment nime to imipooee the aligamment of output valuea with the target runge (1 to 8).
Iuput:
(11.)
Please give your output scrietly in the following format:
Explanations: [Your step-by-step analyses resulte]
Outpot:
output MUST be in REAL NUMBER ROUNDED TO TWO DECIMAL POINTS; make neceusary coumptions nceded: MUST be the the laput)
Please ONLY reply according to this farmat, don't
other words.
Forward Paua Output
Explanations:
1. Idencity the trandformation fanction: Ontgut 1.2 sin(0.9 tnpus) 1.5.
2. Input value: 1.9.
8. Calculate 0.5 11.0 - 1.71.
Calculate sin(1.71) x 0.90,
Caleslate 1.2 0.59 1.19.
Final Output: 1.19 + 1.5 2.69.
Outpat: [2.69]
Many outputs
points
Forward Pess Output
Explanations:
The given inpet is 1.3.
2. According so the tranuformatio function: Output 1:2 * sin(0.9 'input) + 1.3.
3. Caicolate the intermediate valus: 0.9 1.3 = 1.17.
4. Calculate the sime of this intermediate values sin(1.17] is 0.92 (uxing exlenlator or sine table).
5. Mulcinly the sine value by 1:2: 1.2 0,92 = 1.104.
Add 1.5 to the result: 1,104 + 1.5 = 3 604.
Roand the final outpot to two decimal places: 2.60.
Output: [2.60]
Overall Low
0.602
Optimization Prompt
produced the optimizer
your goal
the descriptions for the modal. The model Current Deseriptions below You given target outputs, please opsimise the Pattern Descriptions
given
prediction.
** Inpute batch of 13.d. data); me
[[:19] 1.2] [1.9] [1.3] [-1.2] [3. [-1:3] [1:]] [+1-7] [-2.5]]
** Current Pattern Deseciptions: w*
You are designed lo do rogesusion, be., to prodict the output of any given input. Both imput and output are roal numbero, Il looka like the data is gouerated
by a pertodic finction. You are designed to do regresston, te, to predict the output of any given isput. Both taput and output are real sumbars. It lonies like the dasta is gonerated by a function that could be a combination of a pertodic function and a limer transformation. Specifically, the model should use the tramuformation
Output = 1.2 = Ingot) + 1-5.
This adjustment atme improve the aligament outpot walee with the target range (1 to 3).
The model outputx:
[[o.3] d) [1:2] [2.0] [0.4] [2. [0.4] [2.6] [0.3] [0.6]]
target outpute:
[[a.2] [2.9] [2.9] [1.1] [2:2] [1.1] [I. ] [1.4]]
model performing well, optimiae the model by the "New Pattera Deseriptionn should predics the targei The model distribation. If previous Opptimizationn Stap' provided, can use the information from your wall the banch 1..d. data from the
model
well,
improvinx 'New
last optimization of te's helpful. NOTE: both the model and you can only on the numarical precssion of one dectmal pointel Please thinks step
by stop and give your outguto steictly in the following Jormat:
Reasoning: be explicat and varbose. improve the Current Pattern Descriptions by yourself; please show your work; note that you don't have acceus to computer] New Putters Descriptions:
(put your new descriptions here; MUST spacific and concrete; SUSS provide the excett value of the paramaterr if the descriptions potentially involve un unknown inarnable parametera!!!!
Please ONLY reply according
tormat, don't give any other words. Optimization Output
Reasniig:
moddel culputa indicate that the autput values
deviating signisiuaaly the volues. This auagests that previow pretominaally
adjusiments ampliiude, frequeucy, valuss, indicating that the output needs higher bins potentially better aligement the stnasoldal component's in amplitude, and pasaibly adapting the frequency farther for effective sramufommation.
opsimal
Specitioally, model
than the desized larget
Considering the previons adjustments and observing the new discrepancy, an increased bis term and slight adjusoment in the amplitude might, offer closer aligammns with the target distrrhation. Therefore, more refined strategy would ba to firther increaee the bine to 1.5, adjust the amplitade slightly to 1.3, and recsllate the frequency to 1.0.
Now Pattern Descriptious:
You are designed so do regecosion, iec, to predict the output of any sivon input. Both ipput and oulput are real numbero. It looko like the data is gouerated by a function that could be combination of pertodic function and linerr transformation. Specifically, the model ahould use the transformation: Output 1.3 "= impat) + 1.8.
This further adjustment nima to better aligu the outpus values with the target range 3) and enhance prodiction accuracy.
Updattd
predict the outpot any given ingut. input onrpot numberz. function could coutination periodic fuaction e input) 1.8.
designed regreacion,
like the genarsted Specifically the model should the tramofonnation:
Output 1.3
This furcher adjussmens
botter the oucpus values with the target range 3) onhance prediction accuracy.
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Training History: Sinusoidal Regression Step
Epoch Step Pass Prompt
the discciptiona below to predict oulput of the siven iaput. ** Pattern Devoriptions: ex
model. You
You designed to do regresion, se,, predict the output of grven Both Input and by periodie function. You are derigned to do regression, to predict the autpot of any given tepus. Both toput and autpot are real sumbers. It looke like the data is gonerated by Tuaction that could be combination of a periodie lunction and a ltacaz tronsformation. Specilioally, the model should use She trausformation
real numbers. It looks like the dats generated
Outpot = 1.3 * sin(1.0 Lo Input)) + 1.8.
This further adjastment nime to better aligu the euspus values with the target range (1 to 3) and enhance prediction accuracy.
Iuput:
[-2.2)
Please give your output scrietly in the following format:
Explanations: [Your step-by-step analyses resulte]
Outpot:
output MUST be in REAL NUMBER ROUNDED TO TWO DECIMAL POINTS; make neceasary coumptions needed: MUST be the the laput)
Please ONLY reply accoriing to this farmat, don't Forward Pasa Output
other words.
Explanations:
The given tapot is D.
Using the trandoomation formula: Gutput =1 1.3 ain(1.D ingot)) 1.8.
Subatitiing the input value: Gulput 1.3 sin(1.0 0) I 1.3.
Calculating the sine value: siu(0) = B.
Applying the transformation: Output 1.3
1.8.
Output: [1.80]
Many outpute
points
Forward Puss Output
Explenations:
(liven the
L.8.
2. Calculate the sine the input: sin(1.8).
3. Multiply the sine value by 1.3: 1.3 * sin(1.8).
1. Add 1.8 to the renlts from step 3.
Using caleulator: sm((1a) is 0.97585
1.3 0.97886 s 1:260003
1,206005 1.8 3.068005
Outpat: [3.07]
Overall Low
0.05000000000000008
Optimization Prompt
your goal
the descriptions for the model. The model Current Deseriptions below You given target outputs, please opsimise the Pattern Descriptions
produced the
given
prediction.
** Inpute batch of 1..dd data): =e
[[:2,]] 2. [1-]] [0.]] [1.5] [1.5] [ 1.8) [-1.]] 0o [1.3]]
** Current Pattern Deseciptions:
You are designed lo do rogestsion, ic., to predict the output of any given input. Both imput and output are roal numbero, Il looka like the data is gouerated
by a pertodic function. You are designed to do regresston, te, to predict the output of any given isput. Both taput and output are real sumbars. It lonies likee the dasa is generated by a function that could be a combination of a pertodic function and a limer transformation. Specifically, the model should use the tramffommaiion
Output 1.3 <in(1_0 Ingat) + 1-8.
This further adjussment aime to better align the curput valuae with the target range (1 to 3) and onhance prediction acmirar..
The model outputa:
[[0.8] [a. [2.0] [1:9] [8.1] [3.1] [3:1] (0.0] [a.5]
targot outpute:
[[1.2] [2.9] [2.1] [s.]] [3. [1.1] [2. [3.
model performing well, optimiae the model by Deseription' should predies the targei use the information from your wall the banch 1..d. data from the diseribation. If prevtous 'pptimizationn Stap' provided,
model
improvinx 'New
The model the "New
of te's helpful. NOTE: both the model and you can only operate on the numarical precssion of one dectmal pointel Please think step by stop and give your outguto steictly in the following ing Jormat: er Reasoning: [be explictt and verhore, Improve the Current Pattern Desoriptions by youneelf) please show your work; note that you don't have access to computer] New Patsern Descriptions:
last optimization
(put your new deseriptions here; MUST be specific and concrete; HUS provide the exast value of the parameters if the deseriptions poteatially involve un unknown lsornable paramecnral!!
Please ONLY reply accerding to this format, don't give me any other words. Optimisation Output
Reasoning:
Notably, model the model outputs and the helped, the adplit frequeney higher than necosury, espccillly for higher inpui values. This indicates that while attempta
curzen
ssill performing optimally, evidenced by discepaanccs
adjastucato. Giveu that and change the frequeccy for better fitt Previous frequency 1.0 might alightly misaligned hence might ueed reducing that to 0.8 and keeping modest
aligned closely
amplitude
amplitude to enaure variailitty remaine within check. New Pattern Descriptions:
You are designed to do regrwesion, Le, to prefict the outpot of any given Inpet. Both input and ourpar are real nombers. It looks like the data le generazed by a function that could be combimation of partodic fimction and a linee trandormation, Specifically, the model should use the traneformation: Oulput = 1.2 sin(0.5 input) | 1.8.
This adjustmeut aims to better olign the output values with the targel cange (1 to 3) and ochance prodiction accurocy.
Updated Theta
You are designed to do rogcession ie., to prodict the output any given input. Both input and output are renl numbers. looks like the data generated function that could be combination of periodic fuaction and lincar trandfoaaation. Specifically, the model ahould use the tramffomaation: Output =xn[D.8 Inpot)
adyuetment
sllge cutput
target range (1 and prodiction accuray
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Training History: Sinusoidal Regression Step
Epoch Step Paza Prompt
the discciptiona below to predict oulput of the siven iaput. ** Pattern Devoriptions: ex
model. You
You designed to do regresion, se,, predict the output of grven Both Input and by periodie function. You are derigned to do regression, to predict the autpot of any given tepus. Both toput and autput are real sumbers. It looke like the data is gonerated by Tuaction that could be combination of a peciodie lunction and ltacez tramsformation. Specilioally, the model ahould use She trausformasion
real numbers. It looks like the dats generated
Outpot = 1.2 <in(D.8 Inpot) + 1.8.
This adjustment nima to better align the output values with the target range (1 to 3) and enhance prodiction accuracy.
Iuput:
[0.3]
Please give your output scrietly in the following format:
Explanations: [Your step-by-step analyees resulte]
Outpot:
output MUST be in REAL NUMBER ROUNDED TO TWO DECIMAL POINTS; make neceasary coumptions needed: MUST be the the laput)
Please ONLY reply according to this format, don't Forward Pasa Output
other words.
Explanations:
1. Tako the inpus value: D.D.
2. Apply the tranzformation: Output = 1.2 (in(D.8 & tapot) + 1.8.
8. Calculate the sine fauctiou: siu 8 0.00 sin(0) -
4. Colculate the product: 1.2
0.
Add the comstant: 0 + 1.5 1.8.
Output: [1.80]
Many outpute
points
Forward Puss Output
Explenations:
Identify the ingut valuei 1.7.
2. Use the given cramuoommaiio formuls to calculate the output:
Output 122 sin(0.5 * L.T) 1.8.
3. Calcnlate the valuie inxide the sine function: D.8 1.7 = 1.36.
4. Compote the zine of 1.36: stn(1.36) is 0.9776.
5. Mulciply the sine value by 1:2: 1.2 0.9776 is 1.173.
Add 1.8 so the result: 1. 1.173 + 1.8 as 2.973.
Roand the recalt to two decimal points: 2.97.
Output: [2.97]
Overall Low
0.158
Optimization Prompt
produced the optimizer
your goad learn the descriptions for the model. The model Current Deseriptions below You given target outpuss, please opsimise the Pattern Descriptions
given
prediction.
** IIppus batch of 1..dd data): =e
[[0.3] [:2.3] [0.7] [-2.3] 0.2) [o. [-2.3) [1:7] [ [0.2] [-2.2]]
** Current Pattern Deseciptions: w*
You are designed lo do rogesusion, ic., to prodict the output of any given input. Both imput and output are roal numbero, Il looko like the data is gouerated
by a pertodic function. You are designed to do regresstan, te, to predict the output of any given isput. Both taput and output are real sumbars. It lonies like the dasa is generated by a function that could be a combination of a pertodic function and a limer transformation. Specifically, the model should use the tramuformation
Output = 1.2 stn(D.8 Ingot) + 1.8.
This adjustment atme better alige the output valuse with the target range (1 to 3) and onhance prediction acmuracy.
The model outputx:
[[2.3] [0.0] [2:4] [0.0] [2. [1,9] [0.6] [3. 1.1.2 [2. |0.6]]
targot outpute:
[12.]] [1.2] [1.1] [2:2] [ [1.2] [.. [2:2] [1:2]]
model performing well, optimiae the model by Deseriptionn should predics the targei wall the banch 1..d. data from the distribation. If prevtous "pptimizationn Stap' provided, can use the information from your
model
improvinx 'New
The model the
last optimization of te's helpful. NOTE: both the model and you can only on the numarical precssion of one dectmal pointel Please thinks step
by stop and give your outguto steictly in the following Jormat:
Reasoning: be explicat and varbose, improve the Current Pattern Descriptions by yourself; please show your work; note that you don't have acceus to computer] New Putters Description::
(put your new descriptions here; MUST spacific and concrete; SUMS provide the excett value of the paramaterr if the descriptions potentially involve un unknown inornable parametera!!!!
Please ONLY reply according
tormat, don't give any other words. Optimization Output
Reasonin::
model cucrently underperfomming, particularly uoticcuble with the lower ther--thn-sppeeed outputs, The discrepancy poiuts towards noed adjutt omplitude and posiily she trequeucy further. Additioually, the moddll's seed bit of fine-tuiing to adjust deviations. Hance, to better approstmate the outpute, will elightly incraase the frequency and adjust the it aligua the partodie component better. The amplitude will be increased slightly
be
the
might still
amplitude. The previous frequency of U.5 will be increased to 0.9 to 1.3, considering the sgaee-than-eeeded target valies.. New Pattern Desorittions:
You are designed sa do regression, se,, to predict the outpot of any given inpet. Both input and outpor are resl numbery. It looks like the data le generazed by a function thot could be coustimation of periodie fuaction and a lincur Leandoramation, Specifically, the model abould use the tramiformation:
Oulput = 1.3 sin(0.9 ingut) + 1.8.
This adjustment sime to better alige the output valums with the target range (1 to 3) and unhance prodiction accuracy. Updated Theta
designed do roscession, ie., prodict the output any given input. Both input and ousput aze aumbers, looks like the data geuerated should traneformation:
pertodic
trandfrmation. Specifically
=in(0.9 inpat)
This timent
olign. output values with cauge (1 and enhance prodiction accuracy.
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Training History: Sinusoidal Regression Step 29
.no.cesuoven Epoch Step recosceccese
Forward Prompt
deveriptione to prodies cutput given toput-
Pattera
**
Both iapui and oulgut numbers, looko like the data keuerated periodic function. You designed de regtesion, to predict the output of any given like the data generated by function that could be combination of pertodie function and linear transformation. Spectficall, the madel should the transformation:
desigued do regeession,
prodict the
Buth and
sumbers, looks
Output = 1 03 * sinç0.95 95 inpot) + 2.0.
This fine-tuningg ls intendted to adidress undorestimatione by elightly increasing the linear blae, matntaining the pariodic nature for better aligament with the target ontpote.
Iupul:
3.6
Please give your output strictly in the following format:
Explanations:
[Your stsp-by-stop analyaca and resulta]
Outpot:
outpot MUST he REAL NUMBRR ROUNDRD the laput]
DECIMAL POINTS; neceeaary socumptione neaded; MUST be
Please ONLY reply according this format, don't Forward Pass Output
any other words.
Explamation::
1. Calcalate the inner function of the sine: 0.98 1.4 1 1.372.
2. Find the sine of the rasult: (1n(1.)72) is 0.9803.
3. Multiply by the factor: 1.05 0.9600 2 1,029.
Add the liner blssi 1.029 + 2.0 = 3.029.
Outpot: [3.00]
Many outputs for the rest date points ...
Overall
Optimisation Prompt
Von the optimiser for model, your goal to learn the best deccriptions for the model. The model ueed the Current Pattern Deseriptione below produced the ontpatz of the given Inpist. You given the target outpose, pleaze opstamize the Pattern Deseriptions for better prodiction.
** Inputs (a batch of Lidd: data):
[[2.6] 2.9) [2.9] 0.1] 1.4 [-2.5] 2.1] (-2.4] [-0.5] [-1.6]]
Current Pattern Descriptions:
You are designed so do rogression, i.c., to prodict the output of any given inpet. Both input and output are rosl numbers. It looks like the data is gunerated
by a periodic function. You are designed to do regresiinn ie., to predict the output of any given ispus, Both iaput and outpot are real sumbers, It looks the data generated by a fosction that could be a combination of a periedic function and as thear transfomaatoon Specifically, the madel should ese she traneformation:
Output L.05* sia(0.95 impul) 2.0.
This fiaeswaing insended to oddress undercatimation by alightly increasing liacar biss, maiutaining the periodic nature for better aligament with the target outpute.
model
**
[[2.6] [2.3] [B.1] [S.
[2.9] [12 [1.3] [I
The
**
[[2,6] [2.3] [2:5] [8.1] [3. 1 [1.4] [2.]] (13) [13]] [I. I]
the current descriptions. However, the model not parforming well, plans optimixe the model by suproving the "New Patsern Descripsiona The model uses the "New Fattern Description' should better prodies the target outputa of the given inputo, as
the model doing well,
well as the next basch of Lid, inpus data from the same distribution. It previous "Optimization Ssep' are provided, you can use the information from your last oprimization step if it's helpfu). NOTE: both the model and you can only operate on the momerical precision of one decimal points! Please think step by step and give your outpata strictly in the following formut:
Reasoning:
be explicst and varhose, improve the Current Pattern Descriptions by yourzelf; please show your works; note that you don't have accers computer]
New Pattern Desersptiona:
pput your desoriptions here;
specific concretaj s*SEMUST provide exact value the parameters the deseriptione potentially parameteratt)-ekg
Please ONLY according this
don't
other words.
Optinisation Output
Reasening:
Analyzing the
of Iapota, model outputa, and target putputs:
1. Bor impus [2.4]. the model outgot is 2.5, matching the target outpot exantly.
2. Far impute [2.9] and [2.9), the model ontputs are both 2.3, exsetly matching the target outputs.
3. Par imput [0.1). the model outpot te 2.1, masching the target outpot exactly.
4. For iupul [1.4] the model output is 3.0, matching the turgel oulput cancily.
5. For iopat .2.6), the model cusput is 1.3, while the tarpes outpat is 1.4 elight understivatiion.
For impat 2.1).
model ousput ts 2.9. while the targes autput is 2.8 slighs overestimation.
For imput (-2.4), the model autpus is 1.2, while the target output is 1.3 alight underestimation.
For inpat -0.8). the medel cusput la 1.3, matching the turges ousput exactly.
tngest [-1.6].
cusput
matching targes output exactly.
analyste: have model catputs matching the
cxucily.
The model alightly inatettmmatos the result for [-2.0] (-2.4),
The model slightly undereatinates
[2.1]. Reviewing she current pattorn decoriptions:
The previow adjustment to the licarr biss from 1:95 to 2.0 heiped mitigate indersatimations by lifting the overall predictions.
However, slight underestimatin immtiini still oceuz for negutive inpato while overestimatio appears for positive input. To balance these observations
propore alightly wijusting the linear hiue down tiny step to corract the overwrtmacters and adjusting the amplitude sealing factor to fine-tune the cerponze to negacive iapots:
New Pattern Descriptions:
You are designed to do regression, Le,, to predict the output of any given inpet. Both input and output are real numbers. It looks like the data is generated by a function that could be combiattio of perlodie fimction and liner transfomation, Specifically the model should use the transformation Output 1.00 sia(0.95 imput) + 1.96.
This fine-tuning is intended to correct minor underestimations for negative inputs and slight ovcreatimation for positive inpata, thereby mointaiaiug overall alignment wish targel oulputs.
Updated
designed
regressin, combinttio perlodio function + 1.95.
the
given input.
and outpot numbers. looks like the data transforaatinn Specifioally the model abould the tramsformation:
function could
Outpot 06 six(0.98
This fine-timnn is Intended to correct minor underestimations for negative topote and elight cvnrectmationz for posttive Inpate, thereby maintaining overall aligament with the Larget oulputs.
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Training History: Sinusoidal Regression Step 30
Epooh Step Paza Prompt
the discciption below to predict oulput of siven iaput. ** Pattern Devoriptions: ex
model. You
You decigned to do regnesion, se,, predict the output of green Both Input and to predict the autpot of any given tepus. Both toput and autpot are real sumbers. It lonks like the data is gonerated by fuaction that could be a combination of a peciodie function and a ltacaz tronsfommation. Specilioally, the model should use She trausformasion
real numbers. It looks like the dats generated
by periodie function. You are derigned to do regression,
Outpot = 1.00 06 sin(0.95 as inpot) + 1.95
Thu finetwning is intended to correct minor underestinatiion for negative isputa and slight overeatimations for positive iapata, thereby maintaining overall niigument with the target oulputs.
Input:
[-0.2)
Please give your output strictly in the following format:
Replanations:
[Your step-by-step analyses resulte]
Outpat:
YYou oulput MUST be in REAL NUMBER ROUNDED TO TWO DECIMAL POINTS; make neceosnry coumplions needed:; MUST be in the the laput)
Please ONLY raply according to this farmat, don't
other words.
Forward Pass Output
Replanations:
1. Calcalate the shne of 0:98 simes the input: stn(0.90 98 1) = win(D. 98) zs 0.830
2. Multiply the resalt by 1.06: 1.06 0. 880 as 0.860 3. Add 1.96 to she product: 0.880 + 1.95 a 2.98
Outpot: [2.83]
Many outputa for the reat data points ...
Overall
0.008000000000000000
Optinisation Prompt
goal to learn she best descriptions for the model. The model used the Current Pattern Deseriptions below produced the outputa of the given inputa. You are given the target outputa, please opstmixe the Pattern Diseriptions for bettez prediction.
opsimiser
model,
** Inputs batch of Li.d. data):
[[-0.2] I 1-1] [ 1199 [ 0.7] -2277] [-0.9] 1-7] [2.8] [1. [-0.3]]
Current Pastern Descriptions:
You are designed to do regression, s.e., to predict the output of any given inpet. Both input and ouppur are rosl numbers. It looks like the data to gunerated by a poriodic Dunction. You are designed to do regresslon, Le, to predict the output of any given iopus. Both iopus and output are real numbers, Is looks like the data generated by a fusotion that could be a combination of a periodio Function and a Thear transfommation Specifioally, the model should use the tramffommaton:
Output 1.0 sia(0.08 iaput) 1.96.
This fuo-taning is intiended to coerect minor undercstination negative isputo and slight ovecatimation for positive inpato, thereby mointaisiug
target outputs..
The modal
**
[[)77] [2.0] [3. [2.6] [1.4] [1.1] [s. [2.4] [2.8] [1:00]
** The
**
[[1.8] [2.9] [S. [2.7] [1.6] [1.2] [B. [2.4] [2.8] [1.7]]
if the model doing well,
keep wing the current desetiptions. However, if the model is not performing well, pleus optimize the model by
improving vig the "New Patsern Deseripiion! The model use the "New Fattern Deseription' should betler prodict the targei outguta of the given inpuls,
wall as the next basch of 1..d. ippue data from the same distribation. If previous Optimization Step' are provided, you can use the informacion from your last optimisation step if is's helpful. NOTE: both the model and you can only operate on the nomerical precssion of one dacimall points! Please thinks step by step and give your outputa strictly in the following format:
Reasoning:
be explicst and varhose, improve the Current Pattern Descriptions by yourzelf; plaase show your work; note that you don't have accees computer) Now Puttern Deseriptions:
[put your new deseriptions here;
specitic and concretej **SEMUST provide the exact value of the parameters the deseriptions potentially imsenable parametara!!
ni
according this Please ONLY Optimization Output
don't
other words.
Reasening:
Amalysing the new act of imputo, model outputo, and targel outputo:
1. For imput [-0 the model output is 1.7, while the target ousput is 1.8 alight underestinasion.
2. Far Input [1:1]: the model ontput is 2.0, masching the target outpot exactly.
8. For imput [1.9). the model output is 3.0. masching the turget output caucily.
4. For imput [. tie model output is 2.8, while the target output is 2.7 alight undereatimnnion.
For Impet -2.7). the model ousput in 1.4, while the target output ie 1.8 notable underrestiattin..
Por Imput (-0.9). the model eusput is 1.1, while the target autput is 1.2 eiihht underwttmmstion.
Far taput []. the model outpat is 3.0. masching the target output exaetly.
Por topat 2.6) model cnepot
model ousput
matching the target outpat exactly.
toget
target ourpot output
For [-0.3). the model catput 1.6, while the Five analysio:
slight underotiantiin.
inpuls have model
matching
outpets exactly.
The model alightly underestimater rucults for inpute [-0.2]. (0.7], [-2.7], (-0.9, and [-0.3]. Revieving the current pattern descriptions:
The previous adjustment to the linerr bins 1.95 simed correct understimation while also adjusting the amplitude scaling factor alightly improved estimacions but alight indereetimationn still perelet.
While wdjusting the Binnr blne further down might correct some underestimatione, the predomimant devlation seems to involve negative Inputs (lose prettive concribatione). Also, encuring pertodic factor sealing stability:
Now Patters Descriptious:
You are designed so do regecosion, ie., to predict the output of any givon input. Both ipput and oulput are real numbero. It looko like the data is gouerated by a function that could be combination
pertodic function and linerr transformation. Specifically, the model abould use the traneformation:
Output 1.07 sia(0.99 imput) + 1977.
This fine-tuning intended correct minor undecstimation for negative laputa und casuring overall alignment with the turget outputs.
Updated
designed
regrescion,
predict autpot any given ingut. input onepot
looks like Specificall, the model ahould the tramfornation:
genarsted
function could cousination periodic
Output 1.07 sla(0.95 imput) 1.37.
This tinestuning intended correct milorr underestimations for negative inpote and ensuring overall alignment with the sarget outpots.
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M.4 Two Blobs Regression 1LLama-3-70B without prior)
Training History: Two Blobs Classification
Step 1
Epoch 0 Step
Forward Pass Prompt
You the model.
Model Descriptions:
are designed to do binary classification. input a point 2-D plane [x y]: you need output vector containing two probabilities such that each corresponds to how likely the data belongs to each class, Le., (class 1 prob, class 2 prob.]. The sum of the vector MUST 1.0.
** Input:
[-2.178 9.99
Please give your output strictly the following format:
Explamations: [Your step-by-step analyses and results]
Output:
[ONLY PURE probability vector, where each value is between 0.0 and 1.0 WITH TWO DECIMAL POINTS; make necessary
assumptions if needed]
Please ONLY reply according to this format, don't give me any other words.
Forward Pass Output
Explanations: Since the model not trained, will make an ascumption that the data is uniformly distributed across the 2-D plane.
Based this sasumption, will assign equal probabilities to both classes.
Output: [0.50 0.50]
Many outputs for the rest data points...
Forward Pass Output
Explanations: Since the model a binary classifier, I'll make an assumption about the decision boundary. Let's assume the decision boundary is linear line with slope of -1 and passes through the origin. This means that points above the line belong to class and points below the line helong to class 2. Given the input (-3.616, 7.818], we can calculate the dot product the imput with the normal vector of the decision boundary (which [I,-I]). The dot product is -3.616 + 7.818 4.202, which is positive, so the point is above the decision boundary. Therefore, the point is more likely to belong to class 1. Output: [0.70, 0.30]
Training Accuracy
0.5
Overall Loss
0.4302008 151 11499229
Optimization Prompt
You are the optimizer for a model, your goal is to learn the best descriptions for the model, The model used the Current Model Descriptions below predicted how likely the given inputs belong class. You are given the target values, please optimiie the Model Descriptions for better prediction.
Inputs (a batch of i.idd data on 2-D planer [x yl):
[[2.178 9.99 [-3.111 10.867] [-4.234 8.452] (-3.616 7.818] [ 5.155 2.487] [-2.625 8.713] [ (3.75 1.157] [-2.523 7.957] [-3.348 8.705] 3.569 2.456]]
** Current Model Descriptions: **
You are designed to do binary classification. The input is a point on 2-D plane [x yl: you need to output vector containing two probabilities such that each corresponds to how likely the data belongs to each class, i.c., (class prob. class prob.]. The sum the vector MUST be 1.0.
** The model predictions ([class prob. class 2 prob.]):
[[0.3 0.7 [0.99 0.01] [0.7 0.3 10.7 0.3 [0.85 0.15) [0.5 0.5| 0.2 0.8 [..2 0.7] [0.65 0.35]]
The targets ((class 1 prob. class 2 prob.]):
0.] [1. 0.] 0.I [1. O. [0. 1.] [1.
[1. 0.) [1. 0.] [0. 1.]]
Please update the model by improving the 'New Model Descriptions', which should have lower classification error both on the current and the next. batch of
data. previous 'Optimization Step' are provided, you use information from your last
optimization step it's helpful. Both the model and you MUST ONLY operate the numerical precision of THREE decimal points. You are bad with numerical calculations, so be extra careful! Please think step by step and give your outputs strictly in the following
format:
Reasoning:
bee explicit and verbuse, improve the Current Model Descriptions by yourself; please show your work; note that you don't have access computers
New Model Descriptions:
[put your new decision rules here; MUST be concise and conerete; MUST PROVIDE THE EXACT VALUE OF THE PARAME- TERS if the descriptions potentially involve unknown or learnable parameters!!!
Please ONLY reply according to this format, don't give me any other words.
Optimixation Output
Reasoning:
The goal is to update the model descriptions improve the classification accuracy. From the current model predictions and targets, it's clear that. the model is struggling to distinguish hetween the two classes. Upon visual inspection of the input data, it appears that the data pointa with x 0 tend belong class 2, while those with send belong class Let's simplify the decision boundary by introduciag a threshold on the x-axis. If x > U, assign high probability to class 2, otherwise assigu a high probability to class
New Model Descriptions:
If 0, output [0.0, 1.0). If < 0, output [1.0, 0.0).
Updated Theta
If x > 0, output [0.0, 1.0). If < 0, output [L.0, 0.0).
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Training History: Two Blobs Classification Step 2
Epoch Step
Forward Pass Prompt
You are the model. ** Model Descriptions: **
You are designed to do binary classification. The input a point 2-D plane [x y]; you need to output vector containing twn probabilities such that each correaponds to how likely the data belongs to each class, lee,, (class prob, class prob... The sum of the vector MUST be 1.0. If x output [0.0, 1.0]. If O, output [1.0, 0.0].
Input:
[-2.252 8,91
Please give your output strictly in the following format:
Explanations: [Your step-by-step analyses and results]
Output:
[ONLY A PURE probability vector, where each value is between 0.0 and 1. WITH TWO DECIMAL POINTS; make necessry
assumptions if needed]
Please ONLY reply uctording to this format, don't give me any other words.
Forward Pass Output
Explanations: Since > 0, will output [0.0, 1.0].
Output: [0.00, 1.00]
Many outputs for the reat data points
Forward Pass Output
Explanations:
Given the input [6.784, 2.007), since > 0, the output will be [0.0, 1.0].
Output: [0.00, 1.00]
Training Accuracy
1.0
Overall Loss
449999999944616000 1664990004411444000000
Optimization Prompt
You the optimizer for model, your goal is to learn the best descriptions for the model. The model used the Current Model Descriptions below predicted how likely the given iuputs belong to a class. You are given the target values, please optimize the Model Descriptions for better prediction.
Inputs (a batch of i.id. data on 2-D plane: [x yl):
[[-2.252 8.94 3.378 2.447] [-2.449 11.478] (-3.211 8.687] 6.784 2.607] 4.9 2.755] 4.653 3.427] [-1.504 9.343] [-2.702 9.316] (-1.044 8.789]]
** Current Model Descriptions:
You designed do binary classification. The input point 2-D plane [x vl; you need output vector containing two probabilities such that each corresponds to how likely the data belongs to each class,, Le., (class prob, class 2 prob.]. The sum of the voctor MUST be 1.0. If x
output [0.0, 1.0]. If 0. output [1.0, 0.0].
The model predictions ([class 1 prob. class 2 prob.l):
[[I. 0.] 1.] 0.] [1. 0.] 1.] [0. 1.] [o. [1. 0.) [1. 0.
[1. 0.]]
* The targets ([class prob. class 2 prob.]):
[]. D. 0. 1. 0.) [L. 0.j [o. 1.] [o. 1.] [o. 1. [L. 0.j [I. o.] [1. o.]]
Please update the model by improving the 'New Model Descriptions', which should have lower classification error both on the current and the next batch of i.idd data. If previous 'Optimization Step' provided, you
the information from your last optimization step it's helpful. Both the model and you MUST ONLY operate on the numerical precision THREE decimaal points.
You are bad with numerical calculations, be extra careful! Please think stop by stop give your outputs strictly in the following
format:
Reasoning:
[be explicit and verbose, improve the Current Model Descriptions by yourself; please show your work; note that you don't have access to computers'
New Model Descriptions:
pout your decision rules here; MUST be concise and concrete; ***T PROVIDE THE EXACT VALUE OF THE PARAME- TERS if the descriptions potentially involve unknown or learnable parameters!!!****
Please ONLY reply sccording to this formst, don't 't give any other words.
Optimization Output
Reasoning:
The current model descriptions are already quite simple and accurate; as they correctly classify all the input data points. However, to further improve the model, let's analyze the data points class 2, while those with y > 8.5 tend to belong to class 1. This observation suggests that the y-axis also contains useful information for classification, To incorporate this Information, we can modify the decision rule to cousider both the x and y axes, x O, we still assign a high probability to class 2. If O, check the y value: if < 8.5, assign high probability to class otherwise assign a high probability to class 1.
closely. We notice that the data points with < tend to belong to
New Model Descriptions:
If x > 0, output [0.0, 1.0). Ifx < 0, y < 8.6, output [0.0, 1.0], otherwise output [1.0, 0.0).
Updated Theta
If x 0, output [0.0, 1.0). x < 0, if 8.5, output [0.0, 1.0), otherwise output [1.0, 0.0).
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Published in Transactions on Machine Learning Research (01/2025)
Training History: Two Blobs Classification
Step 3
Epoch Step
Forward Pass Prompt
You are the model. ** Model Descriptions: **
You are designed to do binary classification. The input a point 2-D plane [x y]: you need to output vector containing twn probabilities such that each correaponds to how likely the data belongs to each class, L.e, (class prob, class prob.]. The sum of the vector MUST be 1.0. If x output [0.0, 1.0]. 1 O, if y 8.5, output [0.0, 1.0), otherwise output [1.0, 0.0).
Input:
[-2.267 7.101]
Please give your output strictly in the following format:
Explanations: [Your step-by-step analyses and results]
Output:
[ONLY A PURE probability vector, where each value is between 0.0 and 1.0 WITH TWO DECIMAL POINTS; make necessryy
assumptions if needed]
Please ONLY reply actording to this format, don't give me any other words.
Forward Pass Output
Explanations:
Siuce x 0, output [0.0, 1.0].
Output: [0.00, 1.00]
Many outputs the data points
Forward Pass Output
Explanations:
Siace x O, we need to check the value of y. y is 7.551 which is less than 8.5. Therefore, the output should J0.0, 1.U/. Output: [0.00, 1.00]
Training Accuracy
Overall Loss
L.8420680703922368
Optimization Prompt
You are the optimizer for a model, your goal is to learu the best descriptions for the model. The model used the Current Model Descriptions below prodicted how likely the given inputs belong class. You given the target values, please optimize the Model Descriptions for better prediction.
** Inputs batch of i.i.d. data 2-D plane: yl): ** [[22.67 7.101] [3.96 2.205] 5.453 0.742] [-2.001 7.551] [-3.186 9.626] [-4.428 8.988) [ 5.7233 3.027] [-3.054 9.125] [ [4.563 2.314] [-3.837 9.211]]
** Current Model Descriptions:
You are designed to do binary classification. The input a point on a 2-D plane [x y); you need to output vector containing two probabilities such that each corresponds to how likely the data belongs to each class, i.c., [class prob. class prob.]. The sum of the vector MUST be 1.0. If x > 0, output (0.0, 1.0]. If < 0, if < 8.5, output [0.0, 1.0), otherwise output [1.0, 0.0). ** model predictions (fclass prob. class prob.]): **
[[o. 1.] [0. 1.] 00. 1.] J0. 1.] [1. 0.] [1. 0.. [0. 1. [1. 0.] [o. 1.] (.])
** The targets ([class prob. class 2 prob.]): **
[[1. 0.] 1.] 00. 1.1 [1. U. [1. 0.]
(0. 1. 0. [0. 1.)
Please update the model by improving the "New Model Descriptions', which should have lower classification error both on the
and the batch of i.i.d. data. If previous 'Optimization Step' provided, you the information from your last optimization step it's helpful. Both the model and you MUST ONLY operate on the numerical precision of THREE decimal points. are bad numerical calculations, be extra careful! Please stop by stop give your outputs strictly the following
format:
Reasoning:
be explicit and verbose; improve the Current Model Deseriptions by yourself; please show your work; note that you don't have access computers
New Model Descriptions:
jput your new decision rules here; MUST concise and concrete; MUST PROVIDE THE EXACT VALUE OF THE PARAME- TERS if the descriptions pocentially involve unknown or learnable parsmetens!!!**es
Please ONLY reply acrording to this format, don't give any other words. Optimization Output
Reasoning:
The current model descriptions are still quite simple, but they make some mistakes in the current batch of data. Let's analyze the data points more closely. We notice that the data points with x > and y < 2.5 tend to belong class while those with < and > tend to belong class 1. This observation suggests that the y-axis contains more useful information for classiftcation, especially when x > O. To incorporate this information, we can modify the decision rule to consider both the x and y axess more carefully. If x 0, chock the value: if < 2.5, assign high probahiitty to class otherwise assign high probabllity to class 1. If x e 0, we also check the y value: if > 9, assign a high probability to class 1, otherwise assign a high probability to class 2. New Model Descriptions:
If x 0, if < 2.5, output [0.0, 1.0), otherwise output [1.0. 0.0). If x O, if
9.0, output [1.0, 0.0), otherwise output [0.0, 1.0).
Updated Theta
If
0, if 2.5, output [0.0, 1.0], otherwise output [1.0, 0.0]. If x if 9.0, output [1.0, 0.0], otherwise output (0.0, 1.0].
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Published in Transactions on Machine Learning Research (01/2025)
Training History: Two Blobs Classification
Step 4
Epoch Step
Forward Pass Prompt
You are the model. ** Model Descriptions: **
You are designed to do binary classification. The input point 2-D plane [x y]: you need to output vector containing two probabilities such that each correaponds to how likely the data belongs to each class, Lee, (class prob. class prob.. The sum the vector MUST be 1.0. If 0, y < 2.5, output [0.0, 1.0], otherwise output [1.0, 0.0]. If < 0, if y 9.0, output [1.0, 0.0],
otherwise output [0.0, 1.0].
** Input: ** [-3.988 8.294]
Please give your output strictly the following format:
Explanations: (Your step-by-step analyses and results]
Output:
JONLY A PURE probability vector, where each value is between 0.0 and 1.0 WITH TWO DECIMAL POINTS; make necessary
assumptions if needed]
Please ONLY reply according to this format, don't give me any other words.
Forward Pass Output
Explanations:
Since x 0 and 2.5, according the rules, the output should be [0.0, 1.0).
Output: [0.00, 1.00]
Many outputs for the rest data points
Forward Pass Output
Explanations:
Since x 5.113 0, check the value of = 1.9 < 2.5, so we output [0.0, 1.0].
Output: [0.00, 1.00]
Training Accuracy
0.7
Overall Loss
2.763102108092855
Optimization Prompt
You are the optimizer for a model, your goal is to learu the best descriptions for the model. The model used the Current Model Descriptions below prodicted how likely the given inputs belong class. are given the target values, please optimize Model Descriptions for better prediction.
** Inputs (a batch of I.I.d. data on 2-D plane: [x yl):
[[-3,988 8.294] (-5.129 9.836] 3,403 0.653] [-3.522 9.329] [ 3.663 2.76 [-2.417 7.027] 5.113 1.9 1-3.318 8.513] [-3.925 8,594] [-3.417 7.602]]
** Current Model Descriptions: **
You are designed to do binary classification. The input a point on 2-D plane [s yl: you need to output vector containing two corresponds to how likely the data belongs to class, i.c., (class prob. class prob.]. The sum the vector MUST be 1.0. If > 0, if < 2.5, output [0.0, 1.0], otherwise output [1.0, 0.0). If < O, if >> 9.0, output [1.0, 0.0], otherwise output [0.0, 1.0].
probabiliites such
** The model predictions (fclass prob. class prob.]): **
[[.. 0.] [1. 0.] 10. 1.] [1. 0.] [0. 1.] [1. 0.] [o. 1.) [0. 1.) [0. 1.) (o. 1.]
** The targets ([class prob. class prob.]): **
[[1. 0.] [1. 0.] jo. 1.] [1. 0.] [0. 1.] [1. 0.] [0. 1.) [1. 0.) [1. 0.] [1. 0.]]
'New Model Descriptions', which should lower classification error both on current and the next batch of i.i.d. data. If previous 'Optlmization Step' are provided,
Ploase update
model by improving
use the information from your last optimization step it's helpful. Both the model and you MUST ONLY operate on the numerical precision THREE decimal points.
You are bad with numerical calculations, be extra careful! Please think step by step and give your outputs strictly in the following
format:
Reasoning:
explicit and verbose, improve the Current Model Descriptions by yourself; please show your work: note that you don't have access to computers
New Model Descriptions:
[put new decision rules here; MUST be concise and conerete; *UST PROVIDE THE EXACT VALUE OF THE PARAME-
TERS if the descriptions potentially involve unknown or learnable parameters!!!
Please ONLY reply according to this format, don't give me any other words.
Optimization Output
Reasoning:
Analyzing the current batch of data, we notice that the current model descriptions are mostly correct, but there are some mistakes. For example, the point [-3.318 8.513) is misclassified, and it should helong to class This suggosts that the threshold for y-axis in the x < 0 region should be lowered. Additionally, we obeerve that the points with x > 0 and > 2.5 tend to belong to class 1, and region should be
points with x
y tend belong class 2. This suggests that the threshold for y-axis
raised.
New Model Descriptions:
If
if 3.0, output [0.0, 1.0], otherwise output [1.0, 0.0). If if 8.0, output [1.0, 0.0], otherwise output (0.0, 1.0]. Updated Theta
if
3.0, output [0.0, 1.0), otherwise output [1.0, 0.0). TF if 8.0, output [1.0, 0.0), otherwise output 00.0, 1.0].
78

