Sloth: scaling laws for LLM skills to predict multi-benchmark performance across families
Authors: Felipe Maia Polo
Seamus Somerstep
Leshem Choshen
Yuekai Sun
Mikhail Yurochkin
Venue: NeurIPS 2025 poster ()
Keywords: scaling law
LLM
benchmark
skill
Primary area: evaluation
TL;DR: We propose a scaling law for LLMs skills (e.g., reasoning, instruction following, etc)

Abstract: Scaling laws for large language models (LLMs) predict model performance based on parameters like size and training data. However, differences in training configurations and data processing across model families lead to significant variations in benchmark performance, making it difficult for a single scaling law to generalize across all LLMs. On the other hand, training family-specific scaling laws requires training models of varying sizes for every family. In this work, we propose Skills Scaling Laws (SSLaws, pronounced as Sloth), a novel scaling law that leverages publicly available benchmark data and assumes LLM performance is driven by low-dimensional latent skills, such as reasoning and instruction following. These latent skills are influenced by computational resources like model size and training tokens, but with varying efficiencies across model families. Sloth exploits correlations across benchmarks to provide more accurate and interpretable predictions while alleviating the need to train multiple LLMs per family. We present both theoretical results on parameter identification and empirical evaluations on 12 prominent benchmarks, from Open LLM Leaderboard v1/v2, demonstrating that Sloth predicts LLM performance accurately and offers insights into scaling behaviors for complex downstream tasks, increased test-time compute, and compute-optimal scaling of skills.

Sloth: scaling laws for LLM skills to predict multi-benchmark performance across families
Felipe Maia Polo', Seamus Somerstep'
Leshem Choshen 2.3 Yuekai Sun' Mikhail Yurochkin
Department of Statistics, University of Michigan
2 MIT-IBM Watson Al Lab, IBM Research
Computer Science and Artificial Intelligence Laboratory, MIT Institute of Foundation Models, MBZUAI
Abstract
Scaling laws for large language models (LLMs) predict model performance based on parameters like size and training data. However, differences in training config- urations and data processing across model families lead significant variations in benchmark performance, making it difficult for single scaling law to gener- alize across all LLMs. On the other hand, training family-specific scaling laws requires training models of varying sizes for every family. In this work, we propose Skills Scaling Laws (SSLaws, pronounced as Sloth), novel scaling law that leverages publicly available benchmark data and assumes L.LM perfor- mance is driven by low-dimensional latent skills, such as reasoning and instruction following. These latent skills are influenced by computational resources like model size and training tokens, but with varying efficiencies across model fam- ilies. Sloth exploits correlations across benchmarks to provide more accurate and interpretable predictions while alleviating the need to train multiple LLMs per family. We present both theoretical results parameter identification and empirical evaluations on 12 prominent benchmarks, from Open LLM Leader- board v1/v2, demonstrating that Sloth predicts LLM performance accurately and offers insights into scaling behaviors for complex downstream tasks, increased test-time compute, and compute-optimal scaling of skills. Our code can be found on https: / /t////ittuuut wwt.cccmmmeleeeeecco /ttp:///ooppamaaaoooooooooo sloth.
Introduction
Large Language Model (LLM) scaling laws for benchmarks and downstream tasks efficiently predict the performance of LLM based on its parameter count and training set size. However, variations in training configurations and data processing across different model families often lead to significant differences in benchmark performance, even for models with comparable compute budgets [Ruan et al., 2024]. Consequently, a single scaling law typically fails to predict performance across all LLMs accurately [Choshen et al., 2024]. In contrast, creating family-spechi scaling laws requires training multiple models of increasing size, which is resource-intensee
In this work, we propose new class of scaling laws called Sloth to solve this dilemma. These scaling laws are fitted using publicly available data (e.g., from LLM leaderboards) across multiple benchmarks, leveraging information shared among benchmarks and model families to improve prediction power and interpretability through parameter efficiency, i.e., fewer parameters without hurting performance. Specifically, we utilize the correlations in benchmark scores to simplify the scaling law in terms of parameter count without harming prediction power by assuming that LLM performance is driven by a set of low-dimensional latent skills, such reasoning and instruction
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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following, which can be easily interpreted. Furthermore, we hypothesize that these latent skills are similarly influenced by computational resources, such as model size and training tokens, across different L.M families, with the key distinction being each family's 's efficiency in converting compute into skill levels-which can be estimated with only one model per family. In summary, our main contributions are
Introducing new class of scaling laws, Sloth, that borrows strength across the available bench- marks and LLM families to make more accurate and interpretable performance predictions of (hypothetical) LLMS in given benchmarks of interest. Specifically, we assume that benchmark performances directly depend on low-dimensional L.M skills, which are influenced by factors such the number of training tokens and the number of parameters.
Providing a theoretical result regarding the identification of Sloth's parameters and empirically demonstrating that our scaling laws can (i) accurately predict the performance of large models in 12 prominent LLM benchmarks and (ii) provide interpretable insights into LLM skills and scaling behavior.
Demonstrating how predicted latent skills and our model can be used to (i) predict model perfor- mance in complex downstream tasks that involve coding and emotional intelligence. (ii) predict LLM behavior with scaled test-time compute, and (iii) derive aptimal-scalingg rules for skills.
1.1 Related work
Scaling laws for deep neural networks: In recent years, researchers have studied scaling laws from different angles. Rosenfeld et al. [2019] provides experimental scaling laws that predict model loss as function of training set size, model width, and model depth. Likewise, Kaplan et al. [2020] establishes scaling laws that primarily measure loss (perplexity) and not accuracy on downstream tasks or benchmarks. Motivated by the presence of hard limits on the size of trainable data sets but hypothetical unlimited ability to scale models, the authors of Muennighoff et al. [2023] establish scaling laws in constrained data settings. They find that perhaps unsurprisingly, increasing computing provides diminishing returns if data does not scale. Gadre et al. [2024] addresses the gap between the assumptions in scaling laws and how training is performed in practice; in particular, they construct scaling laws that both perform well in the over-training regime and predict performance on downstream tasks. In a similar but distinct direction, some works try not only to estimate scaling laws but also respond to the following strategic question: "Given a fixed FLOPs budget, how should one trade-off model size and the number of training tokens?" For example, Hoffmann et al. [2022] provides partial answer, introducing the celebrated family of Chinchilla scaling laws and finding that training tokens and parameter size should roughly scale together. This contrasts with the older work of Kaplan et al. [2020] that provides series of power laws that imply that simply increasing parameter count will provide good returns. Each of these referenced works trains models with a particular pretraining setting (e.g., architecture) at various sizes and ultimately seeks to predict test loss. Our focus is distinct, we fit scaling laws on existing benchmark data of multiple model families and predict LLM benchmark performance with minimal amount of data on the new family being predicted. Even though Allen-Zhu and Li [2024], study how LLMs can retain knowledge depending on their scale, the closest related works are Owen [2024], Ruan et al. [2024], Gadre et al. [2024]; we will provide detailed comparison with their work throughout the paper.
LLMs latent skills: Given that the performance of large language models (LLMs) in different and diverse benchmarks is correlated, it makes sense to think that those models have some low- dimensional latent skills that are reflected in downstream tasks. In this direction, Ilié [2023] extracts a general intelligence factor ("g-factor") for LLMs using the Open LLM Leaderboard [Beeching al., 2023] and GLUE [Wang et al., 2018] using factor analysis. They also verify that this "g-factor" positively correlates with model size. In a similar direction Burnell et al. [2023] uses HELM [L.ang et al., 2022] data to reveal that LLM intelligence may be constituted by three distinct, yet correlated factors. They also verify a positive correlation between model size and these latent skills, yet they do not propose a formal scaling law. In their study, the authors do not account for the training set size or model family information, leading to poor fit of the regression model; this leaves good extrapolation as an open problem we address. In Kipnis et al. [2024], a unidimensional item response theory model is applied to each one of the 6 (filtered) benchmarks of the Open LLM Leaderboard. A factor analysis on the skill parameters shows that the main factor (carrying 80% of the data variability) is highly correlated with the "grand" (average) score of L.MMs. In related but different direction, Maia Polo et al. [2024a.b] show that inferring low-dimensional latent skills of L.MS can make model evaluation much more efficient, saving up to 140x in computing power. In this work, explicitly model LLM
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skills as a function of computing resources, which enables the creation of accurate and interpretable scaling laws for benchmark performances.
2 Scaling laws for benchmark data
2.1 Problem Statement
In this section, we describe the setup we work and what our objectives are. Within family of LLMs i (e.g., LLaMA 3), our objective is to estimate the performance of a big LLM, e.g., with 70 billion parameters, in benchmark j, e.g., MMLU, given evaluation data from smaller models in the same family. Let s represent the size of the LLM, defined as the number of parameters, and let I denote the number of training tokens. We define Ysj(s, t) [0, 1] the score of an LLM from family& i, with size and trained on t tokens, on benchmark j. Our goal is to approximate:
Myj = m[Xix (s,
(2.1)
Here, ml should be interpreted as a central tendeney summary measure of random variable, such as the mean or median'. Ideally, the model for Pyj will be simple and some of its parameters will be shared among model families and benchmarks; in this case, the model becomes more interpretable and more data can be used in the fitting process, making the model better estimated. From now on, we denote the set of model families as I = (1,..., , and the set of benchmarks as J = (1....,,,, J}.
2.2 Previous approaches to scaling laws for benchmarks
The closest works to ours that propose models for Pty(e, 1) (2.1) are Owen [2024], Ruan et al. [2024], and Gadre et al. [2024]. While Gadre et al. [2024] indirectly model the quantity of interest via the LLMs perplexity in specific datasets, which might not be readily available, Owen [2024] and Ruan et al. [2024] model (s, t) directly through a regression model connecting compute and benchmark performance. One assumption they made that the performance on benchmarks only depends on s and through the total amount of training FLOPs, which can be approximated by c(s, t) = 6st. That is, if a: R + [0, 1] denotes a fixed activation function, e.g., the standard logistic (sigmoid) function, and % E [0, 1]. then it is assumed that
(sy(s, t) = 73 + (1- (nn,s t)),
(2.2)
where May R² + R denotes linear predictor such that (s, asj + Big log c(s. t), which can be easily interpreted. Here, ?/y adjusts the lower asymptote of Paj and accounts for the probability of LLMs scoring correctly by chance. Owen [2024], in their best performing models, considers the case in which y 0 (or adds a similar offset parameter to the model) and the parameters Ory and Big are independent of the model family i. On the other hand, Ruan et al. [2024] consider both ouj and Bij to be family- dependdt and, in their most general model, 7/ can assume values in [0, 1].
The biggest issue with previous approaches when modeling Mig is that they are either too restrictive or too flexible. From the restrictive side, they assume that (i) Piy depends on s and t only through FLOPs, (ii) there are no family-dependent parameters, or (iii) the activation function is fixed and well-specified. From the flexibility side, Ruan et al. [2024] assume both aij and Big to be family dependent making estimation hard (or impossible) depending on the number of models we see for each family. From Ruan et al. [2024]: "(...) fitting such a scaling law can be tricky, as each model family f and downstream benchmark has its own scaling coefficients B, and ay. This means that scaling experiments, especially for post-training analysis, are often fitted on very few (3-5) models sharing the same model family (..)." Thus, in their experiments, they consider a different problem setting, where a large LLM has been trained and evaluated on some benchmarks and use their method to predict its performance on other benchmarks.
At the end of the day, Owen [2024] and Ruan et al. [2024] end up working in different setups: Owen [2024] does not use family information at prediction time, making their scaling law less accurate but more generalizable, and Ruan et al. [2024] assume families are important at prediction time but consider that the target model has already been trained, making their scaling law less applicable in practice and more interesting from an interpretability point of view. In this work we wish to instead predict the performance of larger LLM without having train it but taking family information into account, thus allowing practitioners to make decisions regarding investing resources into training
Because minimize the Huber loss in this paper, we aim approximate the median
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large LLMs. Moreover, our formulation also allows interpretable insights from the data. Despite different setups, we make comparisons with Owen [2024] and Ruan al. [2024] throughout this work by considering their application//ddaatttionss as baselines.
3 Scaling laws for LLMs skills with Sloth
3.1 Model architecture
We present novel scaling law called Sloth, which introduces several modifications to (2.2). The key innovation of Sloth lies in its explicit modeling of the correlation structure between benchmarks. resulting in improved predictive accuracy and interpretability. Moreover, Sloth proposes that (i) LLM capabilities should scale with computing resources similarly across families up to an efficiency factor, (ii) benchmark performance can depend on and not only through the total number of FLOPs, and (iii) that the function a can also be learned in cases in which predictive performance is important. We detail these points.
Inspired by the latent skills (e.g., reasoning, language modeling, instruction following) inferred from benchmark data in Burnell et al. [2023], Ilić [2023], Ruan et al. [2024], Gor et al., Kipnis et al. [2024]. Maia Polo et al. [2024a.b]. we propose creating a scaling law for LLMs skills by leveraging the correlation structure of the benchmarks: for example, we model how the construct "reasoning" scales with compute instead of modeling benchmarks scores directly. The two major advantages of this approach are better performance prediction since we have fewer parameters to fit (reducing overfitting) and extra interpretailliyyIniigtts Concretely, we model mh (s, t)'s simultaneously for benchmarks j E I as each being linear combination of the same low-dimensional latent skills 0c(s, 1) E Rd plus bias term R where d < J = |J]. Denote (8, L) E R as the vector of (J)e Mathematically, we have
(s, AO,(s,
(3.1)
One can see that A E R 'xd encodes the correlation structure hetween the benchmarks; in particular, it tells which benchmark measures overlapping (or distinct) skills. Interestingly, our model has strong connection with factor analysis (FA) models, which we elaborate on in detail in Appendix C. In FA, the matrix known as factor loadings while (s, t) are known as factors.
Next, we propose a model for (s, t). Inspired by models used in Economics, we use the family of translog production functions from stochastic frontier analysis [Kumbhakar and Lovell, 2003]:
(n(sst) = a. + (s 1 k S d, x(s,t)
(3.2)
log(s) log(t)).
Note that (i) the intercept parameter ag is indeed family-dependee while each skill slope is shared across families and (ii) 0 can depend on s and not only through c(s, t). In economic terms, the intercept term auk can be interpreted an efficiency measure of the family in converting compute to performance for skill & and, in practice, will absorb all hidden factors specific to family i such as data quality, post-training factors, etc.. We note that the interaction term in (log(s) log(L) accounts for the fact that the impact of log(s) and log(!) on skills might depend on each other; in Appendix D, we show some evidence that this indeed the case. Additionally the changes in mhy we propose making the activation function a trainable and specific to each benchmark j if needed. To that end, parametric single-index model using neural networks [Bietti al., 2022]. To make the results more behaved if ofupoot-support generalization is needed, we assume o, R - [0. 1] is given by a monotonic (increasing) neural network, which can be achieved by constraining its weights to be non-negative [Sill, 1997] and its last activation function to be sigmoid. We note, however, that one can always forgo training of the link function and instead assume a sigmoid structure as this simplifies model fitting and may make the model more interpretable. We give more details about model fitting Section 3.2. Since SI Sloth a simple neural network, both model fitting and prediction are done within seconds by commercial laptop.
we adopt

