A Taxonomy of Transcendence
Authors: Natalie Abreu
Edwin Zhang
Eran Malach
Naomi Saphra
Venue: COLM 2025 (2025)
Keywords: language models
data diversity
composition
knowledge graph
TL;DR: We propose a controlled setting in which to study how properties of the pretraining data influence the model's ability to transcend the performance of the sources that generated the data.

Abstract: Although language models are trained to mimic humans, the resulting systems display capabilities beyond the scope of any one person. To understand this phenomenon, we use a controlled setting to identify properties of the training data that lead a model to transcend the performance of its data sources. We build on previous work to outline three modes of transcendence, which we call \textit{skill denoising}, \textit{skill selection}, and \textit{skill generalization}. We then introduce a knowledge graph-based setting in which simulated experts generate data based on their individual expertise. We highlight several aspects of data diversity that help to enable the model's transcendent capabilities. Additionally, our data generation setting offers a controlled testbed that we hope is valuable for future research in the area.

Published as a conference paper at COLM 2025
A Taxonomy of Transcendence
Natalie Abreu, Edwin Zhang' Eran Malach & Naomi Saphra Kempner Institute for the Study of Natural and Artificial Intelligence Harvard University feabre,, @g harvard. edu, (emalach, (saphara @saphra) haaraaarrddd edu
Abstract
Although language models are trained to mimic humans, the resulting systems display capabilities beyond the scope of any one person. To under- stand this phenomenon, we use a controlled setting to identify properties of the training data that lead model to transcend the performance of its data sources. We build on previous work outline three modes of tran- scendence, which we call skill denoising, skill selection, and skill generalization. We then introduce a knowledge graph-based setting in which simulated experts generate data based on their individual expertise. We highlight sev- eral aspects of data diversity that help to enable the model's transcendent capabilities. Additionally, our data generation setting offers controlled testbed that we hope is valuable for future research in the area.
1 Introduction
Language models are trained to mimic human behavioral data. This mimicry makes it tempting to anthropomorphize system-to think of it like a person. However, not only is the model not a person, is not even trained to mimic a person. Instead, the model has been trained to mimic group of people with individual capacities, predilections, and biases. In some circumstances, this combination of multiple people augments their shared biases (Bender et al., 2021), but we also see the enormous advantage of training on data from diverse set of people: often, it is possible to outperform any individual member of that group. The capacity of generalist model exceed individual ability is evident chatbot that can converse with equal competence about cryptography, international law, and the work of Dostoevsky. Our goal is describe the circumstances which model, trained to mimic multiple people, is capable of transcending its sources by outperforming each individual.
For group of humans, there are several ways to outperform any one individual in the group. One path is to allow the group to vote, counting each individual vote equally, thereby averaging out the bias of each individual. Another path is to route disparate knowledge by allowing a cryptographer to handle issues of cryptography and a lawyer to handle issues of international law. Finally, group of specialists can use their shared understanding of the world as an anchor point to combine their knowledge, as when a cryptographer and lawyer reason together through the legality of an algorithmic embargo. Like these human examples, an artificial model trained as a generalist can outperform the capabilities of the individual specialists it is trained emulate, under the following conditions:
Skill denoising: When the biases of each specialist cancel out in their average this is classically described as "wisdom of crowds" or a voting ensemble. This scenario applies when all specialists produce data relevant to an input context, but each may make independent errors.
Skill selection: When different specialists are experts on different parts of the context space. A given input may be common for one specialist and rare for another,
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and reasonably assume that the specialist who frequently encounters such inputs is better equipped to handle them. By routing to the appropriate expert, the generalist model can address wider range of topics than any individual alone. This scenario applies when the input context familiar to at least one specialist.
Skill generalization: When specialist expertise can be combined by using assump- tions of shared underlying structure. Prior assumptions about how to generalize from training data allow the generalist model to interpolate, reason, and compose knowledge across domains by representing inputs in shared semantic space. This scenario applies when the input context is unfamiliar to all individual specialists but can be understood through generalization.
In this paper, we examine the conditions under which a model can transcend its training sources, organizing them into taxonomy of modes of transcendence. Furthermore, connect each type of transcendence to conditions in the training data, showing that the model can transcend its sources only through sufficiently heterogeneous set of experts. Our contributions are as follows:
Building on definitions from Zhang et al. (2024), we formalize three modes of transcendence: skill denoising, skill selection, and skill generalization.
We propose conditions that the training data must hold in order to achieve each mode of transcendence, and build intuition using simple theoretical settings.
Using a synthetic knowledge graph-based setting, we empirically confirm the necessary conditions for each mode of transcendence.
We hope this will provide useful framework for further theoretical and empirical work studying the ability of imitative models to achieve transcendent capabilities.
2
Definitions
2.1
Preliminaries
We borrow much of our notation from Zhang et al. (2024) to align with their definitions. Let X' 7 denote the input space and y 7 denote the output space over set of tokens T. Let F be a class of functions mapping X' to P(Y)) where P(.) denotes probability distribution over the given space. Each function f F then defines a conditional probability distribution of E y given E X' that we denote by fy/xx)
Let there be k experts, each associated with conditional probability function F. Let Pi....,Pk Pk denote the respective input distributions over X for each expert. Define the average input distribution as p(x) Ek_ Pi()), and let supp(p) denote its support. Define the mixture model f(ylx) = E fi where denotes the conditional probability that input is observed under expert
Then, each expert i induces distribution D; over X' x y, defined by Dx(x,) px(x)fi(yxx) Let D be the mixed distribution (xx,y)
We measure the quality or skill level of each expert with reward function r: r: - R. Specifically, for distribution p over X' and some F, define the average reward of under p as:
Rp(f) Ex-p [xx(f), where rx(f) Eyys(11)) [r(x,y)]
For some fixed hypothesis class H C (h: X + P())), the learner chooses some function ho E H that minimizes the expected cross-entropy with the mixture model f:
hD arg min Ex-p(H((()) (x) [x)]
REH
where H is the cross-entropy function.
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We use the definition of "transcendence" from Zhang et al. (2024), as follows: For some test distribution Prest over X, we say a model achieves transcendence
R Rrest (h > max R Prest (fi)
ic(R]
We now proceed formalize three modes of achieving transcendence.
2.2 Skill Denoising
Trained on noisy experts who make occasional errors, model may outperform the experts by simply denoising their outputs. This setting characterized by the following simplifying assumptions:
1. Single imput distribution: All experts share the same distribution over X;; that is, for all i, we have pi p.
2. In-domain test distribution:
The support of the test distribution Prest is contained in the support of p, i.e. we have supp C supp(p). In other words, any example that has non-zero probability under the test distribution has non-zero probability under the training distribution.
