Compositional Generalization from Learned Skills via CoT Training: A Theoretical and Structural Analysis for Reasoning
Authors: Xinhao Yao
Ruifeng Ren
Yun Liao
Lizhong Ding
Yong Liu
Venue: ICLR 2026 Poster (2026)
Keywords: Information-Theoretic Bounds
Compositional Circuits
Reasoning Generalization
CoT Training
Primary area: neurosymbolic & hybrid AI systems (physics-informed, logic & formal reasoning, etc.)
TL;DR: Through a theoretical and structural analysis, we demonstrate that CoT training fundamentally enables compositional generalization by systematically combining simpler learned skills to solve novel, complex problems.

Abstract: Chain-of-Thought (CoT) training has markedly advanced the reasoning capabilities of large language models (LLMs), yet the mechanisms by which CoT training enhances generalization remain inadequately understood. In this work, we demonstrate that compositional generalization is fundamental: models systematically combine simpler learned skills during CoT training to address novel and more complex problems. Through a theoretical and structural analysis, we formalize this process: 1) Theoretically, the information-theoretic generalization bounds through distributional divergence can be decomposed into in-distribution (ID) and out-of-distribution (OOD) components. Specifically, the non-CoT models fail on OOD tasks due to unseen compositional patterns, whereas CoT-trained models achieve strong generalization by composing previously learned skills. In addition, controlled experiments and real-world validation confirm that CoT training accelerates convergence and enhances generalization from ID to both ID and OOD scenarios while maintaining robust performance even with tolerable noise. 2) Structurally, CoT training internalizes reasoning into a two-stage compositional circuit, where the number of stages corresponds to the explicit reasoning steps during training. Notably, CoT-trained models resolve intermediate results at shallower layers compared to non-CoT counterparts, freeing up deeper layers to specialize in subsequent reasoning steps.  A key insight is that CoT training teaches models how to think—by fostering compositional reasoning—rather than merely what to think, through the provision of correct answers alone. This paper offers valuable insights for designing CoT strategies to enhance LLMs' reasoning robustness.

Published as a conference paper at ICLR 2026
COMPOSITIONAL GENERALIZATION FROM LEARNED SKILLS VIA COT TRAINING: A THEORETICAL AND STRUCTURAL ANALYSIS FOR REASONING
Xinhao Yao Ruifeng Ren', Yun Liao', Lizhong Ding², Yong Liu
'Gaoling School of Artificial Intelligence. Renmin University of China. Beijing, China *Beijing Key Laboratory of Research on Large Models and Intelligent Governance 'Engineering Research Center of Gext-Gennration Intelligent Search and Recommendation, MOE *Tianjin University of Science and Technology. BBijing Institute of Technology fyaoxinhac021978) liuyonggsai @rucc edu. on
ABSTRACT
Chain-of-Thought (CoT) training has markedly advanced the reasoning capabili- ties of large language models (LLMs), yet the mechanisms by which CoT train- ing enhances generalization remain inadequately understood. In this work. we demonstrate that compositional generalization is fundamental: models system- atically combine simpler learned skills during CoT training to address novel and more complex problems. Through a theoretical and structural analysis, we formal- ize this process: Theoretically, the through distributional divergence can be decomposed into in-distribution (ID) and out-of-distribution (OOD) components. Specifically, the non-CoT models fail on OOD tasks due to unseen compositional patterns, whereas CoT-trained models achieve strong generalization by composing previously learned skills. In addition, controlled experiments and real-world validation confirm that CoT training accel- erates convergence and enhances generalization from ID to both ID and OOD sce- narios while maintaining robust performance even with tolerable noise. @ Struc- turally, CoT training internalizes reasoning into a two-stage compositional circuit, where the number of stages corresponds the explicit reasoning steps during training. Notably, CoT-trained models resolve intermediate results at shallower layers compared non-CoT counterparts, freeing up deeper layers to specialize in subsequent reasoning steps. A key insight is that CoT training teaches models how to think-by fostering compositional reasoning- think, through the provision of correct answers alone. This paper offers valuable insights for designing CoT strategies to enhance LLMs' reasoning robustness
generalization bounds
than merely what to
INTRODUCTION
Training large language models (LLMs) to generate explicit reasoning chains marks pivotal shift in Al development. This Chain-of-Thouuht (CoT) [109] paradigm moves beyond mere data scaling [39, 32] toward cultivating advanced reasoning strategies [55], resulting in more effective learning outcomes [132, 129, 105, 30, 85, 128, 43, 33, 31, 55]. Leading organizations-suchh as OpenAl, through reinforcement fine-tuning (RFT/ReFT) [96] of its O1 models [67], and DeepSeek, via the incorporation of long CoT cold-start data in DeepSeek-RI [25]-are are increasingly treating step-by- step reasoning annotations not merely as a prompting tool, but as fundamental training objective.
Despite its empirical success, the underlying mechanisms of CoT remain a widely debated topic [75]. Recent studies demonstrate that CoT enhances transformers' [97] expressiveness and com- putational capacity, enabling them to solve problems in higher complexity classes when processing sufficienlly long sequences [17, 63, 53, 73). Other prior efforts have advanced understanding through case studies of functions learnable in-context with CoT [50, 5, 48, 116] and analyses of multi-step
"Corresponding author.
Code is available https: / / thub.hhubbhe wtw.th.ww..cccc..ccccoccoeteeeee 1 23Ct

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model estimation errors [35, 19]. However, these approaches do not explain how such capabili- ties emerge during the training of transformers to generate reasoning chains precisely when core model competencies are being formed [54, 138]. In light of this concern, Kim & Suzuki [42], Wen et al. [110] focus on the parity problem and highlight that CoT improves the sample efficiency of transformers. Meanwhile, Wang al. [106] assess different data collection strategies and point out that the granularity of CoT data strongly correlates with model performance. Crucially, it remains unclear: (QI) Does CoT training improve reasoning generalization across both in-distribution (ID) and out-of- distribbitt (OOD) scenarios- and if so, what theoretical principles govern this ability? (Q2) How is this generalization capability realized within the model's internal representations?
In this paper, we propose that compositional generalization is fundamental: models systematically combine simpler learned skills during CoT training to address novel and more complex problems. We formalize this process through a unified theoretical framework and validate it via detailed structural analysis of the model's internal computational circuits.
Generalization Error Analysis (Section 2, for Q1). We provide an in-depth theoretical analysis to substantiate this proposition. Specifically, we first examine the KL divergence (xi between the training and test distributions. We then employ information-theored methods to derive generalization bounds based on distributional divergence. Theorem (Remark 1) shows the generalization enror naturally decomposes into ID and OOD components: (i) For ID, the compositional patterns of the test problems exactly match those seen in the training set. The model only needs recognize the problem type and retrieve the corresponding compositional pat- tern learned during training. As a result, the generalization error approaches zero under sufficient training (i.e., DxL. (Piel Puniin) = 0), regardless of the use of COT. (ii) For OOD, models trained without CoT fail to generalize, as the OOD test problems feature novel compositional patterns of learned, simpler skills. In contrast, CoT-trained models excel in these settings by leveraging ex- posure to the necessary simpler skills during training, achieving near-perfect OOD performance (Theorem 2). (iii) Additionally, adding noise to training data raises the generalization error upper bounds for both ID and OOD settings, and the error bound grows with the level of noise.
[114, 78, 86, 108, 107]
Our experiments serve as empirical support for the theoretical claims above. We begin with con- trolled data distributions to mitigate the complexity inherent in real-world training and enable precise mechanistic analysis. We demonstrate that CoT training accelerates convergence and substantially improves generalization, extending it from ID to both ID and OOD settings (Section 3.2). This con- firms that CoT enables compositional generalization, whereas non-CoT models only rely on fixed input-output mappings. Next, Section 3.4 shows that CoT generalizes robustly to noisy reasoning steps within a tolerable noise level, underscoring the importance of reliable compositional gener- alization. Finally, validation on real-world datasets (e.g., math word problems) shows that learned skills and compositional patterns remain applicable in complex scenarios.
Internal Circuits of CoT vs. Non-CoT Training (Section 3.3, for Q2). Using logit lens and causal tracing experiments, we find that CoT training induces a two-stage compositional circuit characterized by sparse token dependencies. This structure reflects the process of composition: the model progressively internalizes the reasoning process-eace each explicit reasoning step during training maps onto a distinct stage within the circuit. Conversely, Wang al. [102] show that similar circuits emerge only during ID) generalization, suggesting that without CoT, reasoning steps are not systematically internalized. Interestingly (Figure 2), CoT training allows the intermediate result to be extracted from lower layer (c.g.. index 3) for ID examples, whereas Non-CoT Training requires a higher layer (e.g., index 5). Intuitively, a smaller layer index implies that more layers remain available for processing the second reasoning step, potentially enhancing model performance.
Main contributions. Briefly, we establish a theoretical and structural framework for understanding the generalization of CoT training. A key point is that CoT training helps models learn how to think-by enabling compositional reasoning-na not just what to think, by simply providing correct answers. These findings offer insights into CoT strategies for LLMs achieve robust generalization.
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2 THEORETICAL ANALYSIS
2.1 PRELIMINARIES
To begin with, we formally introduce the data distribution setting, clarify the distinctions between ID and OOD generalization in the context of compositional reasoning tasks, and give the definition of expected generalization error. These form the basis for our subsequent theoretical analysis.
(Data Distribution). Let X and Y denote the input and output random variables, re- spectively. The data distribution and its underlying relationships are characterized the condi- tional distribution P(Y X). Let CoT be a sequence C = =(C,, (C1,C2," C2...... Ck), where each Ck denotes the reasoning state the k-th step, then
Definition 1
get the following decomposition:
K DRRXI,, II P(C
P(Y (x) Σ PXYIIXC))(CII) EPVIX, C)
where P(CE|Cecs X) is the probability of the k-th reasoning step given all previous steps and the input. Each unique sequence of corresponds to distinct compositional pattern. ID and OOD are thus defined by whether these patterns are shared with the training data or are novel and unseen. Representative examples are provided in Appendix C.
Recently, information-theoreee generalization bounds [114, 78, 86, 108, 107] have been introduced to analyze the expected generalization error of learning algorithms. key benefit of these bounds that they depend only the data distribution but also the specific algorithm, making them an ideal tool for studying the generalization behavior of models trained using particular algorithms.
Definition 2 (Expected Generalization Error (informal).. From a traditional statistical learning perspective, the expected generalization error is defined the difference belween the population risk and the empirical risk. Intuitively, the empirical risk reflects performance on the training data, while bounds on the generalization error allow us estimate the population risk on unseen test data. We denote the expected generalization error by error, see Appendix B.2 for detailed definition.
2.2 INFORMATION- CHORMATTIO-THEOORTICC GENERALIZATION BOUNDS
In this part, P Pariin (X) and (Y) denote the conditional distributions of the training and test data, respectively. Based on this, we provide theoretical analysis of the divergence between the training and test distributions, along with the resulting generalization error bounds.