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Published in Transactions on Machine Learning Research (01/2025)
Training History: Two Blobs Classification
Step 5
Epoch Step
Forward Pass Prompt
You are the model. ** Model Descriptions: **
You are designed to do binary classification. The input a point 2-D plane [x y]: you need to output vector containing two probabilities such that each correaponds to how likely the data belongs to each class, Lee, (class prob. class prob.. The sum of the vector MUST be 1.0. If 0, y < 3.0, output [0.0, 1.0], otherwise output [1.0, 0.0]. If < 0, if 8.0, output [1.0, 0.0],
otherwise output [0.0, 1.0].
** Input: ** [4.698 0.83
|
Please give your output strictly the following format:
Explanations: (Your step-by-step analyses and results]
Output:
JONLY A PURE probability vector, where each value is between 0.0 and 1.0 WITH TWO DECIMAL POINTS; make necessary
assumptions if needed]
Please ONLY reply according to this format, don't give me any other words.
Forward Pass Output
Explanations:
0, y 8.0, output [1.0, 0.0].
Output: [1.00, 0.00]
Many outputs for the rest data points...
Forward Pass Output
Explanations:
Since x 0 (4.698 0), check y 3.0 (0.83 < 3.0). This is true, the output [0.0, 1.0].
Output: [0.00, 1.00]
Training Accuracy
0.8
Overall Loss
1.8 1.8200680733952368
Optimization Prompt
You the optimizer for model, your goal is learn the best descriptions for the model. The model used the Current Model Descriptious below predicted how likely the given inputs belong to class. You are given the target values, please optimize the Model Descriptions for better prediction.
Inputs (a batch of I.I.d. data on 2-D plane: [x y)):
[14.998 0.83 [-2.3 7.055] [ 5.462 3.87 5.594 2.625 [6.162 2:27 ] [-1.697 10.371] [ [.199 2.83 ) [-3.11 8.723] [ (3266 1.035] [-2.122 9.983]]
Current Model Descriptions:
You designed do binary classification. The input point 2-D plane [x v]: you need output vector containing two probabilities such that each corresponds to how likely the data belongs to each class, ie., (class 1 prob. class 2 prob.]. The sum of > 0, if < 3.0, output [0.0, 1.0]; otherwise nutput [1.0, 0.0). If < O, if 8.0, output. [1.0, 0.0].
vector MUST be
otherwise output [0.0, 1.0].
** The model predictions ([class prob. class prob.]):
[[o. 1.] [u. 1.] [1. 0.] [0. 1.] [0. 1.] [1. [0. 1. [1. 0.] [0. 1.]
0.1l
** The targets ([class prob. class 2 prob.]):
[[o. [l. 00. [0. 1. [0. 1.]
00. 1. [1. 0.) [0. [1.
Please update the model by improving the 'New Model Descriptions', which should have lower classification error both on the current and next. batch of optimization step it's helpful. Both the model and you MUST ONLY operate the numerical precision of THREE decimal points.
data. previous 'Optimization Step' are provided, you can use information from your last
You are bad with numerical calculations, so be extra careful! Please think step by step and give your outputs strictly in the following
format:
Reasoning:
[be explicit and verbose, improve the Current Model Descriptions by yourself; please show your work; note that you don't have access to computers
New Model Descriptions:
pput your new decision rules here; MUST be concise and concrete; MUST PROVIDE THE EXACT VALUE OF THE PARAME TERS if the descriptions potentially involve unknown or learnable parameters!! ***
Please ONLY reply according to this format, don't 't give any other words. Optimization Output
Reasoning:
Analyzing the current batch of data, we notice that the current model descriptions mostly correct, but there For example, the point [-2.3 7.055] is misclassffed, and should belong to class This suggests that the threshold for y-axis in the < region should be lowered. Additionally, we observe that the points with > 0 and y 2.5 tend belong to class 1, and the points with x >> 0 and y < 2.5 tend to belong to class 2. This suggests that the threshold for y-axis in the x > 0 region is correct.
mistakes.
New Model Descriptions:
If x 0, if < 3.0, output [0.0, 1.0), otherwise output [1.0. 0.0). If x 0, if > 7.5, output [1.0, 0.0), otherwise output [0.0, 1.0].
Updated Theta
If
0, if
3.0, output [0.0, 1.0], otherwise output [1.0, 9.0]. If x if y 7.5, output [1.0, 0.0], otherwise output (0.0, 1.0).
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Published in Transactions on Machine Learning Research (01/2025)
Training History: Two Blobs Classification
Step 81
Epoch Step
Paus Prompt
medel..
** Model Desoriptione:
You decigned do hinary clsctifccttoon The
paint on 2-D plane [x yl: need to
contatning prohsbilitier each that
each corresponda how libely the data belonge mach cluse, t... iclase prob. class 2 prob... The of the vector MUST be 1.0. If 0,
1.7.
ouiput [.0] osherwise y 2.0 and > 330,0 output (0.0, 1.0,, otherwise i < 2.8 and > 4.0, output [0.0, 1.0. otberwise it 2.5 and
3.5.
ousput [0.0, 1.0]. osherwise y 3.5 and
4.5, output [0.0, 1.0] otherwise i < 3.5.and
> 5.0, output [0.0. 1.0) otherwise it < 3.0 and
3.5.
output [0.0, 1.0), otherwise if < 4.3 and x <= 4.0. outpot [1-0, 0.0). otherwise y < 4.3 and > 4.0. outpat 0.0. 1.0) otherwine if > 9.5, ontput [1.0.
0.0) otherwise if x > 4.5 and y < B.0, ewspus [0.0, 1.0). otherwise a x.> 5.0 and y < 2.2, ousput [0.0, 1.0]. osherwiae if x > 4.0 and y 1.5, output [0.0.
1.01. olberwise if x > 6.0 and y 2.5, outpat 0.0. 1.0. otherwise a > 3.0 end y < 2-8, oatput [0.0. 1.0). osberwise i > 4.0 und
2.5, oulput (0.0.
1.0) otheraiee if x 5. 1.5 and y 2.8. autpat [0.0. 1.0.. othersien ix 5 5.0 and y < 3-5, output [0.0. 1-0]. ocherwtse x > 3.5 and y < 2:2; ousput (0.0.
1.0) otherwie if x > 4.0 and y 1.5, autpat [0.0. 1.0). otherwien
4.0 and y < 2.5, output [0.0. 1.0]. ocherwtse if > 4.0 and < 1:2; output [0.0.
1.0) otherwise if x > 5.0 and y 2.3 outpat 10.0. 1.0). otherwise if > 4.0 and y < autppt [0.0, 1.0]. otherwise if x > and y 2.0, ousput [0.0.
1.0, otherwinn outgut [1.0, 0.0) < xx and y> 8.5, outpot [1.0, 0.0). otherwise
x < -2.0
y > outpot [1.0, 0.0). ocherwise
> outpet
0.0)
ontput [1.0,
> ousput
[1.0,
output 0.0). otherwise
[1.0. etherwise
2 9.5. [1.0, otherwise 2.0
output 0.0). otherwise it
[1.0. otherwise
0.0]. otherwise if aud
ouiput
otherwise
<. -2.0 and
7.2
output (1.0. 0.0], otherwiee if -2.0 9.2, [1.0, 0.0), otherwies outgut [L.D, 0.0).
** Input:
(2:115 0.99
Please give your outpor scriatly in the following format:
Explanations: (Your siep-by-stey analyses and resulto]
Outgut:
[ONLY A PURE probability vector, where each value between 0.0 and 1.0 WITH TWO DECIMAL POINTS; make necersrry nacumptions needed]
d1
Please ONLY ceply accerding to format, don't give me any other words.
Output
Rxplanation::
1.7, the output [o.0, 1.0].
Output: [0.00, 1,00]
Many outputa for the reat data points
Forward Pacs Outpot
Eeplanation::
Since x > 0. we check the conditions for x >> o, x 3.560 > 3.0 and y 2.456 < 2:2, so the outpus in [0.0, 1.0).
Outpot: [0.00. 1.00]
Training Accuracy
1.0
1.99899941161260 Overall Loss Optimization
optimiser model, your goal in learn the best descriptions for the model. The model used the Current Model Descriptions below predicted how likely given
belong clase. You given target values, please optimisee the Model Descriptions for better predlction. ** Inpute batch of L.d. data 2-D plane: [x me [[:2.178 9.90 [-3.111 10.867] [-4.234 8 8.452] [-8.610 7.818) [ (5.165 2.487] [-2.625 8.718) [3.76 1.107] (-2.023 7.957] [-3.345 8.703] [ (3.569 2:456] ** Current Model Deseriptions:
You are designed to do hinary discafication, The input ts a point on a 2-D plane [x y]: you need to output a vector contaiming two probsbiltttes euch that each corresponde to kow libely the data belonge to mec chuse, Le., jelass 1 prob, class 2 prob.). The sum of the vector MUST be 1.0. ff > 0, if 1.7.
ousput (0.0, 1.0). osherwise y < 2.0 and x > 3.0, output (0.0. 1.0), otherwise if < 2.8 and x > 4.0, output [0.0. 1.0/, otherwise y < 2.5 and
5.5.
output [0.0. 1.0) otherwlse if y <, 3.5 and x > 4.5, ousport (0.0. 1.0]. otherwise y < and x > 5.0, outpot [0.0. 1.0.. otherwise y 3.0 and
3.5.
outpat [0.0. 1.0], otherette y < 4.3 and x <== 1.0, outpot [1.0, 0.0), othergiee y
and x 4.0. ontgot 0000 1.0], otherwies y 9.5, outpot [1.0.
0.0% othereten if x > 4.5 and y < 3.0, emepus (0.0, 1.0). atherwien if x 3.0 and < 2.2. output (0.0. 1.0). otherwise if x > 4.0 and < 1.5, ousput (0.0.
1.0), otlacriis if > 5.0 and y 2.5, outpat 0.0. 1.6. otherwine if
and < 2.0. outpat (0.0. 1.0). otherwine if
and y 2.5, oulput (0.0.
otherwise x > 4.5 and y 2.5, outpat
1.0, otherwine if 5.0 and y < 3.5, output (0.0. 1.0]. oshorwine > and y 2:2, output (0.0.
output
if 6.0
1.0),
[0.0, 1.0]. otherwise
output
-2.0 otherwise autput 110, 0.0). otherwise -2. and > 10.0. output [1.0, 0.0), othersiee
xxx x.x.x---.85 -3.5
5.5. output
otherwise -2.0
output [L.0, 0.0).
9.2, outpot [1.0, 0.0). otherwiss -1.0 and
9.0,
[1 otherwisn
-2.0 and 2 7.5. ostput [1.0, 0.0). otherwsse if x -2.0 and
-3.5 and
9.5, outpur 1.0. 0.0]. etherwise If
< -2.5 and > 8.0, outgut [1.0, 0.0]. otherwise it .4.0
4400 and
9.0, output [1.0, 0.0). otherwise it -4.5 and
> 9.5, oulput 1.0. 0.0). otherwine It
-2.0 and > 0.5. oatput [1.0, 0.0]. otherwise xx .2.0 and y 9.4, outgut [1.0, 0.0]. otherwise if -1.5 and y > 9.0, ousput [100. 0.0), otherwine It
<.0 and y > 9.0, output [1.0. 0.0]: otherwinn if < 220 and
The model predictiona ([claxs 1 prob. class 2 prob.]): Y, 9.2, output [1.0, 0.0), otherwise ontgot [1.D, 0.0).
[[3. o.] [2. 0.] [1. 0.] [a. 0.) [0. 1.) [1. 0. [0. 1.) 0. [1. 0. [0. 1.1
The targets ([elaex 1 prob. class probb])s
[[.. o.] [3. 6.] [3. b.] [a. 0.] [o. 1.) [s. 0.] [o. 1-] [a. 0. [t. 0. [o. 1.]
Please update the model by improving the 'Now Model Descrsptiona', which should have lower classticaaion error both on the current and the next batch
data prestous "Opsimisasion Step' are provided, you use Information from your last optioisation step helpfel.
ONLY opezate
momertcal precleton THRRE
palate.
culiiiooo,,, extrs
think
stop and give your outputa stricily the following format:
Reasoniug:
bee explicit and varhore, improve the Current Model Deseriptions by yoursalf pleaze show work: note that don't have neceRs computers)
New Model Descriptions:
[pur your new decision roles here: MUST be concise and conerete; ****USST PROVIDE THE EXACT VALUE OF THE PA RA METERS If the descriptione poteriially oyy involve unknown or baaraable parametera!!
Please ONLY roply occording to this tocmat, don't give me any other words.
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Published in Transactions on Machine Learning Research (01/2025)
Training History: Two Blobs Classification Step 81 (contimued)
Optimization Output
Reasoning:
The model performing well, hot noticed that the points with < and > 8.5 are more Hilely belong clase 1. By analysing the data,
that the points x and y 5.6 are cousistently clasified cluss So,
new rule to output [1.0. 0.0] case. Additionally. 4.0 and y < 2.5 more likely belong to claso 2. I'll add new rule outpui [0.0, 1:0] these poiuts.
obaerved the points
1f Descriptions:
1.0). otherwise if
otherwae if
4.0,
otherwise
output
otherwise if
osherwiac and 4.0, ouspus (0.0, 1.0 otherwiie
3.0 [1.0, 0.0). otherwtte
ostput 00.0, 1.0], otherasiee if 4.3
<c 4.0, ouspot [1.0, 0.0.; othergtee If
3.0,
[0.0, 1.0). otherwise If x 3.0 and 2.2.
[0.0, otherwiee
4.0 and
1.5. output (0.0. 1.0). otherwies
and
output [0.0. 1.0. otherwis
3.0 and
2.0.
output [0.0. 1.0) otherwiee
4.0 and
2.6. outgut 10.0, 1.0, otherwine x > 4.5 and < 2.8. outgut [0.0. 1.0) otherwise if > 5.0 and < 3.5. outgut [0.0. otherwine if 8.5 and
2.2 output [0.0. L.), otherwise x 2 4.0 and
1.5, output 10.0. 1.0). otherwine if > 4.0 aod
2.5. outgut [0.0, 1.0) otherwise if 4.0 and
2.5. curpus [0.0, 1.0), osherwise output [1.0, 0.0). 1f i -3.5 and y 8.5, output 11.0 0.0), otherwise if autou
1.2. outpost (0.0. 1.8,, otserwise if > 5.0 and y < 2.3, outpot [0.0. 1.0,, otherwise if > 4.0 and y 2.0. output [0.0, 1.0). otherwise if 4.0 and
-2.0 und y > 9.0, outpet [1.0, 0.0)
otherwise if x < -2.0 aud y > 10.0, output [1.0. 0.0), olactwise it a -2.5 ead y > 10.0, output 11.0, 0.0|, otberwise H -1.0 uad. y 9.0, ouiput [1.0.
0.0: otheronse if < -2.0 and 5 7.5, output [1.0, 0.0). otherasse
-2.0 and y >> 9-2, output [1.0, 0.0). otherwsse
4. -3.5 and 9.5,
ontpet 1.0.
0.0 otheronee
0.0 otherwine < -2.8 and
astput [1:0. 0.0). otherasse
-1.0 and
y > 9.0, output [1.0, 0.0). otherwase
-4.5 and
8.5, ontpet 1.00.
-2.0 and 5.0... output [1:0, 0.0), otherwise
and
y
output 1.0, 0.0). otherwise
2-..2.000 outget
otherwise cutpott
-2.0
2.0.0 output [1.0, 0.0). otherwise
and
y 0.2, outpot [100, 0.0). otherwise
outpet [1.0.
0.0).
Updated Theta
y < 1.7, output
1.0). otherwie i8
8.0,
10.0. otberwiac if
4.0, (0.0, otherwise i2
<.2.
3.5. outgut (0.0, otherwise
3.5
4.5, outget [0.0. otherwise
[1.0, 0.0,, othersse if 3.5
osherwise
3.0 and > 3.5, astpet 00.0. L.), otherwise 1f
<= 4.0, otherwise if x 3.0 and
and 4.0, outpus 10.0. 1.0), otherwse if
9.5. ouspus
[1.0, 0.0). otherwiee if 4.5 and 3.0,
[0.0, 1.0].
2.2. putput [0.0, 1.0). otherwiee
4.0 and
1.5, outpot 0.0. otherwise xxx 5.0 and y 2.5, outgot [.00 1.0. otherwise if x 3.0 and y < 2.0, ontget 0. 1.0. otherwiee if > 4.0 and
2.5, ourgor [0.0. 1.6] otherwiss If 2.3 and < 2.8, ontgos (0.0. 1.0). othersiee if 2.5. 5.0 and y
3.5, outgor [0.0, 1.0 otherwien If 3.5 and
2.2. ourpur [0.0. 1.6). otherwies ix 2.0 and < 1.5, ontpot [0.0. 1.0). otherwies if 4.0 and < 2.5. ontpor [0.0. 1.0. otherwise if > 4.0 and
1.2, outgrtt 0..0. 1.0), otaerwise Ex 5.556. 5.0 snd y < 2.3, output [0.0. 1.0. othecwisc if 5 4.0 and y 2.0, output outpp
[6.0, 1.0). otherwise if 4.0 and
outpus [0.0, 1.0]. oileerise outpat [1.0, 0.0). xf 0. -3.5 and y 8.5, output 1.0, 0.0), otherwise if < -2.0 and > 9.0, output [1.0, 0.0
otherwise
-2.0 and v 10.0, output
0.0], otherwiee
-2.5 and
10.0, output 1.0. 0.0]. otherwise
-1.0 and > 9.0, ousput [1.0.
0.0) otherwise if -2.0 and
7.5. autput [1.0, 0.0). otherwise x -2.0 and y 9:2, patput [1.0, 0.0). otherware if
and
outpet 1.0.
otherwise if -2.6 and
output [1.0, 0.6), otberwise
-4.0 and y 9.0, oatput [100 0.0). otherwise if -4.5
putpet 1.0.
outpot 0.0).
y outpot 0.0).
ontpet
outpot [1.0, 0.0).
and
outpot [1.0, 0.0).
9.5, ontpet [1.0.
otherwise
81

