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
Venue: ICLR 2025 Poster (2025)
Keywords: LLMs
Representation Learning
Reasoning
Representation Collapse
Primary area: foundation or frontier models, including LLMs

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.

Published as a conference paper at ICLR 2025
SEQ-VCR:PREVENTIIE COLLAPSE IN INTERMEDIATE TRANSFORMER REPRESENTATIONS FOR ENHANCED REASONING
Md Rifat Arefin 1.233 Gopeshh Subbaraj*., Nicolas Gontier", Yann LeCun Irina Rish1.2, Ravid Shwartz-Ziv", Christopher Pal
I Université de Montréal, Mila, @ServiceNow, *Polytechnique Montreal, Meta FAIR, New York University
ABSTRACT
Decoder-only Transformers often struggle with complex reasoning tasks, particu- larly arithmetic reasoning requiring multiple sequential operations. In this work, we identify representation collapse in the model's intermediate layers key fac- tor limiting their reasoning capabilities. To address this, propose Sequential Pariance-Covarian Regularization (Seq-VCR which enhances the entropy of intermediate representations and prevents collapse. Combined with dummy pause tokens as substitutes for tf-thouhht (CoT) tokens, our method signifi- cantly improves performance in arithmetic reasoning problems. In the challenging 5 x integer multiplication task, our approach achieves 99.5% exact match ac- curacy, outperforming models of the same size (which yield 0% accuracy) and GPT-4 with five-shot CoT prompting (44%). We also demonstrate superior re- sults on arithmetic expression and longest increasing subsequence (LIS) datasets. Our findings highlight the importance of preventing intermediate layer representa- tion collapse enhance the reasoning capabilities of Transformers and show that Seq-VCR offers an effective solution without requiring explicit CoT supervision.
INTRODUCTION
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Figure 1: Position-wise number of operations needed for 5x5 digits integer multiplication task. Middle tokens in the output sequence need more operations than the peripheral ones, making their prediction much harder (as shown in Figure (). Example of 12345 67890 is shown here.
Large Language Models (LLMs) based on Transformer architectures have achieved remarkable suc- cess across wide range of tasks, positioning them as foundational models in artificial intelligence [Bommasani et al. 2021) Despite their impressive capabilities, LLMs often struggle with tasks
Trifat.arefin @ mila mila.quehec) Work was completed during internship at ServiceNow.
https://gthbb..om/rrrhnniicce ver

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requiring complex reasoning, particularly arithmetic reasoning that necessitates multiple sequen- tial operations (Bubeck et al., 2023). These challenges are attributed to the models' limitations in handling tasks that involve deep cognitive abilities and multi-step reasoning processes.
One of the key obstacles is the representation collapse in the intermediate layers of Transformer models. Representation collapse occurs when internal representation diversity diminishes, leading to less informative features and hindering the model's ability to solve complex tasks (Jing et al, 2021). For arithmetic reasoning, transformers struggle with successive carryovers and storing inter- mediate results Qiu et al. (2024), which are essential for solving complex sub-tasks, requiring more computations (Figure D.. We hypothesize that representation collapse prevents the model from ef- fectively performing sub- sub-tasks by calculating successive carryovers and storing intermediate results, which are essential for accurate prediction.
To address this limitation, we introduce Sequential Marrance-Covariane Regularization (Seq- VCR), regularization technique designed to enhance the entropy of intermediate representa- tions and prevent representation collapse. By increasing the diversity of representations within the model's layers, Seq- VCR enables the Transformer to maintain richer and more informative features throughout the computation process.
Furthermore, we incorporate dummy pause tokens as substitutes for chain-of-thought (CoT) tokens. While CoT prompting has been shown improve reasoning by breaking down tasks into inter- mediate steps (Wei et al., 2022), it often requires explicit supervision and can be computationally expensive. Our approach leverages pause tokens to simulate the effect of CoT without the need for explicit intermediate reasoning steps.
We validate our method challenging arithmetic reasoning tasks. Notably, the 5 5 integer multiplication task, our approach achieves 99.5% exact match accuracy. surpassing models of the same size (which yield 0% accuracy) and GPT-4 with five-shot CoT' prompting (44%). We also demonstrate significant improvements on arithmetic expression and longest increasing subsequence (LIS) datasets.
Our contributions are summarized as follows:
We identify representation collapse in intermediate layers as a key limitation affecting the reasoning capabilities of Transformer models.
We propose Seq- VCR, a regularization technique that enhances the diversity of intermedi- ate representations and prevents collapse.