Since compositional patterns ID test data (and training data) do not appear OOD test data, we make a plausible assumption" about Poea P (YlX):
Assumption (Mixed Test Distribution). Assume Pustt ( ( is a mixture of two distributions with disjoint support: Pid (X) and poon (Y Then
Piea(Y|X) = (1-a) P
+ a tea
pOOD (YIX),
where a E [0, 1] is the mixing coefficient of the OOD data.
Consequently, the distributional divergence between the test and training sets can be assessed quan- titatively, which helps us characterize the expected generalization error as follows:
Theorem (Generalization Bounds via Distributional Divergence). Under the conditions specified in Assumption and Definition 2, we assume that the loss E(w, Z) R-subGaussian for any w e W E Rd, then the expected generalization error is bounded by:
2R (1- a1 (PIP (Y [X] (raii())) + aDLL (POO (Y (Y (rxx)))))))
error SV SVN INN
where N the training data size, Z (X and W the space of hypotheses related to the model. denotes the mixing coefficient of the OOD data in test distribution and DRL(.) represents the KL divergence between ID/OOD test and training distributions.
2 Lnnarr combining rule is good starting point for target mixture distribution [60, 16, 90, 58,
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Remark (Theoretical Analysis of ID/OOD Generalization with/without CoT). Theorem 1 shows the generalization error naturally decomposes into ID and OOD components with coefficient a. ! For ID, the compositional patterns C in the test problems exactly match those encountered during training, since the ID test data comprises instances derived from known compositions in training data. Thus, under sufficient training- or without Co'l-1he tte ID generalization ervor approaches zero (i.e., ( (ealllPmain = 0), consistent with the results in Figure I (Left Part). @ For OOD, considering the data generation process, we have: PXY -Ec P(Y|X, C))(())) where C is the CoT reasoning sequence: (a) when trained without CoT (Figure 1 (Left)), the model does not explicitly learn the distribution P(C (i.e.. never appears during training). the P(C be- comes a uniform prior, making P(Y|X, C) reduce to P(Y|X). Moreover, since OOD test problems involve unseen compositional patterns, training without CoT struggles to generalize to OOD set- lings. (b) In contrast, with CoT training, both P(C|X) and P(Y|X, C) are explicitly modeled (that is, the learned, simpler skills), aligning precisely with the two stages of the compositional circuit ( (Figure 2, Right). CoT training enables models to generalize near-perfectly to OOD examples by exposing them to simpler sub-tasks during training (Figure (Center RightJ). When the composi- tional structure aligns between training and OOD tasks, the model can effectively apply its learned reasoning skills. CoT training teaches models how to think through compositional reasoning- noot merely what to think (matching the answers). For specifics, refer Theorem 2.
Theorem 2 (OOD Generalization Error for Training with CoT). Let the conditions specified in Theorem I hold, and assume" sufficient training such that Dxa. ( (eall) + 0. Define X(Y =
Ec P P(C))),) then the OOD generalization error bounded by:
2R2a N
error S
(POOD (KKI(POOO (CIX) || Prain (CIX))
P (CIX) (DRL 1 (Y CX,X,C)1 Prrain (Y ()III
where C denotes the CoT intermediate reasoning steps, see Appendix B.4 for more extensions.
Remark 2 (Robustness Discussion of Training with CoT). For CoT training in the presence of erroneous reasoning steps, let the distribution after adding noise to Proin ( ( ) be denoted as Pnew (X). Briefly, adding noise leads to an increase in both D (new)) (> Dxu (Pea |Prrii)) (
and DEa ( [ [|Pee>)(>> DEL ( || P This corresponds to an increase in the upper bounds
of the generalization error for both ID and OOD cases, and the larger the noise ratio &, the higher the error bounds. A more detailed theoretical explanation is provided in Theorem 3 (Appendix B.6).
SYSTEMATIC GENERALIZATION VIA A TWO-STAGE CIRCUIT
This section further illustrates the compositional generalization, with our experiments serving as empirical support for the aforementioned theoretical claims. We first show that CoT significantly accelerates convergence and improves generalization compared training without it (Section 3.2), then examine the internal mechanisms of CoT training, focusing on a two-stage compositional cir- cuit (Section 3.3). Next, we analyze the robustness of CoT training in the presence of erroneous reasoning steps (Section 3.4). Finally, we validate our findings on real-world datasets (Section E).
3.1 GENERAL SETUP
Here, we describe the core components of our study by reviewing some basic notations. Briefly, we have the model learn all atomic facts as "simpler learned skills", while treating multi-hop facts as "complex problems". Relational patterns (r1,".. ,rn) are used to simulate compositional patterns.
Atomic and Multi-Hop Facts. The atomic (one-hop) fact describes two entities and the relationship between them, which can be represented as triplet (c1,r1, r1, C2). For example, "The United States' capital is Washington, D.C." More specifically, éi refers to the head entity (e.g., the United States), is the relation (e.g.. Capital), and e2 refers to the tail entity (e.g.. Washington, D.C.). Based on atomic facts, a two-hop fact can be derived by combining two atomic facts, such as (e1, 71, T1, 62) @ (e2, e3)
(c1,T1, r2, e3), where eg serves as bridge entity. Similarly, multi-hop fact can be constructed recursively from more atomic facts, resulting in (e1, e2) ( (62, r2, 6a) @ e... +)
This correlates what architectures and tasks we considering, see Figure and Section 3.1 for details.
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"2,..., Cn+1), where is the number of steps. In scenarios, 62 corresponds to the intermediate inference in a two-hop fact, while e2. C3 ...... On are intermediate results in more complex multi-step reasoning.
(en; Fri Cn+1) = (e1, F1, (1,
Training Data. Firstly, following Wang et al. [102], we define the sets of entities & and relations R. from which the set of atomic facts is constructed as S (( eg)ler, E &, E R). Our training set of atomic facts, denoted as S, is sampled from S (i.e., S C S). Then, we partition S into two subsets, STD and SOOD (STD U SooD S, STD n Soon = 0). which are used to form two sets of two-hop facts, s(2) and S where S = {(e1, T1, 72,
T2, e3) (e1. e2), (e2, r2, Ea) E -SD, e1 * ea), and S00b is formed in similar manner. To evaluate the model's ID and OOD generalization ability, the training dataset T excludes $ OOD that is, T = Sw U SoOD U S
Dimin' where ((2)
iDiii
( Dig = s(2) Notice that Snn n SOOD = 0, so S(2) n GODD - 0 and the relation compositions (ri, in S will not appear in SOOD. Specifically, s(2) and s(2)
1Do, are disjoint subsets whose relation compositions are sampled from the same space, enabling evaluation on unseen instances s(2) within known compositions. In contrast, SOOD involves entirely novel relation compositions, drawn from a disjoint set of relation types, thus testing generalization to unseen compositional patterns. Let 1 I/SSpl where sse|| denotes the number of samples in the set. More ID/OOD evaluation details and representative examples are provided in Appendix C.
Training without/with COT. For atomic facts (S = SiD U SUOD), training and evaluation are per- formed by having the model predict the final tail entity (e1,71 prodict > é2, V(e1, 11, ez) E S, é2 is the prediction of C2, input
predict > output). As for two-hop facts, we consider whether to use CoT annota-
tions during training (f,,,T,T2: (2, 63) E S ). (1) Training without CoT: T1,
predict
é3. only
predict ê2 and e1,
predict
predict the final tail entity e3. (2) Training with CoT: 71, predict both the bridge entity e2 and the final tail entity e3-
03
Definition 3 (Training without/with CoT). Let è denotes the output generated by the model M, which is different from the ground truth. Taking two-hop facts (e1, (e,,T1,22 63) as an example, the test loss is defined as E ggg [E(e3, M(eI,T1, r2))], where L denotes the loss function, For training, (i) Training without CoT, the loss is E g [C(es, (1,1,1,722)l (ii) Training with CoT, the loss + M(e,, F1, (2))) In our analysis, we use the next-token prediction via autoregression [80, 52, 4] rather than next-token prediction via teacher-forcing [4].
E s(0) [C(es, M(e1,
3.2 COT TRAINING BoOSTS BOTH ID AND OOD GENERALIZATION
Although chain-of-ther -ffttuughth (CoT) reasoning can improve task performance [44, 109, 101, 134, 56), it is absent during large-scale
when core model capabilities are developed [54, 138). Prior work has extensively examined whether transformer- based language models can perform im- plicit composition, consistently reporting negative results [74, 117. 140]. Specifically, for CoT prompting, a persistent "compositionallty gap" exists [74], where models often know all basic facts but fail to compose them-regaute reggrddesss of the model scale. Regarding training without CoT, Wang et al. [102] further demonstrate that transformers can learn to perform implicit reasoning under ID generalization, but this ability does not transfer to OOD settings. It naturally raises the question: how does using explicit reasoning steps during training impact generalization ability?
Controllable setup. In order to control the data distribution, we utilize the data data-generation process introduced in Section 1. Specifically, S is constructed with |3| = 2000 entities and | = 200 relations (Appendix D) One-hop facts correspond to the (e1; rı, e2) triplets in S. These triplets are partitioned into two disjoint sets, that is. Spp (95%) and SOoD (5%), which are used to deduce the
ID and OOD two-hop facts ( The candidate set of À is (0.001, 0.9. 3.6, 12.6). Besides, the model employed is a standard decoder-only transformer as in GPT-2 [76], with a con- figuration of 8 layers, 768 hidden dimensions, and 12 attention heads, and the tokenization is done by having a unique token for each entity/relation for convenience*.
Details regarding the model, tokenization and optimization we used are provided in Appendix D.2.
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14
10
94-
JAI2A
Accortty
64.
Accumoo
ACC
2005 203030 Bou Optiibation Step 110' withutt CoT)
20
a re se te
Optimizarton Step (10% with COT)
Optimication Step (500, impact)
Optimizationn Slep (500, scale impact)
Figure 1: The model generalization ability under controllable data settings, Left Part: the accuracy comparison on two-hop reasoning between training without CoT (Left) and training with CoT (Cen- ter Left), À = 7.2. CoT training significantly accelerates convergence and improves generalization from ID to OOD reasoning. Right Part: the impact of two-hop/one-hog ratio (Center Right) and model scale (Right) OOD generalization. Ratio À correlates with OOD generalization speed, while larger models converge more quickly without altering reasoning behavior.
Results. Figure/ (Left Part) illustrates model accuracy on the training and testing two-hop rea- soning tasks during optimization with À = 7.2. (1) Training without CoT (Left). We mirror and observe the same phenomenon (grokking [72]) as Wang et al. [102], namely that the model even- tually generalizes to ID examples ($ only after extensive overtraining, while showing no OOD
generalization (OOD) even after millions of steps, suggesting delayed and non-systematic general- ization likely based memorized patterns (relation compositions). (2) Training with CoT (Center Left). Compared to training without CoT, incorporating CoT annotations significantly accelerates convergence and improves generalization. The model reaches near-perfect accuracy on SiD within
4, 000 steps, and also shows substantial gains SOOD. This indicates that explicit reasoning guidance during training shifts the model from nonsystematic (only ID) to more systematic (ID & OOD) generalization by enabling it to learn underlying reasoning patterns (relation compositions).