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Published in Transactions on Machine Learning Research (01/2025)
Training History: Two Blobs Classification
Step 82
Epoch Step
Prompt
medel..
** Model Descriptione:
You decigned do hinary clsctifccttoon The
paint on 2-D plane [x yl: need to
contatning prohsbilitier each that
each corresponds how libely the data belonge mach cluse, t... iclase prob. class 2 prob... The of the vector MUST be 1.0. If 0,
1.7.
ouiput [.0] osherwise y 2.0 and > 3,0, output (0.0, 1.0,, otherwise i < 2.8 and > 4.0, output [0.0, 1.0. otberwise it 2.5 and
3.5.
ousput [0.0, 1.0]. osherwise y 3.5 and
4.5, output [0.0, 1.0]. otherwise i < 3.5.and
> 5.0, output [0.0. 1.0) otherwise it < 3.0 and
3.5.
outpot [0.0, 1.0), otherwise if < 4.3 and x <= 4.0. outpot [1-0, 0.0). otherwise y < 4.3 and > 4.0. outpat 0.0. 1.0) otherwine if > 9.5, ontput [1.0.
0.0) otherwise if x > 4.5 and y < 8.0, ewspus [0.0, 1.0). otherwise a > 5.0 and y < 2.2, ousput [0.0, 1.0]. osherwiae if x > 4.0 and y 1.5, output [0.0.
1.01. olberwise if x > 6.0 and y 2.5, outpat 0.0. 1.0. otherwise a > 3.0 end y < 2-8, oatput [0.0. 1.0). osherwine i > 4.0 und
2.5, oulput (0.0.
1.0) otheraiee if x 5. 1.5 and y 2.8. autpat [0.0. 1.0.. othersien ix 5 5.0 and y < 3-5, output [0.0. 1-0]. ocherwtse x > 3.5 and y < 2:2; ousput (0.0.
1.0) otherwies if x > 4.0 and y 1.5, autpat [0.0. 1.0). otherwien ix > 4.0 and y < 2.5, output [0.0, 1.0]. ocherwtse if > 4.0 and < 1:2; output [0.0.
1.0) otherwise if x 5.0 and y 2.3. outpat 10.0. 1.0.. otherwise if > 4.0 and y < autpp [0.0, 1.0]. otherwise if x > and y 2.0, ousput [0.0.
1.0, otherwinn outgut [1.0, 0.0) < x -3.5 and y> 8.5, outpot [1.0, 0.0). otherwise
x < -2.0
y > outpot [1.0, 0.0). ocherwise
> outpet
0.0)
2.0 > octput [1.0,
> ousput
[1.0,
output
0.0). otherwise
[1.0. etherwise
-2.6 >
0.0). otherwise it
ouiput
otherwise
-2.0
9.5.
[1.0,
9.4,
0.0]. otherwise it
[1 otherwise
-2200 and > 9.0. output [1.0. otherwise if 220 and 9.2, output [L.O, 0.0]. otherwise if < -3.0 and > 9.5. cutput [1.0. 0.0). otherwise
output [1.0. 0.0]-
IIpput:
(.22522 8.84 ]
Please give your outpur scrictly in the following format:
Explanation:: (Your siep-by-step analyses and resulto]
Output:
[ONLY A PURE probability vector, where each value between 0.0 and 1.0 WITH TWO DECIMAL POINTS: make necemntry nacumptions nceded)
di
raply according
don't give any other
Output
Explunation::
Since check the conditions for -3.5 and 8.5, which trne. Output: [1.00. 0,00]
Many outpute for the rest data points
Forward Prss Output
Explanations:
Since x <. 0, we check the conditions for x < x -2.449, y 11:478.
The condision < -2.0 and y 10.0 satisfied So, the oulput [1.0, 0.0,.
Outpot: [1.00.
Accuracy
1.0
Overall Loss
11988999111126155-91
Optinisation Prompt
You are the uptimiser for model, your goal is to loarn the best deseriptions for the model. The model used the Curreat Model Deseription below predicted how likely the given ipputs belong to a clase. You are given the target valuee, please optinise the Model Descrsptions for better prediction. Inputa (a batch of L..d. data on 2-D plane: [xt y)):
[62:252 8.94 3:875 2.447] (:2.440 11.478| [-3.211 6.687] (6.784 2,507] | 4.9 2.756] [ [4,853 9.427] [-1.504 0.543] (-2.702 9.310] [-1.044 8.780]]
Current Model Desariphione:
You are designed
hinary alsediicattinn The Input is a point on a 2-D plane [x y) you need to outport a vector contatning two probshilittee each that each corresponde how libely the data belonge to mnch chuce, Le., |clase prob. class 3 prob.). The sum of the vector MUST be 1.0. If 0, y
1.7.
ouiput [0.0, 1.0). osherwise y
and 3.0, oulput [0.0, 1.0,, otherwise y < and > 4.0, output [0.0, 1.0), otherwise y and 3.5.
output [0.0. 1.0]. osherwine y and 4.5, outpat [0.0, 1.0], otherwise y < 3.5
> 5.0, output [0.0. 100, otherwise y and 3.5,
output [0.0.
0.0).
y
y cutpus
[0.0, 1.0].
otlaeriie if
6.0
1.0.
if
output
otherwise
1.0, otherait if > 1.5 and
ontpat 0.0. 1.0). otherwies if 5.0 and
3.5,
[0.0. 1.0). ocheretee if 3.5 and
(0.0.
otherwie if x > 1.0 and
autpat 1000 1.0). otherwie # 4.0 and 2.5. output [0.0. 1.0). ocherwtee if 4.0 and 1.2, output (0.0.
1.0, otherwise xx 5.0 5.0 and y 2.3.
outpat 0.0. 1.0) otherwine Ix 4.0 and < 2.0, [0.0, 1.0]. otherwise xf 4.0 and
2.5, output (0.0.
1.0, otherwise output [1.0, 0.0] 1tx O. < -3.5 and y > 9.5, outpot [1.6, 0.0), otherwise xf < -2.0 and y > 9.0, output [t.0, 0.0), ocherwine If
-2.0 and > 10.0, ontpot 1.0, 0.0). otherwise xt < 225 and y 10.0. octput [1.0, 0.0: otherwise If <1.0 and > 9.0, ouspat [1.0. 9.0). otherwien I2
< -2.0 and > 7.6. ostput [1.0, 0.0). otherwise if -2.0 and y 9.2, ostput [1.0, 0.0). otherwise if -3.5 and y > 9.5, ouiput [1.0. 0.0]. etherwise if
< -2.6 and > 8.0. estput [10, 0.0). otherwise if < 4.00 and y > 5.6, output [100, 0.0). otherwise if < -4.5 and y > 8.5, ouiput [1 0.0), etherwise if
< -2.0 and y 2 9.8. outpot [1.0, 0.0]. otherwie if 4, -2.0 and y 9.1, outpat [1.0, 0.0]. otherwise if x < -1.5 and y S. 9.0, outpur 0. 0.0], otheretee
< -2.0 and y > 9.0. ousput [1.0, 0.0), otherwise if -2.0 and y > 9:2, output [L.0, 0.0). otherwien if < -3.0 and y > 9.5, cusput [1.0. 0.0). otherwise
output [1.0. 0.0).
[[). model prodictions (Joloss prob. class peob.])
[b.
1.] 1.] [o.
0. [1. 0:] [1. 0.]
The
(Jclass
prob.l): *.
Please [1.
by 1.) J0. [o. [1. 0. 0. [1. 0.1
classifiaatio bash the current and the next basch can use the information from your lust optimization step it's halpful, Both the modal and MUST ONLY operate the numerical preciaton of THREE decimal points. You are bad with numerical enlculations, be extra careful! Please
the 'New Model Deacsiptions, which should
s.d. data. previous "Optimisattoon Step' provided,
shink by step and give your ousputs striesly in the tollowing format:
Reasoning:
(e explica and varbuse, impeove the Current Model Desoriptiono by yourself, pleas show your work; note that you don't have access to computers)
Now Model Descriptions:
[put your new deciston rales here: MUST be concise and concrate: MUSU PROVIDE THE EXACT VALUE OF THE PARAMETERS if the descriptions potentially involve unknown or learnable parameteral!!
Please raply according format, don't give any
82

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Published in Transactions on Machine Learning Research (01/2025)
Training History: Two Blobs Classification Step 82 (continued)
Optimization Output
Reasoning:
model le performing well, hut noticed that the pointe with > 4.5 and y < 2-8 are more likely to belong clase By analystng the data, rne that the points witk 4.5 < 2.8 are comsisteatly clusified us clus 2. Su, add new to outpat 10.0. shis case. Additionally, Lobeerved are more likely to belong closs I'll
the pointo
-2.5 and
new rale to outgut [1.0, 0.0) these points.
Descriptions:
1.0). otherwise if
otherwae if
4.0,
(0.0, otherwise
output
osherwine
3.0 [1.0, 0.0). otherwtte > and
00.0, 1.0], otherasiee if 4.3 and <c 4.0, ousput [1.0, 0.0, othergtee if
and 4.0, ouspus (0.0, 1.0, othersiie
3.0,
[0.0, 1.0). otherwie If x 3.0 and 2.2.
[0.0, otherwiee
4.0 and
1.5. output (0.0. 1.0). otherwies
output [0.0. 1.0. otherwis
3.0 and
2.0.
output [0.0. 1.0) otherwiee
4.0 and
2.6. outgut 10.0, 1.0, otherwine x 4.5 and 2.8. outgut [0.0. 1.0) otherwise if > 5.0 and < 3.5. outgut [0.0. otherwine if 3.5 and
2.2 output [0.0. L.), otherwise x > 4.0 and
1.5, output 10.0. 1.0). otherwine if > 4.0 aod < 2.5. outgot [0.0, 1.0) otherwise if 4.0 and
2.5. cucpus [0.0, 1.0), otherwise outpot [1.0, 0.0). 1f if -3.5 and y 8.5, output 11.0 0.0), otherwise if autpu
1.2. outpost (0.0. 1.8,, otserwise xx> > 5.0 and y 2.3, outpot [0.0. 1.0., otherwise if x > 4.0 and y 2.0. output [0.0, 1.0). otherwise if 4.0 and
-2.0 und y > 9.0, outpet [1.0, 0.0)
otherwise if x -2.0 aud y > 10.0, output [1.0. 0.0), olaerwise it a -2.5 ead y > 10.0, output 11.0, 0.0|, otberwise H.x -1.0 und. y 9.0, ouiput [1.0.
0.0: otheronse if < -2.0 and 5 7.5, ostput [1.0, 0.0). otherasse
-2.0 and y >> 9:2, output [1.0, 0.0). otherwsse
4. -3.5 and 9.5,
ontpet 1.0.
0.0 otheronee if < -2.5 and
astput [1:0, 0.0). otherasse
-1.0 and y > 9.0. output [1.0, 0.0). otherwase
-4.5 and
8.5, ontpet 1.00.
0.0 otherwine if
and 2.8.8. output [1:0, 0.0). otherwise
-2.0 and
y
output 1.0, 0.0). otherwise
outget
otherwise
2.000 outpot [3.0, 0.0).
otherwise
and
y
outpot [100, 0.0). otherwise
outpet [1.0.
0.0).
ontgot [1-0, 0.0).
ousput [1-0, 0.0]-
Updated
output [0.0. 1.0). otherwie i
and
10.0. otberwaa y
4.0,
(0.0, otherwise i2
<.2.
(0.0, otherwise
[t.0, 0.0,, othersse if 3.5
[0.0. otherwise
5.0,
osherwise
3.0 and > 3.5, astpet 10.0. L.), otherwise if
<= 4.0, otherwise if x 3.0 and
and 4.0, outpus 10.0. 1.0), otherwse if
9.5. cuspus
[1.0, 0.0). otherwiee if 4.5 and 3.0,
[0.0, 1.0].
2.2. putput [0.0, 1.0). otherwiee
4.0 and
1.5, outpot 0.00 otherwise xxx 5 5.0 and < 2.5, outgot [000 1.0. otherwise if 3.0 and y < 2.0, ontget (0.0. 1.0. otherwiee if > 4.0 and
2.5, ourgor [0.0. 1.6] otherwies V and y 2.8, ontgost (0.0. 1.0). otherwiee if 2.5 5.0 and y 3.5, ontgor [0.0, 1.0 otherwien If 3.5 and y
2.2. ourpur [0.0. 1.6). otherwies 1-x2210 and
1-5, ontpot (0.D. 1.0). otherwiss x 4.0 and y 2.5. outpor [0.0. 1.0. otherwise if > 4.0 and
1.2, outprt (0.0. 1.0), otaerwise Ex 5..56. 5.0 snd y < 2.3, output [0.0. 1.0. otherwise if 5 4.0 and y 2.0, output ouppu
[6.0, 1.0). otherwise if 4.0 and
outpus [0.0, 1.0]. oileerise outpat [1.0, 0.0). If 0. -3.5 and y 8.5, outpat 1.0, 0.0), otherwise if < -2.0 and > 9.0, output [1.0, 0.0
otherwise
-2.0 and v 10.0, output
0.0], otherwiee < -2.5 and y 10.0, output 1.0, 0.0]. otherwise
-1.0 and > 9.0, ousput [1.0.
0.0) otherwise
-2.0 and
7.5. output [1.0, 0.0). otherwise
-2.0 and y 9:2, pstput [1.0, 0.0). otherware if
and
outpet 1.0.
otherwise if -2.6 and
output [1.0, 0.6). otberwise xxx
and y 9.0, oatput 1100 0.0). otherwise if -4.5
putpet 1.0.
output [1.0, 0.0).
y outpot [10, 0.0).
ontpet
0.0).
y
outpot [1.0, 0.0).
9.5, ontpet [1.0.
otherwise
output
0.0).
output 0.0].
83

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Published in Transactions on Machine Learning Research (01/2025)
M.5 Two Circles Regression 1aamm---77BB without prior)
Training History: Two Circles Classification without Prior Step
Bpoch Step
Forward Pass Prompt
You the model.
Model Descriptions:
designed to do binary classification input point on plane y): you need output the classs label, ie., an integer in the set. O, 1.
** Input: **
[-0.146 -1.01]
Please give your output strictly in the following format:
Explanations: (Your step-by-step analyses and results]
Output:
[ONLY the integer class label; make necessary assumptions if needed]
dis
Please ONLY reply according to this format, don't give any other words.
Forward Pass Output
Explanations:
Since I don't have any information about the decision boundary, I will make an asumption that the decision boundary is the x=y
line.
The point (0.89, 0.397) is below the x=y line, so it belongs to class 0.
Output:
Many outputs for the rest data points....
Forward Pass Output
have any prior knowledge about classification model, I'll assume a simple linear docision boundary. Let's assume the decision boundary x O. Then, I'll plug in the Imput values to determine the class label. Plugging in x -0.146 and y = -1.01, we get -0.146 + (-1.01) = -1.156 < O. Since the result negative, the point lies below the decision boundary.
Explanations: Since
Output:
Training Accuracy
0.4
Optimization Prompt
You the optimizer for model, your goal is to learn the best descriptions for the model. The model used the Current Model Descriptious below predicted the class labels for the given inputs. You are given the target labels, please optimize the Model Deseriptions for better prediction.
Inputs (a batch of I.I.d. data on 2-D plane: [x y)): [[-0.146 -1.01 [0.224 -0.929] 0.89 0.397] [0.811 -0.56 [-0.873 0,475 [0.27 0.172] 0.008 0.261] [0.518 0.943] [-0.502 -0.788] [-0.069 -0.252]]
** Current Model Descriptions:
You designed to do binary classification. The imput is point 2-D plane yl: you need output the class label,
integer in the set 0,
** The model predictions: **
[[0] [i] [0] [i] [1] [1] [o] [1] [0] [O]]
** The targets: *
[[o] [o] [o] [o] [o] [1] [1] [0] [o] [1]]
the model doing well, you can keep using the current descriptione. However, if model not performing well, please update the model by improving the 'New Model Descriptions' which should have lower classifiaaiion
both the current and the next batch of Li.d. data. If previons 'Optimization Step' are provided, you can use the information from your last optimization stop if it's
helpful. DON'T symbolic representation for the model! Please think step by step and give your outputs strictly in the following format:
Reasoning:
[ee explicit and verbose, improve the Current Model Descriptions by yourself; please show your work: note that you don't have access to computers'
New Model Descriptions:
pput your deciaion rules here; MUST be concise and concrete; MUST PROVIDE THE EXACT VALUE OF THE PARAME- TERS if the descriptions potentially involve unknown or learnable parameterg!!! .**
Please ONLY reply according to this format, don't, give me any other words. Optimization Output
Reasoning:
Analyzing the current model predictions and targets, I notice that the model is not performing well, especially for points with positive y-values. Upon visual inspection the input data, observe that points with y-values greater than send belong class while points with y-values less than 0 tend to beloug to class U. This suggests that the y-coordinate is strong feature for classification. Therefore, will update the model descriptions incorporate New Model Descriptions:
insight.
I 0, then class label
Otherwise, class label Updated Theta
If y > 0, then class label
I. Otherwise, class label 0.
84

