Seq-VCR: Preventing  Collapse in Intermediate Transformer Representations for Enhanced Reasoning

Authors: Md Rifat Arefin, Gopeshh Subbaraj, Nicolas Gontier, Yann LeCun, Irina Rish, Ravid Shwartz-Ziv, Christopher Pal
Primary area: foundation or frontier models, including LLMs
Keywords: LLMs, Representation Learning, Reasoning, Representation Collapse

Abstract:
Decoder-only Transformers often struggle with complex reasoning tasks, particularly arithmetic reasoning requiring multiple sequential operations. In this work, we identify representation collapse in the model’s intermediate layers as a key factor limiting their reasoning capabilities. To address this, we propose Sequential Variance-Covariance Regularization (Seq-VCR), which enhances the entropy of intermediate representations and prevents collapse. Combined with dummy pause tokens as substitutes for chain-of-thought (CoT) tokens, our method significantly improves performance in arithmetic reasoning problems. In the challenging 5 × 5 integer multiplication task, our approach achieves 99.5% exact match accuracy, outperforming models of the same size (which yield 0% accuracy) and GPT-4 with five-shot CoT prompting (44%). We also demonstrate superior results on arithmetic expression and longest increasing subsequence (LIS) datasets. Our findings highlight the importance of preventing intermediate layer representation collapse to enhance the reasoning capabilities of Transformers and show that Seq-VCR offers an effective solution without requiring explicit CoT supervision.

Decision: accept (poster)
Reviewer scores: 8, 6, 6, 5

Reviews and discussion:
<thread>
creation date:
1730536778110

thread_type:
official_review

<thread_object>
creator: reviewer_lszk

<content>
summary:
This work identifies representation collapse in the LLM intermediate layers as a key factor limiting their arithmetic reasoning capabilities. The paper proposes sequential variance-covariance regularization (Seq-VCR). It then combines Seq-VCR with pause tokens to prevent the representation collapse. Experiments on GPT-2-small and minGPT demonstrate the effectiveness in improving accuracy on arithmetic reasoning.

strengths:
1. The identified representation collapse is quite interesting

2. The proposed method, including Seq-VCR regularization loss and pause tokens, demonstrates novelty and effectiveness based on the experimental results.

weaknesses:
1. The representation collapse experiment was conducted only on GPT-2. I am curious whether this phenomenon occurs in more recent and larger LLMs, such as LLaMA 3 or LLaMA 3.1. The authors should either include additional experiments or provide a theoretical analysis to demonstrate that this is not an isolated case.

2. While the proposed Seq-VCR regularization loss has been shown to be effective in arithmetic reasoning tasks, I wonder whether adding this loss after the next token prediction loss would impact the LLM's performance on other tasks  (e.g., math reasoning and general MMLU). If it does have an effect, then this method may not be widely applicable. I encourage the authors to discuss this point.

questions:
see the weaknesses section.
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<thread>
creation date:
1730588517506

thread_type:
official_review

<thread_object>
creator: reviewer_ysnl

<content>
summary:
Background: variance-covariance regularization (VICReg/VCReg) is a technique that was pioneered in vision models.  Given a batch of inputs, the technique uses an NN to encode those inputs to a batch of embedding vectors, and then computes a covariance matrix for the embedding vectors.  It introduces two losses based the covariance matrix: (a) the variance loss ensures that every dimension of the embedding vector has different values, distributed across the batch, and (b) the covariance loss ensures that different dimensions are not correlated.  In vision models, these two losses guard against representational collapse.  

The authors of this paper adapt VICReg from the vision domain to transformer-based language models.  They show that when combined with pause tokens, VICReg (now renamed to Seq-VCR) produces large improvements in several tasks that LLMs are usually very bad on -- multidigit arithmetic, arithmetic expressions, and a longest-increasing-subsequence task.

strengths:
The application of VICReg to language models is novel as far as I know, and the experimental results are very compelling.  This could potentially be a high-impact paper in improving the reasoning capabilities of LLMs.

weaknesses:
Unfortunately, the paper itself is hastily and sloppily written, and difficult to follow in places.  I had numerous questions when reading it that are not addressed by the text.  The current draft does not contain all of the information necessary to replicate these experiments, and does not discuss many of the crucial design decisions.  The authors claim that there is an "Appendix A", but neglected to provide any supplementary material.  See "questions" section below for specific questions.

One of the author's central claims is that transformers suffer from representational collapse, but I do not think that they adequately make that point based on the experimental evidence.  There are only two entropy charts in Figure 2, which cover only two narrow tasks.  On one of those charts (a) the collapse seems minimal at best, while on the other (b) the addition of pause tokens (the second key technique that the authors propose) actually increases collapse, rather than decreasing it.  I would need to see a much larger set of studies, over a variety of different tasks, including general language modeling tasks (translation etc.) to fully buy the author's argument about collapse.  If the authors did such a study, however, it would be a significant breakthrough.