As studied in Zhang et al. (2024), transcendence in this setting can be achieved by low temperature sampling as long as expert errors are uncorrelated. Low temperature sampling returns the mode of the predicted output distribution, which corresponds to a "wisdom of the crowd" approach.
2.3 Skill Selection
In this setting, drop Assumption 1 from the denoising setting. Without this assumption, experts may have differing distributions over the input space. The resulting specialist experts each have specific expertise, subset of inputs These subsets vary across experts. This setting maintains Assumption that the support of the test distribution is contained in the support of the train distribution.
which they will predict correct output.
We claim that transcendence can be achieved when a given input context is more likely to be observed for experts that achieve a higher expected reward on that context. Specifically, the probability g(ixx) of sampling expert i on context x is function dependent on x rather than constant function. Intuitively, this reflects scenario in which experts are more likely than non-experts to encounter and respond to inputs within their domain for example, lawyer is more likely comment on questions of law.
As simple example, we analyze the case of two experts.
Theorem 2.1. Let and b be two experts. For transcendence to hold, we must have that
E Ex-pust [(xx(a)) - rx ((6))(((()))) g(blx))]
That is, the excess probability of seeing given example under expert compared to expert is correlated with the excess reward of expert a on that example. We provide the proof in Appendix A.1.
2.4 Skill Generalization
In this setting, we drop Assumption 2, that the support of the test distribution is con- tained in the support of the train distribution. More strongly, will now assume that (upp(P n (p) = Ø that is, the inputs at test time are never seen during training.
How model answer correctly if no individual expert can? If expert knowledge is representable in a shared latent space, the model can compose knowledge from different experts, producing knowledge outside the scope of any one expert's knowledge.
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Denoising
Selection
Generalization
Figure 1: Illustration of the expert distributions in our knowledge-graph setting. Edges in blue and orange denote facts that the experts will respectively get correct, and the remaining edges they will get wrong. Edge opacity represents the probability of an expert generating sample based on that edge. From left to right: (1) Denoising setting: experts have unbiased errors and uniform probability of generating each fact. (2) Selection setting:
specialized experts are more likely to generate data within their expertise. (3) Generalization setting: subcase of the setting of specialized experts. For simplicity in our generalization experiments, we set the probability of an expert generating an incorrect fact to
One hop
Nerythra
The sibiing Norythra Glyndorath. country citizenship Glyndorath Crystalia. The occupation of Glimmerdsww Zephynwaver.
xhilng
dittenn
Clyndorath
aftiemn
Zophyyweaver
Crystalia
Two facts:
Trasidany
Within expertise:
Indrlath
The country of citizenehip of the sibling of Nerythrs Crystalia. occupation the spouse
Zephyrweaver.
SporTo
Across expertise:
president Crystalio Glimmerdawn.
Figure 2: Example visualization of the knowledge graph with fictional entities. The blue and orange edges represent the knowledge of two different experts. A subset of nodes and edges are labeled with example entities and relations. Two-hop facts, as used the skill generalization experiments, are split into "within expertise" (known by least one expert) and "across expertise" (known by no expert) categories.
We evaluate this idea in a two-hop fact completion task over a knowledge graph. Each expert observes and labels set of one-hop facts, and is able answer certain two-hop queries composed entirely from their own knowledge. However, the test queries require combining information from multiple experts: no single expert has access both necessary one-hop facts.
learner biased toward simple solutions, and if the compositional structure of the task is simpler than memorizing all two-hop inputs, the model may generalize by composing reusable one-hop components. This enables to answer novel two-hop queries that no individual expert could answer. We explore this mechanism more formally in Appendix A.2.
Knowledge graph setting
Our experiments use a synthetic knowledge graph and corresponding corpora of natural language sequences describing the relations in the graph. Consider the toound-truth knowledge graph G consisting of a set of nodes (representing entities) and edges (representing relational facts between entities). Each fact is represented by tuple (head,
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relation, tail), where the head and tail are entities in the graph. We create our knowledge graph by taking the structure of the WIKIDATA-based graph from Cohen et al. (2023) and using GPT-4o-mini (OpenAI, 2024) to replace the entities with fictional names. The result is a realistically structured knowledge graph populated by facts unseen during pretraining (See Appendix B.1 for further details).
In a set of n, experts, expert i has a personal knowledge graph Gj that contains their knowledge of the world. Each expert is assigned predefined amount of correct knowledge from the ground-truth graph, along with a number of incorrect beliefs.
Incorrect beliefs are modeled by introducing corrupted edges into the expert's graph. Specif- ically, for a fact (head, relation, tail) from the ground truth knowledge graph, we generate corrupted version by replacing either the head or the tail with a different entity of the same type. Each entity in the graph is assigned semantic type (e.g,, country, person, occupation), and substitutions are restricted to entities of the same type preserve syntactic plausibility.
For skill selection and skill generalization, we use spectral clustering over the edges to define potential areas of expertise for our synthetic experts. We split the graph into 5,000 clusters of edges. Each expert is then assigned knowledge of one or more clusters. Further details on how experts are generated in each setting are provided in the methodology sections specific to each experiment.
Training data: Each sample is a paragraph about a specific entity, emulating an expert writing about a particular topic. To generate a dataset of N samples from a set of n1 experts, we have each expert generate N/ne samples unless otherwise specified. To generate each sample, we sample node uniformly at random from the expert's personal knowledge graph and generate a templated sentence for each edge (fact) connected to that node. Templates follow a simple pattern such as: "The (relation) ) of (head) is (tail). "" The sentences are written in random order. See Figure 2 for examples of the templated sentences.
We evaluate each model on its query completion accuracy, computed follows. For each fact in the ground-truth set - represented as a triple (head, relation, tail) we construct a query by removing the tail and prompting the model with the remaining components (e.g., "The (relation) of [head} is "). A prediction is considered correct if the model's output exactly matches the ground-truth tail. For queries with multiple correct tails, we mark the output correct if it matches any correct tail. The query completion accuracy is then defined as the percentage of facts for which the model correctly outputs the tail. In other words, we evaluate the percentage of the ground truth facts the model has memorized.