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Published in Transactions on Machine Learning Research (01/2025)
Training History: Two Circles Classification without Prior Step
Epoch Step
Forward Pass Prompt
You are the model.
** Model Descriptions: **
You are designed to do binary classification. The input is point on a 2-D plane [x yl: you need to output the class lahel, i...,
integer in the set 0, y 0, then class label Otherwise, class label
** Imput: e* [-0.305 0.936]
Please give your output strictly in the following format:
Explanations: (Your step-by-step analyses and results]
Output:
[ONLY the integer class label; make necessary assumptions if needed]
Please ONLY reply according to this format, don't give me any other words.
Forward Pass Output
Explanations: Since > 0, the class label is 1.
Output:
Many outputs for the rest data points
Forward Pass Output
Explanations: Given the imput point [-0.735 -0.614], we need to determine the label based on condition provided. The condition states that if y 0, then the class label is 1, otherwise it is O. In this case, = -0.614, which is less than O.
Output:
Training Accuracy
0.3
Optimization Prompt
You are the optimizer for a model, your goal is to learn the best descriptions for the model. The model used the Current Model Descriptions below predicted the class labels the given inputs. You are given target labels, please optimiie the Model Descriptions for better prediction.
** Inputs (a batch of I.I.d. data on 2-D plane: [x y)): **
[[-0.305 0.936 [0.298 -0.037] 0.396 0.931] [-0.735 -0.614| (-0.339 0.006] [-0.253 -0.015] 0.096 -0.248) 1.005 0.018) [-0.329 0.016] 10.051 -0.372]
** Current Model Descriptions:
You are designed to do binary classification. The iuput is point on a 2-D plane [x yl: you need to output the class label, i.e. an label Otherwise, class label
integer the set 0, y
The model predictions:
[[1] [o] [1] [0] [1] [0] [0] [1] [1] [o]]
** The targets:
[[o] [1] [o] [a] [1] [1] [1] [o] [A]]
[A]]
If the model is doing well, you keep using the current descriptions. However, if the model not performing well, please update
the model by improving the 'New Model Descriptions' which should have lower classification error both on the current and the next
i..dd
previons 'Optimization Step' are provided, you use information from your optimisation stop
helpful. DON'T use symbolic representation for the model! Please think step by step and give your outputs strictly in the following format:
Rensoning:
[be explicit and verbose; improve the Current Model Descriptions by yourself; please show your work; that you don't have to computers]
New Model Descriptions:
(put your new decialon rules here; MUST be concise and concrete; MUST PROVIDE THE EXACT VALUE OF THE PARAME- TERS if the descriptions potentially involve unknown or learnable parameters!!!**>g
Please ONLY roply according to this format, don't give me any other words. Optimization Output
Reasoning: Analyzing the current model predictions and targets, I notice that the model is not performing well, especially for points with x-values greater than 0. Upon visual inspection the input data, observe that points with x-values greater than tend to belong to class while points with x-values lees than 0.3 tend belong to class 0. This suggests that the x-coordinate is also strong feature for classification. Therefore, will update the model descriptions to incorporate this insight. New Model Descriptions:
If x > 0.3, then class label 1. Ifx <= 0.3 and y > 0, then class label = 1. Otherwise, class label 0. Updated Theta
If x 0.3, then class label = 1. Ifxx <= 0.3 and y 0, then class label = 1. Otherwise, class label U.
85

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Published in Transactions on Machine Learning Research (01/2025)
Training History: Two Circles Classification without Prior Step
Epoch Step
Forward Pass Prompt
You are the model. ** Model Descriptions: **
You are designed to do binary classification. The imput is point on a 2-D plane [x yl: you need to output the class lahol, i.c.,
integer in the set O, If 0.3, then class label Ifx <= 0.3 and y then class label Otherwise, class label
Input: [0.275 0.163]
Please give your output strictly in the following format:
Explanations: (Your step-by-step analyses and results]
Output:
[ONLY the integer class label; make necessary assumptions if needed]
Please ONLY reply according to this format, don't give me any other words.
Forward Pass Output
Explanations: Since = 0.403 > 0.3, we can directly determine the class label.
Output:
Many outputs for the rest data points
Forward Pass Output
Explanations:
1. The imput is 00, -1.012].
Since x not greater than 0.3, move the next condition.
3. Since <= 0.3 and = -1.012 la not greater than 0, the class label la O.
Output:
Training Accuracy
0.6
Optimization Prompt
You are the optimizer for model, your goal is to learn the best descriptions for the model. The model used the Current Model Descriptions below predicted the class labels the given inputs. You are given target labels, please optimize the Model Descriptions for better prediction.
Inputs (a batch of i..dd data on 2-D planer [x yl); 0.275 0.163 -1.012] [-1.065 -0.126] 0.133 0.22 [-0.317 -0.931] 0.403 0.96 0.013 0.996] [0.147-00) -0.178] [-0.028 0.422] (-0.233 -0.186]]
**
Model Descriptions: **
You are designed to do binary classification. The input is point on a 2-D plane [x yl: you need to output the class lahel, i.e., 0.3, then class label 1. I <= 0.3 and y then class label Otherwiae, class label
integer in the set 0,
** The model predictions:
[[]] [o] [0] [1] [0] [1] [0] [o]]
** The targets:
[[]] [o] [o] [1] [0] [o] [o] [a] [1]]
not performing well, please update the model by improving the 'New Model Descriptions', which should have lower cleasification error both the current and the next
model doing well, you can keep using current descriptions. However,
batch L..d. data. If previous 'Optimization Step' are provided, you can use the information from your last optimization step it's
helpful. TON'T use symbolic representation for the model! Please think step by step and give your outputs strictly in the following format:
Reasoning:
explicit verbose; improve Current Model Descriptions by yourself; please show your works note that you don't access to computers]
New Model Descriptions:
[put decision rules here; MUST be concise and conerete; MUST PROVIDE THE EXACT VALUE OF THE PARAME-
TERS if the descriptions potentially involve unknown or learnahle parametors!!
Please ONLY reply according to this format, don't give me any other words.
Optimization Output
Reasoning:
Analyzing the current model predictions around Upon visual inspection of the imput data, I observe that points with y-values greater than 0.2 tend to belong class while points with y-values less than 0.2 tend to belong to class 0. This suggests that the y-coordinate is also a strong feature for claseification. Therefore, will update the model descriptions to incorporate this insight. New Model Descriptions:
targets, I notice that the model not performing well, especially for points with x-values
If 0.3, then class label If x <= 0.3 and y 0.2, then class label Otherwise, class label
Updated Theta
If x 0.3, then class label 1. Ifx <= 0.3 and y > 0.2, then class label 1. Otherwise, class label
86

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Published in Transactions on Machine Learning Research (01/2025)
Training History: Two Circles Classification without Prior Step
Epoch Step
Forward Pass Prompt
You are the model.
** Model Descriptions: **
You are designed to do binary classification. The imput is point on a 2-D plane [x yl: you need to output the class lahel, i.c.,
integer in the set 0, If 0.3, then class label Ifx <= 0.3 and y 0.2, then class label Otherwise, class label
** Input: e* [-0.811 0.529]
Please give your output strictly in the following format:
Explanations: (Your step-by-step analyses and results]
Output:
[ONLY the integer class label; make necessary assumptions if needed]
Please ONLY reply according to this format, don't give me any other words.
Forward Pass Output
Explanations: Since = 0.943 > 0.3, the class label is 1.
Output:
Many outputs for the rest data points
Forward Pass Output
Explanations:
Since x = -0.995, which is less than or equal to 0.3, we need to check the value of y.
Since y -0.025, which is than or equal to 0.2, we conclude that lahel
Output:
Training Accuracy
0.2
Optimization Prompt
You are optimizer for model, your goal Descriptions below predicted the class labels for the given inputs. You are given the target labels, please optimise the Model Descriptions for better prediction.
the descriptions the model. The model used Current Model
** Inputs (a batch of i.i.d. data 2-D plane: [x yl):
[[-0,811 0.529] (-0.036 0.323] [-0.63 0.698] [-0.173 -0.246] 0.943 -0.297] 0.1 -0.288] [-0.995 -0.025] 0.666 -0.666] 0.24 -0,039] 0.342 0.887]]
Current Model Descriptions:
plane yl: you need output classs label, i.c., an integer in the set. 0, 1. Tf > 0.3, then class label 1. Ifx x <= 0.3 and y 0.2, then class label = 1. Otherwise, class label 0.
are designed to binary classification. input point
** The model predictions: **
[[3] [1] [1] [0] [1] [0] [0] [1] [0] [1]]
Thee targets:
[[o] [I] [0] [0] [1] [0] [o] [1] [o]]
If the model is doing well, you can keep using the current descriptions. However, if the model is not performing well, please update
classification error both the current and the batch of L.ddd data. If previous 'Optimization Step' are provided, you use the information from your last optimization step if it's
model by improving the 'New Model Descriptions', which should
helpful. DON'T use symbolic representation for the model! Please think step by step and give your outputs strictly in the following
format:
Reasoning:
[be explicit and verbose, improve the Current Model Descriptions by yourself; please show your work; note that you don't have access computers'
New Model Descriptions:
pput your new decision rules here; MUST concise and concrete; MUS PROVIDE THE EXACT VALUE OF THE PARAME- TERS if the descriptions potentially involve unknown or learnable paramctend!!
Please ONLY reply according to this format, don't give any other words. Optimization Output
Reasoning:
Anslyzing the current model predictions and targets, I notice that the model is not performing well, especially for points with x-values around Upon visual inspection the input data, observe that points with x-values less than -0.5 tend to belong to class O, while points with v-values greater than 0.5 tend belong to class relationship with the class label. also notice that points with y-values greater than 0.5 tend to belong to class 0 while points with y-values less than -0.2 tend to belong class 1. This suggests that the y-coordinate also important for classification. Therefore, will update the model descriptious to incorporate these insights.
well. This suggests that the cocordinatte has non-linear
New Model Descriptions:
If
-0.5
0.5, then class label 0. If > 0.5, then class label If y -0.2, then class label 1. Otherwise, class label
Updated Theta
If < -0.5 or x > 0.5, then class label = 0. If > 0.5, then class label = 0. If < -0.2, then class label = 1. Otherwise, class label
87

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Published in Transactions on Machine Learning Research (01/2025)
Training History: Two Circles Classification without Prior Step
Epoch Step
Forward Pass Prompt
You are the model.
** Model Descriptions: **
You are designed to do binary classification. The input is point on a 2-1 plane [x yl: you need to output the class lahel, i...,
integer in the set If < -0.5 or x 0.5, then class label 0. If y >> 0.5, then class label y -0.2, then class label
Otherwize, class label
Input: [-0.99 -0.325]
Please give your output strictly in the following format:
Explanations: [Your step-by-step analyses and results]
Output:
[ONLY the integer class label; make uecessary assumptions if needed]
Please ONLY reply according to this format, don't give any other words. Forward Pass Output
Explanations:
Since x -1.059 < -0.5, class label 0.
Output:
Many outputs for the reat data points
Forward Pass Output
Explanations:
Cbeck Ifx < -0.5 or > 0.5. Since 0.167 is between -0.5 and 0.5, this condition not met.
Check if > 0.5. Since 0.235 less than 0.5, this condition is not met.
Check if < -0.2. Since 0.235 greater than -0.2, this condition is not met.
4. Since none of the above conditions are met, the class label is 1. Output:
Training Accuracy
Optimization Prompt
You are the optimizer for a model, your goal is to learn the best descriptions for the model. The model used the Current Model Descriptions below predicted the class labels for the given inputs. You the target labels, please optimize the Model Descriptious for better prediction.
plane: yl): *= [[-0.99 -0.326] [-0.825 0.438] 188 0.308) [-1.059 0.217) [-0.188 -0.184] (-0.322 0.084] 00.892 0.515) [-0.287 0.017, [ [0.167 0.235 [-0.573 -0.845]]
** Inputs batch i.i.d. data
Current Model Descriptions:
You are designed to do binary clussification. The input is point on a 2-D plane [x yl: you need to output the classs label, i.e., an If -0.5
integer in the set
0.5, then class label 0. If >> 0.5, then class label If < -0.2, then class label 1.
Otherwise, class label = 1.
** model predictions: **
[[o] [0] [0] [0] [i] [1][[] [0] [1] [X]]
e* The targets:
[[o] [o] [1] [1] [1] [o] [1] [1] [o]]
the model is doing well, you can keep using the current descriptions. However, if the model not performing well, please update both the current and the next batch of Li.d. data. If previous 'Optimization Step' are provided, you can use the information from your last optimization step if it's model! Please step by step and give your outputs strictly following
the model by improving the 'New Model Descriptions', which should have lower classification
helpful. DON'T use symbolic representation format:
Reasoning:
be explicit and verbose; improve the Current Model Deseriptions by yourself; please show your work; note that you don't have access computers
New Model Descriptions:
concise and concrete; MUST PROVIDE THE EXACT VALUE THE PARAME- TERS if the descriptions pocentially involve unknown or learnable parameters!!!**ee
jout your new decision rules here;
Please ONLY reply acrording to this format, don't give any other words. Optimization Output
Reasoning:
Analyzing the current model predictions and targets, notice that the model is still not performing well. Upon visual inspection of the loput data. I observe that points with y-values around tend to misclassified. This suggests that the y-coordinate threshold of 0.5 too high. I also notice that points with x-values around -0.2 tend belong to class 1, while points with x-values around 0.2 tend to belong to class O. This suggests that the x-corrdinate threshold of 0.5 is too high as well. Therefore, will update the model drscriptions to incorporate thrse insights.
New Model Descriptions:
x then class label y > 0.3, then class label If -0.2, then label 1. Otherwise, class label
= 1.
Updated Theta
If -0.4 0.4, then class label O. If 0.3, then class label If y -0.2, then class label 1. Otherwise, class label
88

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Published in Transactions on Machine Learning Research (01/2025)
Training History: Two Circles Classification without Prior Step 40
Epoch 3 Step 10
Forward Pass Prompt
You are the model.
** Model Descriptions: **
You are designed to do binary classification. The input is point on a. 2-D plane [x yl: you need to output the class lahel, i...
integer in the set If < -0.75 x 0.95, then class label = If y 0.78, then class label O. y < -0.42, then class label
0. Otherwize, if 0.11, then class label 1, else class label O.
Input:
[-0.101 0.309]
Please give your output strictly in the following format:
Explanations: [Your step-by-step analyses and results]
Output:
[ONLY the integer class label; make necessary assumptions if needed]
Please ONLY reply according to this format, don't give me any other words. Forward Pass Output
Explanations:
Since 0.919 > 0.95, class label 0.
Output:
Many outputs for the reat data points
Forward Pass Output
Explanations:
Cbeck If < -0.75 x 0.95. Since -0,655 is not in this range, move to the next step, Check if > 0.78, Since -0.649 not in this range, move the next step.
Check y -0.42. Since -0.649 in this range, class label 0.
Output: 0
Training Accuracy
1.0
Optimization Prompt
You the optimizer for model, your goal is to learn the best descriptions for the model. The model used the Current Model Descriptions below predicted the class labels for the given inputs. You are given the target labels, please optimize the Model Deseriptions for better prediction.
Inputs (a batch of ii.d. data on 2-D plane: [x yl):
[[-0.101 0.309) [-1.011 -0.165] 0.166 -0.229] 1.02 -0.061] 0.225 0.076] [-0.024 -0.333) (-0.655 -0.649) [-1.058 0.15 0.366 00.998 [ 0.919 0.08
** Current Model Descriptions:
You designed to do binary classification. The imput is point 2-D plane [x yl: you need to output the class label, Lee. integer in the set 0, If < -0.75 or x > 0.95, then class label y > 0.78, then class label O. y < -0.42, then class label Otherwise, x 0.44, then class label else class label 0.
The model predictions:
[[] [o] [1] [0] [1] [1] [0] [0] [o]]
** The targets:
[]] [o] [1] [0] [1] [1] [0] [0] [1] [O]]
If the model is doing well, you keep using the current descriptions. However, if the model not performing well, please update
the model by improving the 'New Model Descriptions' which should have lower classification crror both on the current and the next
batch of i..d. data. If previous 'Optimization Step' provided, you the information from your last optimisation step it's
helpful. DON'T use symbolic representation for the model! Please think step by step and give your outputs strictly in the following
format:
Reasoning:
be explicie and verbose, improve the Current Model Descriptions by yourself; please show your work; note that you dou't have to computers]
New Model Descriptions:
[put your new decision rules here; MUST be concise and concrete; MUST PROVIDE THE EXACT VALUE OF THE PARAME- TERS descriptions potentially involve unknown or learnable parameters!!!
Please ONLY reply according to this format, don't give me any other words. Optimization Output
Reasoning:
Analyzing the eurrent model predictions and targets, observe that the model performing well, but there still room improvement. Upon visual inspection of the imput data, I notice that the current x-coordinate threshold of 0.44 seemus to be effective, and the y-coordinate thresholds of 0.78 and -0.42 are ressonable. However, observe that some points with x-values around -0.6 and y-values around -0.65 misclassified, which suggests that the model not capturing the class boundary accurately in this region. also notice that points with x-values around 0.2 and y-values around 0.1 are correctly classified, which suggests that the model is performing well in this region. Additionally, observe that the current x-cnordinate threshold of 0.95 seems he aggressive, it misclasaifies some points with x-values around 0.9.
New Model Descriptions:
If x < -0.75 or x > 0.92, then class label = O. If y 0.76, then class label O. If < -0.41, then class label = 0. Otherwise, if 0.43, then class label = 1, else class label = 0.
Updated Theta
If x < -0.75 or x > 0.92, then class label = 0. If y 0.76, then class label O. y < -0.41, then class label = 0. Otherwise, if < 0.43, then class label = I, olse class label 0.
89

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Published in Transactions on Machine Learning Research (01/2025)
Training History: Two Circles Classification without Prior Step 41
Epoch Step
Forward Pass Prompt
You are the model.
** Model Descriptions: **
You are designed to do binary classification. The input is point on a. 2-D plane [x yl: you need to output the class lahol, i... an
integer in the set I < -0.75 x 0.92, then class label = If y 0.76, then class label O. y < -0.41, then class label
0. Otherwize, if 0.13, then class label 1, else class label O.
Input:
[-0.146 -1.01]
Please give your output strictly in the following format:
Explanations: [Your step-by-step analyses and results]
Output:
[ONLY the integer class label; make necessary assumptions if needed]
Please ONLY reply according to this format, don't give me any other words. Forward Pass Output
Explanations:
Given input 1-0.873, 0.475).
Siuce x < -0.75, class label = 0.
Output:
Many outputs the data points
Forward Pass Output
Explanations:
Check if < -0.75 or x > 0.92. Since 0.08 is not in this range, move to the next step.
Check y > 0.76. Since 0.261 is not in this range, to the next step.
3. Check if < -0.41. Since 0,261 is not in this range, move to the next step.
Since x < 0.43, class
Output:
Training Accuracy
1.0
Optimization Prompt
You the optimizer for model, your goal is learn the best descriptions for the model. The model used the Current Model Descriptious below predicted the class labels for the given inputs. You are given the target labels, please optimize the Model Descriptions for better prodiction.
** Inputs (a batch of I.I.d. data on 2-D plane: [x y)):
[[001166 -1.01] [0.221 -0.929| 0.899 0.397] [ [0.811 -0.56 [:0.873 0.475] [0.2 0.172] 0.08 0.261] 0.518 0.943] (-0.502 -0.788] (-0.069 -0.252]]
** Current Model Descriptions:
You designed to do binary classitication. The input is point 2-D plane yl: you need output the class lahel, Le., integer in the set 0, 1. If -0.75 or x > 0.92, then class label = 0. If > 0.76, then class label = 0. If < -0.41, then class label
Otherwise, if 0.43, then class label 1, else class label
** The model predlctions: **
[0]] [o] [0] [0] [0] [1][]] [1] [0] [0] [A]]
** The targets:
[[o] [o] [o] [o] [o] [1] [1] [o] [0] [1]]
the model is doing well, you keep using the current descriptions. However, if the model performing well, please update
the model by improving the 'New Model Descriptions', which should have lower claasifcaaiinn error both on the current and the next
i..dd
previous 'Optimization Step' are provided, you can use information from your optimisation stop
helpful. 'ON'T use symbolic represeutation for the modell Please think step by step and give your outputs strictly in the following format:
Rensoning:
[be explicit and verbose, improve the Current Model Descriptions by yourself; please show your work: that you don't have to computers]
New Model Descriptions:
[put new decision rules here; MUST be concise and concrete; *UST PROVIDE THE EXACT VALUE OF THE PARAME- TERS if the descriptions potentially involve unknown or learnable parameters!!!**e>
Please ONLY reply according to this format, don't give me any other words. Optimization Output
Reasoning:
Analyzing the current model predictions and targeta, observe that the model performing well, but there still room for improvement. Upon visual inspection of the input data, I notice that the current x-coordinate threshold of 0.43 seems be effective, and the y-coordinate thresholds of 0.76 and -0.41 are reasonable. However, observe that some points with x- x-values around -0.5 and y-valurs around -0.7 misclassified,, which suggests that the model is capturing the class houndary accurately in this region. also notice that points with x-values around 0.2 and y-values around 0.2 are correctly classified, which suggests that the model is x-coordinate threshold 0.92 seems
performing well in this region. Additionally, it misclassifees some points with x-values around D.8. New Model Descriptions: If x -0.75 or > 0.88, then class label = O. If y 0.74, then class label = 0. If < -0.39, then class label = O. Otherwise, x 0.42, then class label = 1, else class labe) Updated Theta
that the
aggressive, as
If x < -0.75 or x > 0.88, then class label = 0. If 0.74, then class label 0. I < -0.39, then class label = O. Otherwise, if 0.42, then class label 1, olse class label
90

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Published in Transactions on Machine Learning Research (01/2025)
M.6 Two Circles Regression (LLama-3-70B with prior)
Training History: Two Circles Classification with Prior - Step 1
Epoch 0 Step
Forward Pass Prompt
You the model.
Model Descriptions:
You are designed to do binary classification, The input point on a plane [x yls you need output the class label, i.., an integer in the set, O, 1. The decision houndary is circle.
** Input: **
[-0.146 -1.01]
Please give your output strictly in the following format:
Explanations: [Your step-by-step analyses and results]
Output:
[ONLY the integer class label; make necessary assumptions if needed]
dis
Please ONLY reply according to this format, don't give any other words. Forward Pass Output
Explanations:
The decision boundary is circle, so we need to find the distance of the iuput point [-0.069, -0.252 from the origin (0,0). The distance (qrt((-0.069) + (-0.252) 2) 0.264. Since the distance than 1, the point is inside the circle.
Output:
Many outputs for the rest data points
Forward Pass Output
Explanations:
The docision houndary circle, A point [x y] is classified as if is inside circle, the circle is assumed to be at [o 0j. The redius of the circle is sasumed to be 1 unit circle). The distance of the point [0.89 0.397] from the center the circle is (0,1(0.89 + 0.397 2) Since the distance is equal to the radius, the poiat is ou the circle. In this it ussumed that the point is classified Output:
outside circle. center
Training Accuracy
0.5
Optimization Prompt
You the optimizer for model, your goal is to learn the best descriptions for the model. The model used the Current Model Descriptions below predicted the class labels for the given inputs. You are given the target labels, please optimize the Model Deseriptions for better prediction.
Inputs (a batch of I.I.d. data on 2-D plane: [x y)): [[-0.146 -1.01 00.221 -0.929| 0.89 0.397] [0.811 -0.56 [-0.873 0.475] 0.27 0.172] 0.08 0.261] 0.518 0.943] (-0.502 -0.788] (-0.069 -0.252]]
** Current Model Descriptions:
You designed to do binary classification. The imput is point a 2-D plane [x yl: you need output the class label, l.e.,
integer in the set 0, 1. The decision boundary is a circle.
** The model predictions: **
[[0] [o] [0] [i] [0] [0] [0] [1] [0] [O]]
** The targets:
[[o] [o] [o] [o] [0] [1] [1] [0] [o] [1]]
the model is doing well, you can keep using the current descriptione. However, if the model not performing well, please update
the model by improving the 'New Model Descriptions' which should have lower clussifiaaiion
both the current and the next batch of Li.d, data. If previons 'Optimization Step' are provided, you can use the information from your last optimization step if it's
helpful. 'ON'T symbolic representation for the model! Please think step by step and give your outputs strictly in the following format:
Reasoning:
[b explicit and verbose, improve the Current Model Descriptions by yourself; please show your work; note that you don't have access to computers'
New Model Descriptions:
pout your new decision rules here; MUST be concise and concrete; MUST PROVIDE THE EXACT VALUE OF THE PARAME- TERS if the descriptious potentially involve unknown or learnable parameters!!!*** .**
Please ONLY reply according to this format, don't, give me any other words. Optimization Output
Reusoning:
The model not performing well, as it misclassifies 4 out of 10 samples. The decision boundary is a circle, but the current model predictions do not seem to follow circular pattern. To improve the model, will try find better circle that separates the classes. I will plot the data points on the 2-D plane and try to find a circle that separates the classes. After visualizing the data, I notice that points (0.27, 0.172) and (0.08, 0.261) are misclassified,
they are close through the midpoint of these two points and is centered at a point that is closer to the origin. Let the center of the circle he (a, b) and the radius be r. The midpoint the two pointa (0.175, 0.2165). I will aet the center the cirele to be (0.175, 0.2165). The radius 0.1.
will try
circle that passes
found by calculating the distance from the center to one of the points, say (0.27, 0. 0.17 The radius approximately
New Model Descriptions:
If (x 0.175) 2 + (y 0.2165) 2 <= 0.01, then output 1, else output O.
Updated Theta
If (x 175) (y 0.2165) 2 <= 0.01. then output I, else output 0.
91