Similarly, I would like to know what the effects of VICReg are on more general language modeling tasks.  If the technique helps the model multiply 5-digit numbers after fine-tuning, but otherwise degrades peformance on most other language modeling tasks, then the technique is useless.  Because the authors do not perform this ablation, it is impossible for me to evaluate whether this is a high-impact advance over SOTA, or a trivial result.

Finally, the use of pause tokens is interesting, but also seems haphazard.  They authors themselves admit that the number of pause tokens is task-specific.  To employ this technique more widely, I would need to see a more comprehensive test of where, how many, and under what circumstances pause tokens should be added.

More specific criticisms:

Equation (3) defines the Seq-VCR loss.  The text of the paper claims that it is "inspired by" prior work, and cites such work appropriately, but it is more than just "inspired".  Equation (3) is lifted almost verbatim from the orginal VICReg (Bardes 2021) and VCReg (Zhu 2023) papers, and the authors need to be crystal clear about the source of that equation.   

(As a minor nit, it is unclear to me whether or not the covariance term in equation (3) should have an additional 1/(d-1) factor; VICReg has the term, while VCReg does not.  I would have appreciated it if the authors explained why they chose one version over the other.)

For further clarity, the authors should also devote a few lines to defining how the covariance matrix C is computed; as is done in other papers.  Otherwise, it can easily be confused with the cross-correlation matrix of the Barlow twins technique, which the authors also cite as inspiration.

questions:
(1) Pause tokens are a crucial part of the author's technique, but at no point do the authors describe where, and how, the pause tokens are added to the input.  

(2) Representational collapsed supposedly happens in the intermediate layers of the transformer, and yet the Lseq-VCR loss term is only applied to the final layer.  (Line 225).  Shouldn't it be applied to the intermediate layers, where you measure the entropy?  Why not?

(3) Equation (3) introduces $\lambda_1$ and $\lambda_2$ as hyperparameters, but the paper fails to say what they are set to. 

(4) What batch size is used for computing the covariance matrix?

(5) Equation 3 computes the covariance matrix only across the batch dimension.  Why?  In a transformer, you could potentially use the length dimension as well, which would drastically increase the effective batch size.  Did you do an ablation study which showed that to be ineffective?

(6) How is the projection layer $f_{proj}$ trained?

(7) For GPT2 on multiplication, you fine-tune a pre-trained GPT2 model, despite the fact that the pre-trained GPT2 has no real multiplication ability to start with.  Why bother with a pre-trained model, instead of just training from scratch, as you do with minGPT on arithmetic expressions?
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</thread>
<thread>
creation date:
1730590408083

thread_type:
official_review

<thread_object>
creator: reviewer_dqgy

<content>
summary:
This paper focuses on the performance of decoder-only models on tasks such as multi-digit mathematical reasoning that require a series of immediate representations. They hypothesize representation collapse of intermediate layers as a key contributor to this poor performance, preventing effective storage of the intermediate steps necessary for these kinds of tasks. While chain of thought reasoning can be effective in counteracting this collapse and performing well on such tasks, the proposed approach seeks to increase entropy among intermediate layers and achieve similar performance with at a reduced computational cost. Formulated in terms of alpha-order matrix-based entropy, the formulate a regularization term which aims are increasing variance and decreasing covariance in the intermediate representations. Additionally, pause tokens are included in the method. Results on three kinds of tests are presented – computing arithmetic expressions, identifying the longest increasing integer subsequence, and performing multiplication of 4 digit or 5 digit numbers. Performance with the regularization term and pause tokens leads to performance which approaches chain of thought on most tests, and regularization performs well on its own for the simpler tasks.

strengths:
The details of the method to seem to be heavily inspired by VICReg, but so far as I can judge, the application of it to the sequence/Transformer is original. The method is, in theory, computationally attractive compared to CoT and the results are fairly compelling. 

The paper is clearly written and the quality of the presentation is moderately high.

weaknesses:
The result in Table 1 naturally provokes a question: This and several previous studies show that GPT-2 with CoT performs remarkably well, but this is actually more difficult to achieve in larger models. What is the evidence/argument that the Seq-VCR approach will scale better with model size than CoT? Figure 8 hints at this but it doesn’t clearly address it.

The speedup vs CoT is intuitively reasonable but it would have been nice to see performance numbers as in the cite Deng 2024 paper.

Similarly, it would be helpful to understand the amount of hyperparameter optimization necessary for, e.g., identifying the number of pause tokens used to obtain the best results. Do the number of pause tokens necessary correlate with, e.g., task complexity?