Evaluation:
Experiment setup: Unless otherwise specified, all experiments finetune a pretrained GPT2 model (Radford et al., 2019). Since the knowledge graph contains entities that are completely fictional, the model's pretraining phase does not include any facts from our knowledge graph. All runs use AdamW (Loshchilov & Hutter, 2019) with learning rate 0.001, weight decay 0.1, 1000 steps of warmup with a cosine decay schedule and batch size 24.
4 Skill denoising
When expert sources commonly make factual strors-butt each makes different errors their collective majority vote is often correct. By leveraging this "wisdom of the crowd", a model trained on their data can transcend the accuracy of any single expert by denoising their mistakes.
Methodology For some number of experts Ne, we assign all experts a shared coverage level e [0, defined as the fraction of the knowledge graph that each expert knows correctly. To create personal knowledge graph G; for expert i, we iterate over each edge in the ground-truth graph: with probability c, we include the correct edge, and with probability 1-c we instead include a corrupted edge as described in Section 3. In our experiments, we vary the number of experts n& proxy for uncorrelated errors. Since errors are sampled uniformly and independently for each expert, increasing the number of experts leads to a more uniform distribution of errors in the resulting dataset.
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Experts=1
1xperts=10
Experts=100
1.0
1.0
0.8
Expert Coverage
Accuracy
0.6
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Sampling Temperature
Figure When trained on noisy experts, the model can leverage low-tenperature sampling to transcend expert skill level.
Results All experiments are run on 10M samples per configuration. For plots where temperature unspecified, greedy decoding (i.e., temperature 0) is used. Accuracy refers to the query completion accuracy defined in Section 3. Consistent with our intuition, we find that with a sufficient number of experts, the model can substantially outperform individual expert skill level. Even for expert coverage scores as low as 0.2, the model achieves over 80% query accuracy when trained with 100 experts. Accuracy across different coverage levels is shown in Figure 4. As seen in Figure 3, low-temperature sampling allows the model to achieve high accuracy, while accuracy at temperature 1.0 remains close to the underlying expert coverage level.
103
Takeaway Data diversity, in the form of uncorrelated errors, enables a model to perform at higher skill level by leverag- ing the "wisdom of the crowd" with low- temperature sampling.
1.0
Accuracy
102 #
Experts
0.5
10
5 Skill selection
0.0
10
In reality, nonexperts often share their mis- conceptions about a topic, so their errors are rarely uncorrelated. We therefore cannot as- sume that a majority vote always provides transcendence through denoising. How- ever, models can still outperform all individ- ual experts across the training set by choos- ing answers from sources with the relevant expertise. For model to transcend the perts by skill selection, it needs training data in which experts comment more on topics within their expertise than topics where they hold common misconceptions.
0.5 Expert Coverage
Figure 4: Expert coverage vs query accuracy one hop facts for the denoising setting. With enough experts, even with low expert coverage, the model is highly accurate.
Methodology As described in Section 3, we partition the edges of the knowledge graph into 5,000 clusters define potential areas of expertise. We set a coverage level c [0,1] shared by all experts, indicating the fraction of the graph each expert knows correctly. For
each expert i, we generate a coverage vector = (s ...., (5000), where each (1) [0,1]
indicates expert i's accuracy on edges in cluster j. This vector is constructed to satisfy:
5000
Σ
S (
|9| =
where denote the set of edges in cluster by C, and |E] is the total number of edges in the graph. To construct expert i's personal knowledge graph, iterate over each edge

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Expert Coverage=0.01
Expert Coverage=0.1
10a
1.0
10a #
Accuracy
Experts
0.5
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0.0
100
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1.0 Generation correctness (a)
0.9
1.0
Figure 5: a vs query accuracy on one hop facts for the selection setting with two different coverage levels. With sufficient number of experts and high probability that experts generate data within their "expertise", the model can achieve high accuracy even when each individual expert knows only fraction of the knowledge graph.
e E E. Let the cluster membership of e, i.e., e E Cj. Then with probability s we include
the correct edge and with probability 1 1 s we replace it with a corrupted version.
To generate training sequences, we sample node uniformly at random from an expert's personal knowledge graph, as in the denoising setting. However, we now introduce another parameter [0, 1] which describes the extent which each expert restricts commentary to their area of expertise. For each fact connected to the chosen node, we write the fact with
as + (1 a), where (1) is the expert's coverage score for the cluster to which the edge belongs (i.e., the probability the edge is correct). In other words, a controls weighted interpolation between the "expertise" scores and the uniform distribution. With 1, fact will be written with probability equal to the expert's level of expertise over the cluster the edge belongs to, and with lower a, we will train more facts outside of the expert's expertise. This provides a mechanism to control the extent to which experts write about what they know versus what they don't know.
probability
Results Each training run is on a dataset 1M paragraphs and is trained for 10 epochs. We set two coverage levels of 0.01 and 0.1 and we vary & for range of values between 0.8 and 1. Increasing the number of experts corresponds with increasing diversity of expertises. As shown in Figure 5, for both levels of coverage, the model can achieve near-perfect accuracy with a sufficient number of experts. Moreover, accuracy consistently improves as increases, Le, as experts are more likely to write about facts within their area of expertise.
Takeaway When non-experts share common misconceptions about subject (i.e. biased errors), transcendence enabled by data diversity in the form of sources with varied expertise. To guarantee this transcendence, experts must write more frequently about topics within their domain of expertise than about those outside it.
Skill generalization
In the skill selection setting, least one expert knows the correct answer question. In skill generalization, by contrast, no single expert knows the correct answer. In order to transcend through skill generalization, a model must compose knowledge from multiple experts by leveraging shared representations of the world. We study this setting using compositional two-hop fact completion task, showing that a model can complete two-hop queries which no expert can by combining diverse expert knowledge in latent space.
Methodology We use a simplified version of the expert generation technique from Sec- tion 5 in which each expert has knowledge of single cluster. For data generation,
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0.5
Standard
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Data Diversity
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Relation Majority
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Direct Connection
Accuracy
Accuracy
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0.34
0.2
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20000
40000
60000
80000
Direct Relation Standard Diversity
Data CoT
Number Two-Hop Examples
ConnectionMajoriy
Figure Left: Across-expertise two-hop query accuracy increases linearly with number of within within-experrtie two-hop examples seen during training. Right: Comparison of methods with full two-hop example training set in terms of across-expertise query accuracy. The standard method refers to training the model paragraphs of one-hop facts well as all 80,000 within-expertise two-hop sentences. For data diversity, we include diverse rephrasings of the one-hop and two-hop samples.
generate samples from each expert proportional to the size of their known cluster. As in previous experiments, the training data consists of paragraphs about entities generated from each expert's personal knowledge graph G;.