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Published in Transactions on Machine Learning Research (01/2025)
Training History: Two Circles Classification with Prior Step 2
Epoch Step
Forward Pass Prompt
You are the model.
** Model Descriptions: **
You are designed to do binary classification. The input is point on a 2-D plane [x yl: you need to output the class lahol, i..., an
integer in the set 0, The decision boundary is circle. If (x 0.175) + (y 0.2166) <= 0.01, then output 1, else output
** Input: e* [-0.305 0.936]
Please give your output strictly in the following format:
Explanations:
Your step-by-step analyses and results]
Output: [ONLY the integer class label; make necessary assumptions if needed]
Please ONLY reply according to this format, don't give any other words. Forward Pass Output
Explanations:
Calculate the value of (x 0.175) + (y 0.2165) 2 (-0.051 0.175) + (-0.372 0.2165) '2 0.1061. Since 0.1061 0.01, the point outside
Output:
Many outputs for the rest data points
Forward Output
Explanations:
The given input point [-0.329, 0.046). determine class label, we need check this point within circle with decision boundary. The circle's center is at (0.175, 0.2165) and the radius ls sqrt(0.01) = 0.1. Let's caleulate the distance of the point from the center:
Distance sqri((x 0.175) (y 0.2165)"2)
aqrt((-0.329 - 0.175) (0.026 0.2165)"2)
sgrt((-0.504)- + (-0.1705) (2)
sQrr(0.253366 0.02914025)
(qrt((0.2823000)
0.532
Since the distance (U.532) greater than the radius (0.1), the point lies outside the circle. Output:
Training Accuracy
Optimization Prompt
You are the optimizer for a model, your goal is to learn the best descriptions for the model. The model used Current Model Descriptions below predicted the class labels for the given inputs. You given the target labels, please optimize the Model Descriptions for better prediction.
** Inputs batch of i.i.d. data 2-D plane: [x y)): **
[[-0.305 0.936] [ 0.2298 -0.037] 0.396 0.931] [-0.735 .0.614| (-0.339 0.006] [-0.253 -0.015] 0.096 -0.248) 1.005 0.048) [-0.329 0.046)
[-0051 -0.372]
Current Model Descriptions:
You are designed to binary classification. The imput is point on a 2-D plane [x yl: you need to output the class label, Lee. an
integer in the set
The decision boundary is circle. If (x u.175) (y 0.2165) 22 <= 0.01, then output else output
The model predictions:
[[o] [0] [0] [0] [0] [0] [0, [0] [o]]
The targets:
[[o] [I] [o] [0] [1] [1] [1] [0] [*]
If the model is doing well, you can keep using the current descriptions. However, If the model not performing well, please update the model by improving the 'New Model Descriptions' which should have lower classifcaaiinn error both on the current and the next batch of ii.d. data. If previous 'Optimization Step' provided, you the information from your last optimization step if it's helpful. DON'T use symholic representation for the model! Please think step by step and give your outputs strictly in the fallowing format:
Reasoning:
[be explicit and verbose, improve the Current Model Descriptions by yourself; please show your work; note that don't have access to computers/
New Model Descriptions:
jput your new decision rules here; MUST be concise and concrete; *UST PROVIDE THE EXACT VALUE OF THE PARAME- TERS if deecriptions potentially involve unknown or learuable parsmeters!!!**eg
Please ONLY reply according to this format, don't give me any other words. Optimization Output
Reasoning:
The model is performing well, it misclassifies
of samples. The current derision boundary circle centered at (0.1 75. 0.2165) with radius of 0.1, but does not seem to separate the classes well. I will re-examine the data and try to find a better citcle that separates the classes. notice that the points (0.298, -0.037) and (-0.339, 0.006) each other. will try to find circle that passes thrrough the midpoint of these two points and is centered at point that is closer to the origin. Let the center of the circle (a. b) and the radius be r. The midpoint the two points is (-0.0205, -0.0155). will set the center of the circle to be (-0.0205, -0,0155). The radius be found calculating the distance from the center to one of the points, say (0.298, -0.037). The radius approximately 0.35.
misclassified, and they are close to
New Model Descriptions:
If (x + 0.0205)* 2 + (y + 0.0155) <= 0.1225, then output 1, elae output 0.
Updated Theta
If (x 0.0205) 2 + (y + 0.0155) 2 <= 0.1225, then output 1, clse output O.
92

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Published in Transactions on Machine Learning Research (01/2025)
Training History: Two Circles Classification with Prior Step 3
Epoch Step
Forward Pass Prompt
You are the model.
** Model Descriptions: **
You are designed to do binary classification. The input is point on a 2-1 plane [x yl: you need to output the class lahel, i...,
integer in the set O, 1. The decision boundary a circle. If (x + 0.0205) + (y + 0.0166) <= 0.1225, then output 1, else output
** Input: [0.275 0.163]
Please give your output strictly in the following format:
Explanations: [Your step-by-step analyses and results]
Output:
[ONLY the integer class label; make necessary assumptions if needed]
Please ONLY reply according to this format, don't give me any other words. Forward Pass Output
Explanations:
Calculate the value denision boundary equation: (0.403 + 0.0205) "2 + (0.96 0.0155) 0.4235'2 0.9755 0.1 + 0.952 1.131. Since 1.131 > 0.1225, the point is outaide the cirele.
Output:
Many outputs for the reat data points...
Forward Pass Output
Explanations:
We given point (0,-1.012) on the 2-D plane. We need to check if the point lies inside the circle defined by the decision boundary. Subetituting the values in the equation: (0 + 0.0205) + (-1.012 0.U155)"2 22 <= 0.1225.
Calculating the values: (0.0205) 2 + (-0.9965) "2 <= 0.1225.
Calculating the values: 0,00012025 + 0.99242025 <= 0.1225.
Calculating the values: 0.9928405 0.1225.
The condition is not satisfied.
Output:
Training Accuracy
0.7
Optimization Prompt
You are the optimizer for a model, your goal is to learu the best descriptions for the model. The model used the Current Model Descriptions below predicted the class labels the given inputs. are given target labels, please optimise Model Deseriptions for better prediction.
** Inputs (a batch of I.I.d. data on 2-D plane: [x yl): **
0.275 0.163 0. -1.012] [-1.065 -0.126] 0.133 0.22 [-0.317 -0.931] 0.403 0.96 ] 0.013 0.996] [0.147 -0.178] [-0.028 0.422] (-0.233 -0.1851 [0.186]]
** Current Model Descriptions: **
You are designed to do binary classification. The input is point on a 2-D plane [x yl: you need to output the class label, i.e. an integer in
0, The decision boundary circle. (x 0.0205) + (y + 0.0155) 2 <== 0.1225, then output 1, olse output
** The model predictions:
[[]] [0] [0] [0] [0] [0] [0] [0] [0] [A]]
** The targets:
[[] [o] [o] [1] [o] [0] [o] [1] [1] [1]]
the model is doing well, you keep using the descriptions. However, if model performing well, please update
the model by improving the 'New Model Descriptions', which should have lower claasification error both on the current and the next
i..dd
previous 'Optimization Step' are provided, you can use information from your optimisation stop
helpful. "ON'T use symbolic represeutation for the modell Please think step by step and give your outputs strictly in the following format:
Rensoning:
[b explicit and verbose, improve the Current Model Descriptions by yourself; please show your work: that you don't have to computers]
New Model Descriptions:
[put new decision rules here; MUST be concise and concrete; ***T PROVIDE THE EXACT VALUE OF THE PARAME- TERS if the descriptions potentially involve unknown or learnable parameters!!!*wey
Please ONLY reply according to this format, don't give me any other words. Optimization Output
Reasoning:
The model is not performing well, as misclassifies out of samples. The current decision boundary a cirele centered at (-0.0205, -0.0155) with radius of approximstely 0.35, but it does not and try to find a better circle that separates the classes. I notice that the points (0.133, 0.22) and (0.147, -0.178) are misclassffied, find cirele that passes throngh the midpoint of these points and is centered at
to separate the classes well. I will re-examine the data
and they close to each other. will point that is closer to the origin. Let the center of the circle be (a. b) and the radius be The midpoint of the two points is (0. 14, 0.021). will the center of the circle one of the points, say (0.133, 0.22). The radius is appruximately 0.235.
(0.11, 0.021). The radius
found by calculating the distance from the
New Model Descriptions:
If 0.14) (y 0.021)"2 2 0.055225, then output elae output 0.
Updated Theta
0.14) (y 0.021) 2 <= 0.055225, then output else output 0.
93

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Published in Transactions on Machine Learning Research (01/2025)
Training History: Two Circles Classification with Prior Step 4
Epoch Step
Forward Pass Prompt
You are the model.
**
Model Descriptions: **
You are designed to do binary classification. The input is point on a 2-1 plane [x yl: you need to output the class lahol, ie.,
integer in the set 0, The decision boundary ls circle, (x 0.14) "2 + (y- 0.021) <= 0.055225, then output else output
** Input: [-0.811 0.529]
Please give your output strictly in the following format:
Explanations: (Your step-by-step analyses and results]
Output:
[ONLY the integer class label; make necessary assumptions if needed]
Please ONLY reply according to this format, don't give me any other words. Forward Pass Output
Explanations:
Caleulate the distanoe from point to center of circle: (0.24 0.14)" (-0.039 0.021) 0.0064 0.0036 0.0100
Since 0.0100 <= 0.055225, the point is inside the circle.
Output:
Many outputs for the reat data points...
Forward Pass Output
Explanations:
We need to check if the point (-0.811, 0.529) lies inside the circle (x 0.14) (v 0.021) <= 0.053225. First, calculate the left side of the inequality: (-U.811 0.14) + (0.529 0.021) '2 = 0.671 0.508 0.449441 + 0.258064 0.707505.
Since U.707505 > 0.055225, the point (-0.811, 0.529) lies outside the circle. Output:
Training Accuracy
Optimization Prompt
You are the optimizer for a model, your goal is to learn the best descriptions for the model. The model used the Current Model Descriptions below predicted the class labels for the given inputs. You the target labels, please optimise the Model Descriptious for better prediction.
** Inputs batch i.i.d. data [-0.811 0.529] [-0.036 0.323] [-0.63 0.698] 170.173 -0.246] 0.943 -0.297] [ 0.1 -0.288] [-0.995 -0.025] 0.666 -0.666) 0.24 -0.039) 0.542 0.887]]
plane: y!):
Current Model Descriptions:
You are designed to do binary classification. The input is point on a 2-D plane [x yl: you need to output the class label, i.c., an The decision boundary is cirele. (x 0.14) + (y- 0.021) 2 <= 0.055225, then output else output
integer in the set
The model predictions:
[[0] [0] [0] [o] [o] [0, [o]]
The targets:
[[]] [1] [0] [1] [0] [1] [0] [0] [1] [O]]
If the model is doing well, you keep using the current descriptions. However, if the model not performing well, please update
the model by improving the 'New Model Descriptions' which should have lower classification crror both on the current and the next
batch of ii.d. data. If previous 'Optimization Step' provided, you the information from your last optimisation step it's
helpful. DON'T use symbolic representation for the model! Please think step by step and give your outputs strictly in the following
format:
Reasoning:
be explicit and verbose, improve the Current Model Descriptions by yourself; please show your work; note that you don't have to computers]
New Model Descriptions:
[put your new decision rules here; MUST be concise and concrete; MUST PROVIDE THE EXACT VALUE OF THE PARAME- TERS descriptions potentially involve unknown or learnable parameters!!!
Please ONLY reply according to this format, don't give me any other words. Optimization Output
Reasoning:
The model (U.14, 0.021) with a radius of approximately 0.235, but it does not seem to separate the classes well. 1 will re-examine the data and try to find a better circle that separates the classes. notice that the pointa (-0.036, 0.323), (-0.173, -0.240), and (0.1, -0.288) are misclassifeed, and they all close to the origin. I will try to find circle that passes through the origin and is centered at point that is closer to the origin. Let the center the circle be (a, b) and the radius be r. I will set the center of the circle to be (0, 0). The radius approximately 0.292.
performing well,
misclassifies
of samples. The current decision boundary circle centered at
he found calculating the distance from the center to of the points, say (-0.173, -0.246). The radios
New Model Descriptions:
If (x) 2 + (y) 2 = 0.085328, then output 1, else output O.
Updated Theta
If (x)"2 2 + ( 0.085328, then output 1, else output
94

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Published in Transactions on Machine Learning Research (01/2025)
Training History: Two Circles Classification with Prior Step 5
Epoch Step
Forward Pass Prompt
You are the model.
** Model Descriptions: **
You are designed to do binary classification. The imput is point on a 2-D plane [x yl: you need to output the class lahol, i...,
integer in the set 0, The decision boundary is circle. If (x)"2 + (x)"2 <= 0.085328, then output 1, else output
** Input: [-0.99 -0.32G]
Please give your output strictly in the following format:
Explanations: (Your step-by-step analyses and results]
Output:
[ONLY the integer class label; make necessary assumptions if needed]
Please ONLY reply according to this format, don't give me any other words. Forward Pass Output
Explanations:
Given input [-1.059, 0.217], calculate (x)'2 + (x) (-1.059)"2 (0.217)'2 1.123. Since 1.123 > 0.085328, it does not satiafy the condition.
Output:
Many outputs for the reat data points...
Forward Pass Output
Explanations:
We given point [-0.825, 0.438]
2-D plane.
We need to check if the point lies fuside the circle or not.
The equation of the circle (x)"2 + (2)"2
0,085328.
Let's calculate the value of (x)"2 + (y) '2 for the given point. (-0.825) 2 - (0.438) "2 0.680625 + 0.191811 = 0.872469.
Since 0.872469 > 0.085328, the point lies outside circle.
Output:
Training Accuracy
0.7
Optimization Prompt
You the optimizer for model, your goal is learn the best descriptions for the model. The model used the Current Model Descriptious below predicted the class labels for the given inputs. You are given the target labels, please optimize the Model Descriptions for better prodiction.
** Inputs (a batch of I.I.d. data on 2-D plane: [x y)):
[[-0.99 -0.325] [-0.825 0.438] 0.188 0,308] [-1.059 0.217) [-0.188 -0.184] (-0.322 0.084] (0.892 0.513) [-0.287 0.017, [ 00.16 0.235 [-0.573 -0.8-15]]
Current Model Descriptions:
You designed to do binary classification. The input is point 2-D plane yl: you need output the class lahel, integer in the set 0, 1. TTe decision boundary is circle. If (xx) 2 (y)"2 <= 0.085328, then output 1, clse output O.
** The model predictions: **
[[0] [0] [0] [0] [i] [0][[0 [0] [1]
[0]]
** The targets:
[[o] [o] [1] [o] [1] [1] [o] [1] [1] [o]]
the model is doing well, you can keep using the current descriptions. However, if the model is not performing well, please update both the current and the next batch of Lidd data. If previous 'Optimization Step' are provided, you can use the information from your last optimization step if it's model! Please think step by step and give your outputs strictly following
the model by improving the 'New Model Descriptions', which should have lower classification
helpful. DON'T use symbolic representation format:
Reasoning:
be explicit and verbose; improve the Current Model Deseriptions by yourself; please show your work; note that you don't have access computers
New Model Descriptions:
jput your new decision rules here; MUST concise and concrete; MUST PROVIDE THE EXACT VALUE THE PARAME- TERS if the descriptions pocentially involve unknown or learnable parsmeters!!!**es
Please ONLY reply acrording to this format, don't give any other words. Optimization Output
Reasoning:
The model is not performing well, as it misclassifies 5 out of 10 samples. The current decision boundary is a circle centered at the origin with a radius approximately 0.292, but does not seem separate the classes well. will re-examine the data and try to find a better circle that separates the classes, notice that the points (0.188, 0.308), (-0.188, -0. 1841,4) and (0.167, 0.235) misclassfied, and they are all close to the origin. will try to find circle that passes through the origin and centered at a point that is closer the origin. Let the center of the circle be (a, b) and the radius be I will set the center of the cirele be (0. 0j. The radius can be found by calculating the distance from the center to one of the points, say (0.167, 0.236). The radius is approximately 0.283. However, this radius is sufficient to separate the classes well. I will try find better radius by examining the other misclassified points. The point (-0.322, 0.084) is also misclassified, and it is closer to the origin than the previous point. The distance from the center to this point is approximately 0.334. will use this distance as the radius. New Model Descriptions:
(x4"2 + (v) <= 0.111556, then output 1, else output 0.
Updated Theta
If (x) 2 + 2 <= 0.111556, then output 1, else output 0.
95

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Published in Transactions on Machine Learning Research (01/2025)
Training History: Two Circles Classification with Prior Step 40
Epoch 3
Step 10
Forward Pass Prompt
You are the model. ** Model Descriptions: **
You are designed to do binary classification. The imput is point on a 2-1 plane [x yl: you need to output the class lahol, i...,
integer in the set 0, The decision boundary is circle. If (x)"2 + (x)"2 <= 0.45, then output 1, else output
Imput: [-0.101 0.309]
Please give your output strictly in the following format:
Explanations: (Your step-by-step analyses results]
Output:
[ONLY the integer class label; make necessary assumptions if needed]
Please ONLY reply according to this format, don't give me any other words.
Forward Pass Output
Explanations:
Given point (-1.011 -0.165), calrulate sum squares: (-1.011 (-0.165) 1.044721 0.45.
Output:
Many outputs for the rest data points
Forward Output
Explanations:
The given input point is [-0.024, -0.333].
To determine the class label, we need to check the point is inside the circle not.
The equation the circle (xx) 2 + (y) <= 0.45. Let's calculate the value of this equation for the given point: (-0.024) (-0.333) 0.011056 0.110889 0.122945 0.45. Since the result is true, the point lies inside the circle.
Output:
Training Accuracy
Optimization Prompt
You are the optimizer for a model, your goal is to learn the best descriptions for the model. The model used Current Model Descriptions below predicted the class labels for the inputs. You the target labels, please optimize the Model Descriptious for better prediction.
** Inputs batch i.i.d. data [[0.100 0.309] [-1.01] -0.165] 061 -0.229) 1.02 -0.061] 0.225 0.076] [-0.024 -0.333] [-0.655 -0.649) [-1.558 0.15 | [03366 -0.998| [ 0.919 0.08
plane: yl):
Current Model Descriptions:
You are designed to do binary classification. The input is point on a 2-D plane [x yl: you need to output the class label, i.e., an The derision houndary circle. IF (x)"2 (x)"2 0.45, then output else output
integer in the set.
The model predictions:
[[] [0] [1] [0] [1] [1] [1] [0] [o]]
The targets:
[]]] [o] [1] [0] [1] [1] [0] [0] [1] [O]]
If the model is doing well, you keep using the current descriptions. However, if the model not performing well, please update
the model by improving the 'New Model Descriptions' which should have lower classification crror both on the current and the next
batch of Li.d. data. If previous 'Optimization Step' provided, you
information from your last optimisation helpful. DON'T use symbolic representation for the model! Please think step by step and give your outputs strictly in the following
it's
format:
Reasoning:
be explicit and verbose, improve the Current Model Descriptions by yourself; please show your work; note that you don't have to computers]
New Model Descriptions:
[put your new decision rules here; MUST be concise and concrete; MUST PROVIDE THE EXACT VALUE OF THE PARAME- TERS descriptions potentially involve unknown or learnable parameters!!!
Please ONLY reply according to this format, don't give me any other words. Optimization Output
Reasoning:
The model la not performing well, misclassifies the sample. The decision boundary circle centered the origin with a radius of 0.45. The 7th sample is far from the origin, but the model still classifies it as 1. This means the radius of the circle too small. will increase the radius to reduce the misclassification error.
New Model Descriptions:
(x)"2 "2 + (y) <= 0.65, then output 1, else output 0.
Updated Theta
If (x) 2 + ( <= 0.65, then output 1, else output 0.
96

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Published in Transactions on Machine Learning Research (01/2025)
Training History: Two Circles Classification with Prior Step 41
Epoch Step
Forward Pass Prompt
You are the model. ** Model Descriptions: **
You are designed to do binary classification. The imput is point on a 2-1 plane [x yl: you need to output the class lahol, i...,
integer in the set 0, The decision boundary circle. If (x) + (x)"2 <= 0.65, then output 1, else output
Input: [-0.146 -1.01]
Please give your output strictly in the following format:
Explanations: (Your step-by-step analyses results]
Output:
[ONLY the integer class label; make necessary assumptions if needed]
Please ONLY reply according to this format, don't give me any other words. Forward Pass Output
Explanations:
First, calculate
(x) (x)" (-0.146) + (-1.01)"2 Since 1.041336 >> 0.65, the point is outside the cirele.
1..44136..
Output:
Many outputs for the reat data points
Forward Pass Output
Explanations:
The luput point is [-0.502 -0.788].
To determine the class label, we need to check the point satisfies the condition (x) + (v) <= 0.65.
Here, -0.502 and -0.788. (x) '2 (-0.502) 2 0,252004, and (y)"2 (-0.788) 00.221944
Now, (x) + (y) 2 0.252004 + 0.621981 = 0.874988.
Since 0.874988 <=
Palse, the point lies outside circle.
Output:
Training Accuracy
1.0
Optimization Prompt
You the optimizer for model, your
learn the best descriptions for the model. The model used the Current Model Descriptious below predicted the class labels for the given inputs. You are given the target labels, please optimize the Model prodiction.
Descriptions for
Inputs (a batch of I.I.d. data on 2-D plane: [x y)): [[0.1146 -1.01] [0.224 -0.929| 0.89 0.397] [0.811 -0.56 [:0.873 0.475] [0.27 0.172] 0.08 0.261] 0.518 0.943] (-0.502 -0.788] (-0.069 -0.252]]
** Current Model Descriptions:
You designed to do binary classification. The input is point 2-D plane vl: you need output the class lahel, integer in the set 0, 1. TThe decision boundary is circle. If (xx) 2 (y)"2 <= 0.65, then output 1, else output O.
** The model predictions: **
[[o] [0] [0] [0] [o] [1] [O] [X]]
** The targets:
[[o] [o] [o] [o] [1] [1] [o] [1]]
the model is doing well, you can keep using the current descriptions. However, if the model not performing well, please update both the current and the next batch of Li.d data. If previous 'Optimization Step' are provided, you can use the information from your last optimization step if it's model! Please step by step and give your outputs strictly following
the model by improving the 'New Model Descriptions', which should have lower classification
helpful. DON'T use symbolic representation format:
Reasoning:
be explicit and verbose; improve the Current Model Deseriptions by yourself; please show your work; note that you don't have access computers
New Model Descriptions:
concise concrete; MUST PROVIDE THE EXACT VALUE PARAME- TERS if the descriptions pocentially involve unknown or learnable parametens!!!**ee
jout your new decision rules here;
Please ONLY reply acrording to this format, don't give any other words. Optimization Output
Reasoning:
The model is performing well, as it correctly classifies all samples. The decision boundary is circle centered at the origin with radius 0.65, which seema New Model Descriptions:
good for the data. changes are needed.
(x)"2 " + (y) = 0.65, then output 1, else output 0.
Updated Theta
If (x) 2 + 2 <= 0.65, then output 1, else output 0.
97

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Published in Transactions on Machine Learning Research (01/2025)
M.7 Text Classification --70m without prior)
Training History: Text Classification Step 1
Bpoch Step
Forward Pass Prompt
You the model.
Model Descriptions:
designed do binary classification. The input term; you need output class label, ic.. integer the
Input:
fatherbood
Please give your output strictly in the following format:
Explanations:
[Your step-by-step analyses and results]
Output:
[ONLY integer class label; uecrssary assumptions if needod]
Please ONLY reply according to this format, don't give me any other words.
Forward Pass Output
Explanations: Since the input is a name; 'Margaret", will assume it's a female name. Therefore, will classify it as female name. Output:
Many outputs
data points
Forward Pass Output
Explanations: The ferm "tombuy" is It does not have any strong connotations or associations that would clearly link it to a sporific class label. However, based on the context and
that refers to girl who exhibits characteristies or behaviors considered typical of boy.
usage of the term, will make an assumption that the class label represents characteristic, and 0 represents a more feminine or girl-like characteristic. Given this sssumption, I would classify "tomboy" as 1. Output:
masculine boy-like
Training Accuracy
0.5
Optimization Prompt
You are the optimizer for a model, your goal is to learn the hest descriptions for the model. The model used the Current Model Descriptions below predicted the class labels the given inputs. You given the target labele, please optimize the Model Descriptious for better prediction.
** Inputs (a batch of I.I.d. text): *8
['stherhood', 'grandfather', 'tomboy', 'Shawn', 'Erma', 'Ada', 'Margaret', 'Caleb' 'Earl', 'Faye']
Current Model Descriptions: **
You designed do binary classification. The input term; you need output the class lahel, Lo., integer in the set The model predictions:
[[1] [i] [1] [1] [1] [0, [1] [0]]
The targets:
[[0] [o] [1] [0] [1] [1] [1] [0] [0] [A]]
If the model is doing well, you keep using the current descriptions. However, if the model not performing well, please update
the model by improving the 'New Model Descriptions' which should have lower classification crror both on the current and the next batch of Li.d. data. If previous 'Optimization Step' provided, you
the information from your last optimisation stop if it's helpful. Please think step by step and give your outputs strictly in the following format:
ns
Reasoning:
[be explicit and verbose, improve the Current Model Descriptions by yourself; please show your work: note that you don't have access to computers'
New Model Descriptions:
[put your decision rules here]
Please ONLY reply according to this format, don't it give any other words. Optimixation Output
Reasoning:
Upon examining the current model predictions and target labels, I notice that the model 10 predictions incorrect. The model scems to be biased towards predicting class lahel 1. I observe that the input terms can he categorized into groups: male
performing well, with out of
female names, and non-name terms related family relationshipe. The target labels suggest that the class label correspouds to male names and non-name terms, while class label 1 corresponds to female names. To improve model,
create new decision rules that take account these categories. will update the descriptions include more sperific patterns and rules to differentiate between male and female names, as well as non-name terms. New Model Descriptions:
(c.g., -so,, -er), prodict class label If the input term a non-name term related to family relationships (o.g.. fatherhood, grandfather), predict class label 0. If the input term is female name (e.g., Erms, Ada, Faye), predict class label Otherwise, predict class label
If the input term ends with suffix commonly associated with male
Updated Theta
the input ends with suffix commonly associated non-name term related to family relationships (e.g.. fatherhood, grandfather), predict class label 0. If the input term is female name (e.g., Erma, Ada, Faye), predict class label Otherwise, predict class label
names (e.g., -son, -er), predict class label the input term
98