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3.2 Model fitting
Assume that for each model family we observe set of tuples (s, t)'s denominated by & Then, we fit the model by solving the following minimization problem
(9, (3,0,
arg min
ΣΣ FET (S.))EE;
Σ
(d(pis(s,t) Yo)
", elo.1], fer
3 R 0. increasing, JE.T
JEJINERR
Oak ER, for iEZ and IskSd
BEER for Igkgd
where ls is given by the Huber loss with hyperparameter 5 = .01 and (s, denotes the most general version of our model. We minimize the loss function via gradient descent using the Adam optimizer [Kingma and Ba, 2017] with a decaying learning rate. We parameterize Y/ using the sigmoid transformation to guarantee the constraints are satisfied. Similarly, we truncate the weights of the awo-hidden-laye -lyer neural network σy to ensure the trainable function is increasing. If one desires, o,'s 's can be set to fixed functions, e.g., sigmoid, and 7/3 can be fixed beforehand. Unfortunately, the minimization problem is not convex as expected when fitting antoorranllyisss models; multiple initializations of the optimizer can be applied to guarantee a better fit.
3.3 Interpretability and practical considerations post model fitting
Ideally, to make reasonable interpretations of models like Sloth, we need its parameters to be identi- fiable, i.e., the map connecting Th (s, t)'s and model parameters should be bijective. Unfortunately, as in all exploratory factor models, this is not the case. However, we theoretically show in Appendix that the model parameters are identifiable up to some parameter transformations. This observation allows us to find configuration that makes the model more interpretable by mirroring standard approach used in factor analysis, e.g., in Chen et al. [2019]'s applications. The main idea is that we fit Sloth without any constraints and then find a configuration of skills (using factor rotation) that makes results interpretable. We detail the applied process in Appendix A.I.
4 Sloth in practice
In this section, we present experimental results that provide evidence of the usefulness of Sloth. We perform experiments
set of twelve benchmarks and satte-ofthh--eee LLM families, including LLaMa 3 [Dubey et al., 2024], Qwen 2 [Yang et al., 2024], and Yi 1.5 [Young et al., 2024]. We explore the following applications: (i) benchmark performance prediction for larger models from a specific L.M family, (ii) interpretability of the scaling of skills (can help practitioners allocate resources based on the skills of interest), and (iii) downstream tasks performance prediction.
4.1 Data
We expand the dataset made available by Ruan et al. [2024], including more models from the HuggingFace Open LLM leaderboard v1 [Beeching et al., 2023] and [Fourrier et al., 2024]. In our extended dataset, we have a total of 30 families?, which 28 are v1 of the Open LLM Leaderboard and 17 families measured on v2 of the Open LLM Leaderboard. Furthermore, there are 15 families at the intersection of the two versions. Furthermore, we collect data and present results on the performance of a variety of instruction-tung versions of the base models we consider. As far as we aware, our dataset is the most comprehensive among prior works benchmark data scaling laws. Please check Appendix for details on the included models.
4.2 Comparing scaling laws in terms of prediction errors
In this section, we compare the predictive power of different scaling laws in predicting LLM performance in all the considered benchmarks; we focus on the two versions of the Open LLM Leaderboard, which include 12 benchmarks: GSM8k [Cobbe al., 2021]. MATH Lvl 5 [Hendrycks et al.. 2021]. MMLU [Hendrycks et al., 2020], MMLU-PRO [Wang et al., 2024], BBH [Suzgun et al., 2022]. GPQA [Rein et al., 2023], MUSR [Sprague et al., 2023]. TruthfulQA [Lin et al., 2021]. HellaS wag [Zellers et al., 2019], Winogrande [Sakaguchi et al., 2019], ARC [Clark et al., 2018]. and IFEval [Zhou et al., 2023]. We apply a leave-one-our cross- validation algorithm to obtain test errors for each family of models. We consider base models and instruct models to belong to distinct
If we consider that instruct and base models are from different families, we end up with 53 families.
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Open LUM Leaderboard v1
Open LLM Leaderboard v2
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HellSwagg
Winegrande
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GRDA
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Averape
Figure 1: The figure shows the average (across LLM families) mean-absse absolute-error (MAE) (within family) for different methods. Sloth performs competitively, with errors similar to or lower than the "Size and Takens" variant, indicating its effective inductive bias.
families (they will share the same intercept in our model, for example), but we do include the instruct (resp. base) family in the training set when the corresponding base (resp. instruct) family is in the test set. Moreover, we do not older versions of recent families if they are available in the training set, e.g., we do not include LLaMa 2 in the set of test families if LLaMa 3 is present in the training set. In this section, we present results for the two leaderboards separately; in Figures 12 and 17 of the Appendix, we also present results for the intersection of the two leaderboards.
In the first set of experiments, consider the case in which only the smallest model of the test family is observed at training time. Because of that reason, we cannot fit the general scaling law in (2.2) which both the intercept and slope family dependent. In this scenario, consider our main baselines to be (i) the model in (2.2) with shared intercept and slope [Owen, 2024] ("FLOPs (shared intercept)"), (ii) the same model but only with shared slope ("FL.Ps")), (iii) version of the PCA idea² proposed by Ruan et al. [2024] in which we predict the principal components using the FLOPs model with shared slope that are then mapped the benchmark scores ("PCA FLOPs"), (iv) and our model with trainable activation function but assuming A is identity ("Size and Tokens"; implies d J). Moreover, we include two versions of Sloth. the "basic" one, we fix a to be sigmoid, and 7 's are given by the 100% over the number of alternatives in the case of multiple-choice benchmarks and otherwise, except for TruthfulQA, which we compute the first percentile of the scores coming from the full Open LLM Leaderboard and fix the lower asymptote to that value. In previous sections, we mentioned that Sloth is parameter efficient; we include parameter count analysis in Appendix where we compare our model with other well-performing baselines.
We include more details about this approach in Appendix E.
"When the benchmark has subsections with a different number of alternatives, we compute the asymptote parameters subsection and then compute an overall asymptote using weighted average in which the weights proportional to the number of examples in each subsection.

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Reasoning
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Figure 2: Needed skills for each benchmark. In this figure, we report the estimated loadings and, based on their values, we give them appropriate names,
Reasoning
Know edge
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Figure 4: In this figure, we plot the skill levels (output) subtracted by the family-specific intereept terms against inputs the and y-axis. From these plots, we can see how each one of the inputs can differently affect the production of skills. For example, "Reasoning" showed to be more affected by model size than tokens when compared to other skills. Moreover, "Knowledge" is more influenced by inputs (level curves are steeper) in general, while the other skills should be more sensitive other family-dependent factors.
Figure 1 gives the results for the first set of experiments. It depicts the average mean absolute error of all methods when predicting LLM benchmark performance, which is measured in percentage points. It shows the competitiveness terms of prediction quality. One important thing notice is that Sloth errors are similar or lower than the "Size and Tokens" " variant, suggesting that the assumed low-dimensional structure between benchmarks results is a good inductive bias. We high- light that the analysis includes recent families like LLaMa Qwen 2. and Yi 1.5. For more details on the tested models and extra related results, including model-specific results, please check Appendix H.2. The extra results are qualitatively similar the ones in Figure 1, in which Sloth often beats the baselines. In a second set of experiments, we consider the case in which the two smallest models of the test family are observed at training time. In this way, we can (2.2) making both parameters family dependent. The results are qualitatively similar to the one presented on Figure and are presented in Appendix H.3. Moreover, we include Figure in the appendix, which is version Figure which Mean Absolute Percentage Error (MAPE) is used instead of MAE; results qualitatively similar.
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Figure 3: Running Sloth with shared intercept offer great way model scaling laws that are family- independent.
In an extra set of experiments, we show that we can still get good prediction results for some benchmarks even if we consider shared intercept between families. The advantage of this approach is that we can claim for general scaling law that holds for all families. Figure 3 shows us a subset of Figure 12 in the appendix and it is built under the same conditions as Figure 1. It is possible to see that, for subset of benchmarks Sloth with shared intercept is strong alternative the FLOPs model used by Owen [2024]. In some cases, it gets similar prediction errors relative to more complete versions of Sloth.
intercept models are not always needed;
'ee selected the best for both versions of Sloth.
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Date model SHIll ce-score)
Bate model sidll Ce-sconel
Figure 5: We compare the skills of base (X-axis) and instruction-tuned models (y-axis); if a model lies on the 45-degree line, it means that the model has the same skill level in its base and instruct versions. Gains from instruction tuning (IT) for different families three latent skills. Findings include a large and positive impact on "Instruction Following" and that provides much larger variations in this skill when compared to inputs seen Figure 4. Moreover, IT had a moderate and negative effect on "Reasoning" and mixed effects on "Knowledge".
4.3 Interpreting the latent skills
In this section, we use the intersection between the two leaderboards, aiming to get more insights from the combined data. Since we have an identifiability result for the "basic" version of Sloth, in which we fix the lower asymptotes 7 and the link function to be sigmoid (see Section A), we opt for interpreting that version of the model. We set d 3 as that model version achieved the best prediction results in Figure 12. Figure 2 illustrates the model loadings, A. from which we assign names to the three dimensions based on our subjective interpretation. We include the loadings for d = 2 and d = 4 in Appendix I. To complement our exploration, we include Figure 4, which gives us the level curves of different skills (disregarding the family-specific intercept term), and Figure 5 in the appendix that compares the skills of base and instruction-tunen -uned models; in this figure, we include LLM families with more number of models. In both figures, the numbers are given in terms of standard deviations as the skills are standardized to have a zero mean and unitary standard deviation.
The first dimension, with strong loadings from benchmarks such GSMSK, MATH, GPQA, MMLU(-PRO), BBH, and MUSR, is labeled "Reasoning." The benchmarks GSM8k and MATH Lv1 5 consist entirely of mathematical word problems while GPQA also contain mathematical and advanced science questions. On the other hand, BBH includes logic deduction and linguistic reasoning puzzles. The strong dependence of BBH on the "Reasoning" skill suggests that in language models, there is an association between logical reasoning, general linguistic ability, and mathematical ability. Finally, MuSR is a benchmark that evaluates "multistep soft reasoning tasks specified in natural language narrative" [Sprague et al., 2023]. Figure 4 shows that Reasoning is primarily a function of model size, with a small dependence on the number of training tokens used. Moreover, the first plot of Figure 5 in the appendix compares base models versus their instruction-tunel versions terms of Reasoning, and we found that there is no clear rule: instruction tuning can either increase decrease the ability of an LLM to reason. These findings robust for different values of d as we can see in the figures of Appendix I.
Reasoning skill:
and
Knowledge skill: The second dimension is positively loaded on ARC, HellaSwag, and Wino- grande. These three benchmarks measure the ability of LLMs to remember common sense and basic knowledge; we denominate this skill as "Knowledge". More specifically, ARC consists of grade school-level science questions, HellaSwag is meant for sentence completion for common scenarios, and Winogrande common sense pronoun resolution problems. Contrasting with Reasoning, Figure 4 shows that Knowledge is highly influenced by both model size and number of training tokens. Moreover, we can see that the range of standard deviations in the middle plot is much greater than in the other two plots, giving us evidence that this skill might be more sensitive to increases in compute resources and less dependent on the LLM families themselves. On the other hand, Figure 5 in the appendix does not show any strong evidence of the effect of instruction tuning on this skill. These findings are similar to the ones reported in Appendix I for different values of d.
Instruction following skill: IFEval, which is positively and heavily loaded in this skill, measures how well language models produce answers that follow verifiable set of instructions; for example, including a keyword r number of times in responses. Therefore, we call "Instruction Following". An interesting fact is that instruction tuning has strong positive effect this skill for all depicted families we can see Figure the appendix. The effect can also be observed in Figure 28 of the appendix. When d = 2, instruction following gets mixed with other skills and we are not able to
8

---

this effect. Regarding Figure 4, we see that Instruction Following depends on both model size and tokens. Unfortunately, this interpretation does hold when as seen in Appendix
4.4 Predicting LLM performance on downstream fasks
Another useful application of Sloth, which is inspired by Ruan et al. [2024], is to pre- diet the performance in downstream task for large model from a relatively small num- ber of prior performance observations from that task. We use Sloth to estimate the la- tent skills of hypothetical LLMs and then use them predict the performance of those LLMs in downstream tasks. With this approach, we expand on the experiments of Ruan et al. [2024], which do not consider performance prediction of hypothetical LLMs.
The basic prediction pipeline is as follows. First, use standard LLM leaderboards to fit scaling law for skills using Sloth. Second, use existing L.LM performance on the downstream task to model how performance can be predicted from latent skills. Third, use Sloth to predict the skills of a (hypothetical) LLM of interest, e.g., a larger version of an existing LLM. Finally, use the model fitted in the second step to predict the performance of the hypothetical model in the downstream task.
Prediction
Bug
05-
Thaiting
metal 3-12b
Mcturary
metal lame 2-1-------- nay
04
We evaluate this pipeline on two downstream tasks, predicting the performance of meta-1lama-3-700 and --1--aam---77--nnse-- tion and meta-llama-- 1allama---7B-instrre intelligence tasks. We fit the same model shown in Section 4.3, but do not include meta-1lama-3-700 or the training set (see Figure 32 for the loadings of the latent skills, which is similar to Figure 2). Next, using either HumanEval [Chen et al., 2021] or EQ bench data [Paech, 2024], we fit regression model with logistic link using latent skills as features and performance on the downstream task as target. Together, this provides us with sufficient information to predict the performance of the held-out models on both tasks with decent accuracy. Figure 6 depicts this logistic curve and the actual values for HumanEval;
on code comple- on emotional
Predicting model per- formance in complex downstream tasks like code completion ("HE") for LLaMa 3 70B (base/instructt. In the first step, we fit Sloth without in- cluding L.MMa 3 70B (base/instruct) in the training set. In the second step. we fit regression model con- necting skills and downstream perfor- mance. Finally, we predict LLaMa 70B (base/instruct) performance from their predicted Sloth skills.
Figure
EQ bench results can be found in Appendix J. Moreover, we can see that "Reasoning" is by far the most important skill in predicting coding ability while a mixture of "Reasoning" and "Knowledge" is needed for emotional intelligence (see Figure 32 for a more accurate interpretation of the loadings).
In Appendix a similar test is provided for agentic capability measured by AgentBench [Liu et al., 2023], although to avoid overfitting, in this case, must fit Sloth with no family-specifff intercept.
4.5 Predicting performance behavior with scaled inference compute
In this section, demonstrate how our method, Sloth, combined with concepts like Item Response Theory (IRT) [Reckase, 2009], can predict how the performance of an LLM scales with increased inference through repeated sampling [Brown al., 2024]. For this experiment, we utilize the MATH dataset [Hendrycks et al., 2021], I]. made available by Brown et al. [2024], which evaluates 10 LLMs also included in our dataset. These models are part of the LLaMa 3 Instruct, Gemma, and Pythia families. The process is:
Train Sloth on our full dataset of 12 benchmarks, excluding the largest models in the LLaMa Instruct, Gemma, and Pythia families.
2. Fit logistic regression model for each MATH question in Brown et al. [2024]'s data, using the skills (s, t) of the 7 training LLMs as covariates to predict the probability of correctly solving those problems.
3. Estimate the skills for the three test models using our scaling law, then predict their probabilities of answering each MATH question correctly via the logistic regressions, and then predict the p metrics for the 3 test models using those probabilities. The predicted pass @ k metric for a certain
L.M is given by the average of 1 - (1 -;)}*' (across j's) ) if pú is the predicted probability of the LLM of interest getting question correct.

