Training Nonlinear Transformers for Chain-of-Thought Inference:  A Theoretical Generalization Analysis

Authors: Hongkang Li, Songtao Lu, Pin-Yu Chen, Xiaodong Cui, Meng Wang
Primary area: learning theory
Keywords: Theoretical Chain-of-Thought, generalization, deep learning theory, In-Context Learning, training dynamics of Transformers

Abstract:
Chain-of-Thought (CoT) is an efficient prompting method that enables the reasoning ability of large language models by augmenting the query using multiple examples with multiple intermediate steps. Despite the empirical success, the theoretical understanding of how to train a Transformer to achieve the CoT ability remains less explored. This is primarily due to the technical challenges involved in analyzing the nonconvex optimization on nonlinear attention models. To the best of our knowledge, this work provides the first theoretical study of training Transformers with nonlinear attention to obtain the CoT generalization capability so that the resulting model can inference on unseen tasks when the input is augmented by examples of the new task. We first quantify the required training samples and iterations to train a Transformer model towards CoT ability.  We then prove the success of its CoT generalization on unseen tasks with distribution-shifted testing data. Moreover, we theoretically characterize the conditions for an accurate reasoning output by CoT even when the provided reasoning examples contain noises and are not always accurate. In contrast, in-context learning (ICL), which can be viewed as one-step CoT without intermediate steps, may fail to provide an accurate output when CoT does. These theoretical findings are justified through experiments.

Decision: accept (poster)
Reviewer scores: 6, 8, 6, 6

Reviews and discussion:
<thread>
creation date:
1730548554699

thread_type:
official_review

<thread_object>
creator: reviewer_6hy5

<content>
summary:
This paper presents a theoretical analysis of training Transformers with nonlinear attention mechanisms to develop Chain-of-Thought (CoT) reasoning abilities. The authors focus on quantifying the requirements (such as sample complexity and iterations) for training a Transformer model capable of CoT generalization on unseen tasks, even under conditions where examples might contain noise and inaccuracies. The study also compares CoT with In-Context Learning (ICL), showing that CoT can outperform ICL when intermediate reasoning steps are essential. The paper provides both theoretical insights and empirical evidence supporting the model's behavior during training and testing.

strengths:
● This paper is one of the first to theoretically explore training dynamics of nonlinear Transformers with CoT capabilities, providing important insights into sample complexity, required iterations, and generalization guarantees.

● The comparison between CoT and ICL is theoretically grounded, with a clear quantitative explanation of why CoT can succeed in contexts where ICL may fail.

● This work addresses an understudied area in Transformer models, specifically how they generalize CoT reasoning abilities across unseen tasks, making it a relevant contribution to understanding advanced prompting methods in LLMs.

weaknesses:
● The authors analyze attention-only models, aligning with previous work by [1] Olsson et al. (2022). Both studies observe that trained models’ attention values tend to focus on context examples with input patterns similar to the test query. While Olsson et al. attribute this to ICL (In-Context Learning) ability, the authors link it to CoT ability and explore how training may facilitate this behavior. A more detailed comparison of contributions between this study and Olsson et al. would be valuable. 

  ○ [1] In-context Learning and Induction Heads.  2022. Olsson et al.

● The paper acknowledges the gap between the one-layer model studied and multi-layer Transformer models. As practical LLMs are complex, the findings might not directly extend to real-world Transformer architectures. Please discuss and analyze the applicability of these conclusions in more complex scenarios.

● The theoretical results assume a linear span between training-relevant (TRR) and testing-relevant (TSR) patterns, limiting applicability to cases with greater diversity in test distributions.

questions:
No further issues.
</content>
</thread_object>

</thread>
<thread>
creation date:
1730610403561

thread_type:
official_review

<thread_object>
creator: reviewer_txg6

<content>
summary:
The paper provides a theoretical study on training Transformers with nonlinear attention to obtain CoT generalization capability, allowing the model to infer unseen tasks when the input is augmented with examples of the new task. They quantify the required training samples and iterations needed to train a Transformer model towards CoT ability and prove the success of its CoT generalization on unseen tasks with distribution-shifted testing data. The paper's major contributions include a quantitative analysis of training dynamics on a one-layer single-head attention-only Transformer, characterizing the conditions for accurate reasoning output by CoT even in the presence of noisy and inaccurate examples. It also provides a theoretical explanation for why CoT outperforms in-context learning (ICL) in certain cases, highlighting the robustness of CoT against noise in reasoning examples. Through experiments, the authors justify their theoretical findings, demonstrating the practical relevance of their theoretical analysis. This work advances the understanding of Transformer models' reasoning abilities and their potential applications in solving complex reasoning tasks.