To evaluate the model's ability to combine knowledge from different experts, we draw on the framework proposed by Yang et al. (2024), which measures the model's latent compositional ability via accuracy on multi-hop facts for which the model knows the relevant single hop facts. The model is said to have latent knowledge if can correctly complete the multi-hop fact without generating the intermediate entity.
To test compositional generalization, we use across-expertise two-hop facts, facts where the two edges are in different clusters. By contrast, we refer to two-hop facts with both edges in the same cluster as withiin-expertic two-hop facts. (See Figure 2.) To teach the two-hop answer format, we include set of within-expertise two-hop facts in the training data. We withhold a validation set of 6,000 within-expertise two-hop queries during training.
The across-expee expeertiee two-hop facts are unseen during training and require the model to combine specialized expertises. addition to measuring accuracy on one-hop queries, we measure accuracy on the unseen validation set of within-expertise two-hop queries and the test set of two-hop across-expertis queries.
We compare two-hop accuracy to baselines based Yang et al. (2024), which describes two potential "shortcut methods" using simple co-occurrence statistics between entities and relations. We refer to these baselines direct connection and majority relation. The direct connection baseline tests if there is direct one-hop connection between the head and tail. The majority relation baseline measures the accuracy achieved from simply assigning the entity of the correct type (e.g,, country, person, occupation) which appears most often with the second relation of the two-hop fact. (See Appendix B.3 for details).
Experiment Details Since this is more difficult task we use the 1B parameter LLaMA 3.2 model (Llama Team, 2024). However, line with prior findings (Yang et al., 2024; Allen-Zhu & Li, 2024b), preliminary experiments in the base setting suggest that increasing model size offers limited improvement in two-hop performance. We train the model on 6M paragraphs containing one-hop facts about given entity, combined with the training set of within-expertise two-hop facts. We repeat the two-hop facts 20 times each epoch in order to saturate accuracy the training set and train for 10 epochs.
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Results Motivated by our analysis in Appendix A.2, measure how increasing the size of the two-hop training set affects the performance on the unseen across-expertise two-hop facts. Our findings reported in Figure 6. We find increasing the size of the training set steadily increases accuracy; by using all 80,000 within-expertise two-hop facts, the model achieves 34% accuracy on across-expertise two-hop facts, which is nontrivial improvement over the 20% baseline. All models achieve near-perfect scores on one-hop query accuracy.
This suggests core challenge in our setting: the number of two-hop facts known by individual experts is limited. We cannot continue to indefinitely increase the size of the training set. Therefore we explore two alternate methods: phrasing diversity and Chain-of- Thought (CoT).
Phrasing diversity Inspired by Allen-Zhu Li (2024a), we consider how augmenting data to increase the diversity of phrasing may improve model performance. Allen-Zhu & Li (2024a) argued that this type of data augmentation allowed the model to retain factual information while encouraging the model to have better latent representations rather than memorizing context. We experiment with adding phrasing diversity both the one-hop fact paragraphs and the two-hop facts. (See Appendix B.3). We find that adding diversity to the one-hop paragraph samples increased performance from 34% to 37,, but adding diversity to the two-hop samples had impact. We leave it to future work to explore whether more principled forms of data augmentation can enhance multihop capabilities.
Chain-of-Thoughh Several prior works have found CoT (Wei et al., 2023) to be essential in solving knowledge manipulation tasks such as solving multihop queries (Allen-Zhu & Li, 2024b; Prystawski et al., 2023). Indeed, model to use CoT as it reaches over 60% accuracy on across across-expertise two-hop queries (Figure 6). We note that within our framework, when the model uses CoT to explicitly name the intermediate node, it reduces the skill generalization problem into a skill selection problem. Further details on the CoT method are given in Appendix B.3.
see an immediate benefit of allowing the
Takeaway Although skill generalization is more challenging than other forms of transcen- dence explored, language models can nonetheless achieve it. By increasing the diversity of surface forms or phrasing, we can promote this form of transcendence. More noticeably, we promote skill generalization by increasing the diversity of compositions provided in the training data.
7 Related work
Recently, Zhang et al. (2024) formally described the phenomenon of tran- scendence, in which model outperforms the individual skills of the humans who generated its data. Zhang et al. (2024) proposed that when model is trained on noisy data, low temperature sampling denoises the data if the noise comes from uncorrelated errors. As an example, they studied a chess model that achieves a 1500 Elo rating despite being trained to imitate players of rating 1000. Our work extends the transcendence framework to addi- tional settings and cases where expert errors are correlated. Cunningham (2023) similarly delineated ways in which an imitative model can transcend human performance.
Transcendence
Data diversity Knowledge acquisition Our setup is similar to Allen-Zhu & Li (2024a), which used a synthetic biography dataset to show that data augmentation enables more flexible knowledge extraction. While they focus on extraction, we also study multi-hop composition. Zhu et al. (2025) suggested adding diverse data formats to improve knowledge acquisition. Allen-Zhu Li (2024b) showed that chain-of-thoughe manipulation; we similarly observe benefits but focus on how data diversity supports implicit representations. Naik et al. (2024) studied how diversity in prompting at inference time can improve model's reasoning ability. Chang et al. (2024) also used fictional entities to study knowledge acquisition, but focused on how knowledge is acquired over the course of training. Our finding that skill generalization relies diverse examples of compositions
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also reflects broader conclusion in the compositionality literature concerned with data diversity (Berlot-Attwell et al., 2024; Levy et al., 2023; Oren et al., 2021; Rahimi et al., 2024).
Knowledge composition Several works have studied failures of knowledge composition in Transformers (Dziri et al., 2023; Press et al., 2023; Wang et al., 2024; Yang et al., 2024; Saparov et al., 2023). Press et al. (2023) measured these failures as compositionatity gap, the fraction of compositional questions which are incorrectly answered despite correct answers to their atomic components. They found that this gap does not shrink with model scale. Wang et al. (2024) studied path finding problem in which transformers must learn adjacency and reachability matrices. They found that transformers fail to learn reachability through transitive relationships, i.e. they cannot deduce a full path after seeing its segmented components. Yang et al. (2024) studied whether language models perform latent reasoning when answering questions that involve multi-hop knowledge. They intervened on each component hop to study how the model's 's recall changes when the prompt changes. Their findings suggest that scaling model size can promote better recall for the first hop of reasoning, but not for the second hop.