We demonstrate that combining Seq- VCR with pause tokens enables models to solve com- plex arithmetic reasoning tasks without explicit CoT supervision.
We provide extensive experimental results showing significant improvements over baseline models and state-of-the art LLMs like GPT-4.
The paper is structured as follows: Section reviews reasoning in large language models, represen- tation collapse, and regularization. Section presents our method, Sequential Porrance-Covariane Regularization (CR)) including its formulation and motivation. Section Sisrioonneesae describes the exper- imental setup, datasets, models, and shows Seq-VCR'S effectiveness arithmetic reasoning and its impact representations. Section summarizes key findings and discusses future directions.
LITERATURE REVIEW
Transformer Reasoning Research in large language models (LLMs) has advanced significantly. Models like BERT (Devlin et al. 2019) and GPT (Radford & Narasimhan 2018) demonstrated im- provements in natural language understanding. However, challenges remain in tasks that require deep cognitive abilities. Integrating external knowledge has shown promise improving reasoning capabilities (Bosselut al. 2019). Recent advances include techniques such chain-of-thought (CoT) prompting, which allow models such as GPT-3 (Brown et al., 2020) to perform complex rea- soning by breaking tasks into intermediate steps (Wei et al. (022) with human supervision, called process supervision (Lightman et al.) 2023). Wang et al. (2022) shows that even incorrect but co- herent intermediate steps improve reasoning performance. Jin et al. (2024) found increasing the
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length of the reasoning steps in the prompts, even without adding new information to the prompt, improves the reasoning abilities of LLMs.
Wang et al. (2024), investigated the addition of dummy tokens such periods or hash symbols to inputs at inference time. They found that this simple modification could improve performance in arithmetic reasoning tasks. Some works show that these dummy tokens, commonly referred to as filler tokens, do not extend transformers' abilities beyond TC circuit complexity, but still significantly enhance problem-solving within this class (Merrill & Sabharwal 2023 Strobl al. 2023). Building on this idea, Goyal et al. 2023a) introduced a more comprehensive framework called
Their approach involves incorporating learnable < pause tokens during both pre-training and fine-tuning of language models. While there are differences between the pause tokens and filler tokens approach, they share the goal of providing additional computation time for transformers which extends the expressive power of transformers within the TC class without changing the fundamental limitations of the model architecture. Chain-of-thought prompting can potentially elevate transformers beyond the TC" complexity class; However, it may not be necessary for all problems. Furthermore, filler tokens have been demonstrated to enhance the expressivity of transformers (Pfau al. 2024). even in cases where traditional transformers, lacking chain-of- thought mechanisms, exhibit insufficient expressive power (Sanford et al. 2024)..
Representation Collapse Representation collapse in representation learning degrades model per- formance by making features indistinguishable. It often occurs unsupervised and self-supervised learning tasks due to the lack of labeled data, leading trivial solutions and low-rank feature space. Research by Jiing et al. (2021) highlights that representation collapse is due to poor optimiza- tion, loss function design, and improper regularization. Recent studies (Grill et al., 2020) suggest the use of contrastive learning, such as SimCLR, to mitigate this by contrasting positive and nega- tive samples. Additionally, Zboonarr et al. (20TT) introduced Barlow Twins minimize redundancy between learned representations, A further advancement is VICReg (Bardes et al.) 2021 Zhu et al.) 2023) which employs variance, invariance, and covariance regularization to prevent collapse, en- suring diverse and non-trivial representations. While these researches focus on images, it is less explored in language models. Recently, Barbero et al. (2024) studied last-layer representations of the next tokens and found representation collapse in pre-trained models.
3 METHODOLOGY
3.1 PRELIMINARIES
Consider a finite set of token vocabulary denoted by (1...u), while x, y E V constitutes sequence of these tokens representing the input = [=],..... x and the output = [v...... vr), where and indicate the respective sequence lengths of the input and output.
We covert the input and output sequence y in an embedded sequence:
.......g) Fremt [ GK. .1,...,
P2. .] RTTKXx of dimension ØT) E Kxd and Fepos P2,,
We can get the language model wise as: f (fass ft fL f1-1... fo)
where fu = [ + PI..... ēK + PK. V1 + PK+1..... .r + PK-Tl and fele E V is the final linear classification layer. the self-attention layer with MLP for layer (Vaswani et al. 2017).
3.2 NEXT-TOKEN PREDICTION LOSS
Large language models like GPT (Radford et al. 2018) (Generative Pre-trained Transformer) trained to auto-regressively predict the next token in a sequence. This minimizes the difference between predicted and actual tokens for coherent and contextually appropriate text. A loss function like cross-entropy for next-token prediction is given by:
T
Enexr(y,(<x)) = Σ I log(f(z;
(1)
t=1
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where yt is the one-hot encoded true token at time step t and input tokens modeled by language model. f.