strengths:
1. Theoretical Advancement: The paper makes a significant theoretical contribution by providing the first analysis of training dynamics for Transformers aimed at achieving chain-of-thought (CoT) generalization capability. This addresses a critical gap in the literature, as prior works have not fully explored how Transformers can be trained to generalize on multi-step reasoning tasks via CoT.
﻿2. Deep Quantitative Analysis: The authors offer a detailed quantitative analysis of the training process, including the number of context examples required for successful CoT reasoning and the conditions under which CoT outperforms in-context learning (ICL). This level of detail is commendable and provides actionable insights for practitioners looking to implement CoT in Transformers.

weaknesses:
1. **Limitations in Model Complexity**: The paper primarily analyzes a single-head attention Transformer, which may not encapsulate the full complexity and performance characteristics of multi-layer and multi-head attention models. The validation was confined to binary classification tasks, thereby restricting the generalizability of the theoretical findings.
2. **Formula Accuracy**: The paper requires a meticulous review of its mathematical expressions. For instance, in Example 1, the function should be denoted as \( f_2(\mu_2') = \mu_1' \) instead of \( f_1(\mu_2') = \mu_1' \), and the second matrix \( A_1^f \) should be corrected to \( A_2^f \). It is essential to verify all formulas throughout the text to ensure accuracy.
3. **Theorem Validity and Clarification**: The theorems presented in the article should be scrutinized for their validity, particularly the sections that substantiate reasoning, as they may induce some ambiguity. Reading the appendix is mandatory for a comprehensive understanding of the article; otherwise, it might lead to misconceptions.
4. **Originality Concerns**: The article's reasoning and writing logic bear similarities to those found in "How Do Nonlinear Transformers Learn and Generalize in In-Context Learning." It raises the question of whether this work is merely an extension of the previous study or if it introduces novel contributions.

questions:
See the section on weakness. We will increase the score based on the answer to the question.
</content>
</thread_object>

</thread>
<thread>
creation date:
1730661520022

thread_type:
official_review

<thread_object>
creator: reviewer_y9x5

<content>
summary:
The paper provides a theoretical characterization of three aspects of in-context learning with examples consisting of chain-of-thought using a simplified data model and a simplified single-layer transformer with non-linear attention. The central result of the paper is a quantitative characterization of the training dynamics that leads to CoT ability. Using this, the authors characterize how the noise and relevance of the in-context examples affect the CoT performance. The final result compares the error rates for ICL with CoT and ICL without CoT with the main conclusion that while the rates are similar, ICL without CoT needs the examples to be dominantly correct, while CoT does not need this. Experiments on synthetic data support the theoretical results.

strengths:
**Originality and Significance**: While previous papers analyze ICL (with and without CoT), their assumptions do not match this paper's. Specifically, this paper does the analysis using a transformer with non-linear attention, with test data coming from a different distribution than the training data, making the setting more general and useful.

**Quality and Clarity**: The theoretical results are presented clearly and are supported by relevant experiments. The discussion and remarks help with understanding the implications of the results.

weaknesses:
It would increase the impact of the paper if the authors could include experiments on some real datasets.

questions:
Are the results applicable, or can they be easily extended to the case where the reasoning steps $z_i, f(z_i)$ are not necessarily of the same sequence length? In other words, $z_1 \in R^{1\times d}$, $f(z_1) \in \mathbb R^{2\times d}$, $z_2 \in \mathbb R^{2 \times d}$, $f(z_2) \in \mathbb R^{1\times d}$. Such a setting might be of greater interest because the steps are typically multi-token in applications involving natural language.
</content>
</thread_object>

</thread>
<thread>
creation date:
1730719197459

thread_type:
official_review

<thread_object>
creator: reviewer_lgxd

<content>
summary:
This work presented a theoretical study on training Transformers to achieve Chain-of-Thought (CoT) reasoning capabilities. The authors explored the training dynamics of nonlinear Transformers and provided a quantitative analysis of the required training samples and iterations for developing CoT ability. They analyzed the model's success in generalizing CoT on new tasks with O.O.D. test data, even in noisy and inaccurate reasoning examples, furthermore, contrasted CoT with in-context learning (ICL) and theoretically characterizes scenarios where CoT outperforms ICL.

strengths:
1. The paper provided a novel theoretical framework for understanding how a simplified Transformer can be trained to perform CoT reasoning, which is important for Transformers-based model reasoning.