Skill generalization Many works have studied the ability of neural networks to generalize beyond their training data, in particular through implicit regularization. Most relevant to our setting of skill generalization are Goldblum et al. (2024), which used information theory to argue that models are biased towards solutions with low Kolmogorov complexity, and Zadrozny (2000), which demonstrates that memorization becomes high-complexity, as the corpus grows, causing compositional behavior to arise in a formal language setting.
Ensembling & Model fusion In our work, model might be seen as building an ensemble of the diverse experts who generated its training data. Because the training data reflects an implicit ensemble of experts, our results often mirror findings in the literature model ensembling. These works have studied how to denoise, select, or combine knowledge from the model perspective that is, how to improve performance in these settings using multiple models or agents. For instance, Wang et al. (2023) and Li et al. (2024) demonstrate how majority voting over model's outputs can improve performance. Work on Mixture-of- Expert (MoE) models (Lepikhin et al., 2020; Fedus et al., 2022), ensembling (Liu et al., 2021; et al., 2022; Gururangan et al., 2023) design architectures explicitly for skill selection by combining diverse expert models, whereas we define experts that generate the training data for a simpler architecture. Work on model fusion (Wan et al., 2024; Mavromatis al., 2024, i.a.) is yet another way to combine diverse models for an interpolated ensemble. Li et al. (2022) and Gururangan et al. (2023) train separate models on different clusters of documents and combine models via ensembling for inference, deliberately exploiting data diversity through model architecture.
8 Discussion
In this work, we outline three types of transcendence in which an imitative model can outperform the sources that generated its data. We analyze what conditions of the data must hold to allow for transcendence in each setting. In particular, we highlight several aspects of data diversity: For skill denoising, low temperature sampling enables transcendence as long as errors are uncorrelated. For skill selecfion, the model can accumulate knowledge from specialized experts as long as experts tend to generate data within their expertise. For skill generalization, we draw on the model's 's simplicity bias to argue that the model will learn to combine expert knowledge once the training data has high enough complexity through its diverse phrasing and examples of knowledge composition.
While valuable for isolating specific phenomena, our controlled experimental setup is lim- ited, and we encourage future work that investigates these ideas in more real-world settings. We also encourage future work developing additional settings in which transcendence may occur - for example, our framework does not capture the idea of skill discovery. We hope our work is useful starting point for broader investigation in the area.
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Acknowledgements
This work has been made possible in part by gift from the Chan Zuckerberg Initiative Foundation to establish the Kempner Institute for the Study of Natural and Artificial Intelligence. Our work was improved by conversations with Annabelle Michael Carrell, Anat Kleiman, Benjamin L. Edelman, Milind Tambe, and Sham M. Kakade as well as with anonymous COLM reviewers. Some experiments were based on code provided by Annabelle Michael Carrell.
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A Proofs and Analysis
A.1 Skill selection analysis
Proof of Theorem 2.1. Let a and b denote two experts. For transcendence hold, have
R Prest (hD) > R Prest (fa) and Pistt (kD) > Rite (f)) Starting with the first inequality, we have
< Rpase (KD) - Rpusa (fa)
Ex-pue Piest
Expp Pestt
(1- - (xa)
Exee (1--g(ab)](s((t))<(()))
Exee Prest ((c(b))))(rs(f))rrr)))
And symmetrically, we have Ex-piss [[g(al=))((((ff)====))) >
Combining these we get the desired result:
Exc-piet [(xx(fa) rx(f6)) (g(4lx) - 8(b[x)) >0
A.2 Skill generalization analysis
We develop a simple case study to provide intuition for the skill generalization setting, based knowledge graph-based experiment setup.
Let G = (V, denote a knowledge graph where the nodes represent entities and the edges represent relational facts between entities. Let R denote set of relation types. Then each edge E is represented by tuple (a,r,b) with a, e V and R. Let F(1) denote the set of one-hop facts in G. That is,
F(1) fe= (R, E}
For simplicity assume there a deterministic mapping of one-hop facts, that is, for a given prefix ra, there is at most one label b. We will refer to this type of prefix (a,r) as a "one-hop input."
A two-hop fuct is represented by tuple (a,r,,r,,)) such that there exists bridge entity b E V where (a, b) F(1) and (b, r2, E F()) Let p(2) denote the set of two-hop facts:
F(2) = (a, 3b E V such that (a, (b,r2,) E F(1)
In particular, G induces function f* V x Rx R V that encodes the two-hop facts in the knowledge graph. We assume ftt is only defined on triples (a,7),72 for which such bridge entity b exists in the graph. We will refer to such triple (a, "two-hop input."
Then f* can be expressed as compositional function:
(((,,1))2) g" (g" (a,rr))))) where g (a,r) b: (a,r,b) e F())
Let the input space contain strings of the type (2) such that V, f1, 12 R and (a,1,,2, c) is a two-hop fact in G for some c E V. Define the output space as ) = Vu {e} where e represents default null label.
Now we will partition the graph according to k "experts." Let (F(") partition of one-hop facts of the graph. Then define the input space of expert as
1...k) k) be
X; (2) 3b ba, (b, r.")
That is, X; is the two-hop inputs where both relevant one-hop facts are in F, Let p; be some distribution over X, such that every element of X, has non-zero probability. Let each expert
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have label function fi X2 + Y. Assume that given an input x E X, expert will always provide the correct label. That is, for any x (a, F,, 12), fi(x) where (2, F(2).
Let D = (x, fi(x)) x X,i ~ [l,...,k]} such that D = N be training set of N unique samples.
We will define a hypothesis class that contains two classes of memorizer functions: one class Hmem that uses single lookup via a lookup table TT22) to map inputs to outputs, and second class Hoomp that uses a compositional function with a lookup table (((1) Formally,
let r I ( be lookup tables where ) is size V R R and ( is of size V x R. Let H = Hmmm U Heomp, where
Hmem =(1 ((11(),1112)) (122 [ for some TT VXRXR-
Hcomp = -hh /h1hha,,,,,22 = (2)), where g(a,r) = ( for some T R R-- y}
Define the complexity of a lookup table to the number of non-null mappings:
x (T)
{x T(x) *c]
Define lookup function f to be function whose only operation is a single lookup with table Ty. Then define the complexity of the lookup function x(f) = x (TT)). Assume there is a fixed overhead associated with composing functions. That is, for a composed function of the form f(((),,,, we define its complexity as:
x(f(f(-),.) )) = =x(f) + Keomp,
where Keomp constant representing the additional cost of composition compared to independent lookups.