3.3
REPRESENTATION COLLAPSE IN TRANSFORMERS
Vanilla
Pause
Seq-VCR
Seq-VCR-Pause
Pretrained
Average Entropy
Average Entropy
0.4
0.6-
0.5
0.2
0.4
Layer
Layer
(a) Training minGPT from scratch on Arithmetic Ex- pression
(b) Fine-tuning GPT-2 Small on 5 5 digit Mul- tiplication
Figure Representation collapse across layers during (a) training (or fine-tuning(b)) for two datasets. The x-axis represents the layer ID, while the y-axis shows the degree of collapse mea- sured by representation the decline in Entropy) experience representation collapse for Pretrained GPT-2 Small 5 digit Multiplication (b) and during Vanilla training or fine-tuning for both datasets, indicating po- tential bottlenecks in information flow feature learning. Tools like Pause Goyal et al. (2023b) tbkend tuning can't fix it, but our proposed regularization VCR can improve collapse.
Entropy. The results highlight how intermediate layers (shown by
Representation collapse in Transformers manifests reduction in the diversity of internal repre- sentations across layers, particularly in tasks requiring complex reasoning. This collapse hinders the model's ability capture and process intricate patterns necessary for tasks like multi-digit mul- tiplication. To illustrate this, we analyze the layer- wise representation entropy of a GPT-2 model fine-tuned on the 5 digit multiplication task. As shown in Figure there is significant drop in entropy in the intermediate layers, indicating collapse in representation diversity. We hypoth- esize that this representation collapse the reason why is very hard for the model to calculate multiplication, carry, and store them, thus resulting in poorer performance, shown in Figure for the output positions, which require more computation (Figure
We quantify the phenomenon of representation collapse in token sequences by computing the matrix-based entropy of representation vectors across transformer model layers. We specifically explore the a-order kernel x applied to layer-wise token representations without assuming the underlying distribution. We linear kernel defined as n(a, b) ab"" in our analysis in (Skean al. 2024). Assuming. z() fi E R'xd be the Ith layer representations of T tokens in sequence with dimension d, we compute the pairwise similarity of T token representations to form the Gram matrix K (Z z0)) 2 z(1i" E RIXT, Then, we determine the matrix-based entropy of order 0:
entropy (Skean et al. 2023 Giraldo al.. 2014) using similarity
S. (z))
1
log Σ
(K) tr(
1
Equation similar to the n-order Rényi entropy eigenvalues of the Gram matrix providing insights into their distribution and the underlying token structure. In particular, we employ lim which corresponds to the Shannon or Von Neumann entropy Bach (2022); Boes et al. (2019). Eigen- values are normalized by the matrix trace tr(K)= = Σ (,()), then raised to the power o, forming
20 7 (Gram matrix) and z 70) (Covariance matrix) have identical nonzero eigenvalues, thus both yield matrix entropy. When d T, the computation of the covariance matrix is generally efficient.
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probability distribution over principal components. Each eigenvalue indicates how much variance the corresponding principal component explains (Scholkopf Smola 2018). Low entropy results in heavy-tailed distribution, with few components dominating, concentrating token information and collapsing the representation space with a lack of representations diversity.
3.4 SEQUENTIAL VARIANCE COVARIANCE REGULARIZATION (SEQ- VCR)
To address representation collapse, we introduce Seq-VCK regularization technique that enforces high variance and low covariance in intermediate representations of Transformer models. Adapted for sequential data, Seq- VCR builds directly on the variance and covariance regularization principles of VICReg (Bardes et al. 2021) and VCReg (Zhu et al. 2023). with Equation 3 re-purposing the formulations from VICReg tackle the challenges sequence modeling.
Seq-VCR is implemented on the final output of the model (X = Sel= The process involves computing a covariance matrix and using a pre-trained model like GPT-2, leading to dimensions such fat E I 2x50257, which can be computationally challenging. So, for model like GPT-2, instead of applying regularization on the output of feiss we apply a linear projection that projects the representation of layer into
manageable space, expressed as X frroz (fi). with foroj E I 22448 This projection layer is inserted over the representation just before the BCQ regularization loss calculation and trained exclusively end-to-end with the Seq- VCR loss defined below.