Ablation studies are also conducted to examine how the two-hop/one-hon ratio (A), model scale influence CoT training performance, with a focus accuracy over the OOD test set. (i) An ap- propriate À accelerates OOD generalization. Figure I (Center Right) shows the OOD test accuracy across different X, which is strongly correlated with generalization speed. A counter-intuitive find- ing is that a smaller ratio can expedite OOD generalization under CoT training', though overly small values may impair the model's ability learn relevant reasoning palterns. Indeed, the model may undergo a process of memorization before gradually learning the underlying patterns. In other words, a higher ratio increases the complexity of the training set, requiring a longer memorization phase, but the model eventually learns the relevant reasoning patterns. (ii) In Figure I (Right), we run the experiments with model layers E (2, 4, 8) and
[3.6, 7.2, 12.6). with other settings as in Figure 1 (Center Left). Scaling up the model does not qualitatively affect generalization, though larger models converge in fewer optimization steps, consistent with previous studies [54, 94, 102]. Additionally, we shift our focus multi-hop scenarios in Appendix D.4, examining whether a model trained solely on two-hop facts during the CoT training phase can generalize to multi-hop facts.
3.3 MECHANISMS OF COT TRAINING: INTERNALIZING STEP- WISE REASONING
Up to this point, we have demonstrated that incorporating explicit CoT training in controlled ex- periments significantly enhances the model's 's reasoning generalization ability- it from ID generalization alone to encompassing both ID and OOD generalization. Key factors such as data distribution (e.g., ratio À and pattern) play a crucial role in shaping the model's capacity for system- atic generalization. However, the internal mechanisms underlying these improvements remain unclear, which we aim to explore this part.
Logit Lens& & Causal Tracing. Logit lens is widely used for inspecting hidden states of LLMs [12, 22, 40, 79, 117, 102). The key idea behind the logit lens is to interpret intermediate hidden states by projecting them into the output vocabulary space using the model's 's output embedding ma-
A key feature of reinforcement fine-tuning (RFT, used in OpenAl 01 models [67]) is its ability to train models for domain-specific tasks with minimal data, similar our findings.

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trix. We adopt the recent practice [117, 102] where the activation first goes through the transformer's final normalization layer before being multiplied by the output embedding, For causal tracing. the transformer can be interpreted as causal graph [70] that propagates information from the input to the output via a network of intermediate states, enabling various causal analyses of its internal com- putations [98, 62, 104, 26, 18, 102]. Specifically, during the normal run, we intervene on the state of interest by replacing its activation with the activation from the perturbed run". We then proceed with the remaining computations and measure whether the target state (the top-1 token through the logit lens) is altered. For simplicity, we denote a hidden state as Ellayer index, token position].
e2
Layer 8
Layer
with CoT
Layer
Layer
Layer 0
Layer
lasst & Ouipat
Erooding Foul
1
F1
F1
e1
r1
F2
Actinations
without CoT
First hop
Second hop
Figure 2: The compositional circuit (layer:8) for two-hop facts. We analyze individual states and assess the strength of connections between hidden states. Left: training without CoT, the circuit emerges only during ID generalization, with the intermediate result e2 being resolved at layer index 5. Right: training with CoT, the model achieves ID/OOD generalization via a two- stage circuit (sparse sequential dependencies between input tokens). It is noted that intermediate result e2 is resolved at layer index where = 3 for ID and 5 for OOD.
We investigate the inner workings of the model during generalization using two widely adopted interpretability methods: logit lens [65] and causal tracing [70]. Specifically, we examine hidden- state representations by projecting intermediate-late the model's output embedding matrix. Besides, we assess the causal contribution of intermediate states by systematically replacing layer-wise representations, enabling us to identify the circuits responsible for generalization. Our analysis is conducted within the experimental setup described in Section 3.2. Here, we use (C1, T1, 62) @ (02,72 T2, es) = (C1, "1,72, es), Yei, 62, C3 E E, Vri, 12 E R to represent two-hop facts, in order to more clearly compare training with CoT and without COT.
activations into the vocabulary space through
Briefly, the circuit reflects the composition process: CoTT-triiedd models learn to systematically compose (both ID and OOD generalization) simpler skills (P) & P(Y|X, C)), whereas those trained without CoT merely match patterns seen in the training data (only ID generalization).
The internal mechanisms of training with CoT: two-stage compositional circuit. We perform set of logit lens and causal tracing experiments after training with COT. Figure 2 (Right) illustrates the discovered compositional circuit, which represents the causal computational pathways after the 8- 8-layer model achieves two-hop ID/OOD generalization. Specifically, pretable causal graph consisting of states in layers 0, 1, and 8, where weak nodes and connections have been pruned. The circuit exhibits two stages, consistent with the explicit reasoning steps in models during training. 1 In the first-hop stage, layer splits the circuit into lower and upper parts: the lower parts retrieve the first-hop fact from the input e1, T1, 72, store the bridge entity (e2) in state Ei11, r2]; the upper parts pass the information of e2 to output state E|8, r2] through the residual connections. Since the data distribution is controllable. can be precisely located (3 for s((2) 5 for
identify highly inter-
GOOD) In the hop stage, the auto-regressive model uses the e2 generated in the first-hop stage. This stage omits the e1, F1 and processes the second-hop from the input e1, F1, 72, to store the tail eg to the output state E(8, e2). CoT training internalizes reasoning steps. with the number of reasoning circuit stages matching the number of explicit reasoning steps during training.
'If perturbing node does not alter the target state (top-1 token through the logit lens), we prune the node. Please method details in Appendix D.5.
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The limits of training without CoT for OOD. For training without CoT, we do the same circuit analysis and reproduce the results of Wang et al. [102]: the circuit emerges only during ID general- ization. Layer 5 splits the circuit into lower and upper layers: the lower layers process the first-hop fact (e1, 71, e2) from the input (e1, F1), storing the bridge entity e2 in F rı]; the upper layers (5-8) gradually form the second-hop reasoning during extensive overtraining. Only when the intermediate result e2 is resolved can the final result é₁ be correctly derived [6], traditional transformer cannot process subtasks in parallel. The difficulty in OOD generalization arises from semantic misalign- ment across layers, due to inconsistent token representations learned by the transformer [100].
How training with CoT mitigates transformer limits in OOD scenarios. (1) As shown above, training with CoT allows the intermediate result e2 to be extracted from a lower layer (index 3) for ID examples, whereas training without CoT requires a higher layer (index 5). From an intuitive perspective, smaller layer index implies that more layers remain available for processing the sec- ond hop, which could lead to better performance. We also demonstrate that a two-layer transformer is sufficient to learn two-hop compositional circuits from CoT training in Appendix D.3. (2) CoT training explicitly processes each task through the model, allowing the model to independently learn each reasoning step. In brief, ID generalization is achievable in both CoT and non-CoT formats after sufficient training, but OOD generalization is challenging without COT. CoT enforces the decompo- sition of subtasks, effectively bringing OOD data closer to the ID, thereby helping the model learn a systematic generalization circuit. This aligns the theoretical analysis in Section 2.
3.4 ROBUST SYSTEMATIC GENERALIZATIO THROUGH COT TRAINING
In practice, not all CoT training data is manually annotated, which means erroncous reasoning steps may be present. We aim understand the source of CoT training's robustness under such imperfect conditions. In this section, we extend our analysis to settings where erroneous reasoning steps exist in the training data.
Acrunnry (0-11
ACC./SCC (C:31
Accuracy (0-11
3
68
6 6 S Optimization Stop (105. (.0.5)
Optinization Step (101. (D)
Optimization Stop (104)000)
Optiization Stop 110 0.1)
Figure 3: Left Part: the impact of only the second-hop noise on ID (Left) and OOD (Center Left) generalization. Noise has significant impact on the final performance
both ID and OOD test data. However, the generalization trends for ID and OOD differ under noisy conditions. Right Part: the model's accuracy training and testing two-hop reasoning facts at different noise ratios (both hops are noisy). It compares the results for & values of 0.05 (Center Right), 0.1 (Right).
o Method. We aim to analyze the robustness of systematic generalization achieved through CoT training when noise is present in the training data. We the same datasets as in Section 3.2, along with a two-layer model described in Appendix D.3. We introduce noise to the ID test examples (2) Si by randomly selecting valid entity (the gold training target is 61, 71, 71, C2, e3): (1) Only the noise e Note that the noise ratio is denoted as &, we explore the impact of different Ç. Nexr, we analyze different noise ratio ( candidate sets for the two situations: 0.05, 0.2, 0.4, 0.6, 0.8 for cases where only the second hop is noisy, and 0.05, 0.1 for cases where both hops are noisy. More results are in Appendix D.6.
second hop noisy, 61, r1, 72, e2, egeise, (2) Both hops are noisy, e1, T1, 72, e2
CoT training still enables systematic generalization even with data containing noisy steps (In comparison to Figure (Left)). (1) Figure 3 (Left Part) clearly demonstrates the impact of only the second-hop noise on ID and OOD generalization. Overall, under CoT training conditions, the model is still able to achieve systematic generalization from noisy training data, but its generaliza- tion ability decreases as the noise ratio increases. More specifically, as training progresses, OOD generalization initially remains constant and then increases, while ID generalization first increases
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and then decreases. The decrease in ID generalization corresponds to the increase in OOD general- ization. However, the final performance of both ID and OOD generalization decreases as the ratio increases. In particular, when the noise ratio (E < 0.2) is small, the model remains largely umaf- fected, highlighting the robustness of CoT training. Additionally, using the method from Section 3.3, we examine the circuits. Since noise is only introduced in the second hop, the first-hop circuit is well-learned, while the second-hop circuit is more significantly impacted by noise. (2) Figure 3 (Right Part) shows a comparison of the results for both hop noise values of 0.05, 0.1. Adding noise to both hops has a much stronger suppressive effect on the model's s generalization compared to adding noise only the second hop. This indicates that the harm caused by CoT training data with completely incorrect reasoning steps is enormous.
(3) To sum up, CoT training enables sys- tematic generalization even with noisy training data, as long as the noise remains within a certain range'. Specifically, when the noise ratio is low, noisy data can still facilitate the model's learning of generalization circuits. The training bottleneck lies in collecting or synthesizing complex long CoT solutions, with some errors being acceptable. Additionally, we analyze the relationship be- tween noisy samples and those with prediction errors, with further details on the effect of first-hop noise provided in Appendix D.7. Validations on real-world datasets (e.g., math word problems & compositional reasoning) are shown in Appendix E.