For completeness, it would be nice to see CoT in figures 7 and 8.

questions:
A discussion that clarified the improvement versus CoT would improve the significance, whether clearly establishing the speedup with Seq-VCR or showing its better generalization/scaling.
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</thread>
<thread>
creation date:
1730604657859

thread_type:
official_review

<thread_object>
creator: reviewer_strw

<content>
summary:
This paper proposes a regularization technique for preventing representation collapse across the intermediate representations of a deep sequence model. Their results show that 1. the regularization technique increases matrix entropy (low matrix entropy = representation collapse) and 2. when pause tokens are added the language model significantly improved in performance for 4x4 and 5x5 arithmetic tasks.

strengths:
- The paper presents a novel regularization technique that improves the model's performance in several reasoning tasks
- The paper presents detailed analysis of the experimental results, showcasing how exactly the regularization techniques affects the diversity of representations, the learning dynamics, as well as the digit-by-digit accuracy on multiplication tasks.

weaknesses:
- The effect of the regularization technique was only studied for a relatively narrow domain of tasks, and it would be interesting to understand its effect on more general language benchmarks as well.
- Slightly more contextualization on how exactly pause tokens are incorporated would assist readers in understanding the work more easily as it is also a core part of what is being proposed in this work.

questions:
Same as the weaknesses section.
</content>
</thread_object>

</thread>
<thread>
creation date:
1732398587037

thread_type:
official_comment

<thread_object>
creator: authors

<content>
title:
General Responses 1/3 & 2/3

comment:
We sincerely thank all reviewers for their time and valuable feedback on our work. In response, we conducted additional experiments using LLaMA 3.2-1B and Code-GPT2, which we are pleased to share.

****1) Representation Collapse in Large Language Models (LLMs)****

**New Experiments**: In response to your feedback, we fine-tuned LLaMA 3.2-1B on the 5x5 digit multiplication task, both with and without our proposed regularization.

**Results**: We evaluate representation collapse across layers in the fine-tuned model, both with and without our regularization. Our results show that the model trained with Seq-VCR exhibits a 15-40% increase in average entropy (indicating reduced collapse) compared to the baseline LLaMA model and solves the task by achieving an accuracy of 97.4%. Full details are provided in Figure 11 in Appendix I of the updated manuscript.

**Additional Scaling Analysis**: We also conducted more analysis on pre-trained LLaMA models of different sizes (1B, 3B, 8B) on the 5x5 digit multiplication dataset. We observed consistent representation collapse across all model sizes (Figure 12 in Appendix J). This indicates that the issue persists at different scales, emphasizing the importance of effective regularization techniques like Seq-VCR to reduce collapse and enhance intermediate reasoning capabilities.

****2) Generalizing Seq-VCR to a Broader Domain of Language Tasks****

**New experiments**: In response to your feedback, we conducted experiments on the CodeXGLUE text-to-code benchmark [1] using CodeGPT2, measuring representation collapse across layers in the fine-tuned model, both with and without our regularization.

**Results**: Our primary results show that the model trained with Seq-VCR exhibits a higher average entropy, indicating reduced representation collapse. Full details can be found in Figure 12 in Appendix J of the updated paper. Representation collapse in intermediate layers presents a significant challenge for Transformers in multi-step tasks, such as multiplication, where precise intermediate computations—like carries—are essential. This issue is also observed in the CodeXGLUE text-to-code benchmark, highlighting the potential of Seq-VCR to address these limitations.  Expanding its application to a wider range of NLP tasks, particularly those involving reasoning and generalization, presents a promising avenue for future research.

**Note:** We would also like to remind the reviewers that the scope we considered for this paper is for multi-step reasoning tasks and understanding the limit of current LLMs on why they fail in these tasks without explicit CoT supervision. While Seq-VCR may not show improvements in pure language tasks, such as translation, we believe it's still valuable to better understand LLMs from a representation point of view and for researchers concentrating on more reasoning-intensive problems. We also think future research along this line could be useful even for further increasing the representational capability of these models before using prompt-based techniques like CoT to fix it.

[1] https://github.com/microsoft/CodeXGLUE/tree/main/Text-Code/text-to-code
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</thread>
<thread>
creation date:
1732402144258

thread_type:
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creator: authors

<content>
title:
General Response 3/3

comment:
***3) Details on pause tokens***

Thank you for asking us to clarify the questions about pause tokens, we added the following clarifications, including the example in Section 3.5 of the updated paper in blue colored text, and added our speedup and accuracy analysis in Appendix E.