Now we can assume that we have N samples in the batch, so N number of X, denoted as X E R Nx'xd, We can calculate the covariance matrix across the batch dimension, C e R Rxxdxd
Leyy-VCR VCR
A1 max(0, 1 v Ci,,, + 1) +d2 ((E
T
Variance Term
Covariance Term
Where for each position in the sequence, covariance matrix can be calculated across the batch di- mension when x is the input for 7th position,
C((X,)) N Σ
x
X (X,, -X,)), where X
X
(4)
where λ and A2 are the coefficients of the variance and covariance regularization terms. 1) is a small constant (set to 0.001 in our experiments), used for numerical stability. Detailed hyper-paramest choices can be found in Appendix A
The Variance Term encourages unit variance in each dimension, while the Covariance Term penal- izes covariance between different dimensions, promoting de-correlation and diversity in representa- tions.
3.5 INCORPORATING PAUSE TOKENS
Increasing the model capacity brings significant accuracy boost for solving n digit multiplication tasks (Qiu et al. 2024). While some previous work increases depth to increase the model capacity, alternative solution is pause tokens that as explicit indicators for the model tem- porarily pause on intermediate states in sequential tasks before proceeding to the next computation step (Goyal al. 2023b).
Similarly, we enhance the model's reasoning capacity by introducing dummy pause tokens, which placeholders for intermediate computation steps. This offers notable benefits over Chain-of- Thought (CoT) reasoning tokens, particularly in reducing inference time (as there are often more
Initial tests indicated optimal performance when regularization was utilized on the final layer. Exper- imentation revealed that additional layers can also be effective. For detailed information on layer-specific regularization, refer Appendix

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CoT tokens to decode than pause tokens) and dependency costly human human-supervinn supervised data. With pause tokens we can solve tasks like multiplication in a fraction of the time compared to CoT, while performing at similar accuracy (5 times faster and close to 100% in experiments. See Appendix E for further details). In all experiments, pause tokens were placed between input and output tokens to emulate CoT reasoning. In particular, the input-output format looked like:
<question> </pause_stare staus <pause>
<pause> </pause_end <answer>.
We tried with 2, 4, 6, and 8 pause tokens on 4 x 4 and 5 x 5 digit multiplication tasks, and we did not find any correlation with task complexity. We believe that this may be due to the fact that all pause tokens share the same embedding. Future work will explore the effect of having different embeddings per pause tokens.
such, we used 2 pause tokens in all our experiments.
3.6 TRAINING OBJECTIVE
The overall training objective combines the standard next-token prediction loss with the Seq- VCR regularization. Our final loss L is:
L Liezt + LCQ
(5)
This encourages the model to not only predict the next token accurately, but also maintain diverse and informative intermediate representations.
EXPERIMENTS
4.1
DATA
Expression
Longest Integer Subsequence
5 x Digit Multiplication
HRUT
[3, 2, 20]
NRUT
Initial dpi1 Array: t1, 1,
INPO1
X 67890
12345
3
5
Idx 8: dp [1, 1, 1,
27
dp
[1, 2,
09876809
1111850
Idx 2: dp
[1, 2, 1,
886415860
748788086
29
dp
[1,
OUTPUT
37
Idx 4: dp [1, 2, 1, 1, 3]
OUTPUT
838182050
[ex((p)
(a) Arithmetic Exprcssion
(b) LIS
(c) 5x5 Multiplication
Figure 3: Illustrations of Input, Output and CoT on the Arithmetic, LIS and mutliplication datasets
We conduct experiments three tasks: we first consider the multi-digit multiplication task from the BIG-bench benchmark (Srivastava et al.) 22022), which is the most challenging among arithmetic tasks (Yang al. 2023). In particular, cation problems, since these two tasks prove very challenging to solve under no CoT, utilizing the training data generated by Deng et al. (2023).
use the four-digit (4 4) and five-digit 5) multipli-
Next, we focus on the Arithmetic Expressions Feng et al. (2024) dataset. This task focuses on evaluating arithmetic expressions. The input sequence for this task is sequence consisting of numbers, mathematical operators such addition (+), subtraction (-), multiplication (x), division (+) and brackets, followed by the equal sign. The goal of this task calculate the input arithmetic expression and generate the correct result. This task is naturally well-suited for Chain-of- Chain-of-Thhught (CoT) method, where each step does part of the calculation, slowly solving one operation at a time while keeping the other parts the same.