---

Figure 7 illustrates that Sloth can accurately predict test-time scaling behavior for these models. Note that unlike scaling law in Brown al. [2024], Sloth can predict inference compute gains for hypothetical L.Mss before committing resources to training them. This highlights practical application where practitioners can estimate the potential performance of hypothetical model at test time, given computational budget.
4.6 Compute-optimal scaling of skills
One relevant question is: given a certain budget in FLOPs, how do we invest it to maximize one skill of interest? This type of analysis novel for skills and has only been carried out for validation loss (e.g., by Kaplan et al. [2020]. Hoffmann al. [2022]). A summary of the mathematical formulation of this question is given in the following and exposed in detail in Appendix B. For each model family and skill k, we have
Coverage (parssg)))
2.6
2.8-
s prediction
0.4
0.2
Gemma-77 78
Pytiib 128
(s, 1) Okkk + 8r0 log(s) + P&1 log(t) + 242 log(s) log((). Define u log(s), simplicity, here we consider a simple, but widely used [Kaplan al., 2020, Hoffmann et al., 2022], compute budget constraint 6st = which is equivalent to u + v = We add extra constraints u and based the training data support to avoid unreasonable out-of-support predictions. We optimize
[k]
log(t). and log(c) log(6). For
Sloth can accurately predict test-time scaling behavior for LLaMa 3 Instruct, Gemma, and Pythia models MATH performance when coupled with Item Response Theory.
Figure 7:
max + 3kou + Bk1V + Bkeuv subject to u u+u=l, 2 E 21, T. u,
V
E (2 Tj.
u reduces this to a quadratic function gu(u), whose maximizer, within the observed range 4 = [maxx(| - ह, u), min(! 2, T)}. determines the optimal allocation. Our analysis indicates that the optimal values for s and do not depend on the model family. Table reports our results for instruction following; table with results for the other two skills is shown in Appendix B.
Substituting v
5 Conclusion
In conclusion, we have introduced the Sloth scaling laws as a novel approach to predicting the performance of large language models across benchmarks and model families. By leveraging the correlations between benchmark scores and assuming that LLM performance is governed by a set of interpretable, cow-dimensional latent skills, our approach offers a more efficient and flexible framework for understanding and pre- dicting LLM behavior. The ability to estimate model performance across a variety of bench- marks and tasks, even with minimal data from individual model families, highlights the practical utility of Sloth scaling laws. Our empirical results demonstrate that Sloth can accurately predict the performance of L.MS across multiple benchmarks while providing insights into the relationship between computational resources and model capabilities.
FLOPs (1e19)
Instruction Following Params (B)
Tokens (T)
100
0.16
1.04
578
0.30
3.24
3346
0.72
7.78
19360
2.15
15.0
112005
12.44
15.0
648000
72.0
15.0
Table 1: Optimal parameter and token allocation.
Limitations. From the predictive side of Sloth, believe that the main limitation is that the model is still dependent, most of the time, on seeing data from at least one LLM from the LLM family of interest. Moreover, we train the link function in the best version of Sloth using flexible neural networks, which can interpolate data very well, but have no guarantee of extrapolation when the (hypothetical) LLM of interest is very different from others in the training set. From the interpretabiliity side, we only understand the identification problems, such as transformations in the latent space, that can arise in simple instance of Sloth: fixed activation function and 7 This fact limits our understanding and interpretability of the most advanced versions of the model.
6 Acknowledgements
This paper supported by the National Science Foundation (NSF) grants 202-7027737, DMS- 2113373, DMS-24 2414918, and gift from OpenAl.
10

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A Identifiability of model parameters and interpretability
To interpret Sloth parameters, we need to guarantee they are identifiable. Given that our scaling law models the function (s, t) = (s, t) that condition is equivalent to the following statement: if two sets of parameters are responsible for characterizing (s, t), then those set of parameters should be the same up to predictable variations such as translations or rotations. To prove identifiability. we work with a fixed and invertible o, as usually done in the literature, and assume Y/ 's are fixed. The last condition is reasonable since these constants are usually known beforehand, e.g. it is well accepted that the lower asymptote " for MMLUU [Hendrycks al., 2020] performance is 25% which is given by 100% divided by the number of multiple-choice alternatives. Denote our fixed design matrix as X E RExp, where each row given by an LLM and p equals 3 plus the number of families, and define
31
Sa
and an
B
E Roxd
:
:
0m1
amd
such that the rows of XB E Raxd give the skills vectors 866 & (XB)O)s of all models in our dataset. Here n denotes the total number of models in the dataset and m is the total number of model families. To prove identifiability, we adopt standard assumptions from the factor analysis literature [Chen al., 2019] or regression literature, which assumes that the skills vectors (10) R are standardized, i.e., their average is null while their covariance matrix is fixed, rank(A) = d, and rank(X)= = p. Assumption A.1 (ldentifiability constraints). Assume that
1 Σ g00) g(i) V
g())
=0,
n2
and that rank(A)= d and xx) p. where ( denotes the i-th row of XB and Ni is positive definite matrix.
One possible choice for the covariance matrix V La [Chen al., 2019], which assumes uncorrelated skills. One implicit implication of Assumption A. is that n 2 p 2 d must be satisfied, otherwise the covariance matrix cannot be full rank. This condition is satisfied in our experiments. Under Assumption A.1, we show the identifiability of the model parameters up to a transformation of A tied a transformation of B, which leaves the outputs of the model unchanged. This means that we can potentially approximate the true values for A and B up to transformation, which is usually the norm within the class of exploratory factor analysis models.
Theorem A.2. Given that the true set of model parameters is (A,b, B), if there is another set of parameters (A, b, B) that satisfy
a (A(XB)O)
= (A(XB)O0 +b) for all forall e [n],
then, under the Assumption A.I, we have h = b, = AM, and B B(MT)-1 for an invertible matrix M E Rd dxd, In particular, M is orthogonal if v = Ia. i.e., M M = TM = Ia-
Proof. We start proving that 6 = b. Because a is invertible, we get
A(XB) (1) + b A(x + for all i in],
and consequently by the standardization of the latent skills
(1
+ À
1
(
+ b=
n
n
Now, we prove that = AM. Given that b = T. we have
(( = =A(XB)(0) for all i [n).
14

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and consequently by the standardization of the latent skills
(XB) (XB)(0) AT = A (xB)0 (xB)() AT AWAT = ÄVÄT.
By Cholesky's decomposition, write v/ LL", for lower triangular matrix L. If we define A AL and A' AL, then
A' = A'TT.
Because rank(A) d, have that rank(A') d and claim that λ A'U for orthogonal matrix U E Rdxd, To see that, first, realize that
rank(AA) =rank(A'A -rank (A'A') -rank(A'). We this by realizing that the null spaces of Ar and T are the same: for arbitrary vector A AATT= 2 = 0 = AT T = 0 = AT = O, where the last implication follows from the assumption that A A/ is full rank (rank(A') d). Because the null spaces of A and A/A the same, their ranks should be the same as well. The same reasoning applies to Ä and A', proving this intermediate result.
A'ATZ 0 and
Because and A TT have the same rank, the column space of these two matrices must be the same as the columns of A' AT are given by linear combinations of columns of A'. Same for Á' and A'ATT. Consequently, the column spaces of A' and λ' are the same.
Because the column spaces of A' and Å' are the same, there must be invertible matrix U such that A' A'U But then
A' AA = NUUTAT = NA = NT
= UU = -UT= =N
(NT NY
and
=I = -
Because A' = A'U, we have that
ÂL ALU = ALUL- AM.
If V La. then L la and M U.
Finally, we prove that B = B(M )- From previous considerations, we can write
AB x AMB x + A = AMB T
mank (nn A)
XB XBM
XT XB X
mank x)=
B B= TM
= B B(M
If V = La, then L = la and (M (MT)-1 U.
From our proof, we see that the matrix M is dependent on the specification of V
A.1 Interpretability
In practical situations, it is hard to fix the covariance matrix of skills to something meaningful before fitting the model, suggested in Section A.1. To make the model interpretable, we mirror standard approach used factor analysis, e.g., in Chen et al. [2019]'s applications. First, we Sloth without any constraints on the covariance of skills obtaining the estimates (Å, b, B). Second, we find the matrix A E Rd dxd such that the skills X B. have covariance identity, update B +- BA, and update (AB
+- Á(A A(AT)-I so the model outputs remains unchanged, because A(XB)T
15

---

Third, we find matrix M Raxd such that AM is easily interpretable (e.g., . it is sparse matrix); there are different methods find M and, in this paper, we use the Geomin [Yates, 1987, Chen et al., 2019] oblique rotation method to find suitable M using the Python package FaalorAnalyzer [Biggs, 2019]. We then update À AM and, to make the model invariant, we also update B +- B(MT) the resulting skills are still guaranteed to have unitary standard deviations, so their covariance equals their correlation. Finally, we standardize the columns of the skills X to have zero mean, while keeping the correlation structure unchanged. This last step implies that must be translated to make the model invariant.
B Compute-optimal scaling
In this section, we derive compute-optimal scaling rules for skills. Specifically, given a language model family and particular skill, our goal is determine the model configuration that maximizes performance under fixed computation budget 6st = c.
Consider model family i with skill & defined as
(8, 8, t) = Ork + Ae log(s) + Sk1 log(t) + 9k2 log(s) log(t).
Letting log(s), , log(t), and log(c) (c) log(6), formulate the optimization problem as: max Oikk + -Bou + BRI0 v + Bu2uu s.t. u -tv=l. u,e
This reduces to simpler optimization problem in terms of u alone: (u)(u) A - 3k2u2 + (3eo 3ai + Be2l)u + (oGa + Bail).
max ( (u),
where
To prevent our compute-optimal scaling method from making unreasonable predictions, we further restrict its solution the ranges of u and observed in the training data. Hence, impose constraints E [u. ul and = - u E [g, u], where bounds u, u, v are set based on quantiles from training data. Combining these constraints yields:
u E - [max(1 - v. u), min(l - n, u)).
Thus, the optimization problem becomes:
max 9Ek (u), wEM/
which is straightforward to solve. Specifically, if A&2 > 0. fak (w) is concave parabola, and the
maximizer is either at the vertex (if falls within U)) or at one the interval endpoints max(1 u)
min(l- 2, T7).
Table 2 extends the results in Table for all skills.
FLOPs (1e19)
Reasoning
Knowledge Params (B)
Instruction Following Params (B)
Params (B)
Tokens (T)
Tokens (T)
Tokens (T)
100
0.93
0.18
0.16
1.04
0.16
1.04
578
5.36
0.18
0.16
6.02
0.30
3.24
3346
30.98
0.18
0.37
15.0
0.72
7.78
19360
72.0
0.45
2.15
15.0
15.0
112005
72.0
2.59
12.44
15.0
12.44
15.0
648000
72.0
15.0
72.0
15,0
72.0
15.0
Table 2: Optimal allocation of parameters (B) and tokens (T) across skills for various compute budgets (FLOPs).
C Connections with factor analysis
Sloth is heavily inspired by (exploratory) factor analysis models. Factor analysis statistical technique used to identify underlying relationships between observed variables by reducing the data's
16

---

dimensionality [Bishop and Nasrabadi, 2006, Chen et al., 2019]. It assumes that multiple observed variables are influenced by a smaller number of unobserved/latent variables called factors (skills p(1), our case). These factors help explain the correlations among the observed variables. The method aims to model the observed variability and reveal the structure behind the data by estimating the factor loadings (A, in our case). The classical model assumes
Y = A0; + Es.
where Y; is vector of variables of interest and E; is am error term. There are versions for the factor model which nonlinear model for Y; assumed, e-g., in item response theory (IRT) [Reckase, 2006, Chen et al., 2019]. It is usually the case that 0 is estimated using random effects model, i.e. practitioners place prior distribution on 0,. In our work, we assume 6, is given by a function of observable covariates and a family-specific intercept, which is fitted using data.
D Motivating the interaction term in Sloth
As shown in Section 3, we include interaction term between log(s) and log(t). In the first place, we consider this as a natural extension of the model that depends on s and only through FLOPs, since recover that formulation if 3&1 Bez and 3k3 0. In the second place, believe that the dependence of benchmark performances on log(s) depends on log(t) (and possibly vice-versa). To motivate this idea we show some plots for two benchmarks these plots, we only keep families with higher number of models. First, realize that in both Figures 8 and 9, the performance within families in the middle plot can be well approximated by a line. Also, the slope of this line has a strong relation with the number tokens in the last plots. For example, Pythia was trained in small dataset and its (hypothetical) slopes the second plot almost in both cases. On the other hand, Qwen2 was trained on more data and its (hypothetical) slope on the middle plots is high. Certainly, this relationship does always exist, but adding an interaction term in our model helps us to leverage this pattern when it exists.
MMLU-PRO and BBH. For
Family
MMUU PRO score
0.5
0.5
MMLU-PR PRO score
0.5
tloom
0.4
MMLU PRO
0.4
0.4
pyth
gwen3.5
0.3
0.3
0.3
0.2
0.2
comrnder?
0.1
0.1
2.5
5.0
log FLOPs
log tokens
Figure 8: Inputs vs MMLU-PRO scores.
Ramily
00H secre
BBH sccre
Bema-2
0.5
0.5
BBH
quard
0.4
0.4
0.4
60
smolm
maremder??
0.3
0.3
V--15
2.5
5.0
7.5
-1
0
2
log LOPP
lag size
log tokens
Figure 9: Inputs vs BBH scores.
17