Increasing model capacity leads to a significant accuracy improvement in solving n × n digit multiplication tasks [1]. While some prior work increases depth [1] and chain-of-thought (CoT) [4] to enhance model capacity, an alternative approach is to use pause tokens[3], which act as explicit signals for the model to temporarily pause at intermediate states in sequential tasks before moving to the next computation step. We adopt pause tokens as a more cost-effective and computationally efficient alternative to Chain of Thought tokens, complementing our Seq-VCR approach. This combination enhances the representation capacity, allowing the model to better utilize the fixed-depth architecture. Below, we address the reviewers' queries regarding the placement, quantity, reason and time complexity considerations of pause tokens:

**Where:** In all experiments, pause tokens were placed between input and output tokens to emulate chain-of-thought (CoT) reasoning. This setup allowed the model to take "pauses," improving its ability to organize intermediate representations and enhance computational reasoning. For example, the input-output format could look like <question> </pause_start> <pause> <pause> </pause_end> <answer>. 

**How many:** We tried 2, 4, 6, 8  pause tokens on 4x4 and 5x5 both digit multiplication tasks, and arithmetic tasks and we did not find any correlation with task complexity. This result is shown in Fig 10 in the Appendix C. We believe it may be due to the fact that all the pause tokens share the same embedding. Future work will explore the effect of having different embeddings per pause tokens.

**Under what circumstances:** When CoT instructions are unavailable or when inference time needs to be reduced, pause tokens provide a simple and effective solution. Unlike CoT, which requires extensive labeled data for multi-step reasoning, Seq-VCR uses a few dummy pause tokens to solve tasks like multiplication in a fraction of the time, while performing at a similar accuracy close to 100% in our experiments (see table below).
Seq-VCR's efficiency in both inference time, data requirements, and accuracy makes it a scalable and robust approach compared to CoT.

**Computation Complexity Analysis:** We provide a detailed analysis below, which we also added in Appendix F of the revised document. Seq-VCR offers notable benefits over Chain-of-Thought (CoT) reasoning, particularly in reducing inference time and dependency on costly human-supervised data. Unlike CoT, which requires extensive labeled data for multi-step reasoning, Seq-VCR uses a few dummy pause tokens to solve tasks like multiplication in a fraction of the time, while performing at a similar accuracy close to 100% in our experiments (see table below).

To compute inference time we utilize the normalized throughput, using the following equation, as introduced the deng2024explicit[2] paper:
$$ T_{\text{norm}} = \frac{T_{\text{target}}}{T_{\text{base}}} $$
Here:
- $$T_{\text{norm}}  \text{is the normalized throughput, which represents the relative inference speed.}$$ 
- $$T_{\text{target}}  \text{is the throughput (number of examples processed per second) when using target method.}$$
- $$T_{\text{base}}  \text{is the throughput (number of examples processed per second) for the baseline model without Chain of Thought or Pause tokens.}$$

Normalized Throughput (the higher, the better) on 4x4 and 5x5 digit multiplication without CoT tokens, with CoT tokens, and with 2 pause tokens.

|  **Method**   | T_{norm}  |   | Accuracy |  |
|------------------------|---------|---------|----------|----------|
|  **Method**   |  (*4x4*) |  (*5x5*)  | (*4x4*)  | (**5x5**)   |
|------------------------|---------|---------|----------|----------|
| No CoT                 | 1.0     | 1.0     | 0.25     | 0.0      |
| With CoT               | 0.17    | 0.14    | 1.0      | 1.0      |
| Seq-VCR + Pause (2)    | 0.95    | 0.91    | 0.992    | 0.995    |


[1] Qiu, Luyu, et al. "Dissecting Multiplication in Transformers: Insights into LLMs." arXiv preprint arXiv:2407.15360 (2024).

[2] Deng, Yuntian, et al. "Implicit chain of thought reasoning via knowledge distillation." arXiv preprint arXiv:2311.01460 (2023).

[3] Goyal, Sachin, et al. "Think before you speak: Training language models with pause tokens." arXiv preprint arXiv:2310.02226 (2023).

[4] Wei, Jason, et al. "Chain-of-thought prompting elicits reasoning in large language models." Advances in neural information processing systems 35 (2022): 24824-24837.
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Metareview:
metareview:
This paper proposes a regularization technique for preventing representation collapse across the intermediate representations of a deep sequence model. Their results show that 1. the regularization technique increases matrix entropy (low matrix entropy = representation collapse) and 2. when pause tokens are added the language model significantly improved in performance for 4x4 and 5x5 arithmetic tasks.

The strengths of this paper are:
* simple technique 
* effectiveness on some simple reasoning tasks the authors experimented on

Weakness of this paper:
* unsure about how this technique would generate to leading LLMs (though the author added experiments to LLama during rebuttal)
* unsure about how this technique performs on more complex reasoning tasks.
* This method requires training, hence should be compared to any other reasoning augmentation method that also requires training.