Finally, we also conduct experiments in more general setting outside arithmetic tasks, called Dy- namic Programming (DP). Dynamic Programming is framework for solving decision-makee problems. The core idea in this setting to break down a complex task into a series smaller sub-tasks that can be solved sequentially. We chose very popular problem in this setting called the

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Longest Increasing Sub-sequence (LIS) as described in the Introduction to Algorithms book (Cor- men et al. 2022). For this task, the goal is to find the length of the longest increasing subsequence of given integer sequence. We generate datasets with different input sequence lengths ranging from (50, 80, 100] as shown in Feng et al. (2024). Moreover, all input sequences, and answers in LIS bounded-range integers and can therefore be tokenized (similar to the multi-digit multiplication task). The CoT tokens for this task consist of the dynamic programming array plus the final answer. Figure shows working example for sample problem in all the datasets considered.
Table 3 in Appendix D provides an overview of the different datasets i.e., Multiplication, Arithmetic Expression and LIS, along with their respective input, chain-of-thoughh (CoT), and output token details.
4.2
MODELS AND TRAINING CONFIGURATIONS
We conduct experiments using two models:
GPT-2 Small Fine-Tuning We fine-tune a pre-trained GPT-2 Small model on the multi-digit mul- tiplication tasks. Fine-tuning is performed for 40 epochs with a learning rate of 5 x 10-4 and a batch size of 32.
minGPT Training from Scratch We train minGPT model from scratch the Arithmetic Ex- pressions and LIS datasets. Training is conducted for 100 epochs with learning rate of x 10-4 and batch size of 128.
We compare five configurations:
Vanilla: Standard training/finetuning without CoT or pause tokens.
With CoT: Training/finetuning with explicit CoT supervision.
Pause: Inserting pause tokens in the input sequence.
Seq- VCR: Applying Seq- VCR regularisation.
Seq-VCR Pause: Combining Seq- VCR with pause tokens.
4.3 RESULTS
4.3.1
REPRESENTATION DIVERSITY
See
Seep PTRte
RaverS
500
Frequency
Doveri
Boyr?
Ixyert
layro
layork
0.5
33
Entropy
Figure 4: Layer-wise entropy distributions for different configurations the 5 x multiplication task. Seq-VCC and Seq- Veg-vCRRPPan Pause maintain higher entropy across layers, indicating greater repre- sentation diversity.

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Seq- VCR significantly enhances representation diversity, evidenced by the layer-wise distribution of representation entropy across different configurations. In Figure 4 we observe that the configura- tions that employ our regularization technique, particularly Seq-VCR VCR and Seq-VCR+Paune higher entropy values the intermediate layers compared to the other configurations (Vanilla, and Pause). This increased diversity in the representation entropy suggests that the model is able to cap- ture a wider range of features and maintain distinct representations. The histograms also reveal that while other configurations show peaks at lower entropy values, indicating potential representation collapse and redundancy in the intermediate representations, our method fosters more uniform dis- tribution across the layers. This uniformity suggests that the model is using richer feature space, enabling it to better generalize to unseen data, thus improving performance in downstream tasks.
exhibit
4.4 LEARNING DYNAMICS
Seq-VCR+Pause
Vanilla
Pause
Seq-VCR
Next Token Prediction Loss
Next Token Prediction Loss
0.9
2.4
1.8
0.3
0.6
Epochs
Epochs
(a) Finc-tuning GPT-2 Small 5 x 5 digit Multipli- cation
(b) Training minGPT from scratch on Arithmetic Expression
Figure 5: Learning dynamics (curves for next token prediction loss) illustrating the phase transition observed across datasets when applying our regularization methods. The x-axis represents training epochs, and the y-axis denotes the model's loss. The phase transition is characterized by sharp reduction loss, marking a distinct shift in the learning regime when using Seq- VCR and Seq- VCR Pause, compared to the gradual decline saturating curves in other configurations.
Seq-VCR
Our regularization method, Seq-VCR or Seq- Pause cause a significant phase transition (im- proving the next token prediction loss) in the performance of the model across datasets compared to other configurations such as Vanilla, and Pause, which do not induce this phase transition as shown in Figure 5 Specifically, this phase transition marks distinet shift in the model's ability capture and generalize representations during training or fine-tuning.
Without regularization, learning is difficult and training loss saturates. Seq- VCR improves the learn- ing curve. Adding Seq-VCR with 2 Pause tokens causes a sharp phase transition (Figure 5 (a)) and solves the 5 5 digit multiplication task. Figure (b) shows similar improvements in the arithmetic reasoning task that is learned from scratch.