Under an ERM learner, for training set we have that
ID arg min ((s,)))(((( heh
This setting is realizable, so we can write hp E Hr where Hi C H is the set of hypotheses that achieve zero loss. Any that correctly memorizes the two-hop examples achieve zero loss. However, this gives us no guarantee as to how the function generalizes to unseen two-hop examples- - namely, two-hop facts in which the first hop and the second hop belong to different expert partitions.
will
Here we will assume simplicity bias to allow us characterize when we will find the generalizing solution:
hD E arg min x(h)
We now argue that, under reasonable conditions, the learner will prefer a compositional solution due to its lower complexity.
Any Hmmm n HD must store a separate mapping for each training example. Thus, its complexity satisfies:
x(h) 2 D
By contrast, for any compositional function E Heomp, the lookup table T(()) only needs to store one-hop facts. Therefore, its complexity bounded by
x(h) < |F()] + Keomp
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where Keomp accounts for some fixed additional cost of composition.
This leads to sufficient condition under which the learner prefers compositional solution:
[D] 2 [FO] + Keomp
To understand when this condition is likely to hold, we analyze the maximum size of the training set [D] for given graph. Importantly, [D] is bounded by the number of two-hop facts where both hops lie within a single expert's domain. Therefore, satisfying the condition above requires that enough such examples exist in the graph.
Let in (v) and dows (v) denote the in-degree and out-degree of node v. Then the total number of one-hop facts in the graph is
|FO1|
Σ dv) DEV
Σ DEV
dout (v)
Each node v induces (v) dout( two-hop facts. However, the training data consists only of two-hop facts in which both hops belong to the same expert partition F.") The total number of such training examples is bounded by:
& Σ Flev
0 (v)
DD
S
R ==
where (v) and d (v) are the out degrees of node v restricted to edges in partition F(")
Combining the complexity bound and the dataset size estimate, we obtain a sufficient condition under which the learner prefers compositional solution:
Σ dv) + Keomp PEV V
in (v)
This condition intuitively requires that enough valid two-hop facts lie within the domain of a single expert. Because expert knowledge is structured in a shared latent space, via reusable hoe representations, these facts can be composed efficiently. When this holds, the training set can be large enough for the compositional function to be simpler than memorization causing the learner, under simplicity bias, to prefer generalization.
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B Training Details
B.1 Knowledge graph generation
For our experiments, we use knowledge graph filled with fictional entities to generate facts the model has not seen during pretraining. We create the knowledge graph by taking the structure of the WIKIDATA- ATA-aaedd graph from Cohen et al. (2023) and we use GPT-4o-mini (OpenAI, 2024) to replace the entities with fictional names. To generate entities, we start by asking GPT-44-mini for set of fictional country names to seed the fictional graph. These fictional country names are randomly assigned to replace the country entities in the original graph. We then run BFS-like procedure in which the updated neighbors of each node are used as a context to generate the fictional name of the current node. We also provide starting letter drawn at random to increase diversity. This guarantees that GPT--4-mini provides a fictional entity name based only on its fictional context. The resulting knowledge graph has approximately 25,000 entities, 39 relation types, and 54,500 edges.
B.2
Compute Resources
All training runs used 1-4 Nvidia H100s for 1-8 hours.
B.3
Skill generalization experiment details
1.0
1.0
Standard
Data Diversity
0.79
0.8
Relation Majority
0.70
Accuracy
Direct Connection
Accuracy
0.6
0.6
0.4
0.4
0.2
0.15
0.2
0.09
0.0
20000
40000
60000
80000
Data Direct Relation Standard Diversity ConnectionMajority
CoT
Number Two-Hop Examples
Figure Left: Number of training samples vs within-expertise two-hop query accuracy on the held-out samples. Right: Comparison of methods in terms of within-expertise query accuracy.
B.3.1 Baselines
We include baselines based on two potential "shortcuts" as described in Yang et al. (2024). For the direct connection baseline, want measure whether the tail entity of the two-hop fact can be predicted from frequent cooccurrences with the head entity in the training data. To do this, we measure the number of two-hop prompts in which the head-tail pair also occur in the one-hop fact training set. That is, the number of two hop prompts (a, (2) + such that E {c: (a, r) E Fone-hop for some relation r} where Fone-hop denotes the one-hop facts in the knowledge graph. This effectively counts all head-tail pairs which co-occur in the one-hop facts. This accounts for 528 out of 6133 validation within-expertise two hop facts, corresponding to an accuracy of 0.086, and 5666 out of 64811 across-expertic two hop facts, corresponding to accuracy of 0.087. We also check head-tail co-occurrences in the training set of two-hop facts and find that this corresponds to an accuracy < 5% both the validation and across-expertise sets, leave this out for clarity.
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For the relation majority baseline, measure the accuracy achieved from simply assigning the most common entity of the correct type (eg.. person, location, occupation) based on the second relation of the two-hop fact.
B.3.2 Scores on validation set
The validation set for two-hop facts consists of roughly 6000 queries of within-expertiie two-hop facts that are held out during training. Scores are reported in Figure 7.
B.3.3
Data diversity experiments
Level
Method
Sample
1
1 template per relation
The place of death of Glimmerhold is Galadron Abyss. The name of the head of government of Galadron Abyss is Morrathis Voidstrider.
2
4 templates per relation
Glimmerhold died in Galadron Abyss. Galadron Abyss's head of govern- ment is Morrathis Voidstrider.
3
GPT generated
Glimmerhold died in Galadron Abyss, which is governed by Morrathis Voidstrider.
4
GPT generated
Glimmerholl met its demise in the treacherous depths of Galadron Abyss, a notable locale governed by Morrathis Voidstrider. This abyss, shrouded in mystery, serves as a poignant backdrop to the tale of Glimmerhold's fate.
Table 1: Comparison of diversity levels in one-hop paragraph generation.
We run preliminary experiments the usefulness of phrasing diversity in improving the model's to answer unseen two-hop facts. For each sample, we generate version at four diversity levels:
Level 1: Facts are written from single template as in the standard setting. Multihop facts are written in a single template.
Level 2: Each relation type has four templates. Each time an edge is written, one of four templates is sampled u.a.r. Multihop facts are written in a single template.
Level 3: We use GPT-4o-mini to generate low creativity entry based on the Level paragraph. The following prompt is used: "You will be provided with list of facts about an entity. Your job is to write a 10-50 word encyclopedia entry about the given entity. You should not make up additional information, just rewrite the facts." "
Level 4: We use GPT-4o-mini to generate a high creativity entry based on the Level 2 paragraph. The following prompt is used: "You will be provided with a list of facts about an entity. Your job to write 10-50 word encyclopedia entry about the given entity. You should not make up additional information, just rewrite the facts. Use creative word choices and phrasing."