2. The authors offered a detailed quantitative analysis of the training process, including the number of context examples, training samples, and iterations required for successful CoT training. This level of detail helps in understanding the complexity and feasibility of training similar models.

3. The paper addresses the generalization of CoT ability to new tasks with distribution shifts, which is a practical concern for real-world applications where training and testing data may differ.

weaknesses:
1. The paper idealized the Transformer as a simplified one-layer single-head architecture. While such a simplification can aid in theoretical analysis, it may limit the generalizability of more complex Transformer architectures.

2. The assumptions made about the data and tasks, such as the orthonormality of training-relevant and testing-relevant patterns, may not hold in all real-world scenarios, which could affect the applicability of the results.

questions:
1. For the CoT training set, this work constructed it with examples + query + (K-1)-reasoning steps, and the K-th step as the label. Why did not adopted more later steps as the label or more flexible label strategies?

2. In the 3rd line of Example 1, should the last affinity function be ?

3. How well do the theoretical results extend to more complex Transformer models with multiple layers and heads?

4. Have the authors considered validating their theoretical findings with real-world datasets, and if so, what are the challenges and potential modifications needed?

5. Could the authors elaborate on how the model's performance is affected by different levels of noise and inaccuracy in the context examples?
</content>
</thread_object>

</thread>
<thread>
creation date:
1732651401456

thread_type:
official_comment

<thread_object>
creator: authors

<content>
title:
A Friendly Reminder

comment:
Dear Reviewers txG6 and 6Hy5,

Since the deadline for resubmitting the updated manuscript is approaching within less than two days, we would appreciate it if you could check our response and see whether your concern has been addressed. If so, we kindly hope that you could consider updating the evaluation of the paper. We are pleased to answer your further questions and update our submission. Thank you very much for your time and effort!

Authors
</content>
</thread_object>

</thread>
<thread>
creation date:
1733280443893

thread_type:
official_comment

<thread_object>
creator: authors

<content>
title:
Summary of Revisions

comment:
Dear Reviewers/AC/SAC/PC,

We appreciate the evaluation and suggestions. Here is a summary of revisions in the uploaded manuscript according to the review.

1. We have corrected typos in Example 1 and other places as suggested by Reviewers LGXd and txG6.

2. We have added Appendix A as a real-world experiment to support our analysis as suggested by Reviewers LGXd and y9x5.

3. We have revised Definition 1 and Remark 1 in Section 3.4, Condition 1 and Theorem 3 in Section 3.5, and Proof Sketch in Section 4.2 for a better readability of the discussion of CoT reasoning as suggested by Reviewer txG6. 

4. We have updated Theorem 2, Remark 1, and related proof to extend our analysis to a wider out-of-domain CoT generalization as suggested by Reviewer 6Hy5.

Thanks,

Authors
</content>
</thread_object>

</thread>


Metareview:
metareview:
Summary: The paper investigates how nonlinear Transformers can be trained to perform Chain-of-Thought (CoT) reasoning, providing a theoretical foundation for this capability. It examines the training dynamics required to develop CoT abilities and demonstrates that Transformers trained in this way can generalize robustly to unseen tasks. Additionally, a comparison with In-Context Learning (ICL) highlights CoT's advantages, particularly in scenarios where ICL struggles due to its dependency on accurate input-label pairs. Simple experiments are used to corroborate theoretical results.

Strengths:
- All the reviewers leaning positive
- Analysis of training dynamics for CoT capability in transformers - an important problem
- Detailed quantitative analysis of sample complexity, iterations, and generalization bounds
- Theoretical framework comparing CoT vs ICL performance

Weakness:
- Analysis limited to simplified one-layer single-head transformer architecture
- Assumptions about orthogonal patterns may not fully represent real-world scenarios
- Experiments primarily on synthetic data rather than real-world applications
- Some presentation issues including typos and formula accuracy that needed correction

Decision:
All the reviewers were in consensus that the paper represents an important step forward in understanding transformer capabilities and limitations, providing both theoretical insights and practical implications for future work. It merits publication at ICLR 2025, with a few minor edits: the contribution section can softened and simplified: currently repitive and over-claiming.