4.5 RESULTS ON 4 X 4 AND 5 X 5 DIGITS MULTIPLICATION TASKS
Table the results on 4 x 4 and 5 5 digits using a fine-tuned GPT-2 Small. Compared to no CoT configurations like (Vanilla, Pause), our method enables solving tasks previously not solvable without explicit COT. We can see the effectiveness of the SeQ on 4 4 digit multiplication task where without pause token improves performance from 25% to 52% and with 2 pause tokens Seq- VCR Pause achieves accuracy 99.2%, which is close to With CoT performance and significantly better than other configuration without COT. It even outperforms the 5-shot prompt(wing or without Co'T) performance of GPT-3.55 and GPT-4.0.
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Model
Configuration
4x4 Mult
5x5 Mult
With CoT
0.43
0.05
GPT-3.5
No CoT
0.02
0.00
With CoT
0.77
0.44
GPT-4
No CoT
0.04
0,00
With CoT
1.0
1.0
Vanilla
0.25
0.0
GPT-2 Small
Pause
0.28
0.0
Seq-VCH VCR
0.52
0.0
Seq-VCHS Pause
0.992
0.995
x digits Multiplication Tasks. GPT-3.5 and GPT-4 results are taken from Deng et al. (2024) which are produced by 5-shot prompt
Table Accuracy (exact match) on
We also see from Thhat GPT-4 with CoT shot prompts achieves 44.0% for 5 x 5 digit mul- tiplication, while Vanilla, Seq-VCR, and Pause achieve 0% without COT. Because, all configu- rations struggle with predicting middle tokens in the output sequence (shown in Figure 6 This happens because those middle tokens require more computations calculate successive carry and store them, since these tokens require more computation (as shown in Figure [. thus making them difficult sub-tasks than predicting the pe- ripheral tokens which require less computations, thus easier to solve. However, Seg- VCg+ Pause with Pause tokens solves the 5 digit multi- plication task, previously unsolved without CoT tokens. Figure (bb) shows our regularization Seq-VCR increases intermediate layer entropy. allowing more exploration the representation space. We hypothesize that along with better rep- resentation by Seq- VCR, Pause tokens increase the computational ability, allowing the model to carry the necessary multiplication values and solve the task. More results by training GPT-2 from scratch are in the Appendix
Vaniil
Pause
SeqUOCR
SeqV/R- Paraee
100
Accuracy
6
6
Index
Figure 6: Position- wise accuracy of Multiplica- tion of 5x5 digits with different configuration of GPT-2 Small. We see that models fail the mid- dle tokens as they require more compute.
4.6
RESULTS ARITHMETIC EXPRESSION AND LONGEST INCREASING SUB-SEQUENCE (LIS) DATASETS
Figure illustrates how the accuracy varies among differentt methods (represented by different bar colors the plot) and with changing levels of task complexity, such the number of operators in the Arithmetic Expressions Dataset or the input sequence length in the case of the LIS dataset.
We can observe that as the task complexity is low, i.e. in case of 4 operators in the arithmetic expression and input sequence length of 50 in the LIS dataset, all the methods perform quite well. But, the number of operators or sequence length increases. the tasks become more complex, which shows the need for models other than the vanilla baselines.
Seq- VCR demonstrates substantial improvement over vanilla models and achieved results compara- ble to the chain-of-thour -hhuugh (CoT)) prompting method as shown in /een et al. (2024). Although CoT outperforms the other methods, there is an increase the serial computation due to its step-by-step reasoning approach. On the other hand, Pause tokens and regularization primarily boost parallel computation, enhancing the model's 's expressivity for these types of problem without significantly increasing serial computation. The results presented in Figure are based on models with 5-layer configuration, optimized using the best best-performin number of pause tokens. We find that the num- ber of pause tokens that works best for a particular model and layer config vary bit. For example,

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Seq
Seq-CR
Vanila
Seq.VER
Seq-VCR
1.0
Arcuracy
Accuracy
0.75
0,4
0.00
50
80
100
Operators
Input Sequence Length
(a) Test accuracy Arithmetic Expressions Dataset
(h) Test accuracy LIS Dataset
Figure 7: Performance of each method across tasks of varying difficulty
Manillaa
Pause
Sey-VC
CaT
Vanila
Pause
Soq
CoT
0.8
Accuracy
Accuracy
0.4
0.2
5
Number of Layers
Number of Lavers
(a) Test accuracy on operator Arithmetic Ex- pression
Test accuracy on LIS Dataset with 100 put Sequence Length
Figure 8: Performance of each method on the complex task w/ varying number of layers
the best working Seq-VCR + Pause model uses 7 pause tokens in the case of the operator task in Arithmetic Expression, while 4 pause tokens work the best for the LIS task with 80 as input
quence length. In Figure g we show how the different models perform in the most complex task in each dataset. We also clearly that more depth of the model helps every method in solving the task better. Another conclusion we can draw from Figures 7 and is that although the combination of Seq-VCR with Pause Tokens works best (almost close to the CoT method), Seq- VCR achieves similar test accuracy without the use of Pause Tokens in both datasets. All results in Figures] 7 and are over seed runs. Detailed ablation studies, exploring variations in the number of pause tokens, layers, and task complexities, can be found in the Appendices B and C
CONCLUSION
Despite the remarkable success large transformer based language models, they exhibit deficien- cies in multi-step reasoning tasks. One proposed remedy is the application of Chain-of ThTThhthh supervision, which, although effective, entails considerable cost. Conversely, there has been lim- ited exploration into enhancing the capacity of the pre-trained model representation itself without explicit supervision. Our analysis indicates that intermediate layer representation collapse detrimen- tally affects the computation of intermediate information essential for solving arithmetic reasoning tasks. Our proposed regularization technique enhances the model's 's representation capacity, thereby improving its ability to solve these tasks. We think that further research is required to enhance the capability pre- training as well, where improved representation learning will enhance the model with greater capabilities.