To increase data diversity of the two-hop sentences, ask GPT-4o-mini to rephrase each templated two-hop sentence. For the models marked with "Data diversity," each model was trained with 1.5 million paragraph samples, each provided at the four levels of diversity for a total of 6 million samples. Models were additionally provided with the rephrased two-hop samples repeated 20x per epoch along with the templated two-hop samples.
B.3.4 Chain-of-Thought
For models that are allowed CoT to answer two-hop queries, we use a QA format and provide the intermediate entity before the final answer: For example, "What is the award received by the screenwriter of Glyndor Aetheralis? Ithryndor Glaciaris; Xyphorian Starblossom. During evaluation, we allow the model to generate an intermediate step proceeding the semicolon and judge accuracy based only on the final answer.
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Reviews:
[{"review_id": "jbmrQEIijn", "reviewer": "Reviewer_tCzd", "summary": "The paper starts with the observation that language models are trained to mimic humans and based on human-produced data but the resulting systems display capabilities beyond the scope of any one human.  For example, they have highly specialised knowledge on such a range and number of topics that no human would master. This is called transcendence.\n\nThe authors concentrate on what properties of the training data lead a model to transcend the performance of its data sources. \n\nTheir taxonomy identifies three modes of transcendence, called skill denoising, skill selection, and skill generalization. The knowledge support of the experts  is formalised as a knowledge graph whose properties can be formally manipulated. \n\nThe results show that for skill denoising, low temperature sampling enables transcendence if errors are uncorrelated. For skill selection, the model can transcend if experts tend to generate data within their expertise. For skill generalization, the authors tentatively conclude that the model will learn to combine expert knowledge once the training data has high enough complexity.", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "Very interesting, thought-provoking topic\n\nGood formalisation that supports identification of precise results. This is of value beyond this specific paper.\n\nVery clearly written.", "reasons_to_reject": "The broader problem and goal is not defined well enough. The authors talk of ‘inhuman capacity’ (line 20). In this definition, they confuse general human capacity as capacity of the species, and capacities of an individual human, which is what they study. This confusion needs to be clarified, as they do not have a solution to specie-wise inhuman capacities.\n\nI would have liked to see, even passingly, a comparison to previous results in ensemble learning and skill denoising, and multi-task learning, and skill generalization, which seem to reach similar conclusions.\n\nThe generalization results are not clear. What is the level of performance that will allow us to say there has been generalization?", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ErjMmnGzqJ", "reviewer": "Reviewer_nXy4", "summary": "This work provides a taxonomy of ways in which language models may be able to perform better in an ensemble than any individual model, namely: skill denoising, skill selection, and skill generalization. It provides evidence of each of these, as well as investigating how factors such as number of experts impact the extent to which they occur.", "questions": "", "limitations": "", "rating": 7, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "* This experimentation is well-designed. While the paper only looks at a very specific task (two-hop reasoning in a knowledge graph), it carefully characterizes different ways in which models could achieve this, and then tests them directly, as well as also evaluating how number of experts might interact with these.\n* The use of a knowledge graph ensures robustness and consistency that may not be present in natural data.\n* The dataset avoids many of the risks of the confounds and contamination found in real data by using new names and generating text using templates (except in the 'data diversity' experiments).", "reasons_to_reject": "Major:\n\n* The main focus of this paper is on how to use an ensemble of language models to perform better than any of the language models individually; however,  the paper does not engage with much of the previous work on language model ensembles (for some recent examples, see, e.g., Bian et al., 2025; Huang et al., 2024; Jiang et al., 2023; Lu et al., 2024; Mavromatis et al., 2024; Xu et al., 2025; Yao et al., 2024; Yu et al., 2024). While the main focus of the present study is different in that it focuses on a specific well-defined capability and how to go about achieving it with multiple models, it is still important to consider that there are many different ways to ensemble models; and some of these works also make inferences about which factors enable ensembles to perform better.\n* Even though they are clearly powerful (especially together), both the ensembling approach and the finetuning approach are straightforward and not novel (as far as I can tell), and so I am not sure that many of the formalizations (especially in Section 3.1) are necessary to put in the main paper - in my opinion, they reduce the readability.\n* The discussion is extremely limited, and so the implications for theory and practical applications (and the scope of these) are unclear.\n* Using GPT-4o to generate names and rephrasings runs a small risk of introducing confounds. For example, the specific process by which the entities were renamed is not stated, but if it used GPT-4o directly (e.g., by asking the model to rename specific entities), there is a possibility that there is some relation between these names and the real names that is also detectable by the GPT-2 models, though this does seem unlikely to have a strong effect.\n\nMinor:\n\n* Based on Appendix A1, Theorem 3.1 in the main body of the text is incomplete.\n* Figure 5 is not referenced in the main text.\n\n\n\n\n\nReferences:\n\n* Bian, Y., Lin, Y., Liu, J., & Ruan, T. (2025). PToco: Prefix-based Token-level Collaboration Enhances Reasoning for Multi-LLMs. In O. Rambow, L. Wanner, M. Apidianaki, H. Al-Khalifa, B. D. Eugenio, & S. Schockaert (Eds.), *Proceedings of the 31st International Conference on Computational Linguistics* (pp. 8326–8335). Association for Computational Linguistics. https://aclanthology.org/2025.coling-main.556/\n* Huang, Y., Feng, X., Li, B., Xiang, Y., Wang, H., Liu, T., & Qin, B. (2024). Ensemble Learning for Heterogeneous Large Language Models with Deep Parallel Collaboration. *Advances in Neural Information Processing Systems*, 37, 119838–119860.