4 MORE DISCUSSION
Theoretical Contribution. According the traditional statistical learning viewpoint, performance can be defined by the sum of optimization error and generalization error. For generalization (Sec- tion 2), the expected generalization error = population risk empirical risk. More specifically, on the one hand, Theorems 1 and 2 show that CoT training helps reduce the expected generalization error. On the other hand, we can evaluate the empirical risk by assessing the model's performance on the validation set (Section 3.2). If the generalization error is known, it is possible to estimate the population risk. Therefore, the most challenging aspect is addressing the generalization er- ror. For optimization, from an intuitive perspective, CoT--rriined models learn to systematically compose simpler skills (P(CIX), P(Y|X, C.). whereas those trained without CoT merely match patterns (P(Y)X)) seen in the training data. That is, the optimization object for CoT training is [IP(Ctarget X)) P(Cprediccl + ||((aargee X, Ctarget) - P(Ypredict) X Cpredict I|² which may be finer- mner-grained (if the snowballing errors [4] are within acceptable range) than optimization object for non-CoTT training || [P((Yargeet X)- - P(Ypredict| X) x)1
Structural Discussion. Our structural analysis (in Section 3.3) leads a discussion the key limitation of transformer models in OOD generalization. For the circuit with two-hop facts (i,,T),))) @ (e2,72, 72, ea) + (e1,71, (2, ea). (i) Training with CoT: the bridge entity (e2) can be extracted from the middle layer hidden states B(layer index, r2). for ID layer index = 3, and for OOD, layer index- = 5 (OOD is more challenging than ID). (ii) Training without CoT: The gener- alization circuit fails to emerge in OOD settings, aligming with the performance drop in Figure (left). In ID settings. layer index 2 5, higher than that in Training with COT. Intuitively, smaller layer index imply that more layers remain available for processing the second hop, potentially leading to better performance. Only when the intermediate result e2 is resolved can the final re- sult e3 be correctly derived, traditional transformer cannot process subtasks in parallel, which is the limitations of traditional transformer architectures.
The generalization circuit we discovered essentially corresponds to the reuse of the model's weights (effective depth increases [17]). Regarding the OOD generalization capability for compositional tasks, what would truly like see is whether the model can accomplish multi-step reason- ing directly within a single computational process. To investigate this, we evaluate models trained with Coinusing twice the number of layers with weight reuse-undes "c1, F1, P2, [mask]". This setup is designed to emulate multiple forward passes typically required in next-token prediction. By examining the intermediate hidden states at the [mask] position (from the last layer), we observe that the model remains capable of correctly predicting e3.
modified input format:
7 Guo et al. [25] collect about 600k long CoT training samples. Errors during the process are acceptable, as is impossible for humans to manually write 600k bigh-quality solutions. Moreover, through reinforcement learning, OpenAl 01 [68] learns to hone its chain of thought and refine the strategies it uses. It learns to recognize and correct its mistakes.

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5 MORE RELATED WORK
Chain-of-Thouun Reasoning and Training. Research has demonstrated that incorporating an intermediate reasoning process in language before producing the final output significantly enhances performance, especially for transformers [97] with advanced and robust generation capabilities. This includes prompting LLMs [109, 59, 139, 41, 81, 136, 93], or training LLMSs to generate reasoning chains, either through supervised fine-tuning [132, 129] reinforcement learning [105, 30, 85, 128, 126]. However, the specific advantages of CoT training remain an open question, which is the focus of our study. There are also theoretical analyses that have shown the usefulness of CoT from the perspective of model capabilities (e.g., expressivity) [17, 63, 53, 73, 127]. For instance, by employing CoT, the effective depth of the transformer increases as the generated outputs fed back into the input [17]. These analyses, along with the practice effectiveness of CoT, motivate our exploration of the core mechanisms underlying explicit Co'l' training. Our findings suggest that two-layer transformer may be sufficient for learning compositional circuits through CoT training, which may explain the origin of CoT's expressivity during the training phase.
Latent Reasoning in Language Models. The investigation of latent multi-hop abilities of LLMs could also have sigmificant implications for areas such as generalization [45, 66] and model editing [137, 10]. To enhance the latent reasoning capabilities of LLMs, it is crucial to first understand the internal mechanisms by which L.MS effectively handle two-hop facts using latent multi-hop reasoning, which is also our focus. While precise latent reasoning pathway has been found in controlled experiments [87, 64, 11. 7, 51, 77, 125, 102], it has not been thoroughly investigated in large pre-trained models. To fill this gap, Yang et al. [117] construct a dataset of two-hop reason- ing problems and discovered that it is possible to recover the intermediate variable from the hidden representations. Biran et al. [6] identify sequential latent reasoning pathway in LLMs, where the first-hop query is initially resolved into the bridge entity, which is then used to answer the second hop, they further propose to intervene the latent reasoning by "back-patching" the hidden represen- tation. Nevertheless, these studies focus only on CoTT prompting and do not consider CoT training. Recently, it has also been found that one can "internalize" the CoT reasoning into latent reasoning in transformers with knowledge distillation [13] or a special training curriculum that gradually shortens CoT [14]. Loop transformers [24, 8, 15] have been proposed to solve algorithmic tasks. However, these works focus more on innovations in training methods and algorithmss, without fully analyzing the underlying mechanisms of CoT training. This is precisely what we aim to uncover.
CONCLUSION AND LIMITATION
Conclusion. In this work, we present a theoretical and structural framework to understand the generalization of CoT training. Our analysis reveals that the key mechanism behind CoT training lies in its ability to foster compositional generalization: models learn to systematically combine simpler, previously acquired skills to tackle novel and more complex problems. Theoretically, demonstrate that the generalization error bound naturally decomposes into ID and OOD components. By breaking down complex tasks into simpler subtasks, CoT training narrows the gap between the training distribution and both ID and OOD test distributions, thereby enabling robust generalization performance. Structurally, we identify that CoT-trained models internalize reasoning into staged compositional circuit-an an architectural reflection of how CoT encourages models to learn "how to think", rather than merely "what to think". The quality of CoT data should be measured not only by the correctness of answers, but more critically, by the clarity and decomposition of key reasoning steps. Our findings have important implications for the design of CoT training strategies.
Limitation. However, further studies are required to (i) explore the potential of LLM reasoning in an unrestricted latent space, specifically through approaches such as training large language models to reason in a continuous latent space [29], (ii) extend our analysis to reinforcement fine-tuning settings [89, 82, 133, 21, 83] and (iii) while the notion of "simpler learned skills" is straightforward in synthetic tasks (e.g., atomic facts), it becomes substantially more implicit and difficult to delineate in real-world scenarios. Yuan et al. [131] propose that one practical approach is to first use a small seed set of SFT data to instill fundamental concepts within target domain. Intuitively, the basic capabilities acquired by the model before CoT fine-tuning can be viewed as such "simpler learned skills". Those will deepen our understanding of the CoT training capabilities.
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ACKNOWLEDGMENTI
This research was supported by National Key Research and Development Program of China (NO. 2024YFE0203200), Beijing Natural Science Foundation (Z250001), National Natural Sci- ence Foundation of China (No.62476277), CCF-ALIMAMA TECH Kangaroo Fund (No.CCF- ALIMAMA OF 2024008), and Huawei-Renmin University joint program Information Retrieval. We also acknowledge the support provided by the fund for building worldelass universities (disci- plines) of Renmin University of China and by the funds from Beijing Key Laboratory of Big Data Management and Analysis Methods, Gaoling School of Artificial Intelligence, Renmin University of China, from Engineering Research Center of Gext-Geeeration Intelligent Search and Recommen- dation, Ministry of Education, from Intelligent Social Governance Interdisciplinary Platform, Major Innovation & Planning Interdisciplinary Platform for the "DoubleFirst Class" Initiative, Renmin University of China, from Public Policy and Decision-making Research Lab of Renmin University of China, and from Public Computing Cloud, Renmin University of China.
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LLM USAGE
Regarding the use of LLMs, they were employed solely for language polishing purposes and played no role research ideation, literature retrieval, or any other academically substantive activities.
A MORE RELATED WORK
Chain-of-Thougee Reasoning. Research has demonstrated that incorporating an intermediate rea- soning process in language before producing the final output significantly enhances performance, especially for transformers [97] with advanced and robust generation capabilities. This includes prompting LLMs [109, 59, 139, 41, 81, 136, 93], or training LLMs to generate reasoning chains, either with supervised fine-tuning [132, 129, 123] or reinforcement learning [105, 30, 85, 128, 126, 124]. There are also theoretical analyses have shown the usefulness of CoT from the perspective of model expressivity and generalization [17, 63, 53, 73, 122, 120, 118, 119, 127, 37, 36]. By employ- ing CoT, the effective depth of the transformer increases as the generated outputs are fed back into the input [17]. This aligns with our findings, suggesting that a two-layer transformer is sufficient for learning compositional circuits through CoT training. While CoT has proven effective for certain tasks, its autoregressive generation nature makes it difficult to replicate human reasoning on more complex problems [46, 27], which often require planning and search [113, 121, 28, 47, 20, 88, 49). These analyses, along with the practice effectiveness, motivate our exploration of the core mecha- nisms underlying CoT training.
Latent Reasoning in Language Models. The investigation of latent multi-hop abilities of LLMs could also have significant implications for areas such generalization [45, 66] and model editing [137, 10J. To enhance the latent reasoning capabilities of LLMs, it is crucial to first understand the internal mechanisms by which LLMs effectively handle two-hop facts using latent multi-hop reasoning, which is also our focus. While precise latent reasoning pathway has been found in controlled experiments [87, 64, 11, 7, 51, 77, 125, 102], has not been thoroughly investigated in large pre-trained models. To fill this gap, Yang et al. [117] construct a dataset of two-hop reason- ing problems and discovered that it is possible to recover the intermediate variable from the hidden representations. Biran et al. [6] identify sequential latent reasoning pathway in LLMs, where the first hop query is initially resolved into the bridge entity which is then used to answer the second hop, they further propose intervene the latent reasoning by bacck-ptchhing the hidden repre- sentation. Shalev et al. [84] discover parallel latent reasoning paths in LLMs. Ghandeharioun et al. [23] introduce a framework called "Patchscopes" to explain LLMs internal representations in natural language. Recently, it has also been found that one can "internalize" the CoT reasoning into latent reasoning in the transformer with knowledge distillation [13] or special training curriculum which gradually shortens CoT [14]. Loop transformers [24, 8, 15] have been proposed to solve algorithmic tasks. These have some similarities to the two-stage compositional circuit we present.
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B OMITTED PROOFS AND DEFINITIONS
B.1 LEMMA 1 AND PROOF
Lemma 1 (KL Divergence Decomposition). Under the conditions of Assumption the KL diver- gence between test and training conditional distributions satisfies:
(Re(Preu ( PX)llPmriii (YIX) - (a) + (1 DKI (Pid (Y|X)I| mainn (Y
+ xz (P ( (Y P (Y))),
where DKL (Y
Prrin (X train (Y ()) Bylzz-Pen Paen
log ()) H(a) ) is the entropy of quanti-
Ponin (Y [X]
fying ID/OOD uncertainty, the other terms capture ID/OOD deviation from the training distribution.