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ACKNOWLEDGMENTS
We acknowledge the support of the Canada CIFAR AI Chair Program and IVADO. We thank Mila and Compute Canada for providing computational resources.
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A HYPERPARAMETERS
Parameter
Value/Details
Learning Rate
0.0001
Batch Size
128
Optimizer
AdamW W
Dropout
0.1
Attn. Heads
4
Epochs
100
Number of Layers
4, 5,
Input Sequence Length
50, 80, 100
# of Operators
4, 5, 6
Total Compute Resources
32GB GPU, 6CPU, GB RAM
Total Job Time
max 24 Hrs
Table 2: Summary of Hyper-parameters, Compute Resources, and Time for Experiments. Each experiment was run on same configurations of GPUs, with the total time dependent on the number of layers and input parameters such as Dataset, task complexity.
We manually searched for hyperparameters, we did not do exhaustive searches such as grid search or random search. We found this was enough find good hyperparameters for our tasks. For the two coefficients Å₁ and A2 in Equation ??, we keep their proportion similar to/Bardes et al. 202 For multiplication tasks \ 1.0 and A2 0.004 and for other tasks we A 0.1 and A2 0.5. For computing the covariance matrix, we use a batch size of 32 for multiplication tasks and 128 for others. Other hyperparams like learning rate and batch size values are similar other related works Deng et al., 2023 Feng et al. 2024).
Regarding the number of pause tokens, we tried 2, 6, pause tokens on 4x4 and 5x5 digit multi- plication tasks, and we did not find any correlation with task complexity. We believe it may be due to the fact that all the pause tokens share the same embedding.
B MODEL COMPLEXITY Vs TASK DIFFICULTY
Operator
Langth
Length
0.96
0.9
Arcurary
C.00
Accuracy
0.80
0.3
0.72
Number of Layers
Number Layers
(a) Test accuracy Arithmetic Expressions Datasct
(b) Test accuracy LIS Datasct
Figure 9: Seq-VCR VCR Pause Accuracy Vs Number of Layers for tasks of different complexity in Arithmetic Expressions and LIS
Both plots in Figure9 demonstrate the trade-offs between model complexity (in terms of layers) and the complexity of the input (number of operators or sequence length). The model performs best when handling simpler configurations, such as fewer operators shorter sequences, and tends to struggle with more complex setups as the number of operators or sequence length increases. Moreover, Increasing the number of layers seems mitigate performance drops some extent, especially in simpler cases, but cannot fully offset the challenges posed by more complex inputs. All results in figure are over seed runs.
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C
NUMBER OF PAUSE TOKENS VS TASK COMPLEXITY
0.8
Vanilla - Low
Accuracy
0.6
Vanilla Medium
Vanilla High
Seq
Seq.VCR Medium
Seq-VCR- High
0.2
0.0
Number of Pause Tokens
Figure 10: Ablation of Varying Pause Tokens and Comparing Vanilla Vs Seq-VCR + Pause Tokens for different Task Complexities in the arithmetic dataset. Low, High, Medium refer to 5, 6 arith- metic Operators respectively, when number of layers is fixed to 5.
Figure shows that Seq- VCR Pause model consistently achieves high accuracy (around 0.8 higher) across all numbers of pause tokens, indicating that the addition of pause tokens does not affect its performance. The Vanilla model, in contrast, exhibits much lower accuracy across all settings except for the low-complexity task with four operators. We hypothesize that this is because five layers are likely sufficient solve such simple task.