\n* Jiang, D., Ren, X., & Lin, B. Y. (2023). LLM-Blender: Ensembling Large Language Models with Pairwise Ranking and Generative Fusion. In A. Rogers, J. Boyd-Graber, & N. Okazaki (Eds.), *Proceedings of the 61st Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)* (pp. 14165–14178). Association for Computational Linguistics. https://doi.org/10.18653/v1/2023.acl-long.792\n* Lu, K., Yuan, H., Lin, R., Lin, J., Yuan, Z., Zhou, C., & Zhou, J. (2024). Routing to the Expert: Efficient Reward-guided Ensemble of Large Language Models. In K. Duh, H. Gomez, & S. Bethard (Eds.), *Proceedings of the 2024 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies (Volume 1: Long Papers)* (pp. 1964–1974). Association for Computational Linguistics. https://doi.org/10.18653/v1/2024.naacl-long.109\n* Mavromatis, C., Karypis, P., & Karypis, G. (2024, August 26). Pack of LLMs: Model Fusion at Test-Time via Perplexity Optimization. *First Conference on Language Modeling*. https://openreview.net/forum?id=5Nsl0nlStc\n* Xu, Y., Chen, J., Wu, J., & Zhang, J. (2025). Hit the Sweet Spot! Span-Level Ensemble for Large Language Models. In O. Rambow, L. Wanner, M. Apidianaki, H. Al-Khalifa, B. D. Eugenio, & S. Schockaert (Eds.), *Proceedings of the 31st International Conference on Computational Linguistics* (pp. 8314–8325). Association for Computational Linguistics. https://aclanthology.org/2025.coling-main.555/\n* Yao, Y., Wu, H., Liu, M., Luo, S., Han, X., Liu, J., Guo, Z., & Song, L. (2024, October 4). Determine-Then-Ensemble: Necessity of Top-k Union for Large Language Model Ensembling. *The Thirteenth International Conference on Learning Representations*. https://openreview.net/forum?id=FDnZFpHmU4\n* Yu, Y.-C., Kuo, C. C., Ziqi, Y., Yucheng, C., & Li, Y.-S. (2024). Breaking the Ceiling of the LLM Community by Treating Token Generation as a Classification for Ensembling. In Y. Al-Onaizan, M. Bansal, & Y.-N. Chen (Eds.), *Findings of the Association for Computational Linguistics: EMNLP 2024* (pp. 1826–1839). Association for Computational Linguistics. https://doi.org/10.18653/v1/2024.findings-emnlp.99", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "d7C6stVEyZ", "reviewer": "Reviewer_76dv", "summary": "Formalization and experimentation with the concept of transcendence, where models trained on data written by experts outperform these experts. Three modes of transcendence are defined, namely skill denoising, selection and generalization. The experiments with synthetic data simulate these conditions in relational knowledge memorization and extrapolation, showing that low temperature sampling results in the phenomenon.", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "Useful theoretical grounding for the types of transcendence, explained well and motivated by previous work.\n\nExperiments demonstrate the phenomenon in a clear model system.", "reasons_to_reject": "The knowledge graph construction uses GPT-4o, which means the \"unseen\" facts are still artificially likely according to GPT, so the experimental setup is biased.\n\nExpert coverage is not clearly defined in section 5.\n\nThe dataset used is cited but no information about it is provided, nor how exactly it is anonymized by GPT-4o, making it difficult to evaluate it.\n\nThe task is somewhat simplistic since it amounts to pure relational knowledge learning + transitive closure. This is very specific but the paper overclaims the generality of the result.\n\nThe paper is very repetitive, stating the definition of the taxonomy again in every section without necessarily saying anything new about it.\n\nExperiments use the same language model, GPT-2, raising doubt about their generalizability.", "questions_to_authors": "Theorem 3.1: missing >0", "extra_scores": {}}, {"review_id": "sxTmfxMyw9", "reviewer": "Reviewer_vsp5", "summary": "This paper studies the phenomenon of transcendence, where a system can show better performance than its knowledge sources, e.g., a set of experts. Building upon previous work studying such behavior on chess, the authors leverage knowledge graphs to sample experts and extend the study of transcendence to textual knowledge acquisition. Specifically, the authors introduce three types of transcendence: skill denoising, selection, and generalization. Experiments on fine-tuning pre-trained language models show interesting findings on how language models can perform better than simulated experts, leveraging the latent knowledge connection.", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "1. The paper studies an interesting topic: transcendence, where the findings might provide interesting intuition on how LLMs can achieve super expert performance, as observed in tasks such as Big-Bench Hard.\n\n2. This paper is well-written and easy to follow. The takeaways are clear to the readers.\n\n3.  The experiments are presented in a neat, concise, and controlled setting with the introduction of experts sampled from constructed knowledge graphs, which provides good evidence supporting the taxonomy.", "reasons_to_reject": "1.  The motivations in the introduction can be better grounded with further citation from previous ML, NLP, or LLM literature, e.g., Line 26, “allow a group to vote” is relevant to the line of research on bagging. \n\n2. Current findings are mostly based on training a pre-trained GPT2 (except Section 7 with Llama 3.2). It would be good to know how model parameters and pre-training design choices will impact the learning and robustness of the skill acquisition (denoising, selection, and generalization). Other newer models with similar sizes (<2B) of GPT2 can also be experimented with, e.g., Phi-1.5, Pythia, Qwen2, and SmolLM.\n\n3. Although the analysis framework is interesting, current observations regarding diversity and CoT seem common and similar to prior findings from beyond Allen-Zhu’s line of research.", "questions_to_authors": "1. There are many citation problems: line 70, no year. Line 141, what is the wisdom of the crowd? Line 301, use {} over the whole team’s name.\n\n2. Lines 208-210: How is this guaranteed? Is there any probing or pre-training data analysis to show that the model does not include any facts?\n\n3. Previous work on knowledge conflicts [1][2] shows that there is a ripple effect during knowledge acquisition or editing – one edge can influence the relevant multi-hop neighbors. Can the current evaluation framework provide extended findings considering the ripple effect?\n\n[1] Evaluating the Ripple Effects of Knowledge Editing in Language Models\n\n[2] “Merge Conflicts!” Exploring the Impacts of External Distractors to Parametric Knowledge Graphs", "extra_scores": {}}]

Author rebuttal:

Meta-review: The paper starts with the observation that language models are trained to mimic humans and based on human-produced data but the resulting systems display capabilities beyond the scope of any one human. For example, they have highly specialised knowledge on such a range and number of topics that no human would master. This is called "transcendence".

The authors focus on on what properties of the training data lead a model to transcend the performance of its data sources.

Specifically, the authors introduce three types of transcendence: skill denoising, selection, and generalization. The knowledge support of the experts is formalized as a knowledge graph whose properties can be formally manipulated.

Experiments on fine-tuning pre-trained language models show interesting findings on how language models can perform better than simulated experts, leveraging the latent knowledge connection.
The results show that for skill denoising, low temperature sampling enables transcendence if errors are uncorrelated. For skill selection, the model can transcend if experts tend to generate data within their expertise. For skill generalization, the authors tentatively conclude that the model will learn to combine expert knowledge once the training data has high enough complexity.
Decision: Accept