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Published in Transactions on Machine Learning Research (01/2025)
Training History: Text Classification Step
Epoch Step
Forward Pass Prompt
You are the model. ** Model Descriptions: **
You are designed to do binary classification. The input is term: you need to output the class label, i.., an intoger in the set 0, 1. Tf the input term enda with a suffix commonly associated with male names (e.g., -son, -er), predict class label O. the imput term is non-name term related to family relationehips (e.g., fatherbood, grandfather), predict class label O. If the imput term is female name (e.g., Erma, Ada, Faye), predict class label 1. Otherwise, predict class label O.
** Input: **
Lydia
Please give your output strictly in the following format:
Explanations: [Your step-by-step analyses and results]
Output:
[ONLY the integer class label; make necessary assumptions If needed]
Please ONLY reply uctording to this format, don't give me any other words.
Forward Pass Output
Explanations: The input term "Marlene" is female name. Output:
Many outputs for the reat data points...
Forward Pass Output
Explanations: The input term "paternal grandma'. The term 'grandma" non-name term related family relationships, and "paternal" is adjective that means related to father. Therefore, sccording to the rules, the class label is O.
Output:
Training Accuracy
0.9
Optimization Prompt
You are the optimizer for a model, your goal is to learn the best descriptions for the model. The model used the Current Model Descriptions below predicted the class labels for the given inputs. You are given the target labels, please optimize the Model Descriptions for better prediction.
Inputs (a batch of Li.d. text): *
['Lydii', 'Steven', 'Ida', 'Lisa', 'Allan', 'Jane', 'patornal grandma', 'paternal grandad", 'Marlene', 'Darrell]]
Current Model Descriptions:
You are designed to binary classification.
label, ins, an integer the set the input term ends with a suffix commonly associated with male names (e.g. -son, -er), predict class label O. If the imput torm is non-name term related to family relationships (e.g., fatherhood, grandfsther), predict class label the input term is a female (e.g., Erma, Ada, Faye), predict class label 1. Otherwise, predict class label U.
input term; you need output
** The model predictions:
[[X] [0] [1] [1] [o] [1][9] [9] [0.
[0]]
The targets:
[D] [o] [1] [1] [o] [1] [1] [0] [1] [o]]
If the model is doing well, you can keep using the current descriptions. However, if the model is not performing well, please update
the model by improving the 'New Model Descriptions', which should have lower classifccation error both on the current and the next batch of ii.d. data. If previous 'Optinization Step' provided, you use the information from your last optimization step if it's helpful. Please think step by step and give your outputs strictly in the following format:
Reasoning:
explicit and verbose, improve the Current Model Descriptions by yourself, please show your work; note that you don't have access to computers.
New Model Descriptions:
[put your new decision rules here]
Please ONLY reply according to this format, don't give any other words. Optimization Output
Reasoning:
Upon examining the current model predictions and target labels, I notice that the model is performing ressonably well, but there are still some errors. The model correctly predicts out of 10 labels, with errors occurring for the termss "paterual graudma' and "paternal grandad". I realixe that the current. descriptions da not acrount for non-name terms that contain leywords related to family relationships, but are not exact matches (e.g. "paternal grandma" vs. 'grandma"). ". To improve the model, I will update the descriptions include more specifie rules for non-name terms that contain keywords related to family relationships. I will also refine the rules for female names to reduce the likelihood of misclassification. New Model Descriptions:
If the imput term ends with sufiix commonly associated with male names (e.g.. -son, -er), predict class label O. If the input term contains keywords related to family relationships (e.g,, 'father", 1, 'mother', "grand", "paternal', "maternal") " predict class label O. If the input term female name (e.g,, Frma, Ada, Faye, Lydia, Ida, Jane, Marlene], predict class label 1. the imput term that does not clearly indicate gender (e.g., Leslie, Jordan), predict class label 0. Otherwise, predict class label 0.
Updated Theta
If the imput term ends with a suffix commonly associated with male names (e.g., -son, -er), predict class label O. If the input term contains keywords related to family relationships (e.g., "father", 1, 'mother', "grand', "paternal", "maternal)) ) predict clase label O. If the input term female name (e.g., Erma, Ada, Faye, Lydia, Ida, Jane, Marlene), predict class label 1. If the imput term name that does not clearly indicate gender (c.g., Leslie, Jordan), predict class label O. Otherwise, prodict class label O.
99

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Published in Transactions on Machine Learning Research (01/2025)
Training History: Text Classification Step
Epoch Step
Forward Pass Prompt
You are the model.
** Model Descriptions:
You are designed to do binary classification. The input is term; you need to output the class label, lee, an integer in the set O, 1. If imput term ends with suffix commonly associated
male names (e.g., -son, -er), predict class label 0. the input contains keywords related to family relationships (c.g., "father", 'mother', 'graadd, "paternal", "maternal") predict class label O. If input torm a female name (e.g,, Erma, Ada, Faye, Lydia, Ida, Jane, Marlene), predict class 1. the input term a name that does not clearly indicate gender (e.g. Leslie, Jordan), predict class label O. Otherwise, predict class label 0.
** Input: *
Marissa
Please give your output strictly in the following format:
"*
Explanations:
[Your step-by-step analyses and results]
Output:
[ONLY the integer class label; make necessary assumptions if needed]
Please ONLY reply acrording to this format, don't give any other words.
Forward Pass Output
Explanations: The input term "Maria", which female name.
Output:
Many outputs for the rest data points....
Forward Pass Output
Explanations:
Check if term ends with suffix commonly associated with male names: Julian does not end with a suffix like son or -rr.
2. Check if the term contains key words related to family relationships: Julian does not contain keywords like "father"; "mother", "grand,, "paternsl", or "maternal'.
3. Check if the term
female name: Julian is not female uame.
Check if the term is a nane that does not clearly indicate gender: Julian is a male name.
Output:
Training Accuracy
0.9
Optimization Prompt
You are the optimizer for a model, your goal is to learn the best descriptions for the model. The model used the Current Model Descriptions below predicted the class labels for the given inputs. You given the target labels, please optimize the Model Descriptions for better prediction.
** Inputs (a batch of I.I.d. text): **
['Marissa', 'Wallace', 'Julian', 'Adeline', 'Jacob', 'Dylan', 'William', 'Anne', 'Maria, 'sorority']
** Current Model Descriptions: **
You are designed to do binary classification. The input is term: you need to output the class label, L.e,, integer in the aet 0, 1. If the input term ends with a suffix commouly associated with male names (e.g. -son, -er), predict class label O. the iuput term contains keywords related to family relationships (e.g., "father', 'mother', 'grand', "paternal", 'maternal") 1) predict class label O. If the input torm is a female name (e.g., Erma, Ada, Faye, Lydin, Ida, Jane, Marlene), prodict class label 1. If the imput term a name that does not clearly indicate gender (e.g., Leslie, Jordan), predict class label O. Otherwise, predict class label 0.
The model predictions:
[[]] [o] [0] [1] [o] [0] [0] (1) [1] [o]]
**' The targets:
[[]] [o] [o] [1] [o] [u] [o] [1] [A]]
[A]]
keep using the current descriptions. However, if the model not performing well, please update the model by improving the 'New Model Descriptions' which should have lower classification error both on the current and the next batch of i.idd data. If previous 'Optimization Step' provided, you use information from your last optimization step if it's helpful. Please think step by step and give your outputs strictly in the following format: is
If the model is doing well, you
Reasoning:
[e explicit and verbose, improve the Current Model Descriptions by yourself; please show your work; note that you dou't have access to computors'
New Model Descriptions:
[put your new decision rules here]
Please ONLY reply according to this format, don't give me any other words. Optimization Output
Reasoning:
Upon examining the error occurring for the term 'sorority'. realize that the current descriptions do not account for non-name terms that are clearly related females or males, are names. improve the model, I will update descriptious include more specific rules for non-name terms that are clearly related to females or males. will also refine the rules for female names to reduce the likellhhood of misclassification. New Model Descriptions:
model predictions and target labels, I notice that the model is performing well, but there still
If the input term ends with a suffix commonly associated with male names (o-g., -san, -er), predict class lahel O. If the input term contains keywords related to family relationships (e.g., "father', ; 'mother', 'grand", "paternal", "maternal"), ),,, prediet class label 0. If the input term is female name (e.g., Erma, Ada, Faye, Lydia, Ida, Jane, Marlene, Marisss, Adeline, Anne, Maris), predict class name that does not clearly indicate gender (e.g., Leslie, Jordan), predict class label If the imput
label 1. If the input term term is a non-name term clearly related to females (c.g.. 'sorrrity', "girl"), predict class label 1. Otherwise, predict class label O.
Updated Theta
If the input term ends with suffix commonly associated with male names (e.g., -son, -er), predict class label O. If the input term contains keywords related to family relationships (o.g., "father', , "mother', "grand", "paternal", "maternal") , prodict class label 0. If the input term is female name (e.g., Erma, Ada, Faye, Lydia, Ida, Jane, Marlene, Marisss, Adeline, Anne, Maris), predict class label 1. If the input term is a name that does not clearly indicate gender (e.g., Leslie, Jordan). predict class label 0. If the input term nom-name term clearly related females (e.g., "sorrrity', "girl"), predict class label Otherwise, predict class label
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Training History: Text Classification Step
Epoch 0 Step
Forward Pass Prompt
You are the model. ** Model Descriptions: **
You are designed to do binary classification. The input is term; you need to output the class label, in., an intoger in the set 0. 1. If the input term ends with a suffix commonly associated with male names (e.g., -son, er)), predict class label O. the input term contains keyworda related to family relationships (e.g., "father', 'mother', "grand", "paternal', "maternal") ) prediet class label 0. If the input term is female name (e.g., Erma, Ada, Faye, Lydia, Ida, Jane, Marlene, Marissa, Adeline, Anne, Maris), predict class label 1. If the imput term name that does not clearly indicate gender (e.g., Leslie, Jordan), predict class label If the imput term is a uon-name term clearly related to females (e.g., "sorority' 'girl"), predict class label 1. Otherwise, predict class label O.
** Input: **
Natasha
Please give your output strictly in the following format:
Explanations: [Your step-by-step analyses and results]
Output:
[ONLY the integer class label; make uecussary assumptions if needed)
Please ONLY reply according to this format, don't give me any other words.
Forward Pass Output
Explanations: Veronica female name, predict class label
Output: 1
Many outputs for the rest data points
Forward Pass Output
Explanations: The input term 'Autonic' is a male name, and does not contain any keywords related to family relationships. It does not end with suffix commonly associated with male names, but is clearly male name.
Output: 0
Training Accuracy
1.0
Optimization Prompt
You the optimizer for model, your goal is to learn the best descriptions for the model. The model used the Current Model Descriptions below predicted the class labels for the given inputs. You are given the target labels, please optimize the Model Deseriptions for better prediction.
Inputs (a batch of i.id. text):
|'Natasha', 'Veronica', 'Dawn', 'Mamic', 'Darren; ; 'Monique', 'gynecologccal' 'Antonio', 'Enla', 'Henry'] ** Current Model Deseriptions:
You are designed to binary classification. The input is term: you need output the class label, L.e,, an integer the set 0, the input term ends with suffix commonly associated with male names (e.g.. -son, -er), predict class label 0. If the iuput term contains keywords related to family relationships (e.g,, "father', ; 'mother', 'grand", "paternal", "maternal") ) prodict class label O. If the imput label 1. If the iuput term is a name that does not clearly indicate gender (e.g., Leslie, Jordau), predict class label 0. If the input term a non-name torm clearly related females (n-g., "sorority"; ; "girl"), predict class label 1. Otherwise, predict class lahel O.
is fomale name (e-g., Erma, Ada, Faye, Lydia, Ida, Jane, Marlene, Marissa, Adeline, Anne, Maria), predict class
** The model predictions:
[[]] [i] [1] [1] [0] [1] [1] [0] [1] [o]]
** The targets:
[[] [1] [1] [i] [o] [1] [1] [o] [i] [o]]
If the model is doing well, you keep using the current descriptions. However, if the model not performing well, please update
the model by improving the 'New Model Descriptions', which should have lower claasiication error both on the current and the next
batch i.i.d. data. If previous 'Optimization Stop' are provided, you can use information from your optimisation stop helpful. Please think step by step and give your outputa strietly in the following format:
Reasoning:
be explicit and verbose; improve the Current Model Deseriptions by yourself; please show your work; note that you don't have accrss to computers
New Model Descriptions:
[put your new decision rules here]
Please ONLY reply according to this format, don't give me any other words. Optimization Output
Reasoning:
Upon examining the current model predictions and target labels, notice that the model performing well, but there is no need for further improvemeut since all predictions match the target labels. The current descriptions are compreheusive and accurate, covering various cases such as male and female names, non-name terms related to females or males, and names that do not clearly indicate gender. Since the model is slready performing perfectly, will not make any changes the descriptions. New Model Descriptions:
If the input term ends with suffix commonly associated with male names (e-g.--oon, -nr), predict class lahel O. If the input contains keywords related to family relationships (e.g., "father', "mother", 'grand", "paternal', 'maternal"), ), predict class label O. the input term
female name (e.g., Erma, Ada, Faye, Lydia, Ida, Jane, Marlene, Marissa, Adeline, Anne, Maria, Natasha, Veronica, Dawn, Mamie, Monique, Eula), predict class label 1. If the input term is name that does not clearly indicate gender (e.g.. Leelie. Jordan), predict class label 0. If the input term is a non-name term clearly related to females (e.g., "sorority", "giel", "yyecologgcall", ) predict class label 1. Otherwise, predict class label U. Updated Theta
If the input term enda with suffix commonly associated with male names (o-g-, -er), predict class label O. If the input contains keywords related to family relationships (e.g.. 'father', "mother', 'grand", "paternal', 'msternal"), ) predict class label O. the input term
female name (e.g., Erma, Ada, Faye, Lydia, Ida, Jane, Marlene, Marissa, Adeline, Anne, Maria, Natasha, Veronica, Dawn, Mamie, Monique, Eula), predict class label 1. If the input term is name that does not clearly indicate gender (e.g., Leslie, Jordan). predict class label U. If the input term is a non-name term clearly related to females (e.g., "sorority", 'giel", 'gynecological") ) predict class label 1. Otherwise, predict class label O.
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Training History: Text Classification
Step
Epoch Step
Prompt
model.
** Model Descriptione:
Yon designed to do binary clarstfication. The
need output the chuse label, Le., intager in the set 0, If the iapus term the
term contains keywards relased family rolationships to.g. 'Tather", "mother"" "grand", "paternal", "maternal'), predict class label o, If the iaput Serm is fouale name (oog. Brma, Ada, Fayo, Lydia, Ida, Jane, Marlene, Marissa, Adeline, Anne, Maria, Natacha, Veronica, Dawe, Mamin, Moniqqee Eula], prediet clase label I. the input term name that done not clearly indiaate gender (e Lealin, Jordan), prodict class labal D. If the in put term ls a son-name term clearly related to females. (eig. "sererityy "sitl", "kyaecological") ". predict clasa label I, Otherwire, predies class label o. Input:
ende with auffi commonly ssrociated with male namer (e.g, -et) predict class label
Tanny
Please give your output scrietly in the following format:
Explanations: [Your step-by-step analyces and resulte]
(ONYY
lakel: make accesurry ssampticn accded)
Please ONLY roply according to this focmat,, don't any other words.
Forward Pass Output
Explanations: The input torm 'Batsy' is fomale Outpot: I
Many outpute for the rest data points
Forward Poss Output
Eeplazations: The ippus term is 'brethran'. It does not end with a suffix commonly associated with male names, nor does it contata losyaards related to family relatiomhipp. It not female name, and does not cleatly indicate gender. However, term related to group of males (brothers), so it more ussoiiated with males than femalex.
Outpot:
Training Accuracy
1.0
Optimisation Prompt optimiser
goal learn the best descriptions for the model. The model used the Current Model Descriptions below predicted the cinss labela for the given inputs. You given the turget labela, please optimise the Model Descriptions for better prediction. ** Imputs (a batch of Li.d. text): **
You
model,
(Daanny' ; 'hatty. 'Calinn, 'Janat', 'Allison', "missis", 'Trandon', 'Carl', 'Ben', "hrethren!] Current Modal Descriptions;
You are designed to do hinary chasification, The input is a term; you need to output the clase label, Le., an integer in the set D, 1, If the impus term ends with sullix commonly associated with male names (e.g, -500, set), prediet clase label O. If the joput term contaios woywords relased to family velationehips [eig., "Iather", "mother", "grand", "paternll" "maternal"), prediet claes label D. the input serm is a female name (evg.. Brma, Ads, Paye, Lydia, Ida, Jane, Marlene, Marisma, Adeline, Anne, Maria, Naturha, Veronica, Dawn, Mamie. Monique, Euls), predict class label 1. name does not eleatly indicate gender (e.k. Lealie, Jordun), predict cluso label the isput term is a aon-nsme term clearly reluted to females (e.g.. "sorority" "girl" "kyaccological"" ). predict claso label 1. Otherwise, predies cluss label O. ax"
isput term
[[o] [t] we
[9] [0] [o]]
The
[[o] [o] [1] 3 [1]
[0] [o]]
keep using the current descriptione. Howwver, the model is not performing well, pleaen update the model by improving the 'New Model Deseripsions, which shculd have lower classitcation crvor both the curreal and the next butch of Lid, data, If previou OOlimization Step' are provided, you oan use the information from your last optimization stop it it's helpfall Please Chink step by step and give your outputs stricily in
model te doing well,
the following format:
Reasoning:
be explicit and varhore, Impoove the Current Model Descriptions by yourself please showe your work; note that you don't have access to computerre] New Model Deveriptione:
[put your new deciston ralez hare]
Please ONLY roply according Optimisation Output
tormat, don't give me any other words.
Reasoning:
The current descriptione model predictions and comprahendive and socurate, covering varions mains, and namer that do clearly indicata gander. However, would like update the famale namer lust to include the new female names ancoustered in this bateh, which are "Betty', "Janet', and 'Allisou'. Additionally, I nolice that the term 'minsis' is a non-name term related to temales, no will add it to the list of non-name serms, Since the term 'brethren'' is nom-name term related to males, I will add it to the list of non-name terms as well. New Model Deceriptions:
eaamining
labela, notice thas the model performing well, with all predictions matching she targel label.
auch male and female
non-name terme relisted famalne
If the impal terai enda with a suflix commonly usscciated with male name (exk. -oo, -er), prodict class label o. If the input term contains waydards related to fumily relationalips (ex.. "fatber', 'motter'', "grand", , "paterual", "materual"), ) prodict cluss label O. If the iaput term is femule nome (c Bema, Ads, Paye, Lydis, Ids, Jane, Marlene, Marissa, Adaline, Anne, Maris, Natadha, Verontea, Down, Mamie, Monique, Rals, Besty, Janet, Allieon); predict clase label I.
ipput term non-name term clearly related to females (t.g. "sorority". 'girl" "synecological" "misatz") ). predict class labal 1. the input term is a non-name term clearly related malen g. "brethren"), prodict class labe) D. Otherwise, predict class labe) D.
name that does not cleary indicate gender (eig., Lestie, Jordan), prediet class label 0. the input term
Updated
prodict class label ussciated with "paterual", 'matermal"), prodict cluss label If the iaput suflis 'falber', 'mother', predict class label 1. If the Jane, Marleue Macisse, Adelise, Aune, Marin, Nalaclan, Veronica, Dawn,
waywarda
related family relationahipa
Fulo, Jeaude Allison), that does not clearly indicate gender (e.g.. Lestin, Jordan), prediet class label O. If the term females (e.x.. "sorority", "giri", "synecological", "misatz)), predict class labal 1. If the input term non-name term clearly related to maten (e (e.gi, "breshren"), ") predict class labe) D. Otherwise, predict class labe) 0.
Eemo, Ado, Faye, dydie,
term non-name term elearly related to
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Published in Transactions on Machine Learning Research (01/2025)
M.8 Medical Image Classification (GPT-4o with prior)
Training History: Medical Image Classification (with Prior)
Step 1
Epoch Step
Forward Pass Prompt
You are the model.
** Model Deseriptions:
You are designed to do binary classification. The input is an image; you need to output the class label, Le., an integer in the set O, 1.
The imput X-ray image identifying paeumonia.
Input:
the given image.
Please give your output strictly in the following formas:
Explanations: (Your step-by-step analyses and results]
Output:
[ONLY the integer class label; make uecessary assumptions if needed]
Please ONLY reply according to this format, don't give me any other words. Forward Pass Output
Explanations:
The X-ray image shows signs of lung opacity and consolidation, which indicative of pneumonia. The presence of these features suggests an abnormality consistent with pneumonia.
Output:
Forward Pass Output
Explanations:
The X-ray image shows signs of lung opacity and consolldation, which are indicative of pneumonia. The presence of these abnormallties suggests an infection in the lungs.
Output:
Many outputs
data points
Forward Pass Output
Explanations:
The X-ray image shows the thest area. analyzed the lung fields for any signs of opacity, consolidation, or other abnormalities that indieative of pneumonia. The image appears to have clear lung fields without signifieant opacities consolidations.
Output: 0
Optimization Prompt
You are the optimizer for model, your goal is to learn the best descriptions for the model. The model used the Current Model Descriptions below predicted the class labela for the given images. You are given the target labela, please optimize the Model Descriptions for better prediction.
Inputs (a batch of i.i.d. images):
See the given
Current Model Descriptions:
binary classification. input image; you need output label, i..., an integer 0, The input is X-rey image for identifying pneumonia.
designed
** The model predictions: **
[[]] [1] [0] [1] [1] [0] [1] [1] [1]]
The targets:
[[o] [o] [0) [1] [0] [0] [o] [0] [0]]
If the model is doing well, you can keep using the current descriptions. However, if the model is not performing well, please update the
lower classification error on the current and the next batch of Li.dd data. If previous Optimization Step' are provided, you can the information from your last optimization step if it's helpful. Limit your 'New Model Descriptions" to be less than 100 words! Please think step by step and give your outputs strictly the following format:
model by improving upon
'Current Model Descriptions', which should
Reasoning:
[be explicit and verbose, update the Current Model Descriptions by youreelf; please show your work and use the features in the images; note that you don't have access to computers] New Model Descriptions:
[put your new decision rules here, the rules MUST based on the features in the imput imagee; make sure it is clear when to output which class labels)
Please ONLY reply according to this format, don't give any other words. Optimization Output
Reasoning:
The current model descriptions fail to adequately differentiate between pneumonis and non- nau-puemmonia cases leading to incorrect predictions. Upon analyzing the images, observed key indicators such as lung opacity, consolidations, and the presence of fluid. Enhanerd model descriptions should include these visual features for improved accuracy. New Model Descriptions:
The imput X-ray image. To classify:
Output 1 (Pueumoniaa) if lung upacity or consolidations are present, or if there is fluid in the lungs. Output (No Paeumonis) lungs appear clear without significant opacity or fluid presence. Updated Theta
The input an X-ray image. To classify:
Output (Pneumonia) lung opacity consolidations present, there fluid the lungs. Output 0 (No Pneumonis) if lungs appear clear without significant opacity or fluid presence.
103