Proof. Original definition of KL divergence:
Prest ( [X)
DKI. (Pest (Y)))| Pmin (Y]X))
E Eyl log
P Pirain! (YI))
Since Pies is a mixture distribution, the expectation can be decomposed into two parts:
log Piett (M(x)
log
P Pieat (X
a) E
Ou E P000
Pirain (Y|X)
P Piaain (YIX)
Due to the disjoint support assumption:
PID (Y
aDODD (X
= (1 - ax) E p
log
+ a E yy poop
log Ptrain (Y(X)
Perain
(Y
By isolating the constant term and the KI. divergence:
=a log a + (1- a) log(1 -1 (1-a) (PID (Y) (X)))
+ aaDi (PO00 (Y [ Prain
Irain (YlX))
H(a) + (1 - DKI. Piesl
(Y\
+ aDxu (POOD (Y x | P Prein (YlX)),
where H(a) ΣK ai log 0i EK 04 1, Va, 2 0. This completes the proof.
B.2 EXPECTED GENERALIZATION ERROR
Recently, information-iin -thoorettcc generalization bounds [1 14, 78, 86, 108, 107] have been introduced to analyze the expected generalization error of learning algorithms. key benefit of these bounds is that they depend not only the data distribution but also the specific algorithm, making them an ideal tool for studying the generalization behavior of models trained using particular algorithms.
Definition 4 (Expected Generalization Error). We let Z X x be the instance space and μ be an unknown distribution on Z, specifying random variable Z. Here, X denotes the feature space and y is the label space. Suppose one observes a training set SN A (7,,...,Zy, ZN) E 2N analysis framework,
with N i...d. training examples drawn from μ. In the information-theores we let W be the space of hypotheses related to the model, and stochastic learning algorithm A which takes the training examples SN as its input and outputs hypothesis W W according to some conditional distribution QwiSy. Given a loss function É W x Z - R++, where E(w, Z) measures the "unfitnes" or "error" of any Z E Z with respect to hypothesis take as continuous function and assume that & is differentiable almost everywhere with respect to w. The goal of learning is to find a hypothesis w that minimizes the population risk, and for any W, the population risk is defined as I.((w) & Ez=a(E(w,, Z)). However, since only can partially observe via the sample Sx, we instead turn to use the empirical risk, defined as Lsw (w) N EN C(w, Z.). Then the expected generalization error of A is defined as
W. We
error & Ew,, (L, -Lw (W)),
where the expectation taken over (SN, W) N 69 QwISx.
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Remark 3. At this point, revisiting Definition and Assumption 1, SN corresponds to the training data distribution P (Y),, while corresponds to the test data distribution Pees(Y|X] Accord- ingly, the equation be reformulated as:
error - E(h W ( W)],
where + (w,
That generalization error=population risk-empirical risk.
B.3 PROOF OF THEOREM 1
Information -hheorrtiic generalization bounds commonly center on input-output mutual information,
with the Donsker-Varadhan representation of KL divergence (Polyanskiy and Wu [71, Theorem 3.5)) serving as a key analytical foundation.
Lemma (Donsker and Varadhan's 's variational formula). Let P1, P2 be probability measures 2,
for any bounded measurable function f Z + I. we have (xi(Pi))Ps) = sup, P=, pf())
log Ez~P,[exp (Z)].
Lemma (Wang [107, Lemma 2.4.6)). Let Pi and P2 be probability measures on Z. Let ~ Pt and Z P2. If g(Z) is R-subgaussian, then,
E ~r, (g( Ez La 2)] $ (2
Proof. Let f =t 9 for any t R. by Lemma 2, we have
( 2 sup E ~ [tg 2
log Ezg eexpt - [()
sup Ez ~ ~ |/0 1
log Bz /x (exp (()(2) (Bz=P<(9(2)) + Ezur, L0( [))
sup Ex' n. [t } Ez~P [tg log Ez-Ps /exp (g(() Ez B [g(()))
2 sup (1 -1. 10 2 - Ez-ry (a( (2)) 12 r²² /2,
where the last inequality is by the subgaussianity of g()).
Then consider the case of > 0 and t < 0 (t = 0 is trivial), by AM-GM inequality (i.e. the arithmetic mean is greater than or equal the geometric mean), the following is straightforward,
Ex'' [o( 2 - (g S Dx This can also be proven by computing the extremum through differentiation with respect to
Below, we provide the proof of Theorem 1.
Proof. Apply Lemma to the generalization error (Definition 4), we let g(s, w) N + Zw) Thus, if I(w, Z) is R-subgaussian for all E W, then g(S W) is R/VN- N- subgaussian due to the i.i.d. assumption on Z₁'s. This implies that 9 W') is also R/VN VN- subgaussian, where Stest w' are independent copies of strain W, respectively, So we have:
2R N
error SV
DKL (Pesell (Prann)
(1)
According to Lemma 1, for (a)) < 0:
(KL(Piesl Puain) = - (((a) + (1 DKL (2DD((PPl)))
(PIID) | Prain) + aD (PrO Purin) <1 (11-a a (Pit) Puria) + aDKL (OKI(PIO (POOD) Puraa)).
Putting everything together, we arrive at:
2R
[1 - DKL (PID [ + aDkL (Panni]
error
N
Under the conditions of Lemma and Lemma 3, we complete the proof.
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B.4 EXTENSIONS OF THEOREM
Recall Theorem I, the expected generalization error is bounded by:
2R N [1 DkI (P.ii((PPt)))
error S
(Pita MO (X)) Pmiin (Y]X)) + (PPE ( (Y]X))]
where N the training data size, 7 (x, Y) and W is the space of hypotheses related to the model. denotes the mixing coefficient of the OOD data in test distribution and DKL (-) represents the KL divergence between ID/OOD test and training distributions.
For ID, since $ consists of unseen instances within known compositions (ri, ry) from $1 the reasoning patterns in ID test examples exactly match those in the training set. Therefore, "Drrg sufficient training-whether whether with or without CoT--ho ID generalization error approaches zero (i.e., DxL ( [Pes Prain) = 0). Consequently, we obtain the following corollary:
Corollary 1. Let the conditions specified in Theorem hold, and assume sufficient training such that Dei (Piell Prrii)) + 0. Then the expected generalization error is dominated by the OOD component:
V 22 N
error S
( (Y ( [X] (Y
This implies that when ID patterns are perfectly learned, the error scales with v DoDD KL. revealing that OOD generalization fundamentally limits performance even under optimal ID training.
For OOD, considering the data generation process, we have: P(Y[X) = Ec P(Y C)P(C)X) where C' denotes the CoT reasoning steps. (a) In training without CoT, due to the model does not explicitly learn P(C|X) (i.e., C never appears during training). the P(C becomes a uniform prior, making P(Y|X,C) C) reduce P(Y]X). Moreover, since OOD test examples SOOD... involve unseen compositional patterns, training without CoT struggles to generalize to OOD settings (the reasoning circuit only emerges during ID generalization). That is, under training without CoT, there has been no improvement in the upper bound of OOD generalization.
(b) In contrast, under training with COT, both P(CIX) and P(Y]X, C) are explicitly modeled, which correspond precisely to the two stages of the compositional circuit. Since the sub-tasks have been observed during training, when the step- wwie decomposition in s and S aligns, OOD generalization can also achieve near-perfect performance. The next result provides an explicit form.
Corollary (Thereom 2, OOD Generalization Error for Training with CoT), Let the conditions specified in Theorem I hold, and assume sufficient training such that DRI (PPl)) P Prmin) - 0. Define X) = Ec P(Y(XXC)P(CCIX C(C))) then the OOD generalization error is bounded by:
2R (DKI((POOO (CIX) |Prrain (CIX))
error <
N
+ BC (C)X) |DRI [DKI (POOP (Y|X, C) (Y|X,
where C; denotes the CoTl reasoning steps, see Appendix B.5 for a proof.
Remark 4. When the intermediate reasoning outcomes (e.g., data patterns and reasoning steps) from explicit CoT training are highly aligned with or closely match the intermediate reason- ing required for final inference during testing, the model's 's generalization ability is significantly enhanced. Specifically, the model explicitly learns both P(C)X) and P(Y|X, C"), such that DxL (C) (Clnin(( + 0 and Dxa ( C (Y]X,C)) + 0. Therefore, OOD generalization can also approach optimal performance under CoT-based training.
Remark 5 (Length Generalization Discussion). Length generalization continues be an indispens- able research avenue in CoT investigations [140, 126, 112, 1, 106), we also experimentally study in Appendix D.4. From theoretical perspective (Corollary 2), models achieve optimal generalization capability when the number of reasoning steps during training (m1) matches that during testing (mz), as this allows the model to independently learn to optimize the DRL(-) for each individual step. The model can still generalize well when > m2 and m2 is prefix structure [106] of my (i.e., the subtasks in the shorter chain have all been encountered in the longer chain). On the other hand, when m, m2 and m2 contains unseen subtasks, generalization becomes more challenging.
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B.5 PROOF OF THEOREM 2
We are given two conditional distributions defined as mixtures:
Pa x) Σ Pully c, Piz(e z), Pa(y x) E Paily | P2((c z).
We aim derive the KL divergence between Pi x)) and P2(y) z):
Pi log P1 | P2 x)
DKL(PI Pn
Substituting the expressions for P1 and P2:
Ev Puily c e1 (x)
DRI((Pi|PP))
Σ
Σ (,,0lc)))P1((l1))
log
Ee Paily P22(c' [))
1/
Ee Pui d, Piz(d |x)
Σ
Σ Piz(e P)P c, log
En P21 d, (22((' 2))
This is the exact form of the KL divergence but is difficult simplify further due to the logarithm over sums. We now derive a useful upper bound using the log-sum inequality (f(z) = x log x is convex function):
a, log di z Σ
Σ ai
log
Σ aj Σ b
b,
Let
a: Pii 1, x) Piz(i [x),
b, = P21 (x x).
Then the inequality implies:
Pi(y x)
Pn c, Pi(ec
Py x) log
S EPil( Pn) c. log
Pa(y x)
Pa c, x, x)
Summing both sides over y:
Pnily c,
Pi2(c x) (c x))
(PL(P)) & Σ Pia(c PI (.,x)
log
+ log
Pzi (,z)
N.c
Pni (,.)
Σ Piz(el x) Σ Pui(1) c,
Ps c,z)
Piale x)) log Pi2(c x)
Pzz(c r)
-E- [KL (Pul c) [,,a)] + D
P2) [=)))((( lx)).
Thus, we obtain the following upper bound:
D (Pr s (Pil [ x)) + Eew Pia(- a) [KI.. (Pu c, Pai [c,)))
Apply it Corollary complete the proof of Theorem 2 (Corollary 2).
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B.6 THEOREM 3 AND PROOF
Given that the reasoning pattern remained unchanged during the noise injection process in Sections 3.4 and E, adopt the following assumption regarding the training data distribution:
Assumption 2 (Noise Injection to Training Distribution). Assume that output noise randomly re- places the original oulputs with probability S. Let the original output distribution be denoted as P ( , then, the oulput distribution after noise injection is given by:
new (Y = Prai ( (YI)) + & (YIX);
where Pa moise (Y represents the distribution over outputs after random replacement.