---

E PCA approach formulation
We follow the ideas of Ruan Moreover, we follow their code and apply PCA with the same set of hyperparameters. Assume we have a matrix of scores Y '0, 1|m*J in which columns represent benchmarks and each row represents a language model. We compute the covariance matrix of benchmark scores using Y and then compute eigenvector matrix U, where the columns give the ordered eigenvectors (from highest eigenvalue to the lowest one). To reduce the dimensions of Y, we keep only the first d columns of YU, resulting in Y Rxd For each column of Y (principal components; PCs), train a linear regression model using logFLOPs as the covariate; in this case, either the intercept or both the intercept and slope can be model and then go back to the original coordinate axis to obtain the final predictions by computing Σj=1 PCJU. E R..
[2024] closely as possible create a prediction method.
At test time, predict the PCs of held-out
Sloth is parameter-fffcient: parameter count analysis
When the number of model families f is moderately large, Sloth actually uses fewer parameters than the top-performing baselines (both of which consider family-specific intercept train the activation function). This is because the assumed latent skill structure of Sloth. For example, with d and we use J 12 benchmarks:
Sloth: 12 (d +) + d (f + +3 = 69 + 3f parameters
"FLOPs" baseline: 12 (f 3) + 12f parameters
"Size and Tokens" baseline: 12 (f 5) 50 + 12f parameters
So for f 2 4, Sloth uses fewer parameters than either baseline.
'See https: http::////.ttbbbboomrrr//////sss
18

---

G Models in our dataset
Table 3 gives a detailed view of our dataset. The column "Family" considers that base and instruct models are from different families, while "OriginalFamily" does not. The column "TestFamily" tells if that specific family considered remaining three columns tell if the data is available for these specific benchmarks. For the EQ data, only the following models are available "gemma-bbitt', 1lama-2-13b-chan 'llama2-70b-chat "llama2--bb-chat"
be part of the test in our experiment while the
'qwen 1.5-1 1--1.88-chat". "qwenn 1.-- 4bb-chat' 'qwenl .5-32b-chat'' 'qwenl -5-bb-chat', qwwen..5-7b-che "yi-1.5-34b-chh
saeiialllmaa--7---nnsss
"eeta-llaaa-3-8--ine
"yi-1-5-bb-chat" "yi-1.5-9b-chh yy-334b-chat.
Model
Family
OriginalFamill
TestFamiil
Leaderbuurdl
Leaderbooud22
HumanEval
bloom
blom
bloom
True
Thue
False
True
bloom- Ib1
bloom
bloom
True
True
True
True
bloom-3b
bloom
bloom
Trme
Trme
Thae
True
bloom- 560m
bloom
bloom
True
True
True
True
True
blossom-v5. 1-34b
blossom-vS.
yi-1.5
False
True
Trae
False
blossom-v5
blosom-v5..1
yl-1.5
False
Falsc
True
False
codegen- 16b-al
codegen-nl
codegen
True
True
False
True
codegen 6b
codegen. nl
codegen
True
True
False
True
codellamm-1
codellama
codellama
Trme
True
False
True
10
codellama-344b
codellama
codellama
True
True
Falsc
Truc
codellama
False
True
12
codellama-76
codellama
codellama
True
Truc
False
Truc
13
deepscek-soder-
doepsock-
deepseek-
True
True
False
True
1.3b base
cuder buse
cuder
14
deepseek-code coder-
deepseek-
despseek-
Tree
True
False
True
33b-hase
coder-hase
coder
15
doepscck-coder-
doepseek-
deepseck-
Truc
Thue
False
Truc
6.7b-base
coder-buse
coder
16
dolly-v2-12b
dolly-v2
pythia
True
Thuc
True
Truc
17
dolly-v2-3b
dolly-v2
pythia
True
False
True
False
18
d2.1y-v2-76
dolly-v2
pythia
True
True
True
False
19
delphin-2.9. 1.9.1---
dolphin-2.9.1-
yi-1.5
Trme
Trme
Thue
Falke
1,5-34b
yi-1.5
20
dolphin--29.11y9
dolphin-2.9.1-
yi-1.5
True
Thue
True
False
1.5-9b
yi-1.5
21
dolphin-2.9.2-
dolphin-2.9.22
True
False
True
False
qwen2-72b
qwen2
22
dolphin-2.9.2
dolphin-2.9.2
True
False
Thue
False
qwen2-7b
qwen2
23
falcon-180b
falcon
falcon
Truc
Truc
False
False
24
falcon-40b
Falcon
falcon
True
Thue
True
False
25
falcon-400-instruct
falcon-instruct
falcon
True
False
True
False
26
Falcon-7b
Falcon
falcon
True
True
True
False
27
falcon 7b instruct
falcom instruct
falcon
True
True
True
False
28
gemma-2-25
gemma-2
Trme
False
Thae
False
29
gemma-2-2b-it
gomma-2-it
gomma-2
True
False
True
False
gemma-2-99
gemmaa-2
gemma-2
True
True
False
31
gemmaa 2 2-9b-it
gemmna 2-it
gemmaa-2 2
True
False
True
Fulse
32
gemma-2b
gemma
gemma
True
True
True
True
33
gemman-2b 2b-it
gemma-it
gemma
True
True
True
True
34
gemma. 7h
gemma
gemma
True
True
True
True
35
gemma-7b-it
gemma-it
gemma
True
True
Thae
True
36
gpt-j-6b
gpt-i-nco-ncox
gpt-nco/j
Truc
True
False
Truc
gpt-nco-1.36
gptej-neo-neox
get-neofj
True
38
gpl neo 125m
gnl jel-j-neoo.a neox
gpt nevfj
True
True
Fulse
True
39
gpt-neo-2.7b
gpt-j-neo-neox
gpt-neofj
True
True
True
True
40
gpt 20b
gpl internlm?
gpt neu/j
True
True
Fulse
True
41
internim2-200
internim2
Trme
True
False
Falke
42
internlm2-7b
intemim2
internim2
True
Truc
False
False
43
laama-133 13b
Ilaaa
Ilama
False
True
True
True
luama
True
True
45
Hama- 2. chat
llama 2-chat
laaa.
False
True
True
True
46
2-mna-2-70b
llama22
Ilama-2
False
True
47
laaa 2. 70t chat
laaa chal
Ilama
Fals
True
Thue
True
48
1lmm--2-7b
laaa-2
Ilama-2
False
True
True
True
1aama-2-catt
Ilama-2
False
True
True
True
50
llama-3-
llam--3-
meta-Ilamas
True
False
True
False
sauerkraullm.
70b-instruct
instruct
llama-3--
llama-3.
meta-llama-3
True
True
Tru
smmrrkruttm-
smmerkrmutlm-
8b-instruct
instruct
19

---

52
Iamaa 30b
laaaa
Ilama
False
True
False
True
53
larma 65b
Iaaaa
Ilana
False
True
Thue
True
54
Ilama-7b
Ilama
laaa
False
True
Thue
True
55
meta Ilama 3 706
meta Ilama-3 3
meta laama 3
True
True
True
True
meta-Ilama-3- 3-ma-
aatta-llama3--
meta-llama-3
True
True
instruct
instruct
57
meta-llama-3-4
meta-llama-3
True
True
True
True
58
meta Ilama-3
meta llama 3. instrnct mpt
llamn-
True
TTree
True
True
instruct
59
mpt-30b
mpt
True
True
False
True
60
mpp-300 306 chat
mpt chat
mpt
True
True
Fulse
False
mpt- 30b-instrret
mpt-instruct
mpt
True
False
62
mpt-7b
mpt
True
Truc
False
True
63
mpp- 7b-chat
mpt-chat
mpt
True
True
False
False
64
7h instract
mpt instruct
mpt
True
True
False
False
65
olmo-11
olmo
olmo
True
True
Thue
Falke
66
olmo-7b
olmo
olmo
True
Truc
True
False
67
open laaaa 13b
open Ilama_
openllama
False
True
False
False
open Ilama 68
open, Hoaaa_
opcallama
False
False
False
69
open_ Bii__llma__000 3b__?
Jiama_
opcallamav2
Palse
True
False
False
70
open Hamma
open, laaa,
opeallana
Fulse
True
False
False
71
open Ilama_ 7h
open llama
openllamav2
False
Tree
False
False
72
openhermes-133
openhermes
Ilama-2
False
True
Thue
False
73
openhermes-76
openhermes
Ilama-2
False
True
False
74
o.t-1.3b
opt opt
opt
True
True
True
True
75
opt-125m
ope
True
True
False
True
76
opt-13b
opi
opt
True
True
False
True
77
opt 2.7b
opt
opt
True
True
False
True
78
opt-30h
opt
Trme
True
Thae
True
79
opt-350m
opt
opt
Truc
Truc
False
Truc
opt-6.7b
opt
opt
TTuee
False
True
81
opt 66b
opt
upt
True
True
False
True
82
orca-2-13b
orca-2
Ilama-2
False
True
True
False
83
orca-2-7b
Ilamna2
False
True
True
False
84
orea, orca_mini__ v3. 13b
orca, mini v3
llama
False
True
Thue
Falke
85
rea_mini_re v3_ 63_70b
orca_ mini v3
Ilama-2
False
False
True
False
86
orca_mini v3_7b
orca, mini v3
Ilama-2
False
True
True
False
orca_ v
orca, mini
gwen2
88
orca mini _______7_ v7_7b
orca, mini v7
gwen2
False
False
True
False
89
pythia-1.4b
pythia
pythia
True
True
False
True
90
Pythia- 12h
pythia
pythia
True
True
True
True
91
1ythia-160m
pythia
pythia
Trme
True
True
True
92
pythia-1b
pythia
pythia
True
True
False
True
93
pythis-2.8b
pythia
pythia
True
Thue
True
True
pythia-410m
py pythia
pythia
True
True
95
pythia-6.9b
pythia
pythia
True
True
True
True
96
pythia-70m 70m
Py
pythia
True
True
False
True
97
qwen- 14h
qwen
qwen
False
Thue
False
True
98
qwen-72h
quen
qwen
False
True
False
True
99
qwen-7b
qwen
qwen
False
Truc
False
True
100
gwen1.5-0.56
qwen1.5
gweal.5
Fulse
Thue
True
True
qwen 1.5.n...5..5... 5.0.5b
qwen1.5
qwen1.5
102
qwenl.5... 1.5.
qwen 1.5
qwen
False
Truc
True
True
qwen1.5. 1.. chat
qwen1.5 chat
qwen1.5
Fulse
True
True
False
104
qwen1.5.1e.
qwen
1.enl.5
False
Thee
Thae
True
105
qwenl.5-16 1--1---hht
qwen 5-chat 5-chat
gwen1.5
False
True
True
False
qwenl.5-32b
qwenl.5
gwea1.5
True
True
107
gwen1.5. 3.b.l..53bbb--. chat
qwen1.5 chat chat
qwenl.5
Fulse
Thue
True
False
qwen 1.5.4h
qwen
qwenl.55
True
109
qwen 1.5-44b-hhat
1-5chat
gwen1.5
False
Truc
Truo
False
110
qwen 1.5.1..5.75
qwen
qwenl.5
False
True
False
True
1.5-72b-cht
qwen 5--chat
1.5
False
True
False
True
112
qwen1.5-70
qwen1.5
gwea1.5
False
True
True
Truc
gwenl.5-7b-chat
qwenl.5-chat
gwen1.5
114 qwen2 0.5b
qwen2
gwen2
True
True
Thue
False
qwen2-0.5b-insmm
qwen2-instruct
qwen2
116
qwen2-1.5b
qwen2
gwen2
True
True
True
Falsc
117 qwen2-1.55 Soen2-1.Sh-inerme i.Sh-instrnct
pwen2-instreet iinstruct
qwen2
Trme
False
True
False
qwen2-726
qwen2
True
True
Thue
False
119
gwen2-72b-instruct
gwen2-instruct
gwen2
True
False
True
False
gwen2-7h 7b
qwen2
ywen2
True
121 qwen2 Th- Th-instract
qwen2 insstrect
qwen2
Thue
False
True
False
10b--ll-4b-pile
rwky-4-pile
rakv
True
False
False
123 rwky-4-166 169m--
rwky-4-pile
rwky
True
True
False
False
124
rwkv-4-1b5 1b5--plle
rwkv-d-pile
rwky
True
True
False
False
mwky-4-3b-pilc
pwky-4-pile
True
True
False
Falsc
126 rwks-4-430m-pp
twkv-4-pile
rwky
True
Tiue
False
False
7h pile
rwky pile
False
128 smerkrauthm-
sanerkrautmm
gemma
Tree
Tric
Thue
False
gemma-2b
gemma
20