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Published in Transactions on Machine Learning Research (01/2025)
Training History: Medical Image Classification (with Prior) Step 2
Epoch Step
Forward Pass Prompt
You the model.
Model Descriptions:
The imput
X-ray image. To classify:
Output 1 (Pucumonia) if lung opacity or consolidations are present, or if there is fluid in the lungs.
Output (No Pueumonia) hungs appear clear without significant opacity fluid presence.
Input: **
See the given image.
Please give your output strictly in the following format:
Explanations: [Your slep-by-step analyses and results]
Output:
[ONLY integer class lahel; necrssary assumptions if needed]
Please ONLY reply according to this format, don't give me any other words.
Forward Pass Output
Explanations:
The X-ray image clear lung fields without significant opacity or fluid presence. There are no visible consolidations or sigus of lung opacity.
Output: Forward Pass Output
Explanations:
The X-ray image shows clear lung fields without significant opacity or fluid presence. There no visible consolidations signs of lung opacity.
Output:
Many outputs for the rest data points
Forward Pass Output
Explanations:
The X-ray image shows significant lung opacity and consolidations, indicating the presence of fluid in the lungs. Output:
Optimization Prompt
You are the optimizer for a model, your goal is to learn the best descriptions for the model. The model used Current Model Descriptions below predicted the class labels for the given images. You are given the target labels, please optimize the Model Descriptions for better prediction.
** Inputs batch of i.i.d. images): *
See the given images.
** Current Model Descriptions:
The input
X-ray image. To classify:
Output 1 (Pneumonis) lung opacity or consolidations are present, or there fluid the lungs. Output (No Pueumonia) if lungs appear clear without significant opacity fluid presence.
The model predictions:
[[]] [0] [0] [1] [U] [1] [1] [1]]
The targets:
[[] [1] [o] [0] [o] [0] [1] [1] [1] [I]]
If the model doing well, you can keep using the current descriptions. However, if the model is not performing well, please update the
model by improving upon the 'Current Model Descriptions', which should results in lower classification error on both the current and the next batch of i..dd data. If previous Optimization Step' provided, you the information from your last optimization step if it's helpful. Limit your 'New Model Descriptions' to be less than 100 words! Please think step by step and give your outputs strictly the following format:
Reasoning:
[be explicit and verbose, update the Current Model Descriptions by yourself; please show your work and use the features in the images; note that you don't have access to computers]
New Model Descriptions:
[put your new decision rules here, the rules MUST based on the features in the imput image; make sure it is clear when to output which class labels)
Please ONLY reply according to this format, don't give me any other words. Optimization Output
Reasoning:
The current model descriptions still leading early sigus of paemmonia. Addition of more granulated sigus of pneumonia such as patchy infiltrates, mild interstitial markings or signs of hyperinflation can improve the performance. Also noting bilateral versus unilateral presentations can a useful signal. New Model Descriptions:
misclassifications, particularly it underestimates that show very mild
The input an X-ray image, To classify:
OOtput 1 (Pueumonia) lung opacity, consolidations, fluid, bilateral infiltrates, or mild interstitial markings present. Output (No Pneumonia) lungs appear clear without significant opacity, flnid presence, or other mentioned indicators of pneumonia.
Updated Theta
input X-ray image. To classify:
Output 1 (Pneumonis) if lung opacity, consolidations, fluid, bilateral infiltrates, or mild interstitial markings are present. Output (No Pneumonia) lungs appear clear without siguificant opacity, fluid presence, or other mentioned indicators of pacumonia.
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Published in Transactions on Machine Learning Research (01/2025)
Training History: Medical Image Classification (with Prior) Step 3
Epoch Step
Forward Pass Prompt
You the model.
Model Descriptions:
The input X-ray image. To classify:
Output 1 (Pucumonia) if lung opacity, consolidations, fluid, bilateral infiltrates, or mild interstitial markings are present.
Output (No Pneumonia) lungs appear clear without significant. opacity, fludd presence, or other mentioned indicators of
paeumenis.
Input:
See the given image.
Please give your output strictly in the following format:
"*
Explanations: |Your step-by-step analyses and results]
Output:
[ONLY the integer class label; make uecessary assumptions if needed]
Please ONLY reply according to this format, don't give any other words.
Forward Pass Output
Explanations:
X-ray image shows clear lung fields without significant opacity, consolidations, fluid, bilateral infiltrates, or mild interstitial markings.
Output:
Forward Pass Output Explanations: The X-ray image shows significant lung opacity and consolidations in both lungs. There are also signs of fluid presence and bilateral infiltrates.
Output:
Many outputs for the rest data points....
Forward Pass Output
Explanations: The X-ray image shows significant lung opacity and consolidations. There are also indications of fluid presence and bilateral infiltrates.
Output:
Optimization Prompt
You are the optimizer for model, your goal is to learn the best descriptions for the model. The model used the Current Model Descriptions below predicted the class labela for the given images. You are given the target labela, please optimize the Model Descriptions for better prediction.
Inputs (a batch of I.I.d. images):
the given images.
Current Model Descriptions:
input X-ray image. To classify:
Output 1 (Pneumonis) if lung opacity, consolidations, fluid, bilateral infiltaaes, or mild interstitial markings are present.
Output (No Pneumonia) lungs appear clear without siguificant opacity, fluid presence, or other mentioned indicators of paeumonia.
The model predictions:
[[] [1] [0] [1] [1] [1] [0] [2]]
The targets:
[[] [I] [o) [0] [o] [0] [o] [1] [0] [I]]
If the model doing well, you can keep using the current descriptions. However, the model is not performing well, please update the model by improving upon the 'Current Model Descriptions', which should results in lower classification error on both the current and the next batch of i..dd data. If previous Optimization Step' provided, you the information from your last optimization step if it's helpful. Limit your 'New Model Descriptions' to be less than 100 words! Please think step by step and give your outputs strictly the following format:
Reasoning:
[be explicit and verbose, update the Current Model Descriptions by yourself; please show work and use the features in the images; note that you don't have access to computers]
New Model Descriptions:
[put your new decision rules here, the rules MUST based on the features in the imput image; make sure it is clear when to output which class labels) "*
Please ONLY reply according to this format, don't give me any other words. Optimization Output
Reasoning:
The current descriptions are leading to over-identiffcation pueumonia (more false positives). Incorporating the presence of specific signs and ruling out other conditions be helpful. Adding features like the presenee of pleural effusion, absemee of heart enlargement, and noting whether the opacity is localized (can be other pathologies) may help. Precision in observing these smaller details will aid classification accuracy.
New Model Descriptions:
The input an X-ray image, To classify:
Output 1 (Pueumonia) if lung opacity, consolidations, fluid, pleural effusion, bilateral infiltrates, mild interstitial markings
present and not associated with heart enlargement.
Output 0 (No Pneumonia] lungs appear clear without significant opacity, fluid presence, pleural effusion, other mentioned indicators of pneumonia, or if opacities are localized and not typical for pueumonia.
Updated
The input
X-ray lmagge. To classify:
Output 1 (Pneumonis) lung opscity, consolidations, fluid, pleural effusion, bilateral infiltrates, or mild interstitial markings are present and associated with heart enlargement.
Output 0 (No Pneumonia) lungs appear clear without significant opacity, fluid presence, pleural effusion, or other mentioned indicators of pneumonia, opacities localized and typical for poeumonia.
105

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Published in Transactions on Machine Learning Research (01/2025)
Training History: Medical Image Classification (with Prior) Step 4
Epoch Step
Forward Pass Prompt
You the model.
Model Descriptions:
The imput
X-ray image. To classify:
Output 1 (Puemmonis) if lung opacity, consolidations, fluid, pleural effusion, bilateral infiltrates, or mild interstitial markings are present and assoriated with heart enlargement.
Output 0 (No Pueumonia) if lungs appear clear without significant opacity, fluid presence, pleural effusion, other mentioned indicators of pnemmonia, or opacities are localized and not typical for pucumonnia.
** Inpuu: **
See the given image.
Please give your output strictly the following format:
Explanations: (Your step-by-step analyses and results]
Output:
[ONLY the integer class label; make necrssary assumptions if needed]
Please ONLY reply according to this format, don't give me any other words. Forward Pass Output
Explanations: The X-ray image shows significant lung opacity and consolidations in both lungs. There is no clear indication of heart enlargement. The presence of
features is consistent with pneumonia.
Output:
Forward Pass Output
The X-ray image shows significant lung opacity and consolidations. There also signs of fluid presence and bilateral infiltrates. These findings are consistent with pneumonia and are not associated with heart enlargement.
Explanations:
Output:
Many outputs for the rest data points ...
Forward Pass Output
Explanations: The X-ray image shows clear lungs without significant opacity, fluid presence, pleural effusion, other indicators of pueumonia. There are no visible lung opacities, consolidations, or bilateral infiltrates.
Output:
Optimization Prompt
You are the optimizer for a model, your goal is to learn the best descriptions for the model. The model used Current Model Descriptions below predicted the class labela for the given images. You are given the target labels, please optimize the Model Descriptions for better prediction.
** Inputs batch of i.i.d. images): **
See the given images.
** Current Model Descriptions **
The input
X-ray image. To classify:
Output (Pnemmonis) lung opscity, consolidations, fluid, pleural effusion, bilateral infiltrates, or mild interstitial markings are present and associated with heart enlargement.
Output 0 (No Pneumouia) if lungs appear clear without significant opacity, fluid presence, pleural effusion, or other mentioned indicators pneumonia,
opacities localized typical for pueumonia. The model predictions:
[[0] [o] [1] [i] [1] [1]["] [1] [1] [o]]
The targets:
[o] [o] [0] [0] [1] [1] [1] [0] [1] [0]]
If the model doing well,
keep using the current descriptions. However, the model not performing well, please update the model by improving upon the 'Current Model Descriptions', which should results in lower classification error on both the current and
batch of i.id. data. If previous 'Optimization Step' are provided, you can the information from your last optimization step if it's helpful. Limit your 'New Model Deseriptions' to be less than 100 words! Please think step by step and give your outputs strictly the following format:
Reasoning:
be explicit and verhose, update the Current Model Descriptions yourself; please show your work and use the features the imagee; note that you don't have access to computers]
New Model Descriptions:
[put your new decision rules here, the rules MUST based on the features in the input image; make sure it is clear when to output which class labels)
Please ONLY reply according to this format, don't give me any other words. Optimization Output
Reasoning:
criteria, focus on more specific: patterns indicative pneammnia lobar consolidation or "ground-glass" opacities. Exclude cases where opacities are likely due to external factors such as overlying soft tissue shadows or slight varistions that are non-dlagnostic. New Model Descriptions:
The model produces positives. To
The input is an X-ray image. To classify:
Output (Pneumonia) if clear indications of lobar consolidation, ground-glass opacities, fluid presence (excluding pleural effusion). bllateral infiltrates, or moderate interstitial markings are present and not related to heart enlargement.
Output 0 (No Pneumonia) lung fields appear clear without significant. opacity, fluid, hilateral infiltrates, other paeumonia. indicators, if features suggest alternative explenations (e.g., overlying soft tissue shadows).
Updated Theta
The imput X-ray image. To classify:
Output (Pneumonia) if clear indications of lobar consolidation, ground-glase opacities, fluid presence (excluding pleural effusion), bilateral infiltrates, moderate interstitial markings present and related to heart enlargement. Output 0 (No Pneumouis) if lung fields appear clear without eignificant opacity, fluid, bilateral infiltrates, or other pneumonia indicators, features suggest alternative explanations (e.g,, overlying soft tissue shadows).
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Training History: Medical Image Classification (with Prior) Step 5
Epoch Step
Forward Pass Prompt
You the model.
Model Descriptions:
The input
X-ray image. To classify:
Output 1 (Pneumonia) if clear indicatious of lobar consolidation, ground-glass opacities, fluid presence (excluding pleural effusion); bilateral infiltrates, or moderate interstitial markings are present and related to heart enlargement.
Output 0 (No Pueumonia) if lung fields appear clear without significant opacity, fluid, bilateral infitrates, other pneumonis indicators, or if features suggest alternative explanations (e.g., overlying soft tisse shadows).
** Inppu: **
See the given image.
Please give your output strictly in the following format:
Explanations: [Your step-by-step analyses and results]
Output:
[ONLY the integer class label; make necrssary assumptions if needed]
Please ONLY reply according to this format, don't give me any other words.
Forward Pass Output
Explanations: The X-ray image shows clear indications of bilateral infiltrates and moderate interstitial markings. There are no signs heart enlargement that could explain these features.
Output:
Forward Pass Output
Explanations: The X-ray image shows clear lung fields without significant opecity, fluid presence, bilateral Infiltrates, or other indicators of pneumonia. There are no clear signs of lobar consolidation, ground-glass opacities, or moderate interatitial markings. Output:
Many outputs for the rest data points
Forward Pass Output
Explanations: The X-ray image shows clear indications lobar consolidation and ground-glass opacities the lung fielde, These fiudings are consistent with pueumonia.
Output:
Optimization Prompt
You are the optimizer for a model, your goal is to learn the best descriptions for the model. The model used Current Model Descriptions below predicted the class labels for the given images. You are given the target labels, please optimize the Model Descriptions for better prediction.
** Inputs batch of i.i.d. images): *
See the given images.
** Current Model Descriptions: **
The input
X-ray image. To classify:
Output (Pneumonis) if clear indications of lobar consolkdation, ground-glass opacities, fluid presence (excluding pleural effusion), bilateral tufiltrates, moderate interstitial markings present and not related to heart eulargement.
Output 0 (No Pueumonis) if lung fields appear clear without eignificant opacity, fluid, bilateral infiltrates, or other pneumonis indicators,
features suggest alternative explanations (e.g., overlying soft tissue shadows). The model predictions:
[[o] [o] [1] [0] [1] 11 [o] [0, [I]]
** The targets:
[]] [o] [1] [0] [1] [0] [o] [0] [1] [A]]
If the model doing well,
keep using the current descriptions. However, the model not performing well, please update the model by improving upon the 'Current Model Descriptions', which should results in lower classification error on both the current and
batch of i.id. data. If previous 'Optimization Step' are provided, you can the information from your last optimization step if it's helpful. Limit your 'New Model Deseriptions' to be less than 100 words! Please think step by step and give your outputs strictly the following format:
Reasoning:
be explicit and verhose, update the Current Model Descriptions yourself; please show your work and use the features the imagee; note that you don't have access to computers]
New Model Descriptions:
[put your new decision rules here, the rules MUST based on the features in the input image; make sure it is clear when to output which class labels)
Please ONLY reply uccording to this format, don't give me any other words. Optimixation Output
The current model descriptions are missing some complexities encountered in the images. We observe some misclassifications indicating need differentiate between the types opacities more clearly. Including signs such as bronchograms pattern of unilateral versus bilateral presentation can aid in improving the description. New Model Descriptions:
Reasoning:
The imput
X-ray image. To classify:
Output 1 (Pnemmonia) if there are signs of lobar consolication, ground-glass opacities, air bronchograms, fluid presence (excluding pleural effusion), bilateral infiltrates,
moderate interstitial markings. typical pneumonia pattorn,
Output (No Pneumonia) if lung fields appear clear without significant opacity or fluld, if opacities are limited to a single lung and
are not
these indicators are due to overlying soft tissue shadows.
Updated Theta
The iuput an X-ray image. To classify:
Output (Pnemmonia) if there signs of lobar consolication, grouud-glass opacities, air bronchograus, fluid presence (excluding pleural effusion), bilateral infiltrates, or moderate interstitial markings.
Output (No Pneumonia) if lung fields appear clear without significant opacity fluld, opacities limited single lung and are not of the typical pueumonia pattern, or if these indicators are due to overlying soft tissue shadows.
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Training History: Medical Image Classification (with Prior) Step 50
Epoch Step 10
Forward Pass Prompt
You the model.
Model Descriptions:
The imput X-ray image. To classify:
Output 1 (Pueumonia) there are lobar consolidations, asyumetric or bilateral ground-glass opacities, air bronchograms, moderate significant alveolar fuid, usymmetric infiltrates, patchy inflammations, moderate severe pleural effusion, central peripheral acute opacities, air-fluid levels, increasing opacities, pueumatocele formations, rapidly changing patterns chronic lesions indicative of acute infoction, or persistent cough-based findings.
Output (No Pneumonia) if lung fields clear, minimal lncalixed fiuid, distinct acute opacities, stable fihrotic searring, solitary caleifications, old bealed lesions, stable localized non-infectious anomalies, chronic interstitial fibrosis, chronic reticulonodular patterus without
changes, stable chronic lesions without signs, purely non-persistent anomalies.
Input:
the given image.
Please give your output strictly in the following format:
Explanations: (Your step-by-step analyses and results]
Output:
[ONLY the integer class label; make uecessary assumptions if needed]
Please ONLY reply according to this format, don't give me any other words.
Forward Pass Output
Explanations:
The X-ray image shows Inbar consolidations and asymmetric ground-glass opacities. There are also visible air bronchograms and moderate alveolar fluid. These findings indicative of pneumonia.
Output:
Forward
Output
Explanations: The X-ray image shows clear lung fields without any significant opacities, consolidations, or fluid accumulations. There are no signs of air bronchograms, pleural effusion, or other acute changes indicative of pneumonia.
Output:
Many outputs the data points
Forward Output
Explanations: The X-ray image shows lobar consolidations and asymmetric ground-glass opacities. There are also visible air bronchograms and moderate alveolar fluid. These findings are indicative of pneumonia.
Output:
Optimization Prompt
You are the optimizer for model, your goal Descriptious below predicted the class labels for the given images. You are given the target labels, please optimize the Model Descriptions for better prodiction.
the best descriptions the model. The model used Current Model
Inputs (a batch of I.I.d. images):
See the given images.
Current Model Descriptions:
The input is an X-ray image. To classify:
Output (Pneumonia) if to significant alveolar fuid, usymmetric infiltrates, patchy inflammations,, moderate to severe pleural effusion, central or peripheral acute opacities, air-fluid levels, increasing opacities, pnermatocele formations, rapidly changing patterns chronic lesions indicative of scute infection, or persiatent cough-based findings.
lobar consolications, asymmetric bilateral grouud-glass opacities, bronchograms, moderate
Output (No Pueumonia) if lung fields are clear, minimal localized fluid, no distinct acute opacities, stable fibrotic scarring, solitary caleifications, old healed lesions, stable localized non-infectious anomalies, chronic interstitial fibrasis, chronic reticulonodular patterns without recent changes, stable chronic lesions without acute signs, or purely non-persisteut anomalies.
** The model predictions: **
[[]] [0] [o] [0] [1] []] [0] [0] [+]]
** targets:
[[1] [o] [0] [1] [0] [1] [o] [0] [1] [A]]
the model doing well, you can keep using the current descriptions. However, the model not performing well, please update the
model improving upon the 'Current Model Descriptions', which should results in lower classifiatiion error both the current and
the next batch of Li.d. data. If previous 'Otimization Step' are provided, you can use the information from your last optimization step if it's helpful. Limit your 'New Model Descriptions" be less than words! strictly in the following format:
step by step and give your outputs
Reasoning:
be explicit and verhose, update the Current Model Descriptions by yourself; please show your work and use the features the images; note that you don't have acress computers]
New Model Descriptions:
pput your drcision rules here, rules MUST based features the input image; make sure clear nutput which class labels)
Please ONLY reply according to this format, don't give any other words.
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Training History: Medical Image Classification (with Prior) Step 50 (continued)
Optimization Output
Reasoning:
The madel missod true of pneumonia and incorrectly identified non-pneumonia To improve the arcurncy. refine the criteria to better identify pneumonia even in less obvious cases and differentiate between chronic stable conditions and infections.
New Model Descriptions:
imput X-ray image. classify:
Output (Pneumonis) if there lobar consolicatious, asymmetric bilateral grouad-glass opacities, bronchograms, moderate to siguificant alveolar fuid, asymmetric infiltrates, patchy inflammations, moderate to severe pleural effusion, central or peripheral opacities, air-fluid levels, increasing opacities, pneumatocele formations, rapidly changing patterns chronic lesions indicative of scute infection, persistent cough-related findings, or recent oneet of ground-glass opacities. Output (No Pueumonia) if lung fields clear, minimal localized fluid, distinct solitary caleifications, old healed lesions, stable localized non-infectious anomalies, chronic interstitial fibrosis, chronic reticulonodular signs,
opacities, stable fibrotic scarring.
pastorns without rerent changes,
chronic
any acute infortion markers.
Updated Theta The input an X-ray image. To classify:
Output (Pneumonia) if there lobar consolidations, asymmetric bilateral ground-glass opacities, bronchograms, moderate to siguificant alveolar fiuid, saymmetric infiltrates, patchy iaflammations, moderate to severe pleural effusion, central or peripheral opacities, air-fluid of acute infection, persistent cough-related findings, or recett onset of ground-laass opacities. clear, minimal localised solitary calcifications, old healed lesions, stable localized non-infectious anomalies, chronic interstitial fibrasis, chronic reticulonodular pasterns without recent changes, stable chronic lesions without acute signs, or absence of any acute infection markers.
increasing opacities, pueumatocele formations, rapidly changing patterns chronic lesions indicative
Output (No Pneumonia) lung
no distinct acute opacities, stable fibrotic searring.
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M.9 Medical Image Classification (GPT-4o without prior)
Training History: Medical Image Classification (without Prior) Step 1
Epoch Step
Forward Pass Prompt
You are the model.
** Model Descriptions: **
You are designed to do binary classification. The input is an image; you need to output the class label, Le., an integer in the set O, 1.
** Input: **
See the given image.
Please give your output strictly in the following format:
Explanations: (Your step-by-step analyses and results]
Output:
[ONLY the integer class label; make necessary assumptions If needed]
Please ONLY reply actording to this format, don't give me any other words.
Forward Pass Output
The image appears to be chest X-ray, will analyze the image for abnormalities signs of disease that could indicate the class lsbel. Output:
Explanations:
Many outputs the data points
Forward Pass Output
Explanations: The image appears to be a medical X-ray of chest. The analyals involves exemining the festures and patterns in the image to determiue the class label. Based on the visual inspection, the image shows certain characteristics that align with class Output:
Optimization Prompt
You are the optimizer for model, your goal Descriptions brlow predicted the class lahels for the given images. You are given the target labels, please optimize the Model Descriptions for better prediction.
the best descriptions the model. The model used Current Model
** Inputs (a batch of 1.i.d. images):
See the given images.
** Current Model Descriptions: **
You are designed to do binary classification, The input is an image; need to output the class label, Le., an integer in the set U, 1.
** model predictions: **
[[o] [0] [1] [I] [1] [0] [9] [1] [X]]
** The targets:
[[o] [o] [0] [1] [o] [o] [1] [0] [0] [o]]
the model is doing well, you can keep using the current descriptions. However, the model not performing well, please update the
model improving upon the 'Current Model Descriptions', which should results in lower classification error both the current and
the next batch of Li.d. data. If previous 'Optimization Stop' are provided, you can use the information from your last optimization stop if it's helpful. Limit your 'New Model Descriptions' be less than 100 words! Please think step by step and give your outputs strictly in the following format:
Reasoning:
be explicit and verhose, update the Current Model Descriptions by yourself; please show your work and use the features the images; note that you don't have access computers]
New Model Descriptions:
[put your new decision rules hern, the rules MUST based the features the input image; make sure it clear when output which class labels)
Please ONLY reply according to this format, don't give any other words. Optimization Output
Reasoning:
The current model's performance indicates significant misclassification. To improve, must explicitly identify and use key visual features the images that are indicative of target classes. For instance, if classification relates medical diagnostics (e.g., chest X-rays for paeumonis), specific features like the presence of fluid, opacity, other anomalies must be emphasized. Similarly, non-medical classifications (typical objects versus atypical ones) could require ideutification of distinct shapes, colors, or textural features. Analysing the given images visually will guide the addition of pertinent descriptive rules that improve classification accuracy. New Model Descriptions:
If the image contains prominent, clearly defined geometric bodily structures and no umusual opacity anomalies, output the class label O.
the image shows abnormal features such as irregular shapes, unexpected opacitles, or non-unlform patterns that suggest pathology anomaly, output the class label 1. 3. When distinguishing medical images specifically, check for fluid accumulation, granular textures, or any signs of abnormality in the expected anatomy; if these are present, output the class lahel
Updated Theta
the image contains prominent, clearly defined geometric bodily structures and no umusual opacity anomalies, output the class label O.
the image shows abnormal features such as irregular shapes, unexpected opacities, or non-unlform patterns that suggest pathology anomaly, output the class label 1.
3. When distinguishing medical images specifically, check for fluid accumulation, granular textures, or any signs of abnormality in the expected anatomy: if these present, output the class lahel
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Training History: Medical Image Classification (without Prior) Step 2
Epoch 0 Step
Forward Pass Prompt
You the model.
Model Descriptions:
the image contains prominent, clearly deflned geometric bodily structures and unusual opacity anomalies, output the class label 0.
the image shows abnormal features such irregular shapes, unexpected opacities, or non-uniform patterns suggest pachology anomaly, output the class label 1.
3. When distinguishing medical images specifically, check for fluid accumulation, granular textures, or any signs of abnormality in the expected anatomy; if these present, output the class lahel
Input:
the given image.
Please give your output strictly in the following format:
Explanations: (Your step-by-step analyses and results]
Output:
[ONLY the integer class label; make necessary assumptions if needed]
Please ONLY reply according to this format, don't give me any other words.
Forward Pass Output
Explanations: The image shows irrogular shapes and unexpected opacities. There are non-uniform patterns that suggest an anomaly. Output:
Many outputs for the data points
Forward Output
Explanations: The image appears to be a medical X-ray. There are irregular shapes and unexpected opacities present in the lung area, which suggest pathology or anomaly.
Output:
Optimization Prompt
You are the optimizer for model, your goal Descriptious below predicted the class labels for the given images. You are given the target labels, please optimize the Model Descriptions for better prodiction.
the best descriptions the model. The model used the Current Model
Inputs (a batch of I.i.d. images):
See the given Images.
Current Model Descriptions:
1. If the image contains prominent, clearly defined geometric or bodily structures and no unusual opacity or anomalies, output the class label O.
2. If the image shows abnormal features such irregular shapes, unexpected opacities, or non-mmiform patterns that suggest pathology or anomaly, output the class label I.
3. When distinguishing medical images specifically, check for fluid accumulation, granular textures, or signs of abnormality in the expected anatomy; if these are present, output the class label 1.
** The model predictions:
[[]]] [i] [1] [i] [i] [1] [4] [1] [A] [A]]
** The targets:
[[]] [1] [o] [0] [o] [0] [1] [1] [1] [X]]
model doing well, you can keep using the current descriptions. However,
model not performing well, please update
model by improving upon the 'Current Model Descriptions', which should results in lower classification error on both the current and
the next batch of Li.d. data. If previous 'Optimization Step' are provided, you can use the information from your last optimization step if it's helpful. Limit your 'New Model Descriptions" to be less than 100 words! Please think step by step and give your outputs strictly in the following format:
Reasoning:
explicit verbose, update Current Model Descriptions by yourself; please show your work and use the features the images; note that you don't 't have access computers]
New Model Descriptions:
[put your decision rules here, the rules MUST based the features in the input image; make sure it is clear when output which class labels)
Please ONLY reply according to this format, don't give me any other words. Optimization Output
Reasoning:
The current model is too sensitive to features that are indicating anomalies, leading to many false positives. A refined approach would include granular analysis distinguish normal variations from actiaal Ihnormalities. Specifically, enhancing the detaction criteria to differentiate between minor variations and siguificant pathological markers is necessary. Reviewing more images, I will refine descriptions include clearer definitions significant anomalies versus natural structures.
New Model Descriptions:
the image contains clear, cousistent geometric or anatomical structures without significant opacities, Irregular patterns, or sigus of fluid, output the class label
If the image shows irregular shapes, significant opacity, or patterns that deviate considerably from typical anatomical structures, output the class label I.
3. Examine for fluid accumulation, granular textures, and other siguificant deviations in expected anatomy. Minor irregularities that do not suggest severe pathoingy should output class label 0, otherwise class lahel 1.
Updated Theta
the image contains clear, cousistent geometric or anatomical structures without siguificant opacities, irregular patterns, or sigus of fluid, output the class label
If the image shows irregular shapes, significant opacity, or patterns that deviate considerably from typical anatomical structures, output the class label 1.
3. Examine for fluid accumulation, granular textures, and other siguificant deviations in expected anatomy. Minor irregularities that do not suggest severe pathology should output class label 0, otherwise class label I.
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Training History: Medical Image Classification (without Prior) Step 3
Epoch Step
Forward Pass Prompt
You the model.
Model Descriptions:
the image contains clear, cousistent geometric anatomical without significant opacities, irregular patterna, sigus
of fluid, output the class label U.
the image
irregular shapes, significant opacity, or patterns
considerably from typical anatomical structures,
output the class label 1.
Examine for fluid accumulation, granular textures, and other significant deviations experted anatomy, Minor irregularities that do not suggest
patholngy should output class label otherwise class lahel 1.
Input:
the given image.
Please give your output strictly in the following format:
as
Explanations: (Your step-by-step analyses and results]
Output:
[ONLY the integer class label; make necessary assumptions if needed]
Please ONLY reply according to this format, don't give me any other words.
Forward Pass Output
Explanations: The image shows significant opacity and irregular patterns that deviate considerahly from typical anatomical structures. Output:
Many outputs for the data points
Forward Output
Explanations: The image shows clear and consistent geometric, and anatomical structures without significant opacities, irregular patterns, or signs of fluid. There are no significant deviations from typical anatomy.
Output:
Optimization Prompt
You are the optimizer for model, your goal Descriptious below predicted the class labels for the given images. You are given the target labels, please optimize the Model Descriptions for better prodiction.
the best descriptions the model. The model used the Current Model
Inpuus (a batch of I.I.d. images):
the given images.
Current Model Descriptions:
1. If the image contains cleat, consistent geometric or anatomical structures without significant opacities, Irregular patterns, or signs of fluid, output
class label
2. If the image shows irregular shapes, significant upacity, or patterus that deviate considerably from typical anatomical structures, output the class label
3. Examine for fluid accumulation, granular textures, and other significant deviations in expected anatomy. Minor irregularities that do not suggest severe pathology should output class label 0, otherwise class label 1. The model predictions:
[[]]] [i] [0] [i] [0] [1] [1] [1] [0] [A]]
** The targets:
[[]] [1] [o] [0] [0] [0] [0] [1] [0] [1]]
model doing well, you can keep using the current discriptions. However,
not performing well, please update
model by improving upon the 'Current Model Descriptions', which should results in lower classification error on both the current and
the next batch of Li.d. data. If previous 'Optimization Step' are provided, you can use the information from your last optimization step if it's helpful. Limit your 'New Model Descriptions' to be less than 100 words! Please think step by step and give your outputs strictly in the following format:
Reasoning:
explicit verbose, update Current Model Descriptions by yourself; please show your work and use the features the images; note that you don't 't have access computers]
New Model Descriptions:
[put: your decision rules here, the rules MUST based the features in the input image; make sure it is clear when output which class labels)
Please ONLY reply according to this format, don't give me any other words. Optimization Output
Reasoning:
The model is still producing numerous false positives. The current descriptions are good at detecting anomalies but not at distinguishing minor, non-critical irreglaarttes from significant pathological fentures. nord to refine the differentiation by including more specific anatomical markers and clearly delineating between slight structures and substantial irregulacities. New Model Descriptions:
1. If the image contains clear, consistent anatomical structures, free from significant opacities, masses, or signs of severe pathology. output the class label
If the image shows substantial opacity, masses, shapes siguificantly deviating from typical structures indicating pathology, output the class label 1.
Examine for clear signs fluid accumulation, large granular textures, significant abnormalities in anatomical structures. Small, inconsequential variations should output class label 0, otherwise class label 1.
Updated If the image contains clear, consistent snatomica] structures, free from significant opacities, masses, or signs of severe pathology, output the class label If the shows substantial opacity, masses, shapes siguificantly deviating from typical structures indicating pathology, output the class label 1.
Examine for clear signs fluid accumulation, large granular textures, significant abnormalities in anatomical structures. Small, inconsequential variations should output class label 0, otherwise class label 1.
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Training History: Medical Image Classification (without Prior) Step 4
Epoch Step
Forward Pass Prompt
You the model.
Model Descriptions:
the image contains clear, consistent anatomical structures, free from significant opacities, masses, or signs severe pathology,
output the class label D.
the image shows substantial opacity, masses, or shapes siguificantly deviating from typical structures indicating severe pathology,
output the class label 1.
Examine for clear signs fluid accumulation, large granular textures, or significant abnormalities in anatomical structures. Small, inconsequential variations should output class lahel otherwise class label I. Input:
the given image.
Please give your output strictly in the following format:
as
Explanations: (Your step-by-step analyses and results]
Output:
[ONLY the integer class label; make necessary assumptions if needed]
Please ONLY reply according to this format, don't give me any other words.
Forward Pass Output
Explanations:
The image shows substantial opacity and shapes significantly deviating from typical structures, indicating severe pathology.
Output:
Many outputs for the rest data points
Forward Pass Output
Explanations: The image shows clear, consistent anatomical structures without significant opacities, masses, or signs severe pathology. There clear signs of fluid accumulation or large gramular textures.
Output:
Optimization Prompt
You are the optimizer for a model, your goal is to learn the best descriptions for the model. The model used the Current Model Descriptions below predicted the class labels the given images. You are given target labels, please optimize the Model Descriptions for better prediction.
** Inputs (a batch of I.I.d. images):
See the given images.
Current Model Descriptions:
the image contains clear, cousistent anatomical structures, free significant opacities, masses, or sigus severe pathology, output the class label D.
the image shows substantial opacity, massrs, or shapes significantly deviating from typical structures indicating severe pathology, output the class label 1.
Examine for clear signs fluid sccumulation, large granular textures, or significant abnormalities in anatomical structures. Small, inconsequential variations should output class label otherwise class label 1.
The model predictions:
[[0] [1] [0] [1] [1][1] [1] [0] [o]]
The targets:
[[o] [o] [o] [0] [I] [1] [o) [0]]
If the model doing well, you can keep using the current descriptions. However, if the model is not performing well, please update the
model by improving upon the 'Current Model Descriptions', which should results in lower classification error on both the current and the next batch of f..dd data. If previous Optimizatiion Step' provided, you the information from your last optimization step if it's helpful. Limit your 'New Model Descriptions' to be less than 100 words! Please think step by step and give your outputs strictly the following format:
Reasoning:
[be explicit and verbose, update the Current Model Descriptions by yourself; please show your work and use the features in the images; note that you don't have access to computers]
New Model Descriptions:
[put your new decision rules here, the rules MUST based on the features in the imput image; make sure it is clear when to output which class labels)
Please ONLY reply according to this format, don't give me any other words. Optimization Output
Reasoning:
The model is generating several false positives, indirating may he conservative and is mistakenly identifying minor irrgularittes as significant pathology. The descriptions need to further clarify the distinction between mild and serious irregularities. Adding specific attention to the size, shape, and delineation opacities and anatomical deviations might improve acruracy. New Model Descriptions:
If the image displays clear, consiatent anatomical structures without significant opacities, masses, or deviations, output the class label U.
If the image highlights substantial, well-defined opacities or masses and shapes deviating significantly from normal structures pathology, output the class label 1.
indicating
3. Check for large, well-delinested fluld accumulations, substantial granular textures, or major abnormalities in anatomy. Minor, non-critical variations should output class 0, otherwise class label I. Updated Theta
If the image displays clear, consistent anatomical structures without significant opacities, masses, or deviations, output the class label O.
2. If the image highlights substantial, well-defined opacities or masses and shapes deviating significantly from normal structures indicating
pathology, output the class label 1.
3. Check for large, well-delineated fluid accumulations, substantial granular textures, or major abnormalities in anatomy. Minor, non-critical variations should output class 0, otherwise class label I.
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Training History: Medical Image Classification (without Prior) Step 5
Epoch Step
Forward Pass Prompt
You the model.
Model Descriptions:
the image displays clear, consistent anstomical structures without significant opacities, masses, deviations, output the class label U.
image highlights substantial, well-defined opacities or masses and shapes deviating significantly from normal structures indicating severe pathology, output the class label 1.
Check for large, well-delineated fuuid accumulations, substantial granular textures, or major abnormalities in anatomy. Minor; non-critical variations should output class label O otherwise class label I.
Input:
the given image.
Please give your output strictly in the following format:
as
Explanations: (Your step-by-step analyses and results]
Output:
[ONLY the integer class label; make necessary assumptions if needed]
Please ONLY reply according to this format, don't give me any other words.
Forward Pass Output
Explanations: The image displays clear, consistent anatomical structures without significant opacities, masses, or deviations. Output:
Many outputs for the data points
Forward Pass Output
Explanations: The image displays clear, consistent anatomical structures without signifirant oparities, masses, deviations. There are no large, well-delineated fluid accumulations, substantial granular textures, or major abnormalities in anatomy. Output:
Optimization Prompt
You are the optimizer for model, your goal Descriptious below predicted the class labels for the given images. You are given the target labels, please optimize the Model Descriptions for better prodiction.
the best descriptions the model. The model used the Current Model
Inputs (a batch of I.i.d. images):
See the given images.
** Current Model Descriptions:
1. If the image displays clear, consistent anatomical structures without significant opacities, masses, or deviations, output the class label
2. If the image highlights substautial, well-defined opacities or masses and shapes deviating siguificantly from normal structures indicating severe pathology, output the class label 1.
3. Check for large, well-delineated fluid accumulations, substantial granular textures, major abnormalities in anatomy. Minor, non-critical variations should output class label 0, otherwise class label 1.
** The model predictions:
[[]] [0] [0] [0] [i] [0] [0] [0] [0] [A]]
** The targets:
[[]] [o] [1] [0] [1] [0] [0] [0] [1] [X]]
the model doing well, you can keep using the current discriptions. However,
model not performing well, please update
model by improving upon the 'Current Model Descriptions', which should results in lower classification error on both the current and
the next batch of Li.d. data. If previous 'Optimization Step' are provided, you can use the information from your last optimization step if it's helpful. Limit your 'New Model Descriptions" to be less than 100 words! Please think step by step and give your outputs strictly in the following format:
Reasoning:
explicit verbose, update Current Model Descriptions by yourself; please show your work and use the features the images; note that you don't 't have access computers]
New Model Descriptions:
[put your decision rules here, the rules MUST based the features in the input image; make sure it is clear when output which class labels)
Please ONLY reply according to this format, don't give me any other words. Optimization Output
Reasoning:
The model still misclassifies some images, both false positives and false negatives. We need to be more precise while idcutifying anomalies, forusing substantial deviations such as the density of opacities masses and their impact surrounding structures. Including the context of tissue appearance and location could help. New Model Descriptions:
1. If the image shows clear, consistent anatomical structures without significant opacities, masses, deviations, ousput class label 0.
If the image highlights substantial, well-deflued opacities or masses and shapes with significant density, deviating significantly from normal structures, output class label 1.
3. Check for large, well-delineated fluid accurulations, substantial gramular textures, or major abnormalities in anatomy. Minor; non-critical variations should output class label 0, focusing also the context and location of tissues, otherwise class label I.
Updated Theta
the image shows clear, consistent anatomical structures without significant opacities, masses, deviations, output class label
2. If the image highlights substantial, well-defined opacities or masses and shapes with significant denaity, devisting significantly from normal structures, output class label 1.
3. Check for large, well-delineated fluid accumulations,, substautial grauulor textures, major abnormalities anstomy. Minor, non-critienl variations should output class label 0, focusing also on the context and loration of tissues, otherwise class label 1.
114