Lemma 4 (Convexity of the KL divergence). For a fixed probability distribution P. the KL diver- gence Dx Pll)) is convex function with respect Q. That is, for any two distributions Q1, Q2, and any E [0, 1], the following inequality holds:
DEL (PI((1 3)Q1 3Q2) S (1 D) (x) (PIIQ)) 3Dxz (Pl))Q))))..
Proof. The KI. divergence is defined as:
P(œ)
(RI((1 Σ P(x) In
Q(x)
Now consider Q = 001 (1-8)Q] + 3Q2. Substituting this into the definition of KL divergence gives:
P(x) DKL (P (1) + + P()))) (1 ()Q1(z) + 3Q2(z)
Using the convexity of the logarithmic function In(1/t) (since In(() is concave, and In(1/t)
-I((t) is therefore convex), we have:
P(x)
P(x) 81 P(x)
In (1 ()Q + 3Q2(z) S 8)
Q1(x)
Q2(x)
Multiplying both sides by P(x) (noting that P(x) > 0) and summing over we obtain:
P(x)
3Q2(z) S (1-8) 3) Σ P(x) In P(x) P(a) P(x)
P(x) (1- (1=3
Q1(x)
Q2(x))
That is, DRI
S (1- (P)))) + DKi. (PI(Q2).
Theorem (Generalization Bounds with Noise Ratio). Under the conditions specified in Assump- tion 2 and Definition we assume that the loss f(w, ) R-subGaussian for any w W RRd, then the expected generalization error is bounded by:
2R² [11 - Dx (Pesel | Praii)) &DKL (Presel [nonise)]
error
N
where N the training data size, Z (X,Y) and W is the space of hypotheses related to the model. demotes the noise ratio in training distribution and DxI(-) represents the KL divergence between two distributions. Pres
a.) PiD aOODD.
Proof. Replace Puiin (Y in Eq. (1) with Prem (Y we have:
2R N
error S DEL Pnew).
To apply Assumption and Lemma 4, we let E, P- Prest, Peew. that is, Q1 Pininn Q2
P Phoise- Therefore, we will get:
2R N [11 DKI. (Prest Purin) + EDI. (Prest Mastl|PPnicc Pnnisse)
error
We complete the proof.
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C EVALUATION AND REPRESENTATIVE EXAMPLES
ID/OOD Evaluation. To better evaluate the generalization capacity of the model, assess its performance on both ID and OOD data. (1) ID generalization aims to determine whether the model has correctly learned the latent patterns by evaluating its ability to complete previously unseen hop facts s (2) OOD generalization aims to assess the systematicity [44] acquired by the model, specifically its ability to apply learned patterns to knowledge irrespective of its distribution. This done by testing the model on facts s(2) SOOD- the model performs well on ID data, it may have memorized learned patterns present in the training data T. However, strong performance OOD data indicates that the model has learned the latent patterns, as 7 contains only atomic facts SOOD. not SO0D Representative examples highlighting the distributional differences between ID and OOD data are presented below.
The key distinction between ID and OOD data lies in whether their distributions are the same. We provide a more prominent example with distribution differences here. The training set composed of Sin, SooD and IDursin"
s(2)
Sip:
(Paris, CapitalOf, France), (France, LocatedIn, Europe), (Berlin, CapitalOf, Germany), (Germany, LocatedIn, Europe).
Soon:
(Lima, CapitalOf, Peru), (Peru, HasNaturalFeature, Andes Mountains).
DD a uniformly random subset of the inferred facts derived from Spp: (Paris, CapiaalOiCounrryyLoaeeeddll Europe).
(2)
1 previously unseen inferred facts derived from Sip (ID generalization):
(Berlin, CopptalOOCCuuttyoooaeeelll Europe).
((2)
DOD previously unseen inferred facts derived from SooD (OOD generalization):
(Lima, CapitalOfCountry WathtttyWWtaaaaaFFFFaaaalla Andes Mountains).
Key Changes. This OOD data setup requires the model to tackle greater knowledge distribution differences and reasoning challenges during evaluation. (1) Change in Relation Types: New relation types, such as HaasaaturalFeatur
are introduced in the OOD data. (2) Complexity of Reasoning Paths: Reasoning paths involving natural features are added, requiring the model not only to reason but also to generalize to new types of knowledge.
D MORE EXPERIENCE RESULTS
D.1 DATA GENERATION PROCESS
Specifically, for one-hop facts, random knowledge graph G is constructed with [E] entities and |R| = 200 relations. Each entity, acting as the head (h), is linked to 20 unique relations, with another randomly selected entity serving the tail (t). One-hop facts
each relation connecting correspond to the (h, r. r.t) triplets in G. S C C S. These triplets are partitioned into two disjoint sets: Sip (95%) and Soon (5%), which are used deduce the ID/OOD two-hop facts (Vh, b, I. E, Vri, E R): (h, ri, b) @ (0,r2,0) = (h, 12,
D.2 TRAINING DETAILS OF CONTROLLABLE DATA
Model. The model we employ is a standard decoder-only transformer as in GPT-2 [76]. We use 8-layer for Figure 1 (Left, Center Left, Center Right) and Figure 2, while 2,4,8-layer for Figure (Right) and 2-layer for Figures! and The other hyperparameters are consistent with those described in Section 3.1.
Tokenization. In our main content (Section 3.2), we assign a unique token to each relation/entity by default. This is because Wang et al. [102] find that different tokenizations affect the results in
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rather expected ways, and do not influence the reasoning generalization findings. We also validate our findings in the real-work entity-tokennn mnity-tokenization in Section E.
Optimization. Optimization is done by Adam W [57] with learning rate 10 batch size 512, weight decay 0.1 and 2000 warm-up steps. Notably, models are trained for a large number of epochs/steps beyond the point where training performance saturates. All runs were done with PyTTrcch [69] and Huggingface Transformers [11 I] on NVIDIA GeForce RTX 3090.
D.3 TWO-LAYER MODEL CIRCUIT
e2
e3
Layer 2
Layer 1
Layer 0
Input Output Embeddings
Position Encoding
Final Layer
e1
T1
T2 e1
r1
T2
e2
Activations
First hop
Second hop
Logit Lens
Figure 4: The two-stage compositional circuit (layer:2) for two-hop facts. We use logit lens to interpret individual states, and use causal tracing to measure the strength of connections between states. It is evident that two-layer model can learn compositional circuits from CoT training. This aligns with Cabannes et [8], who describe how a certain distribution of weights within the first two attention layers of transformer, referred solve iterative tasks with CoT reasoning with relative ease. We identify CoT reasoning circuits in transformer with only 2 layers. The output of first stage is autoregressively used for the second.
"iteration head," " enables transformer to
For CoT prompting, the effective depth of the transformer increases because the generated outputs are looped back to the input [17].
D.4 TWO-HOP HOP TO MUTI-HOP
In Sections 3.2 to 3.3, primarily focus on two-hop facts. In this subsection, shift focus multi-hop scenarios: Can model that has only encountered two-hop facts during the CoT training phase generalize to three-hop facts
Experiment Setup. The rule of (three-hop) composition is (E1, 62) + (e2, r2, (2,,72, 63) @ In CoT training, we only
(e3, P3, (q) (61, r2, T3, ea), Yei, 62, Cs, E VV1. r3 prodict
use one- mne-hop/tw--hop facts (e1, prodict model can generalize reasoning three-hop facts (e1, F1, r2,73 are same in Section 3.1.
62; 11,11
C2 C2, 63).
and we whether the predict
E2, E3, E4). Other settings
II naturally extend multi-hop scenarios thoroughly analyzed.
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Results. A model that has only encountered two-hop data during CoT training cannot directly gener- alize to three-hop scenarios. Intuitively. if we consider 61, 12 - e2. e3 and 62. T2, T's - E3: e4 as new atomic facts and e,, P2.7a + 62: E3, ea their compositional combination, models trained exclusively on atomic facts exhibit limited reasoning capability for compositional facts. However, when artificially split three-hop fact into two two-hop facts for testing, the model is also able predict
prodict C3, 64 62, e3, 64 to be correct. These
to generalize effectively. In other words, test c1, 72.
E2, C3 and e2: 72, L3 predict
separately, and when both are correct, consider 61, 72: 13 align with [8]: CoT relates to reproducing reasoning patterns that appear in the training set.
Discussion. For cite references, Zhu et al. [140] emphasize that the model fails to directly de- duce A, + C; when the two intermediate steps A, + B; B; - Cj are trained separately, this corresponds to that the absence of three-hop instances in the training data significantly limits its generalization capability to three-hop queries during evaluation. Moreover, when we include three- hop facts in the training set ([three-hop one-hopl 3.6), the model achieves over 0.9 accuracy on both ID and OOD test sets. Building upon this, we further conduct circuit analysis on the model trained with three-hop facts, revealing three-stage mechanism: the first stage activates e1, 7', the second stage activates r2, Eg. and the third stage activates r3, e3. Besides, we agree with [140] that "the reversal curse is a consequence of the (effective) model weights asymmetry". autoregressive models struggle with parallel generation of subtasks (see Section 3.3 for details). The generalization circuit we discovered essentially corresponds to the reuse of the model's weights. We also note the emergence of "Loop-transformee" Reasoning" [130] works. Our results further support the validity of this research direction.
Insight: When the intermediate reasoning results (data patterns) from explicit CoT training are highly aligned with or closely match the intermediate reasoning required for final infer- ence during testing, the model's generalization ability is significantly enhanced. This could explain why the researchers behind DeepSeek-RI [25] construct and collect small amount of long Coll data fine-tune the model in the Cold Start phase.
D.5 CAUSALITY ANALYSIS DETAILS
The methodological descriptions here serve to clarify the approach we used, not to attribute these methods to our own work. We mirror the recent practice [104, 102], where the causal tracing process consists of three steps (for every stage in Section 3.3):
(1) In the normal run, we capture the model's hidden state activations for regular input (e1, (2)/(2))) (1, e2). Notably, as the model maintains perfect training accuracy throughout the CoT training process, the final prediction invariably aligns with the ground truth" bridge entity e2 for the first-hop stage and tail entity e3 for the second-hop stage.
r2,
(2) During the perturbed 10, the model given a slightly modified input that alters its prediction, and the hidden state activations are recorded once again.
Specifically, for the hidden state of interest, we modify the input token at the corresponding position by substituting it with a random alternative of the same type (e.g.. Ti - ri ). which results a different target prediction (e.g., C2 >> 62 C3 eg )
(3) Intervention. During the normal run, we intervene the activation of the state of interest by sub- stituting it with its corresponding activation from the perturbed run. After completing the remaining computations, we check whether the target state (determined as the top-1 token via the logit lens) has changed. If perturbing a node does not alter the target state (top-1 token through the logit lens), we prune the node.
For simplicity, when refer to state token, indicating the top-ranked token of that state determined by the logit lens. "Frr the perturbation, studies have explored adding noise to the input [62] and replacing key tokens with semantically close ones [98, 18). We adopt token replacement which avoids unnecessary distribution shifts [135].