---

129
smmerkrautim-
sauerkrautim-
True
Thue
True
False
gemmaa 7b
genmma
130
smallm-1.70
smollm
smollm
True
False
Thee True
False
131
smollm-1133 1.7b.
smollm-
smollmn
True
False
False
132
mollm-133m
smollm
smollm smollan
True
False
True True
False
smollm-133 1355m
smollan-
True
False
False
instruct
instruct
134
smallm- 360m
smollm
smollm smollm
Trme
False
Thue
Falke
smolln-360m-
smollm-
True
False
Thue
False
instrcct
instruct
stablelm-base-
stablelm-hans
stablelm
True
False
alpha-3b
alpha
137
stablelm-base base-
stableim- base-
stableim
True
TTree
False
False
alpha 7h
alpha
138
strredde2-115h
staronder2
starcoder2
Trme
True
Thue
True
starcoder2-3b
stareoder2
starcoder2 starender2 starcoder
True
Truc
True
True
140
starender-77b
starender2
True
True
True
True
stareoderbase
stareoderbase
False
True
142
staroderbase-1
stareoderbase
starcoder
False
True
False
True
stareoderbase 3b
starcuderbusee
slarcoder
Fulse
True
False
True
144
starenderhase-77
starenderbase
starcnder
False
Thie
False
True
145
wizardIm- 13b-v1.0
wizardlm-v1.0 1.0
Ilama-2
False
False
Thue
False
wizardIm-70b-v4.
wizardlm-v1.0
Ilama-2
False
True
False
xglm-1.7b 147
xelm
xglm
True
True
False
True
xglm-4.5b
xglm
xglm
True
Truc
False
True
149 xgim-564mm
xglm
xglm
True
Trre
False
True
150 xglm. 7.5b
xglm
xglm
True
True
False
True
yi-1.5-34
yi-1.
yi-1.5
True
True
Thae
False
152 yi-1. 5-344b-chat
yi-1---hhat
yi-1.5
Truc
True
Thue
False
yi-1.5-bb
yi-1.5
154 1..1...bb.o chat
ya-1.5 chat
yi-1.5
True
True
True
False
yi-1.5-bb
yi-1.5
yl-1.5
True
True
True
False
156 1i-1.5 cha
ya- --chatt
yi-1.5
True
True
True
Fulse
157 yi.34h
yi
y1
False
True
Thue
True
158 yi-34b-200k
yi-200k
y1-200k
False
Truc
False
False
159 yi-34bbchat
yb-chat
y1
False
Truc
False
False
yi-6b
True
True
161 6b. 200k
200k
yi 20-200k
False
Thue
False
False
162 yi-6b-chat
yi-chat
yi
False
False
True
False
163 yi.90 9b
Tabe
True
True
False
21

---

H Extra performance prediction results
In this section, we present the full versions of the figures presented in the main text and some other extra results.
H.1 Mean Absolute Percentage Error (MAPE) plot
Open LUM Leaderboard
Leaderboard
FLOPS
intertap))
22.7
14.1
11.7
7.9
14.5
78.5
24.9
FLOPS (shared
32.3
71.9
25.6
14.2
4.4
3/2
22.6
32.9
13.9
FLSP
16.3
54.3
8.2
11.3
19.5
Sizs and Takens
19.5
Tidwns
17.2
9.9
59.6
9.9
7.0
42.1
24.3
(0-1)
23.0
12.2
11.2
6.7
10.8
211.3
45.8
FCA FLOPs
(0-1)
PCA + FLDPs
23.1
(6-2)
20.6
19.4
173.3
0.8
7.4
88.1
47.9
FLOPs (d-3)
26.4
6.6
6.7
2.8
14.0
747.6
50.7
PCA FLOPS
20.7
56.7
FLOPs
25.2
6.8
2.9
14.7
252.1
51.4
110Ps
(0-4)
20.6
18.5
240.6
9.7
54.9
Slith basic (0-1)
51.9
(d-1)
22.7
14.2
74.0
7.1
5.8
37.1
26.8
(d-2)
15.4
8.5
10.5
5.0
10.0
84.2
22.3
Slath basic (0m22
Skith basic
11.4
(5:33
16.9
10.1
60.2
6.9
7.6
28.0
21.6
Sloth
(0-4)
11.2
9.9
9.8
4.7
11.0
37.8
14.1
Sleth
16.8
67.4
28.4
(d=11)
19.5
6.8
7.0
3.7
10.3
57.0
17.4
Slath (d-1)
31.7
75.3
21.9
Sloth (0-2)
(d>2)
16.55
5.8
71.8
8.3
5.5
10.7
19.8
(6-3)
10.4
6.4
6.4
4.1
11.3
37.3
12.6
(0-3)
Slath
Slith
11.7
(5m4)
16.0
5.5
91.5
6.3
7.2
16.8
23.9
MORL
ABC
HellSSpug
Winigrande
COMBK
butrage
Whml
MATH L015
ORGA
MIBR
Figure 10: MAPE version of Figure 1. The results are given in percentage points, and that the results are qualitatively similar the MAE version, with Sloth producing best predictions for both Open LLM Leaderboards.
22

---

H.2 Test families have exactly one model in the training set
H.2.1 Average prediction loss across models
Open Leaderboard v1
LLM Leaderboard v2
TLOPs (shared Intercpc)
Intercept)
9.4
4.5
4.2
2.6
2.8
5.6
4.9
FLON
FLOIS
5.1
3.6
2.6
2.5
4.4
4.5
3.8
Size and
4.9
3.8
3.3
3.0
2.8
6.8
4.1
POA FLOP
[d=1]
(d-1)
9.3
7.3
4.4
2.7
3.2
9.4
6.0
POA FLOP%
(0+2)
(d-2)
6.1
7.4
4.3
2.7
2.8
9.3
5.4
POA FLOP
[0=3)
(d-3)
6.1
7.1
5.3
2.7
2.6
9.1
5.5
PCA FLOPs
(d+4)
6.1
7.1
5.5
2.7
3.7
8.9
5.7
Sloth basic (d=1)
10.5
5.1
5.2
2.8
2.8
6.2
5.4
(Shored Intercepe)
Sloth basic (d=2)
11.3
5.0
5.5
3.0
3.0
5.6
5.6
(Shored Intercept)
Sloth basic (d=3)
11.2
5.1
5.6
3.1
2.9
5.5
5.6
(shored Intercepe)
Sloth basic [d=4)
(shard interept))
11.3
5.2
5.6
3.0
3.0
5.4
5.6
Sloth (d=1)
(shared interrept)
10.0
4.9
3.4
2.5
3.9
5.6
5.0
Sloth (1=2)
(shared interrept)
10.2
4.2
3.0
2.3
2.4
5.0
4.5
Sloth (d=3)
(shared interrept)
10.2
3.4
3.1
2.1
3.3
5.3
4.6
Sloth 4d=4]
(shared interrept)
10.1
3.4
3.5
2.2
3.6
5.2
4.7
Slath basic
Sloth (dh11
9.2
6.0
3.8
2.3
2.3
7.5
5.2
Slath basic
Sloth (4"22
4.8
4.1
3.2
2.2
1.9
5.6
3.6
Slath basic
Sloth (1"11)
4.8
4.3
2.9
2.2
3.0
5.7
3.8
Slath basic
Sloth
(11-4)
4.8
5.0
3.8
2.2
3.3
5.9
4.2
Slath
(1111
9.9
4.2
2.6
1.9
2.0
4.2
4.1
Slath
(11
4.9
2.6
2.3
2.5
2.2
2.6
2.9
Slath
(d+3)
4.8
2.5
4.1
2.4
4.0
3.1
3.5
Slath
5.3
(d+4)
Slath
4.8
2.4
4,3
2.0
2.9
3.5
3.3
ABC
Winopande
GSMBK
Lats
OPGA
MUSR
Figure 11: The figure shows the average (across LLM families) mean-absolute-eroo (MAE) (within
family) for different methods. This complete version of Figure 1, in which we include Sloth versions with shared intercept.
23

---

Open Leaderboard v1/v2
FLOPs (shared
Interoept)
FLOPs
5.6
3.9
2.6
2.7
7.1
5.3
4.2
3.5
2.2
2.2
4.7
5.4
4.1
Size and
4.8
PCA FLOPs
[d-1)
11.5
6.1
7.9
4.8
4.9
14.7
7.2
4.5
3.3
2.1
2.7
5.9
6.3
PCA FLOPs
PCA FLOPs [0m3)
11.0
4.5
4.1
2.5
5.3
15.0
5.7
4.4
3.0
2.1
2.8
6.0
5.6
PCA + FLOPs (dm4)
12.0
Sloth (shared intercopt)
9.9
5.7
5.5
4.7
6.4
11.8
7.9
5.5
5.3
2.6
3.0
8.0
6.4
Sloth basic (d- intercept)
Sloth [shared Intercopt)
9.1
6.4
5.8
3.8
6.7
8.2
7.6
6.3
6.3
3.0
3.4
7.8
6.2
Sloth bask Id intercept)
[shared Intercopt)
8.0
5.2
6.7
4.7
5.9
10.7
7.8
4.5
3.2
2.3
3.1
6.7
5.7
Sleth Idw
interoupt)
[shared intercopt)
(d
7.4
4.7
4.9
4.1
5.6
9.8
8.6
4.0
3.2
2.4
2.2
6.9
5.3
Slath () (shared interoept)
Slath (d-1)
11.3
7.0
7.7
5.6
6.6
15. 9
8.6
5.0
3.0
2.3
2.4
6.7
6.8
Sleth basic (dm2)
Slath
7.5
5.8
6.1
3.8
4.6
8.6
4.0
3.4
2.3
1.9
2.2
4.3
4.5
(dm33) 3)
Sleth basic
8.6
5.1
7.2
5.3
4.3
12.3
6.7
2.8
2.9
2.0
1.7
4.4
5.3
Idm1)
Sloth
[dm3) 3)
5.4
3.0
4.4
3.1
5.0
6.4
4.7
2.5
1.6
2.2
1.5
2.5
3.5
Sloth (0+4)
5.4
5.0
6.5
MALU
ABC
HelaaSweg
Whetrannd
TrathfRA
GRMOK
IPEsel
BBH
LATH
OPGA
MUSR
MIAL-PRO
Avarape
Figure 12: The figure shows the average (across LLM families) mean-absolute-emmo (MAE) (within family) for different methods when fitting only one scaling law for both leaderboards.
24

---

H.2.2 prediction losses
Open
Ulverage acness benchmarss)
RoPss (stare inteopp0
13:3
2.5
520 an1
PCA FLOPs
10.2
10.8
KCA FLOR:
PCA TLOPs
PCA FLOPs
Sith bacc [p-3]
Sioth basic (0=2)
Siuth basic (0
South bazic ((wd)
Sant Interepp
Skch (0m
(sha Interept
2.6
16.4
14.9
1.5
11.7
10.0
Sarth Inta Intereppt
16.9
8.6
5.0
1.1
10.4
12.3
12.6
(0-2)
3.5
2.2
8.3
5.5
7.3
13.22
7.4
7.5
4.4
5.
6.7
(c=2)
3.2
2.6
6.5
9.2
5.3
10.2
6.2
4.4
2.3
6.8
5.6
6.1
3.0
6.5
4.8
5.0
5.0
[0=3)
3.2
4.2
8.0
9.6
5.4
10.5
20
2.4
4.2
5.8
13.2
1.7
8.3
6,5
1.5
3.0
3.1
6.5
2.7
(0
2.8
2.1
5.a
9.5
7.6
6.8
6.6
5.4
3.6
4.6
4.0
2.6
2.0
4.7
1.3
7,3
1.2
18
(0
6.0
10.2
7.4
6.5
6.1
1.3
3.5
3.7
5.1
2.6
2.0
5.8
cndegerrel
tatelana
Impprenk-ccoeee Damm
Poitica
Gdian
ammma
BRt Brbiewweeet
intanind
Garador'
ataiiiim alota
Figure 13: The figure shows the average (across benchmarks) mean- mean-absolluee-rroo (MAE) for each family considering only Open LLM Leaderboard v1.
25

---

Open LUM Leaderboard (Average ener acresa benchmaric))
FLOPs Shares
1.4
TLOPs
3.
2.5
3.3
10015
4.1
4.3
3.0
4.5
13
63
(3m1)
4.1
9.0
19
82
9.1
5.0
102
9.6
22
67
533
19
POX + FLORS
POL FLDPS
(2m4)
2.5
5.3
5.6
2.2
7.4
othrd interapit
1.8
3.1
6.9
148
22.0
4.1
18
6-5
4.7
geth (5-2)
42
Sloth busic (de3)
(stured interpapi
3.2
20.0
othred interpppi
2.0
1.1
2.3
7.
1.3
18.9
4.4
11
6.0
8.5
1.3
7.8
Shared intercapii
(d=2)
4.0
Slath cdwd)
(shared intercapi
Sluth (3m4)
1.8
Slich basic
(g=1)
1.8
4.8
4.4
8.9
11.4
2.1
5.3
11.3
3.5
2.4
5.3
Sloth basic
3.0
3.6
5.7
4.1
1.9
2.7
15.2
1.7
2.2
4.2
Slich badic
Sloch bagic
(d=1)
2.1
1.2
4.7
6.5
8.1
5.0
13
4.2
2.3
5.3
(9=2)
1.3
4.2
4.2
4.8
6.7
3.4
1.6
3.9
Sicch
5.0
1.8
5ath (d+4)
Mloo
Pyetia
mala Mamaa
anvodder
mmadon
Figure 14: The figure shows the average (across benchmarks) mean- araa-aaboluute-mnd (MAE) for each family considering only Open LLM Leaderboard v2.
26

---

Leaderbeard (Average benchmarks)
FLOIS (shated nieccap))
2:2
3.0
4.8
4,9
144
14.4
6.9
4.2
9.2
9.2
2.0
4.0
6.8
5.0
1-5
2.3
3.4
5.3
3.4
(d-1)
5.2
8.8
6.1
6.5
3.1
5.7
9.1
4.7
4.3
4 FLOPs
(d-2)
4.1
9.4
3.6
PCA FLOPs (d=c)
5.1
9.5
4.8
6.1
2.5
4.3
9.3
92
3.0
3.8
bassc basic (d-1)
(shered ineecapU
Sloth bask intercopti
2.5
(Shared intercopti
Sloth basic
2.4
2.6
2.5
5.1
1.6
19.9
6.9
4.5
8.8
8.8
bloth basic (d-4) (shered intecpp
(d-1)
2.6
(sharea intercopti
2.5
2.4
5.3
6.7
1:3
15.2
6.4
2.1
5.8
12.7
6.4
(shored intercapti
Sleth (1
2.3
Sorth (d-4)
(d-1)
3.9
4.4
5.4
7.9
2.5
3.2
4.6
23.5
13.0
4.2
Sloth bask:
(8-2)
Sleth basic
(d=4)
3.9
5.7
7.7
6.4
1.5
6.0
1.7
6.1
7.8
2.8
3.7
(d-1)
Sluth
3.7
3.7
6.8
1.2
6.1
3.7
8.2
4.0
Sloth
4.5
(d-3)
2.1
3.3
5.0
1.3
2.4
1.4
3.7
3.7
Slih (d4))
5.6
4.3
2.1
1.1
5.7
3.4
bitem
Pydha
Islan
pemma
und Poifeenmen mett
3amma
dime
owent2
H455
Figure 15: The figure shows the average (across benchmarks) mean- rean-abbolute-errd (MAE) for each family considering Open LLM Leaderboard v1/v2.
27