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Published in Transactions on Machine Learning Research (01/2025)
Training History: Medical Image Classification (without Prior) Step 50
Epoch Step 10
Forward Pass Prompt
You the model.
Model Descriptions:
anatomical structures, uniform texture, symmetry, and normal lung fields without visible opacities, well-defined masses, streaks, irregular granular patterns, or any deviations, output class label O.
If the image shows
image shows any distinct opacitics masses, interstitial patterns, visible fluid accumulations, asymmetry, irregular heart shapes, multiple small opacities, output class label 1.
For granular or streaky opacitics over any lung region, indistinct or uneven lung fields, subtle masses, minor deviations in symmetry, any lung irigglaaritiss output class lahel 1. For minimal irregularities slight deviations, output class label
Input:
the given image.
Please give your output strictly in the following format:
Explanations: (Your step-by-step analyses and results]
Output:
[ONLY the integer class label; make necessary assumptions if needed]
Please ONLY reply according to this format, don't give me any other words.
Forward Pass Output
Explanations:
image shows clear anatomical structures uniform texture aymmetry. lung fields appear normal without visible opacitics, well-defined masses, streaks, irregular granular patterns. There deviations observed.
Output:
Many outputs for the rest data points
Forward Pass Output
Explanations: The image shows clear anatomical structures with uniform texture and aymmetry. The lung fields appear normal without visible opacitics, well-defined masses, streaks, irregular granular patterns. There deviations observed. Output:
Optimization Prompt
You are the optimizer for a model, your goal is to learn the best descriptions for the model. The model used the Current Model Descriptions below predicted the labels given images. You are given target labels, please optimize Model Descriptions for better prediction.
** Inputs (a batch of I.I.d. images):
See the given images.
Current Model Descriptions:
clear anatomical structures, uniform texture, symmetry, and normal lung fields without visible opacities, well-defined masses, streaks, irregular gramular patterns, or any deviations, output class label O. any distinct opacities masses, interstitial patterns, visible fluid accumulations, asymmetry, irregular shapes, or multiple small opacities, output class label I.
If the image
image
For granular or stresky opacities over any lung region, indistinct or uneven lung fields, subtle masses, minor deviations symmetry. any lung irregularities, output class label 1. For minimal irregularities or slight deviations, output class label O.
The model predictions:
[[]] [0] [0] [1] []][]] [U] [0] [o]]
The targets:
[[] [o] [o] [o] [o] [1] [I]]
If the model doing well, you can keep using the current descriptions. However, if the model is not performing well, please update the model by improving upon the 'Current Model Descriptions', which should results in lower classification error on both the current and the next batch of i.idd data. If previous Optimizatioo Step' provided, you the information from your last optimization step if it's helpful. Limit your 'New Model Descriptions' to be less than 100 words! Please think step by step and give your outputs strictly the following format:
Reasoning:
[be explicit and verbose, update the Current Model Descriptions by yourself; please show your work and use the features the
images; note that you don't have access to computers]
New Model Descriptions:
[put your new decision rules here, the rules MUST based on the features in the imput image; make sure it is clear when to output which class labels)
Please ONLY reply according to this format, don't give me any other words.
115

---

Published in Transactions on Machine Learning Research (01/2025)
Training History: Medical Image Classification (without Prior) Step 50 (continued)
Optimization Output
Reasoning:
The model misclussified several images where minor opacitics very light irregular patterns present. We nend to be precise in detecting these subtle features while examining the overall lung fields more thoroughly to reduce false negatives for label 1. Specifically, minor irregularities granular patterns that inconsequential should still be marked potential sbuormalities they occur in specific regions, such as the periphery. New Model Descriptions:
symmetry, and normal lung fields without any visible opacities, well-defined masses, streaks, or gramular patterns, output class label O.
If the image shows clear anatomical structures, uniform
interstitial patterns, visible fluid accumulations, asymmetry, irregular heart shapes, or multiple small opacities across the lung fields, output class label 1.
If the image shows distinct opacities
For granular streaky opacities over any lung region, indistinct synueetry; or any lung irregularitiss or granular patterus, including faint ones, in the periphery, output class label 1. For very minimal irngulariiies slight. deviations
lung fields, subtle mases,, minor deviations in
regions only, nutput
Updated Theta
the image shows clear anatomical structures, uniformn texture, symmetry, and normal lung fields without any visible opacities, well-defined
streaks, granular patterns, output class label 2. If the image shows any distinct opacities or mases,, Interstitial patterns, visible fluid accumulations, aaymmetry, irregular heart shapes, multiple small opacities across lung fields, output
label
3. For granular or streaky opacities over any lung region, indistinct or uneven lung fields, subtle masses, minor deviations in symmetry; any lung irrogularities granular patterns, including ones, the periphery, output class label I. very minimal irregularities slight deviations in central regions only, output class label
116


Reviews:
[{"review_id": "p5OjdOwCba", "reviewer": "Reviewer_NRYu", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "### Strengths\n\n1. **Innovative Concepts**: The paper introduces the novel framework of Verbalized Machine Learning (VML), which leverages natural language as a model parameterization method. This unique approach redefines function approximation and highlights the potential of large language models (LLMs) in machine learning.\n\n2. **Enhanced Interpretability**: VML offers improved interpretability by encoding decision rules and learned patterns in natural language. This enables human-understandable explanations of the model's behavior and the reasoning behind updates.\n\n3. **Effective Incorporation of Inductive Bias**: The framework provides a straightforward and unified way to incorporate inductive bias via natural language. This mirrors the way humans learn, allowing for seamless integration of prior knowledge into the learning process.\n\n---\n\n### Weaknesses\n\n1. **Insufficient Explanation of the Differences Between APE and VML**:\n   - The distinction between Automatic Prompt Engineering (APE) and VML is inadequately articulated, particularly in Sections 2 and 4.8. While the authors argue that VML recognizes numerical patterns from the data, this reasoning is vague and could be misleading.\n   - A more effective comparison would focus on the mechanisms and methodologies of APE and VML rather than their results. For example, both frameworks utilize LLMs for prompt optimization, but VML explicitly requires prompts to incorporate numerical evaluations. From this perspective, the key difference is that VML mandates explicit numerical reasoning in its prompts.\n   - Additionally, the authors should detail the experimental setup for APE. The original APE paper predominantly used older models, such as GPT-3, and it is critical to specify which LLMs were used in these experiments for a fair comparison. Including the prompts employed for APE would further clarify the distinction. Without this, I highly doubt that APE cannot achieve comparable results under similar conditions.\n   - Finally, attention to detail in the presentation is lacking. For instance, Figure 10 has both subfigures labeled as (a), which can confuse readers. Moreover, the prompts for APE and text classification are incomplete, which undermines the credibility of the comparison. It is unlikely that APE, even with GPT-3, cannot generate complete sentences.\n   - This comparison with APE is very important, as the only other comparison with existing methods in this paper is the one with in-context learning (ICL), which is a very basic method.\n\n2. **Limited Scope of Experiments**:\n   - The experiments focus exclusively on low-dimensional and small-scale search spaces, which is a known limitation of most LLM-based optimizers.\n   - For instance, in the conducted experiments, the search spaces were limited to fewer than five dimensions and involved no more than 100 iterations of optimization. While this is suitable for the study’s scope, it is unrealistic for many practical machine learning scenarios involving higher-dimensional spaces and more complex models.\n   - Although this does not diminish the utility of VML in optimization tasks, the authors should explicitly acknowledge this limitation. Many other works using LLMs as optimizers address similar constraints, and acknowledging this upfront would enhance the paper’s transparency and credibility.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "- The authors propose an innovative way for using LLMs as prompt optimizers. In this approach, the LLM-based optimizers perform on numerical evaluations. \n- The authors propose several innovative concepts for using LLMs, such as using LLMs as function approximation", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "The authors should focus on addressing the primary weakness regarding the insufficient explanation of the differences between Automatic Prompt Engineering (APE) and Verbalized Machine Learning (VML). The following actions are recommended:\n\n1. **Provide Details on APE**:\n   - Clearly specify the details of APE, including the LLMs and prompts used in the experiments. This would allow for a more direct and fair comparison between the two methods. For instance, if older models like GPT-3 were employed in the APE setup, explicitly stating this is essential to contextualize the results.\n\n2. **Explain Differences at the Algorithmic Level**:\n   - The explanation of differences between VML and methods like APE should be rooted in the algorithmic or methodological level rather than being limited to a comparison of results. For example, the authors could emphasize that both methods optimize prompts for LLMs but highlight that VML explicitly requires the incorporation of numerical evaluations in its prompts. This distinction would help clarify the unique methodological contributions of VML.\n\n3. **Include a Pseudo-Code Formulation for VML (Optional)**:\n   - To enhance clarity and alignment with existing literature, the authors could present a pseudo-code formulation of VML, similar to how Algorithm 1 is used in the APE paper. This addition would provide a concrete, step-by-step understanding of the VML framework, making its differences from APE more transparent and actionable.\n\n4. **Fix Presentation Errors in Figure 10**:\n   - Address inconsistencies and errors in Figure 10, such as both subfigures being labeled as (a). Ensure that all prompts and visual elements are complete and accurately reflect the experimental setup. For instance, incomplete prompts for APE and text classification should be rectified to avoid misrepresentation.\n\nAlso, the authors could adjust claims regarding VML and APE: Moderation in claims about the superiority of VML over APE would enhance the paper’s consistency between claim and evidence, a critical criterion of TMLR.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "gJhD8i8bac", "reviewer": "Reviewer_EmjK", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "# Strengths\n\n- The aim of the paper is to describe a new framework (VML) to harness the abilities of LLMs to solve machine learning tasks. Using language allows the user to naturally specify problems, requirements etc to obtain a solution with intermediate steps also specified in language. \n\n- Using text to describe data and parameters is innovative at least to this reviewer. This approach may open up new ways to  use LMs for machine learning that includes interpretability as a nice side-effect\n\n- Several basic examples are provided in the paper and the appendix to show one possible implementation of VML\n\n# Weaknesses\n\n-W1- VML relies on numerical machine learning by design. This appears to be a significant limitation of the proposed VML framework.\n\n-W2- Since VML requires natural language, the paper would benefit from having an analysis in languages other than English. (Acknowledgment: This itself opens interesting issues for paper review)\n\n-W3- The examples used in the analysis are basic ML problems which can be seen as a limitation. It's not clear how well this approach would scale up to larger problems.\n\n-W4- The discussion on prompt optimization vs VML could benefit with other baselines like APO which is reference [37] in the paper.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper proposes Verbalized Machine Learning (VML), an approach to solve machine learning tasks by using language models (LM/LLM is rest of the review) to parameterize the solution space  and (possibly another) LLM as an optimizer. A natural consequence of this approach is that the solution space is described via text/language that's interpretable to human users. Furthermore, VML allows the user to specify inductive bias via natural language as additional parameters. The paper conducts an empirical analysis with open LLMs to demonstrate how VML might be a viable approach to solve machine learning tasks.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "- VML can be considered a position paper on how to use the features (\"capabilities\") of LLMs to solve problems of interest to users in machine learning. However, as noted in W1, the paper could benefit from having a more persuasive argument on why VML is better than numerical ML\n\n- The paper would benefit from an analysis in other languages as it can highlight strengths or weaknesses of existing models\n\n- What are the scaling properties of VML in terms of dataset sizes the model can solve? Can VML currently be used to solve MNIST image classification task for example? Can VML solve the problem in a more sample efficient manner than using numerical ML?\n There is a relationship to context window so a discussion on what LLM builders should do to support this framework in the future would be useful\n\n- The final weakness listed above could be due to the reviewer's limited exposure to prompt engineering research literature. So, a broader discussion of existing prompt optimization vs VML-based approach would be useful to the reviewer and perhaps other readers.", "broader_impact_concerns": "Since VML proposes to use natural language the paper would benefit from including a discussion on how a problem can be solved by users of languages outside English.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "JSqvxnjPt5", "reviewer": "Reviewer_HL6p", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "### Strengths:\n\n1. Novel Concept: The paper introduces Verbalized Machine Learning (VML), an innovative approach where Large Language Models (LLMs) function as function approximators. This represents a significant conceptual advancement in the field.\n\n2. Architectural Parallel: The authors draw an insightful parallel between their proposed model representation and the von Neumann architecture, where programs are represented as data. This analogy provides a familiar framework for understanding the novel approach.\n\n3. Enhanced Explainability: The VML design potentially increases the explainability of machine learning tasks, addressing a critical need in the field for more interpretable models.\n\n4. Clear Presentation: The authors present their ideas with clarity, and the experimental procedures appear to be well-described, facilitating replication by other researchers.\n\n### Weaknesses:\n\n1. LLM Hallucination Concerns: The potential impact of LLM hallucinations on numerical determinations is not adequately addressed, raising questions about the reliability of the approach in quantitative tasks.\n\n2. High-Dimensional Data Handling: The paper lacks a thorough discussion on how the VML approach performs with high-dimensional data, which is a common challenge in many real-world machine learning applications.\n\n3. Limited Discussion of Failure Cases: The authors do not provide a comprehensive analysis of potential failure modes, which would be valuable for understanding the limitations and robustness of the proposed approach.\n\n4. Comparative Performance Analysis: The paper lacks a detailed comparison of VML's accuracy against traditional machine learning techniques, making it difficult to assess the practical advantages of this new approach in terms of performance.\n\n5. Limited LLM Diversity: The study only uses three LLMs to demonstrate results, which limits the generalizability of the findings. Including results from a wider range of open-source LLMs would provide a more comprehensive evaluation of the VML approach.\n\n6. Performance Gap Analysis: The paper does not address potential performance differences between API-based LLMs and open-source ones when applied to the VML framework. This comparison would be valuable for understanding the practical implications of choosing different types of LLMs for implementation.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper presents Verbalized Machine Learning (VML), which introduces a novel approach to leveraging large language models (LLMs) for machine learning tasks. The key contributions of the paper are as follows:\n\n1. **Framework for Function Approximation**: VML establishes a framework in which pretrained LLMs serve as function approximators by utilizing natural language prompts. This allows the LLM to actively engage in the learning process, rather than merely providing explanations for its decisions.\n\n2. **Iterative Training Algorithm**: The authors propose a concrete algorithm that facilitates iterative training of LLMs through verbalized updates, thereby extending the capabilities of traditional single-step learning found in in-context learning (ICL) paradigms.\n\n3. **Interpretability and Explainability**: VML enhances interpretability by enabling LLMs to generate human-readable explanations for their updates and predictions, thereby increasing transparency in the decision-making process.\n\n4. **Encoding of Prior Knowledge**: The framework allows for the straightforward encoding of prior knowledge (inductive bias) through natural language, enabling automatic model selection based on both data and verbalized insights.\n\n5. **Empirical Validation**: The authors provide empirical evidence demonstrating that VML effectively encodes prior knowledge and performs well across various applications, underscoring the practical applicability of this approach.\n\nOverall, VML reimagines traditional machine learning tasks by treating both data and models uniformly as text prompts, which may significantly enhance the interpretability and flexibility of machine learning systems.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. Please provide discussion about how VML will handle LLM hallucinations.\n2. How VML will handle high dimensional data needs to be presented.\n3. Failure cases needs to be discussed.\n4. Comparison between accuracy of traditional ML and VML needs to be presented.\n5. More diverse LLM and their performance needs to be presented. How open source LLM performance in comparison to Api based one needs to be discussed.", "broader_impact_concerns": "No concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The initial manuscript had several concerns such as potential similarity to naive prompt engineering and optimization, small-scale low-dimensional experiments, and hallucination and diversity issues. However, through the rebuttals, the authors addressed these issues and revised the manuscript accordingly. Therefore, I recommend the acceptance of the current manuscript as it is.
Decision: Accept (poster)