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D.6 COT TRAINING WITH NOISE
Accunery (0-3)
Accurly (0 11
Accuraty 10 30
Accuracy (0-31
Optimization Stap (100.gm0.05)
Optimiation Step (16), (m0..)
Optimization Stap [10).
Optimicatin Step 119P, (-44
Figure 5: The model's accuracy on training and testing two-hop reasoning facts at different noise ratios (both hops are noisy). This figure compares the results for ç values of 0.05, 0.1, 0.2, and 0.4.
The model's accuracy (100%) on all reasoning facts at different noise ratios (only the second hop is noisy). This table compares the results for & values of 0.05, 0.2, 0.4, 0.6 and 0.8 with 20,000 optimization steps in Figure (Left Part).
Table 1:
E
Sin
SooD
Si()) Daaa
(x2)
11PTI
S
0.05
99.9
100.0
99.3
99.0
97.8
0.2
100.0
99.9
98.9
95.1
77.6
0.4
100.0
99.9
98.0
86.6
49.1
0.6
100.0
99.5
96.8
70.4
30.9
0.8
100.0
99.2
97.3
42.6
14.8
Table 2: The model's accuracy (100%) all reasoning facts at different noise ratios (both the first and the second hops are noisy). This table compares the results for & values of 0.05, 0.1, 0.2 and 0.4 with 20,000 optimization steps in Figure
SID
SooD
(
(2)
GO0D
0.05
98.3
99.7
93.7
55.4
53.7
0.1
97.5
97.4
85.4
29.8
27.1
0,2
95.0
92.2
73.6
11.5
8.4
0.4
84.5
72.4
65.2
2.1
0.3
Results. We analyze different & (noise ratio) candidate sets for the two situations: (0.05, 0.2, 0.4, 0.6, 0.8) for only the second hop is noisy and {0.05, 0.1, 0.2, 0.4) for both hops are noisy. The comparison results are as follows:
(1) Figure 3 and Table 1 clearly demonstrate the impact of only the second-hop noise on ID and OOD generalization (s(3) and s Overall, under CoT training conditions, the model is still able to achieve systematic generalization from noisy training data, but its generalization ability decreases as the noise ratio increases. More specifically, as training progresses, OOD generalization initially remains constant and then increases, while ID generalization first increases and then decreases. The decrease in ID generalization corresponds the increase in OOD generalization. However, the final performance of both ID and OOD generalization decreases as the ratio increases. In particular, when the noise ratio ( < 0.2) relatively small, the model is almost unaffected, demonstrating the robustness of CoT training. Furthermore, the circuits. Since we only add noise in the second hop, the circuit the first-hop stage is learned relatively well, while the circuit in the second-hop stage is more heavily affected by noise.
also use the method in Section 3.3 to examine
(2) Figure and Table 2 show a comparison of the results for both hop noise & values of 0.05, 0.1, 0.2, and 0.4. Adding noise to both hops has much stronger suppressive effect the model's 's generalization compared to adding noise only to the second hop. A noise ratio greater than 0.2 is enough nearly eliminate both ID and OOD generalization ability.
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(3) Table 1 and Table 2 demonstrate that adding noise to both hops exerts significantly stronger suppressive effect on the model's generalization compared to introducing noise solely to the second hop. When the noise ratio is 0.2, the accuracy of S in Table is 95. %, and the accuracy of SoOD is 77.6%. However, in Table 2, the accuracies of $ and SooD are only 11.5% and 8.4%. This indicates that the harm caused by CoT training data with completely incorrect reasoning steps is enormous.
(4) To sum up, even with noisy training data, CoT training can still enable the model to achieve systematic generalization when the noise is within certain range"! Especially when the noise ratio is small, these noisy data can still help the model learn generalization circuits. Moreover, we further analyze the connection between noisy samples and the samples with prediction errors from the model, and give the impact of only the first-hop is noisy in Appendix D.7.
Insight: CoT training still enables systematic generalization with noisy data, highlighting that data quality outweighs the method itself. The training bottleneck lies collecting or synthesizing complex long CoT solutions, with some errors being acceptable.
D.7 NOISE DISCUSSION
In Section 3.4, we demonstrated that adding noise to both hops exerts a significantly stronger sup- pressive effect on the model's generalization compared to introducing noise solely to the second hop. Here, we further investigate the impact of scenarios where only the first hop is noisy, We introduce noise to s(2) by randomly selecting a valid entity, where the gold training target e1, T1, r2, e2, 63- In this case, only the first hop is noisy, resulting in 61, F1, T2, ehoise C3.
Only the Forst-Hop is Noisy. Table 3 highlights that & increases, performance generally declines across all datasets, with the most significant drops observed 5(2) and s at higher noise levels. For instance, when E 0.4, the accuracy for s(2) falls to 0.3%, indicating a substantial degradation in model generalization under noisy conditions. This degradation indicates that errors intermediate steps within the CoT training data have a greater impact. However, the model still retains certain level of generalization capability under low noise conditions.
Table 3: The model's accuracy (100%) all reasoning facts different noise ratios (only the first hop is noisy). This table compares the results for & values of 0, 0.05, 0. 0.2, and 0.4. Regarding the optimization (20,000 steps) and model settings, we remain consistent with Section 3.4. All values are averaged over 3 random seeds.
E
Sw
SOOD
S DiyRAAN
(2)
DDass
(2))
GOOD
0
100.0
99.9
99.9
100.0
99.4
0.05
98.4
99.4
94.3
53.3
51.3
0.1
97.9
98.2
86.3
29.7
29.5
0.2
97.0
94.5
74.9
10.6
7.5
0.4
89.6
77.0
68.7
1.8
0.3
Noisy Samples Affect the Probability Distribution of the Output Vocabulary Space. As shown in Table 3, adding noise to s(2) can even impact the model's basic knowledge (Sjp and SOOD). We further analyze the relationship between noisy samples and the samples with prediction errors from
e1002; tu44 validate the model after gold label
the model. Namely, we use sample 61567; T32, T23
wrong predict
CoT training with E 0.05. The model's erroneous output e1567, T32, T23
E1640; 6605.
noisy label
and identify the noisy sample in the training set C1567, P32, 7'132
C1640 C851-
11 Guo et al. [25] collect about 600k long CoT training samples. Errors during the process are acceptable, as it is impossible for humans manually write 600k high-quality solutions. Moreover, through reinforcement learning, OpenAl 01 [68] learns to hone its chain of thought and refine the strategies it uses. It learns to
recognize and correct its mistakes.
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We check the probability distribution of the output vocabulary space: (1) For one-hop fact e1567, r32 gold label
61002), the token with the highest output probability is e0002, followed by 61640 as the second highest. Adding excessive noise can bias the previously learned one-hop facts. (2) For
two-hop fact 61567: r32, Fanyl,
10002; Cany2), the token with the highest output probability is e1640. followed by 61002 as the second highest. The structure of such noisy data can indeed con- tribute to the learning of generalization performance. However, the addition of noise can negatively impact the reasoning of related facts.
golld label
D.8 DETAILS OF REAL- WORLD DATA VERIFICATION
Data Deseriptions. The test data is sourced from GSM8K [9] (containing 1,319 samples), while the training data is derived from MetaMathQA [129] (comprising 395K samples), which enhances data quality and improves model performance. Considering training cost and following Meng et al. [61]. we use only the first 100k samples for 1 epoch of training.
Training Formats. The data has been converted into alpaca format [91] for case use. Unlike the non-CoT format, which simply outputs "The Answer is ...", the CoT format provides reasoning steps alongside the answer.
Hyperparameter Configurations We employ LoRA [34] with rank 16 and alpha 16, without dropout (0). The model is trained for 1 epoch with a maximum sequence length of 512 tokens. The training uses batch size of 4 per device without gradient accumulation (steps=1). The opti- mization employs a learning rate of 2e-5 with no weight decay (0) and a warmup ratio of 0.03. The framework and hardware information we used is provided in Appendix D.2.
We randomly inject noise with a probability of noise ratio. Notably, as shown in Table 4 (Right), the model was trained for only 1,000 steps.
Noise Types. Only two types of noise are added: (1) Step skipping (replacing intermediate steps with " "..."). (2) Incorrect step results (keeping the left-hand expression while modifying the right- hand result), Below we show an example comparing the results before and after noise injection.
sample_cata
x oouputt: 'At first, there were 10 snowflakes. AnEvery 5 minutes, an additienal x snowflakes fell, so after minutes, there were 10 + it /5) xx snowflakes. \nwe are given that there were 58 snowflakes, so we can write: 10 + (t/5)xx 58. .a.olvvvvg for t, we get: (t/5) - 48.\ get: t/5 48/x.AnMule AnMultiplyin both
nDividing both sides by x, sides by 5, get: t (5x48)/x. AnWe are given that t 60 minutes, so we can write: (5x48) = 60. AnSolving for x, we get: x (5*48) /60. \nSimplifying the right side, we get: x = 4, AnTh value of x is AnA4444 4AnThe answer is: 4',
'instruction': : x There were 10 snewflakes at first...', type'' 'GSMLFOBAR', 'input':
Roise_data {
x 'output' : 'At first, there were 10 snowflakes. An'
'Every minutes, an additional x snowÉlakes fell, so after minutes,
there were 10 + (t/5)*x snowflakes.
'we are given that there were 58 snowflakes, so we can write: 10 + (t/5)*
x An 8 Incorrect step results (58->57)
SSolvnn for t, we get: (t/5)*x +x ...An' step skipping
'Dividing both sides by x, we get: t/5 = 49/x. An' # Incorrect step results (48->49)
MMittppyyng both sides by 5, we get: = (6x48) /x. Ant $ Incorrect step results (5->6)
'We are given that 60 minutes, so we can write: (5x48) /x 59, An' Incorrect step results (60->59)
Solving for we get: x (5+48) /61, An & Incorrect step results (60->61)
Simplifying the right side, we get: An. Step skipping
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x The value of is 4. An'
FHFF 4\n
'Th answer is: 4', insttuction' There were 10 snewflakes at first...', ttyee' 'GSM_FOBAR,
"input': (
D.9 LAYER-WISE GRADIENTS CHANGE
Weight Changes for Each Layer
Layer
Weight Change
4-
att W
2-
mlp f2
Checkpoint
Layer
Weightt Change
att w
mlp f2
Checkpoint
Layer
Weight Change
att w
4
mlp
Checkpoint
Layer
Weight Change
att w
mip
Checkpoint
Figure 6: The nuclear norm of gradients across difffernnt layers when training with COT. This aligns with the Figure (Right), we plot the setting with layer 4, X = 7.2. The checkpoint-4 means the optimization steps=4,000, when the OOD test accuracy begins to grow. The ID generalization phase (0 4) leads to larger gradients across layers compared the OOD generalization phase (5 8). This aligns with the results [49], indicating that slow thinking with CoT results in smaller gradients and enables the gradients to effectively distinguish between correct and irrelevant reasoning paths.