---

H.3
Test families have exactly two models in the training set
H.3.1
Average prediction loss across models
Open LLM Leaderboard v1
Open LLM Leaderboard
FLOPs
9.2
7.9
5.5
5.0
8.8
20.0
9.4
13.9
7.4
26.6
4.8
9.8
8.6
11.9
Size and Tokens
7.5
3.1
3.4
1.9
5.6
7.9
4.9
7.1
2.2
4.6
4.2
3.9
1.8
4.0
PCA RLOPs
(d-1)
10.8
9.0
13.1
7.2
5.5
13.8
9.9
FLOPs
(d-11
9.0
8.1
4.1
1.7
3.9
8.2
5.8
PCA FLOPs
(d=2)
10.8
8.3
12.2
6.0
5.2
12.8
9.2
(d-2)
11.3
6.8
4.5
1.4
3.5
6.0
5.6
PCA FLOPs (dm3)
11.7
8.4
11.6
6.0
5.0
12.2
9.2
FLOPs
11.2
7.0
3.3
1.4
3.8
6.1
5.5
(d-3)
PCA + ELOPs (d=4)
12.7
8.3
11.3
5.8
8.1
12.1
9.7
11.2
6.9
3.2
1.3
10.6
6.5
6.6
(d-4)
Sloth basic (d-1) Ishared intercepti
8.4
6.0
7.2
5.5
6.6
9.7
7.2
Slath [d=1) [shared intercepti
9.2
5.1
5.0
2.4
3.3
4.9
5.0
Sloth basic (d=2) Ishared intercept)
7.6
6.1
6.5
4.7
4.8
8.8
6.4
Sleth intercepti
9.5
5.2
6.0
2.5
3.3
5.1
5.3
Sloth basic (d=3)_ (shared intercept)
7.7
6.2
6.5
4.9
6.4
7.5
6.5
Slath [d-3) intorcepti
9.2
5.2
5.9
2.7
4.1
5.0
5.4
Sloth basic (d=4)__ Ishared intercept)
7.6
6.2
6.6
4.9
6.3
7.3
6.5
Sloth intercepti
9.2
5.2
5.9
2.8
4.0
4.8
5.3
Sloth (d=1) (shared intercopt)
8.9
5.2
6.9
5.5
7.7
13.1
7.9
(d-1)
9.6
5.3
4.3
2.4
3.3
4.7
4.9
Sloth (d-2) Ishared intercept)
8.0
5.2
5.9
4.9
6.3
10.9
6.9
9.3
5.1
4.5
2.6
3.4
4.9
5.0
intercepti
Sloth (d=3) (shared intercept)
7.7
5.6
5.8
4.9
5.8
10.8
9.5
5.1
4.5
2.5
3.6
4.9
5.0
Sloth (d-4)_ Ishared intercept)
8.0
5.1
5.7
4.6
5.8
11.5
6.8
9.4
4.8
4.2
2.5
3.6
4.8
4.9
intercepti
Sloth basic (d-1)
11.9
6.4
7.9
4.8
5.3
19.1
9.2
8.3
5.7
3.3
1.4
3.2
5.0
4.5
Slath basic (d-2)
5.3
4.9
6.9
3.3
4.1
10.2
5.8
4.9
4.8
3.2
1.4
3.2
4.5
3.7
Sleth basic (d-3)
4.9
4.8
4.6
6.4
2.0
3.6
4.2
4.3
Slath basic (d-4)
4.5
4.7
3.9
6.8
2.5
3.1
4.7
4.3
Sloth (d=1)
8.5
5.5
8.1
4.4
7.9
1.5
2.8
4.3
4.8
Slath (d-2)
5.1
4.3
8.2
2.0
2.8
2.3
4.1
Sleth (d=3)
4.7
3.5
4.3
6.7
2.4
3.8
1.7
5.4
1.8
3.6
Slath (d-4)
5.0
8.0
4.3
5.8
2.1
5.1
1.6
3.3
1.6
3.2
MMLU
ARC
HelaSwag
Winograndd
FruthtuDAA
GSMBK
Avera Average
IFEval
69H
LA
GRDA
MUSR
PRO
Average
Figure 16: The figure shows the average (across LLM families) mean-absolute-erro (MAB) (within a family) for different methods.
28

---

Open Leaderboard v1/v2
3.2
3.1
3.3
3.2
7.1
12.2
5.8
4.7
8.6
4.3
3.4
3.9
5.2
Takens
PCA FLOPs
14.3
[0+1]
PCA FLOPs
15.1
8.6
10.3
5.6
6.2
19.7
10.8
7.2
4.1
2.0
2.2
7.6
8.3
(dm2)
PCA FLOPs
14.7
PCA FLOPs (dm4)
13.6
9.8
12.7
6,9
7.1
20.6
10.3
7.2
3.8
2.0
3.1
6.3
8.6
Sloth basic Id-11 interoupt)
Sloth (shared intercopt)
6.9
4.9
5.9
4.2
7.1
10.2
8.4
5.2
6.5
2.7
2.5
5.9
5.9
Sloth basic Idw 3) intercept)
Sloth [shared intercopt)
6.8
5.3
5.6
4.1
6.7
10.3
8.7
5.3
7.4
2.8
2.9
5.7
6.0
Sleth Ide 1) interoept)
5.7
[shared Intercopt)
6.6
3.8
4.8
3.6
6.8
15.7
9.1
4.8
5.3
2.3
2.7
6.0
6.0
Sleth Ide 3) interoupt)
5.9
Ishared intercopt)
5.7
3.9
5.2
3.9
6.0
11.6
10.0
4.9
4.9
3.2
2.8
5.8
5.7
Sleth basic [d-1)
Slath
5.4
5.8
7.8
4.3
5.8
12.7
8.7
5.0
4.4
1.5
2.2
4,9
5.7
(d-2)
Sloth basic 3)
Slath (dm4)
3.0
5.8
5.8
3.2
6.8
10.7
4.2
5.2
6.5
1.8
2.9
4.3
5.0
Sloth 1)
5.9
3.1
3.0
2.1
4.9
14.9 9
8.2
4.1
3.6
1.9
1.9
4.0
4.8
2)
Sloth 3)
3.1
3.4
(d+4)
2.3
2.7
2.4
1.9
6.2
9.5
5.5
4.6
4.1
2.3
3.6
2.5
4.0
MHLU
HellaSeng
Winegranie
TrahtuKA
GSMAK
FEaa
GPGA
MUsR
HILL-PRRO
Average
Figure 17: The figure shows the average (across LLM families) mean-absolute--ee (MAB) (within a family) for different methods using the intersection of both leaderboards.
29

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H.3.2 prediction losses
Open LLM Leuderbbend v1 (Average erier across benchmeresi
RLPP
6.5
13.3
30.1
20.3
23
8.8
4.5
11.9
0.9
10.8
18.5
7,9
79
2.1
6.5
72
2.0
4.2
355
1.4
3.8
1.3
1.0
KCA RSK
17.2
5.
84
2.3
10.9
PEA RAP
P0A RLG
REA RAP
18.3
2.4
6.7
2.7
0.3
2.5
Glored iooccpii
10.3
24.5
11.3
3.4
5.6
5.A
33.2
3.3
Odare intrrppi
4.
11.1
13.5
12.5
2.3
4.7
4.2
3.0
1.4
7.0
3.9
2.2
34.0
4.3
6.5
Ghorce mnoocppi
11.5
14.3
13.4
2.2
4.4
4.3
3.0
1.7
7.4
5.3
2.1
23.7
4.1
5.7
Sath bace 26-41 (dhmme ennrrett
13.4
2.2
2.1
13.0
Sucn 11
mooocppi
12.4
3.3
32.5
Son 14-21
estam
can (e:d)
12.3
2.2
4.3
1.0
15.4
4.6
8.5
4.2
5.6
5.7
37.6
3.6
21
3.4
9.0
6.0
7.0
3.5
6.6
4.3
3.0
7/2
10.
45
5.4
73
2.5
4.5
31
3.8
8.0
4.1
9.3
3.1
5.7
4.3
3.2
6.8
103
6.7
4.5
6.8
2.4
333
Soth
2.5
4.9
7.
2.5
11
Swith
233
24.9
2.8
5whh
Swit
Such
3.4
1.5
2.5
5.0
1.4
coneltan
snsppeak Cadertaate
Rami-2
ownt Bamma.
cantarierd
Figure 18: The figure shows the average (across benchmarks) mean- www.--absouuterrrroooo (MAE) for each family considering only Open LLM Leaderboard vl.
30

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Leaderboard (Average benchmarkss)
FLOPS
10.5
5.3
29.4
13.3
11.4
2%.1
1.9
4.8
36.2
Size and
Toxens
3.9
6,4
2.2
6.0
2.3
3.5
32.4
(d=21
4.8
3.9
5.8
3.0
5.6
9.6
2.0
3.5
22.2
PCA FLO
(d-3)
5.6
8,4
2.0
(shared intercepti
Sloth basic
2.3
7.3
5.0
1.6
3.2
10.3
1.8
6.5
Sloth basic (d=2) (shaned intercep4t
(Shaeed intercopti
2.6
7.2
4.7
1.5
3.3
13.0
2.0
6.6
7.4
(shared intercepti
Sloth basic
2.6
7.1
1.5
3.3
13.0
2.0
6.7
7.3
Sloth (d=11
(shaned intercep4t
(shared intercopti
2.5
6.9
4.5
1.5
4.5
7.4
1.4
8.8
7.3
(shared intercept)
Sloth (d-3)
2.5
6.9
5.2
1.4
4.4
1.4
9.0
Sloth (d=4)
(shaned interceptt
1.4
5.7
5.3
1.6
8.0
6.9
2.0
3.0
6.6
Sloth basic
(d-21
1.6
3.7
2.5
6.4
1.5
2.3
Sloth basic
(1=3)
(1-4)
1.5
3.6
4.0
2.5
5.7
12.6
1.7
2.3
4.6
(d-11
Sloth
1.8
6.0
1.2
6.7
11.4
3.5
1.0
Sloth
1.6
(d 31
2.3
4.2
4.2
2.0
3.9
6.4
1.9
2.6
5.0
Sloth (1
Hlem
Hambe2
orea mini v3
Dnhha
Deren
smalim
earceder2
Figure 19: The figure shows the average (across benchmarks) mean-absoluut--erro (MAE) for each family considering only Open LLM Leaderboard v2.
31

---

Open
(Awerage
benchmarks)
FLOPs
6.7
7.3
19.7
9.2
17.2
6.1
11.6
Size and
3.0
Tokens
3.0
6.7
4.0
5.5
26.6
5.5
10.5
PGA + FLOPs
[d=2)
3.7
5.9
3.1
5.5
24.9
6.0
9.9
PCA + FLOPs
5.9
[d=4]
3.0
7.8
4.2
5.6
6.0
10.3
6.1
intercept)
Sloth basic (d=2) (shand intercept)
10.9
Sleth (d=3) intercept)
2.5
6.6
2.6
5.5
8.1
10.8
6.2
Sloth basic (d=4) (shared intercept)
5.6
10.9
2.8
7.0
4.7
6.6
6.1
10.6
6.6
intercept)
Slath (d=2) (shared intercept)
6.1
6.2
13.0
Id=3) intercept)
5.4
6.3
12.8
Sleth (d=4)
5.4
6.2
11.3
(shared intercept)
basic
6.8
10.5
6.6
Sloth basic (d=2)
6.2
8.7
4.7
5.4
9.0
4.3
Sloth basic (d=4)
2.1
4.9
9.3
4.6
Sloth
7.7
6.7
7.5
Sloth (d=2)
3.3
6.9
6.6
4.0
4.2
Sloth
3.9
7.5
2.7
Sloth (d-4)
1.6
4.4
6.8
4.1
2.5
bloom
llema-2
Rythla
qwen2.5 5
Swen2
starcader2
44..5
Figure 20: The figure shows the average (across benchmarks) cean-absolute-error (MAE) for each family considering only Open LLM Leaderboard v1/v2.
32

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I
Extra interpretability results
I.1
Results for d = 2
Reasoning
1.74
0.62
0.11
Knowledge
-0.07
0.28
0.27
0.81
0.82
Instruction following
GSMBK
MATH 5
GPOA
MMLU
MALL-PRO
BBH
MUSR
TruthduiAA
ARC
HellaSwag
Winegrande
iFEval
Figure 21: Needed skills for each benchmark. In this figure, we report the estimated loadings and, based on their values, we give them appropriate names.
Reasoning
1.00
0.61
Knowledge Instruction following
0.61
1,00
Reasoning
knowledge Instruction foilowing
Figure 22: Needed skills for each benchmark. In this figure, we report the estimated loadings and, based on their values, we give them appropriate names.
Knewledge Instruction following
Reasoning
Instrctt mosel skill score)
1.5
Farriles
0.6
gemma
0.1
metaiilams-
carent
-0.8
xi
-1.6
0.1
Base model skill (2.cooe)
skill
Figure 23: Gains from instruction tuning for different families on three latent skills. Major findings include large and positive impact on instruction following and negative impact on mathematical reasoning.
33

---

Reasoning
Instractin folowing
Traning loge trillons
Tookring tokwers (togu trillonst
0.90
26
0.85
6.25
-2.00
-1.2
0.5
05
15
Size 1o illonn parameters!
Size llogs billons of parameters)
Figure 24: Level curves producing different latent abilitics from parameter count and training tokens.
1.2
Results for d = 3
Reasoning
1.00
0.12
0.43
Knowledge
0.12
1.00
0.20
Instruction
0.43
0.20
1.00
Fallowing
Reasoning
Knowledge
Figure 25: Needed skills for each benchmark. In this figure, we report the estimated loadings A and, based on their values, we give them appropriate names.
L.3 Results for d
Reasoning
4.82
4.85
2.61
2.03
2.03
1.86
0.75
0.78
0.46
0.29
0.45
0.21
Knowledge 1
-1.44
-1.04
0.02
0.11
0.10
0.43
0.70
0.73
0.09
-0.04
-0.04
0.08
Instruction following
Knowledge
-0.27
-0.04
0.67
0.64
GSMBK
MATH 5
GPOA
MMLU
MMILL-RR
BAH
MUSR
TruthduiAA
ARC
HellaSwag
Winogrande
IFEval
Figure 26: Needed skills for each benchmark. In this figure, we report the estimated loadings and, based on their values, we give them appropriate names.
34

---

Reasoning
Knowledge
Instruction tollowing
0.06
0.30
1.00
0.31
0.34
0.26
0.31
1.00
Reatoning
Knowledge
Instrcction tallowing
Knowledge 2
Figure 27: Needed skills for each benchmark. In this figure, report the estimated loadings A and, based on their values, we give them appropriate names.
Reasoning
Knowledge 1
Instruction following
Knowledge
Instruct model skill (e-core)
1.1
1.6
0.7
-0.1
moma
1.2
-1.6
-1.2 -0.5 -0.1 0.5
1.1
-1.6 =0.8 0.0
0.8 1.6
-2. +14 +0.1 1.3 2.6
-19 =1.0 -0.1 0.7
Base model skill (2-score)
Base model skill (2-ccore)
Base model skill (2-Score)
Base model skill (2-score)
Figure 28: Gains from instruction tuning for different families on three latent skills. Major findings include large and positive impact instruction following and negative impact mathematical reasoning.
Reasoning
Knowledge:
Instruction following
Knowledge 2
Triring Inllonn
Thipiing bolerne coogra triliven))
3.90
Thebiing tolwens Deego triibben))
0.77
0.7
6.75
8.75
(lgu
0.22
6:2
6.29
8.35
-2.8
-8.25
-6.23
-0.23
00
Size Dogua biliens ofparameters)
Size Ooqua billions of parameters)
Size (kogo billions parameters)
Size (loguc billors of parametar
Figure 29: Level curves producing different latent abilities from parameter count and training tokens.
35

---

1.0
Namaltzd S.cor
0.8
0.99
2.6
1- 0.44
2.4
0.07
Data
Figure 30: Predicting model performance complex downstream tasks like emotional intelligence ("EQ"" for LLaMa 70B (basc/instruct). In the step, we fit Sloth without including L.LaMa 70B (hase/instruct) the training set. In the second step, we fit regression model connecting skills and downstream performance. Finally, predict L.LaMa 3 70B (instruct) performance from their predicted Sloth skills.
Extra downstream task plots
J.1
Emotional Intelligence
J.2 Agentic Capabilities
Traning Colo
Ixms-2-775 2-28-a-7
AperBonch Acarocy
+0.57
-0.6a
4.76
60.67
-8.36
-0.45
0.04
-0.28
MeLL
AfCiC
Hollswway
GDMM
Win de
HomannFrl
-3.75
-359
-3.25
-1.00
.035
-025
0.00
Figure 31: Predicting Agentic Capabilities of Llama-2-70B-chan
3.44
3.87
0.10
0.20
0.01
-0.25
0.34
0.41
0.38
0.42
0.17
0.08
0.62
0.73
0.77
0.02
2
0.03
-0.00
-0.10
0.01
0.01
0.03
0.06
0.31
0.05
0.04
-0.02
0.53
GSMAK
MATH 5
A
MMU
MHLLLPRRO
BSH
MUSA
nuthfuugA
ARC
HellaSwag
Winggande
iFEva
Figure 32: Loadings for downstream prediction tasks.
K Insights from the different link functions
In this section, we visually compare Sloth considering trainable and logistic link function a,, Owen [2024]'s model ("FLOPs (shared intercept)") and our adaptation of Ruan et al. [2024]'s observational scaling law ("PCA + FLOPs") deseribed in Appendix For this experiment, we study the two Open LLM Leaderboards separately and consider LLaMa-3 and Yi-1.5 families as the test families; we make this choice because both families are popular ones and the training set size the same models in each family, making comparison between models possible (in the x-axis, we use model size). For LLaMA-3, we just include one model from that family in the training set and not train a family-specific slope for PCA+FLOPs. For Yi-1.5, we include two models in the training
36

---

set and train function can produce much more flexible scaling law that can better predict performance saturation (e.g., the performance of Yi-1.5 in ARC, HellaSwag efc.), (ii) training no family-specific parameters all ("FLOPs (shared intercept)") usually produce poor prediction results, and (iii) PCA+FLOPPs often produces flatter curves that underestimate the performance of bigger models, 6.g. see Yi-1.5 in TruthfulQA, GSM8k, and MMLU.
slope for PCA+FLOPs. In summary, we see that: (i) training the link
Family: meta-llama-3
80
TrutffuiA
GSMBK
50
50
80
100
HellaSwag
Winagrande
ARC
60
80
40
10a
10a
10
10a
10
Size (billions of parameters)
Size (billions of parameters)
Size (billions parameters)
train
test
[d=3)
intercept)
Figure 33: Prediction curves for different methods considering Open LLM Leaderboard benchmark and the LLaMa-3 as the test family.
Family: meta-llama-3
MATH Lvi.5
GPDA
40
MMLU-PRO
50
20
30
25
10a
10
102
10a
10a
10a
10"
104
10
75
50
BBH
MUSR
IFEval
50
25
107
10'
10a
10a
10a
10a
10"
10a
104
Size (billions parameters)
Size (billions of parameters)
Size (billions of parameters)
train
test
Sloth basic (d-3) 3)
Sloth (d=3)
(Shared intercept)
FLOPs
PLO::
PCA+ FLOPs (d-3)
Figure 34: Prediction curves for different methods considering Open LLM Leaderboard 2 benchmark and the LLaMa-3 as the test family.
37

---

Family: yi-1.5
GSMBK
MMLU
75
Trushnfuli
50
50
10
10
10'
80
HellsSwag
Winegrande
90
ARC
60
80
10
10
Size (billions parameters)
Size (billions parameters)
Size (billions of parameters)
Sloth
Sloth
OLOPs
FLOPs
train
test
[d-3)
3)
(shared intercopt)
(d-3)
Figure 35: Prediction curves for different methods considering Open L.M Leaderboard benchmark and the Yi-1.5 as the test family.
Family: yi-1.5
MATH LVF5
40
MILU PRD
60
40
20
40
0
10
10
10a
10'
50
BBH
60
MUSR
45
IFEval
40
40
20
10
10
Size (billions parameters)
Size (billions metem))
Size (billions of parameters)
Sloth
Sloth [dl33) 3)
OLOPs
PCA+FLOPs
train
test
[d-3)
(chared intercopt)
(d-3)
Figure 36: Prediction curves for different methods considering Open LLM Leaderboard 2 benchmark and the Yi-1.5 as the test family.
L Comparing against Ruan et al. [2024] in their observational scaling law setting
In this section, we compare Sloth with Ruan et al. [2024]'s observational scaling law; that is, extract abstract skills using of benchmark scores and then those skills to predict the performance of models of interest in target downstream task. For this experiment, we use the same data and tasks explored in Section 4.4. For our method, we fit Sloth using benchmark data from all models, including performance data of LLaMa-3-70B models, and extract the skills of each model. For Ruan al. [2024]'s method, we fit PCA on the benchmark data extract the skills. For both methods, we set 3 and then fit a regression model with logistic link to predict downstream performance from skills. Figures Figures 39 and 40 give the loading of both approaches. In both plots. out-sample prediction has similar prediction error. At the same time, in-sample fit is better for Sloth in the coding task and for Ruan et al. [2024]'s observational scaling law in the emotional intelligence task. Regarding the loading, possible to draw some similarities, e.g., the presence of instruction following skill, but there is no one-to-one correspondence between skills.
and present the prediction results for both methods and
38

---

6+
Pras
1.05
NE (EAcooo
1.52
0.44
6.2
Bl. 4.C
Flg18roa) +
Figure 37: Predicting code completion of L.LMa 70B (base/instruct) with Sloth vs Obs Scaling Law [Ruan et al., 2024].
Auo
Nomatzed EQ Score
1.04
0.0
Traiiing
Mommabed
1.34
.1.99
6.-
Pig( +
Figure 38: Predicting emotional intelligence of LLaMa 3 70B (base/instruct) with Sloth Obs. Scaling Law [Ruan al., 2024].
0.03
-0.02
-0.21
-0.01
-0.00
0.02
-0.02
0.29
0.06
0.03
-0.02
0.57
1
-0.52
-0.63
0.06
0.11
0.07
0.15
0.08
0.01
0.44
0.56
0.57
0.05
3.54
3.56
2.24
1.73
1.80
1.69
0.85
0.75
0.68
0.52
0.65
0.33
CSM9K
MATH L
5
GPOA
MeLU
MALL-PRO
BBN
MUSA
muthfurQA
ARC
HellaSwag
Winogrande
iEevl
Figure 39: Loadings for downstream prediction tasks (Sloth).
39

---

0
0.62
0.13
0.07
0.43
0.30
0.24
0.05
0.16
0.29
0.26
0.19
0.23
0.64
0.12
0.04
-0.09
0.00
-0.02
-0.02
-0.06
-0.37
-0.53
-0.27
-0.27
-0.05
0.04
-0.07
-0.11
-0.01
-0.04
-0.03
0.26
-0.16
-0.29
-0.20
0.87
GSMBK
MATH Lui
GPOA
MALU
HALLLPRO
BBH
MUSR
TrttfuOA
ARC
HellaSwag
Winograrde
iFeval
Figure 40: Loadings for downstream prediction tasks (bss. Scaling Law [Ruan al., 2024)).
40