Nuclear Norm:
The nuclear norm of gradient (G) is defined as the I norm of the singular values, which reflects the sparsity of the spectrum and serves as a convex surrogate of the matrix rank. Hence, it does not only quantify the gradient magnitude but also the concentration of the spectrum on its top few singular values, which vital to understand the gradient patterns each layer, i.e., -E -)zi((m))) 1a,1. oj 1,...) are singular values of G.
||G'] Inslear
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E REALISTIC DATA VERIFICATION
At a high level, our study so far paves the way for a deeper understanding and enhancement of the transformer's generalization abilities through CoT training controlled data distributions. How- ever, real-world training data distributions are often more complex. Accordingly, we empirically validate the insights presented in Sections 3.2 and 3.4 on real-world datasets.
Experiment setup with real-world data. Following prior work [129, 61], we perform LoRA- based fine-tuning [34] (rank=16) LLaMA3.1-8B [3] and Qwen2.5-3B [92] using MetaMathQA [129], then evaluate their mathematical reasoning performance on the GSM8K [9] test set. (1) For training without CoT, the model's 's target consists solely of the final answer, whereas training with CoT incorporates the reasoning steps. (2) To avoid alterations to the reasoning format, we only inject noise into the mathematical expression components of reasoning steps, randomly selecting between two noise types: step skipping only; b) incorrect step results. The details of data descriptions, training formats, noise types and hyperparameter configurations are provided in Appendix D.8.
Table 4: Left: GSM8K performance with/without CoT fine-tuning (bold: best results). Right: the impact of mathematical expression noise ratio. The Qwen2.5-3B is fine-tuned for only 1000 steps to reduce training overhead. Even with the noise ratio=1, the model maintains improved reasoning generalization through fine-tuning with COT.
Model
Method
GSM8K (100%)
Qwen2.5-3B
before fine-tune
33.13
Llama3.1-8B
Noise Ratio
GSM8K (100%)
fine-tuning without CoT
22.79
fine-tuning with CoT
74.01
0
69.37
0.4
69.06
before fine-tune fine-tuning without CoT fine-tuning with CoT
14.56
Qwen2.5-3B
0.8
68.98
19.21
1.0
68.83
78.81
Verification Results. (1) Table (Left) shows that CoT fine-tuning significantly improves accu- racy on GSM8K benchmark compared to answer-only fine-tuning and pre-trained baselines, even fine-tuning without CoT may degrade the original reasoning capabilities of the base model. For instance, LLaMA3.1-8B performance dropped from 33.13 to 22.79 after answer-only fine-tuning. (2) Table 4 (Right), since the original reasoning patterns in the training data are preserved and noise is injected solely into the mathematical expression components of the reasoning steps, the model achieves improved reasoning generalization through CoT fine-tuning. Notably, fine-tuning with CoT (noise ratio 1) results in significantly higher accuracy of 68.83%, compared to only 19.21 % for fine-tuning without COT. Together, these findings substantiate the claims made Sec- tions 3.2 and 3.4 with solid empirical evidence. Additional real- world validations are also provided in Appendix E.
'"Considering training cost, this part only the first 100k samples for only one epoch of training like [61].
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E. MORE REALISTIC DATA RESULTS
Experiment Setup. We extend our experiments to Llama2-7B [95] to verify whether the insights (that the model can achieve systematic composition generalization through CoT training) hold true for real-world datasets. We use the dataset provided by [6], which contains 82,020 two-hop queries based on data from Wikidata [99]. To exclude the influence of the model's inherent knowledge. we filter the data and split the training and test set, the filtering process can be seen as follows (the methods described clarify our approach and data, without implying originality):
Referring to Biran et al. [6], we aim to filter out cases where no latent reasoning occurs. Given a two hop query ((e,,Ti, 62), (e2, T2, 63) we test two prompts constructed to detect and filter out cases where the model performs reasoning shortcuts [115, 103, 38). (1) The first prompt is ((, ([rr,,2), e2); (e2, r2, E3) (i.e., the query without e1), aimed at filtering out cases where the model pre- dicts generally popular entities.
(2) The second prompt is ( (e2, T2, e3) (i.e., the query
without rı), aimed at filtering out cases where the model predicts correctly due to a high correla- tion between é1 and Ez. We perform this filtering for the model (Llama2-7B(-chat) using greedy decoding creating a subset of the dataset.
We specifically interested in the model's 's reasoning performance, therefore, we further generate two dataset subsets: (1) The first subset is made up of 6,442 cases where the model correctly answers both the two-hop query and the first hop. (2) The second subset includes 2,320 cases where the model correctly answers both the first and second hop in isolation, but fails to answer the full two- hop query. Notice that we only use the first subset for CoT training and the second subset for generalization evaluation. When it is about to training and testing data: (I) The training set is made up of 6,442 cases where the model correctly answers both the two-hop query and the first hop. (2) The test set includes 2,320 cases where the model correctly answers both the first and second hop in isolation, but fails answer the full two-hop query.
We force the model to only output the answers to both hops, for example, the model input is "the capital of the country of the base of OpenAl is" and the model target "the United States. Wash- ington, D.C." To be more specific, the first hop of the example is "the country of the base of OpenAl is the United States." and the second hop of the example is "the capital of the United States is Wash- ington, D.C.". Due to resource limitations, using LoRA [34] (rank=32) and test it on the test set. The framework and hardware information we used is provided in Appendix D.2.
fine-tune the model for 100 epochs on the training set
In more detail, we examined the outputs of the model after fine-tuning. Excitingly, when the model receives the input "the capital of the country of citizenship of Jenna Jameson is" (in test set), the first token it outputs is "Washington.", ", which is also the first token of "Washington, D.C This indicates that CoT training does indeed provide the model with some generalization ability.
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Reviews:
[{"review_id": "Vof9KAS1eL", "reviewer": "Reviewer_L9CC", "summary": "This paper provides a theoretical explanation, using generalization bounds to explain why training with CoT data is helpful. The idea is that when the model needs to predict y given x, then the probability can be decomposed as $P(y|x) = \\sum_C P(y|X,C)P(C|x)$. If the model has seen the intermediate contexts $C$ then generalization is possible, which does not happen for training without CoT. Experimentally, the authors use a synthetic set-up; the model is trained on 1,2-hop facts 1)without CoT (trained on 1-hop and some 2-hop, evaluated on 2-hop and 2) with CoT (trained on 1-hop and 2-hop together with the intermediate 1-hop facts to perform the 2-hop facts). Their experiments show improved performance and generalization when the model is trained with CoT.", "questions": "1. Did the authors tried to perform CoT prompting to the models that were trained without CoT data and see if the performance was improved? The models know each individual component, CoT training basically teaches how to combine these components.\n\n2. Could the authors provide na example from their experimental set-up that matches the theoretical results and shows how CoT training can help predict the correct answer while non-CoT training is unable to do so?\n\nTypo: line 173 the sum should be over C.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The authors provide theoretical results to explain why it can be the case that CoT training leads to better generalization. To my knowledge these results are novel.\n2. The paper is clearly written.\n3. The experiments also consider noisy data to represent more closely real training data.", "weaknesses": "1. The experimental set-up is not particularly novel. I think it is quite equivalent to other experimental set-ups with synthetic datasets like for example addition. In addition, for example, models are trained with for example one digit addition and some two digit and it has observed that when CoT is also provided the models generalize and perform better (see [1] for example).", "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": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "uhJOhIBs1R", "reviewer": "Reviewer_pBps", "summary": "The paper \n- theoretically proves that the information-theoretic generalization bounds through distributional divergence can be decomposed into ID and OOD components. \n- empirically proves that CoT training accelerates convergence and enhances generalization from ID to both ID and OOD scenarios. \n- also shows interesting findings that CoT-trained models resolve intermediate results at shallower layers and learns how to think by CoT training.", "questions": "Q1: In your experiments, why you split the dataset into 2 parts where one part is to output thought given prompt and the other part is to output final answer given prompt+thought. I think it is equivalent to do SFT on thought+final answer given the prompt as input. Why do you formalize the dataset like that?\n\nQ2: grammar error at line 474, should be \"an architectural\"", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "The papers empirical finding is interesting that CoT training can help OOD generalization. The paper also combines theory with empirical findings, which is a good try.", "weaknesses": "I think the biggest issue is that you cannot directly assume \"under suffcient training, the ID generalization error approaches zero\" without any justifications. This correlates to what architecture and tasks you are considering. For example, the model can't perform well on the train dataset on the star-graph problem by next-token prediction even if you trained the model suffciently[1]. Thus, in my opinion, it is better to have the view from optimization, which might be hard to analyze. Or you can make more assumptions on the model/task.\n\n[1] Bachmann, G., & Nagarajan, V. (2024). The pitfalls of next-token prediction. arXiv preprint arXiv:2403.06963.\n\nI think the work itself is great, but I think this weakness matters a lot to me. I will increase my score depending on your explanation.", "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": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "cUXziBe5Ta", "reviewer": "Reviewer_ECGg", "summary": "This paper proposes that Chain-of-Thought (CoT) training enhances out-of-distribution generalization by enabling models to systematically combine learned skills—a process termed compositional reasoning. This claim is buttressed by both theoretical generalization bounds and controlled experiments. Furthermore, a circuit analysis uncovers the internal, multi-stage computational structure that underlies this ability.", "questions": "See weakness.", "limitations": "", "rating": 4, "confidence": 5, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. This paper provides both theoretical analysis and experimental results to support their claims and a mechanism analysis is also completed, which makes the paper comprehensive.\n2. The use of a controlled synthetic task allows for a crisp demonstration of the core principle—compositional generalization.\n3. The presentation of this paper is clear and easy to understand.", "weaknesses": "1. In my view, the results achieved by the CoT training paradigm proposed in the paper still essentially rely on the model performing single-step reasoning. The model merely learns to recognize the first two tokens in the pattern \"e1, r1, r2\" and complete the first step of reasoning, and then, given the input \"e1, r1, r2, e2\", it learns to recognize the last two tokens and complete the second step. What the model has actually learned are two distinct single-step reasoning tasks under different input formats, rather than demonstrating genuine compositional reasoning ability. Moreover, this two-step prediction approach is essentially no different from performing single-step reasoning on atomic data and then explicitly executing two separate single-step inferences. Regarding the OOD generalization capability for compositional tasks, what we would truly like to see is whether the model can accomplish multi-step reasoning directly within a single computational process.  Figure 1 (center right) may also be interpreted by this concern, that there is a trade-off between learning two-hop formations and the atomic reason.\n2. There is a discernible disconnect between the theoretical framework and the experimental work. The theoretical results remain without a concrete experimental counterpart for their verification.\n3. The robustness analysis feels somewhat detached from the preceding sections. Furthermore, the findings presented in Figure 3 lack a control group for comparison, and the observed trends are intuitively self-evident, which collectively diminishes the insightfulness of this investigation.\n4. The related work section discusses latent reasoning, yet I fail to see which part of this paper actually relates to it. If the proposed CoT method is indeed performed in a latent space, the authors should explicitly clarify this point in method part. In that case, I would reconsider my first weakness and the overall contribution of this work.", "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": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)