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Reviews:
[{"review_id": "ZYJttaq4T4", "reviewer": "Reviewer_TtJM", "summary": "This paper proposes Sloth, a new class of scaling laws that predicts large language model (LLM) performance across benchmarks and model families. The core idea is to represent LLM capabilities using a small set of interpretable latent skills (like reasoning and knowledge), which scale with compute. Sloth leverages leaderboard data and allows for family-specific efficiency without requiring multiple models per family. It supports flexible, optionally learned activation functions to improve prediction accuracy. The method is validated across 12 benchmarks and used to forecast downstream task performance for unseen, larger models.", "questions": "Please see Weakness part", "limitations": "Please see Weakness part", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. The paper is very well written, with clear structure and logical flow throughout.\n\n2. The authors conduct thorough ablation studies, which help isolate the contribution of each component in the proposed method.\n\nWeaknesses:\n\nI did not find major weaknesses in this work. \n\n1. I’m particularly interested in the design choices behind the proposed scaling law formulation. It would be helpful if the authors could share more motivations and potentially failed alternatives. Why is each skill slope shared across families in Equation (3.2)?\n\n2. I’m not fully convinced by the interpretation of benchmark-skill associations in Section 4.3. In particular, I believe MMLU might be more appropriately categorized under knowledge-based skills rather than reasoning.", "was_revised": true, "final_justification": "The authors have addressed my concerns.\n\nI keep my score.", "rebuttal": "Thank you for your work on our paper! We have addressed your comments and questions below. Please let us know if you have further questions or concerns.\n\n------- \n\n### **Design choices behind our scaling law**\n\nOur scaling law is informed by the shortcomings of previous benchmark scaling models, such as those proposed by Owen (2024) and Ruan (2024). We build on these approaches by explicitly decoupling the effects of training tokens and model size, which were previously conflated through the use of FLOPs. Furthermore, we introduce an interaction term, discussed in Appendix F, which further improves the flexibility of the model. While we also explored higher-order (quadratic) terms, we found these unnecessary for capturing the relevant trends.\n\nAs highlighted in Section 2 and in our experimental results, earlier models tended to be either overly flexible, making both intercept and slope family dependent, which hinders reliable estimation when only a few models are available per family, or too constrained, failing to capture differences across LLM families. In our setting, allowing slopes to vary by family would be even more challenging given that our model incorporates more covariates than previous work. By choosing to make only the intercepts family-specific while keeping the skill slope shared, we achieve a balance: Our approach enables robust extrapolation to larger models based on observations from just a few (or even a single) smaller model(s). Beyond these empirical findings, there is also a theoretical rationale for favoring family-specific intercepts with shared slopes, which we outline in the following section.\n\n**More intuition behind our formulation with family-specific intercepts (and shared slopes):** This formulation is inspired by classical models in economics (as mentioned in the text), where intercepts represent the efficiency of a firm or production unit. Similarly, we interpret the family-specific intercept as a measure of 'model family efficiency', directly affecting how input levels (such as training tokens or model size) translate into performance. This 'model family efficiency’ term could absorb things like the quality of the training data or changes in architecture.\n\nTo make this concrete, consider a simplified case, assuming the benchmark performance is affected by only one skill in Eq 2.2, that $\\gamma=0$, and in which we assume no interaction term of $\\log s$ and $\\log t$. Concretely, assume that the expected performance in that benchmark is given by\n$$\n\\sigma(\\alpha_i + \\beta^\\top x(s, t)) = \\frac{e^{\\alpha_i + \\beta_1 \\log s + \\beta_2 \\log t}}{1 + e^{\\alpha_i + \\beta_1 \\log s + \\beta_2 \\log t}}\n= \\frac{A_i s^{\\beta_1} t^{\\beta_2}}{1 + A_i s^{\\beta_1} t^{\\beta_2}},\n$$\n\nwhere $A_i = e^{\\alpha_i}$ captures the efficiency of family $i$.\nThis highlights how the impact of model size ($s$) and training tokens ($t$) on performance fundamentally depends on the family intercept $\\alpha_i$: for families with higher efficiency ($\\alpha_i$), increases in $s$ and $t$ yield a bigger impact on performance. This intuition first guided our choice to make only the intercept family-specific.\n\nIn summary, our design choices are grounded in both empirical considerations (robustness and generalizability across families) and theoretical intuition (interpreting intercepts as efficiency), resulting in a model that is both interpretable and practically effective.\n\n--------- \n\n### **Benchmark-skill associations**\n\nThank you for this observation. When we refer to \"Knowledge,\" we specifically mean “common-sense knowledge,\" (as explained from line 287 onwards), and when we refer to \"Reasoning\", it can also incorporate advanced STEM knowledge (which is the case of MMLU). To clarify, we will label \"Knowledge\" as “Common-sense knowledge\"  and \"Reasoning\" as \"Reasoning and advanced knowledge\" in the text. Thank you!"}, {"review_id": "q3DJvSE9W7", "reviewer": "Reviewer_9m8f", "summary": "Due to the inherent differences across model families, existing scaling laws often fail to generalize universally across LLMs. Constructing separate scaling laws for each model family, however, incurs substantial computational cost. To address this challenge, this paper proposes Skills Scaling Laws (SSLaws, Sloth), a method that leverages publicly available benchmark data and assumes LLM performance is driven by a set of low-dimensional latent skills—such as reasoning and instruction following. By exploiting the correlation structure across benchmarks, Sloth enables more accurate and interpretable performance prediction, while alleviating the need to train multiple models per family. Empirical results demonstrate that Sloth accurately predicts LLM performance and provides valuable insights into downstream scaling behavior, test-time compute efficiency, and compute-optimal skill development.", "questions": "1. Are the identified latent skills strongly correlated with the choice of benchmark datasets? How does the number of these skills influence the model’s predictive performance?\n2. If certain benchmarks are potentially contaminated during model training, to what extent can their evaluation results be considered trustworthy?\n3. Beyond predicting model performance on benchmarks, can this scaling law offer guidance for model training or the adjustment of specific capabilities? Given that benchmark datasets may not always faithfully reflect the full spectrum of a model’s abilities, it would be valuable to understand whether the proposed method can inform capability refinement or training strategies.", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths:\n1. By leveraging publicly available benchmark data and the concept of latent skills, this work proposes a resource-efficient approach to model performance prediction. In addition to reducing the need for training multiple models, the method offers valuable insights into the interpretability of model capabilities.\n\nWeaknesses:\n1. The main method proposed in this paper is quite similar to that of Ruan et al., but the fundamental differences between the two are not clearly explained in the related work section or in the subsequent content.\n\n\n[1] Yangjun Ruan, Chris J. Maddison, and Tatsunori Hashimoto. Observational Scaling Laws and the Predictability of Language Model Performance, July 2024.", "was_revised": true, "final_justification": "The author’s response provided clarification regarding the similarities with Ruan’s work, which helped resolve some of my concerns. As a result, I have decided to raise my original score.", "rebuttal": "Thank you for your thoughtful review of our paper. We have addressed your comments and questions below. Please let us know if you have any additional questions or concerns.\n\n----------------\n\n### **Similarity with Ruan (2024)**\n\nWe respectfully disagree in this point. Our approach differs substantially from Ruan (2024), primarily in three aspects (*all of them explained or explicit in Sections 2.2, 3, 4*):\n1. **Different goals and setup.** While Ruan focuses on only interpreting the skills of trained LLMs (i.e., with no predictive aspect), our method is designed to predict the skills/performance of hypothetical LLMs that have not yet been trained, while retaining the interpretability aspect. *Please check our Section 2.2 for a detailed discussion.*\n2. **Different methodology.** Our scaling law has several difference from theirs: (i) we decouple FLOPs into size and tokens, incorporating an interaction term (explained in Appendix F); (ii) we make the link function $\\sigma$ trainable (if needed); (iii) we propose that slopes do not need to be family-dependent for a good performance; (iv) we propose a new fitting algorithm that allow skills being different from principal components, allowing more flexibility and better interpretability (since skills do not need to be orthogonal, for example).\n4. **Different set of experiments.** All experiments in Sections 4.2, 4.4, 4.5, and 4.6 are not possible within Ruan's setup (and not the focus of their paper) either because they do not focus on the predictive aspect of scaling laws (4.2, 4.4, 4.5) or because they do not decouple model size and tokens (4.6) in their work. Regarding Section 4.2, we attempted to adapt Ruan's model for prediction in our experiments and find that it does not perform well (see, for example, the PCA method in Figure 1).\n  \n---\n\n### **Questions**\n\n**Q1: Are the identified latent skills strongly correlated with the choice of benchmark datasets? How does the number of these skills influence the model’s predictive performance?** Indeed, the latent skills we identify are influenced by the set of benchmarks used; for example, if none of the benchmarks require reasoning, reasoning skills cannot be identified. However, the number of skills primarily reflects the diversity of the benchmarks and LLM abilities, not predictive accuracy per se. Regardless of the true number of skills, we can fit models with any number of skills (see Figure 1). Too few skills will underfit, while too many will overfit.\n\n---\n\n**Q2: If certain benchmarks are potentially contaminated during model training, to what extent can their evaluation results be considered trustworthy?** Contamination is a concern for skill estimation, as it is for any application involving AI evaluation and benchmarks. Our approach does not eliminate this challenge, and it is an important caveat for interpreting results.\n\n---\n\n**Q3: Beyond predicting model performance on benchmarks, can this scaling law offer guidance for model training or the adjustment of specific capabilities? Given that benchmark datasets may not always faithfully reflect the full spectrum of a model’s abilities, it would be valuable to understand whether the proposed method can inform capability refinement or training strategies.** Yes, our scaling law can be used to guide training strategies. As described in Section 4.6, we provide a concrete example of how the model can be used to optimize resource allocation when training LLMs, maximizing desired skills within a fixed compute budget."}, {"review_id": "sJfuEFgC0S", "reviewer": "Reviewer_mbUa", "summary": "This paper proposes to include model family to estimate scaling laws across data and models.", "questions": "- How much data/benchmarks are needed to make predictions for a new model family? Is this exacerbated given the heavy tailed nature of the data? How does this relate to the diversity of benchmarks? \n- How much does the predictive performance change as you add more samples? Given that \n- In Figure 6, sloth overestimates the performance of pythia, can you elaborate for which scenarios the proposed model  over vs under estimates the performance?", "limitations": "yes", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 1, "strengths_and_weaknesses": "The paper is well written, and provides a through analysis across benchmarks. \nThe authors claim that family specific intercepts are not always needed (Figure 3), and add some worst case analysis to their results. However, these conclusions should be highlighted in the main test\nThe method sees little benefit from adding more samples within a model family as long as there is enough data available for a wide variety of datasets; however, outside of the model family the model can under/over estimate performance. It is unclear when it is biased towards either failure model.", "was_revised": true, "final_justification": "As I mentioned, I appreciate the proposed framework in the paper, which leverages information from multiple benchmarks to predict model performance within families. However, as it stands “under/overestimation is sample-dependent and can vary by scenario.” This currently limits the conditions under which users can reliably apply the method.", "rebuttal": "Thank you for your work on our paper. We have addressed your comments and questions below. Please let us know if you have further questions or concerns.\n\n---\n\n### **Questions**\n\n**Q1: How much data/benchmarks are needed to make predictions for a new model family? Is this exacerbated given the heavy-tailed nature of the data? How does this relate to the diversity of benchmarks?**\n\n- For prediction, our approach can make reliable predictions for a new model family using as little as a single benchmark (the \"Size and Tokens\" results in Figure 1 illustrates this), and a single model per test family (full Figure 1).\n- For skill estimation, once the Sloth weights are fitted, it’s sufficient to observe scores on $d$ (the number of skills) benchmarks, assuming they’re reasonably diverse, to estimate the family-specific parameters ($\\alpha_i$) for the new family. This flexibility even accommodates missing data, though in practice, gathering benchmark results is not a significant bottleneck for the type of benchmarks we use.\n- Importantly, we do not see how heavy-tailed data can affect negatively the results.\n\n\n**Q2: How much does predictive performance change as you add more samples?**\n\nAs expected for most machine learning methods, increasing the number of samples improves accuracy. In practice, however, we find that having just one model per family in the training set is often sufficient (please see Appendices J.2 vs J.3).\n\n\n\n**Q3: In Figure 6, sloth overestimates the performance of pythia. Can you elaborate on scenarios where the model over- vs. under-estimates performance?**\n\nThere is no general rule: under/overestimation is sample-dependent and can vary by scenario, as is common in machine learning.\n\n---\n\n### **Clarifying points** \n\n- From your review, we understand that there may be some misunderstanding about our work. Our contribution is not simply proposing “to include model family to estimate scaling laws across data and models”; we propose a **scaling law for LLM skills** that **decouples model size and training tokens** and uses **family‑specific intercepts with shared skill slopes** to enable reliable extrapolation both within a family (from small to large models) and to **new families with minimal observations**. We are happy to clarify in more detail if needed. Please let us know if there is still anything unclear. \n\n- From your review ratings, you are mostly concerned with novelty and significance of our work while this is not reflected by your written review. For an explanation on novelty, please see our response to Reviewer 9m8f. For significance, you do not mention any serious limitation of our work in your review that could justify it; moreover, we have provided a diverse and practical set of five realistic applications in which our method could be applied to. Could you clarity these two points, please? Thank you.\n\nWe are open for discussion and happy to clarify any misunderstanding about our work."}, {"review_id": "2bTyJYE6ba", "reviewer": "Reviewer_vnid", "summary": "The core contribution of this paper is the proposal of Skills Scaling Laws (SSLaws), or \"Sloth\". This new class of scaling laws aims to address the limitations of existing scaling laws, which often struggle to generalize across different LLM families due to variations in training configurations and data processing. Sloth posits that LLM performance on various benchmarks is driven by a set of low-dimensional, interpretable latent skills, such as reasoning, knowledge, and instruction following. Instead of directly modeling benchmark scores as a function of computational resources (like model size and training tokens), Sloth models how these underlying latent skills scale with compute.\n\nA key innovation of Sloth is its ability to leverage publicly available benchmark data across multiple LLM families. It assumes that while the efficiency of converting computational resources into skill levels may vary across families, the fundamental influence of resources on skills is similar. This allows Sloth to \"borrow strength\" across benchmarks and families, leading to more accurate and interpretable predictions. Furthermore, Sloth provides theoretical results on parameter identification and demonstrates practical applications, including predicting performance in complex downstream tasks, understanding behavior with scaled test-time compute, and deriving optimal-scaling rules for skills given a compute budget.", "questions": "Q1: why is theta considered to be latent skills or `reasoning'? What's your clue and what's the justification?\nQ2: If I understand correctly, your alpha is family-dependent variable. How do you model the family with only the smallest model given, as described in L223-236? \n\nQ3: l252, Which figure shows the results of the two observed models? Your link directs to a bunch of figures.", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 4, "originality": 3, "strengths_and_weaknesses": "Strength: \nThis paper has several clear strengths. \n1. Address important limitations of previous work: I appreciate the authors' taking Section 2 to compare against previous studies and better position the paper's contributions. Previous work either ignores the family-dependent feature of the scaling law or has the model first then predict the performance on non-evaluated benchmarks. This paper addresses these limitations by encoding the correlation structure between benchmarks. \n\n2. Practical experiments and setups: The Authors evaluated their methods in a practical setting where only one or two models with the smallest sizes are trained, and predict the performance of the large models. \n\n3. Improved predictive accuracy and generalizability: Experimental results consistently show that Sloth performs competitively, often achieving similar or lower mean absolute errors than baselines, even for recent and diverse LLM families like LLaMa 3, Qwen 2, and Yi 1.5. \n\nWeakness: \n\nI found no critical weaknesses.", "was_revised": true, "final_justification": "I think this paper is of good quality and the author's response to my question looks good to me. Thus, I retain my score.", "rebuttal": "Thank you for taking the time to review our paper. We have responded to your comments and questions below. If you have any further feedback or concerns, please feel free to let us know.\n\n-----------\n\n### **Questions**\n\n**Q1: Why is theta considered to be latent skills or “reasoning”? What is the justification?**\n\nThe $\\theta$ parameters are low-dimensional embeddings for each LLM, capturing what the models are capable of. We demonstrate that model performance can be predicted using just these tiny embeddings. Referring to them as “skills” or “abilities” follows conventions in psychometrics, which studies latent traits like human intelligence using models like ours (e.g. factor analysis, item response theory, etc).\n\n---\n\n**Q2: If I understand correctly, your alpha is a family-dependent variable. How do you model the family with only the smallest model given, as described in L223-236?**\n\nWhen fitting Sloth (see the optimization problem in Appendix D), we only need to observe a single LLM from the target family to estimate its family-specific parameter ($\\alpha_i$). This is analogous to fitting a linear regression with group-specific intercepts, often referred to as fixed effects in statistics. Naturally, for some families (not necessarily the target of interest), we do need to observe models of different model/training set sizes to estimate the shared slope parameters ($\\beta_k$’s). We give you a brief intro to linear regression with group-specific intercepts below, so it can solidify the intuition. If anything remains unclear, please let us know.\n\n*Linear Regression with Group-Specific Intercepts (Fixed Effects):* In a standard linear regression, we model the outcome $y$ as\n\n$$\ny_{ij} = \\beta_0 + \\beta_1 x_{ij} + \\epsilon_{ij}\n$$\n\nwhere $i$ indexes groups (e.g., model families) and $j$ indexes observations within group $i$.\n\nTo account for group-specific baseline differences, we extend the model by allowing each group to have its own intercept:\n\n$$\ny_{ij} = \\alpha_i + \\beta_1 x_{ij} + \\epsilon_{ij}\n$$\n\nwhere $\\alpha_i$ is the intercept specific to group $i$. Here, $\\beta_1$ is shared across all groups, but each group can start from a different baseline level via $\\alpha_i$. The fixed effects $\\alpha_i$ can be estimated for a target group by only observing one individual in that group as long for some training groups we observe more than one individual (needed to estimate $\\beta_1$).\n\n\n---\n\n**Q3: l252, Which figure shows the results of the two observed models? Your link directs to a bunch of figures.**\n\nThank you for catching this! Figure 15 is directly analogous to Figure 1 and presents the results for the two observed models. We will clarify this in the revision."}]

Author rebuttal:

Meta-review: # Summary
This paper proposes skills scaling laws (SSLaws) that trains a model to predict the performance of a model from a specified family on a specified benchmark, given the model size and the amount of training tokens. The SSLaws model is trained on a predefined set of model families and a set of benchmarks, and can generalize to untrained models associated with an unseen configuration of (model size, training token number) from one of the model families. Compared to existing scaling law models, different benchmarks in SSlaws share the same representation space defined by low-dimensional latent skills, and the effect of increasing model sizes and training tokens is reflected on the skill representations. Experiments demonstrate that SSlaws accurately predicts LLM performance and provides valuable insights into downstream scaling behavior, test-time compute efficiency, and compute-optimal skill development. 

# Strengths
- The paper is well-written with a clear logical flow.
- The ablation studies and experimental results are comprehensive and show the advantages of SSlaws.
- The generalization from one or two small models per family to other model configurations is non-trivial.
- The method captures the correlation within the same model family and the shared skills, which are novel compared to existing works.

# Weaknesses
- The paper can be improved by providing more discussion and clarification about the motivations, other alternative approaches, and benchmark-skill associations.
- It is not fully clear how the approach can predict other specific capabilities of LLMs and guide training. 
- A more detailed comparison to some related works is needed.
- Insufficient analysis about the main factors affecting the prediction accuracy, e.g., the number of training data. 

# Reasons to Accept
- The paper proposed novel parameter-sharing strategies (latent skills, skill slopes, etc.) to improve the generalization of the scaling law prediction model. 
- The proposed method is effective across different model families and benchmarks in the experiments.
- Most reviewers support the acceptance. 

# Discussion Summary
- In the rebuttal, the authors provided detailed responses with additional details to answer the questions of the reviewers.
- All reviewers confirmed that they are satisfied with the responses.
Decision: Accept (poster)