For this task, where the Vanilla model performs better, we observe notable drop in accuracy when pause tokens are added. We speculate that this occurs because, in scenarios where the model has suf- ficient capacity to solve the task, pause tokens introduce distractions that hinder performance rather than enhancing it. However, tasks with insufficient depth, such as the 5x5 digit multiplication task on GPT2-Sman pause tokens prove beneficial, as shown in Table [
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D DATA STATISTICS
# Input
# CoT
Output
Dataset
Task
Size
Tokens
Tokens
Tokens
4 X 4 Mult
808k
47
8
Multiplication
x Mult
808k
11
75
10
4 Operators
1M
19
24
1
Arithmetic Expression
Operators
1M
23
40
6 Operators
1M
27
60
Seq Length 50
1M
50
50
1
LIS
Seq Length 80
1M
80
80
Seq Length 100
1M
100
100
Table 3: Dataset statistics. Size refers to the training set. The number of input, output, and inter- mediate chain of thought tokens are median values on the validation set. The number of tokens are based on the GPT-2 tokenizer, and special ending symbol is counted for both intermediate tokens and output tokens.
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E
SPEEDUP AND ACCURACY TRADE-OFF FOR PAUSE AND COT TOKENS
Seq-VCR offers notable benefits over Chain-offThhouge (CoT) reasoning, particularly in reducing inference time and dependency on costly human-supervised data. Unlike CoT, which requires exten- sive labeled data for multi-step reasoning, Seq-VCR uses a few dummy pause tokens to solve tasks like multiplication in fraction of the time (5 times faster), while performing at similar accuracy close to 100% in our experiments (see table below). Seq-VCR* efficiency in both inference time, data requirements, and accuracy makes a more scalable and robust approach compared CoT.
To compute the normalized throughput for inference, we use the following equation, as in the Deng al. (024) paper:
Turget
T
T Thase
Here:
Torm is the normalized throughput, which represents the relative inference speed.
Turgu is the throughput (number of examples processed per second) when using target method.
Thase is the throughput (number of examples processed per second) for the baseline model without Chain of Thought or Pause tokens.
T Rorm
Accuracy
Method
4x4 Mult
5x5 Mult
4x4 Mult
5x5 Mult
No CoTT
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
Table Normalized Throughput (the higher the better) and Accuracy measures on 4 and 5 x digit multiplication without CoT tokens, with CoT tokens, and with 2 pause tokens.
Note, that it is expected that models with CoT tokens perform the best as CoT tokens carry more useful information than simple dummy pause tokens. However, they are more expensive to get (human labor) and require more compute to generate at inference time (since there are more of them). In that sense, SeQ-VCR will always scale better than CoT, no matter the model size.
F COMPUTATION OF THE COVARIANCE MATRIX ACROSS BOTH THE BATCH AND LENGTH DIMENSIONS
Here we present an ablation study done by fine-tuning GPT2-small on the 5 x 5 digit multiplication. We compared the performance of calculating the covariance matrix across the batch dimension vs both the batch and the length dimensions and present results in the Table below.
GPT2-small (fine-tuned)
Accuracy
Compute Complexity
baseline
Seq- VCR batch dim
0.99
O(b-d))
Seq-VCR VCR batch+length dim
0.98
Olb n dd)
Table 5: Accuracy and compute complexity (when b. and d are batch size, # tokens and feature dimension respectively) difference between computing the covariance matrix across the batch vs length+batch dimensions.
We find that there are no significant differences (between 98 and 99% accuracy 5x5 digit multi- plication). However the computation time n times larger on average when using both the batch and length dimensions, which grows with the increase of sequence lengths.
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G ABALATION OF LAYERS
We apply Seq- VCR to all layers while fine-tuning GPT-2-small on a 5x5 multiplication task. The results indicate improved performance when Seq- VCR is applied the last layer.
Layer
L0
L11
L12
Accuracy
0.97
0.98
0.99
Table 6: Performance of Seq- VCR + Pause across different layers.
With À 1.0, A = 0.004, and pause = 2, we achieve these results, where applying Seq-VCI on the first and last two layers almost solves the task, but for the rest of the layers it does not work. However, hyperparameter tuning may be required for other layers.
H PRE-TRAINED MODELS VS TRAINING FROM SCRATCH
Here, we run more experiments to emphasize the advantage of fine-tuning pre-trained models. We train a GPT2-small model from scratch on digit multiplication tasks over multiple hyperpa-
rameter settings and present the results in the table below:
GPT2-small
From Scratch
Vine-Tunee
Layer
L12
L0
L12
LO
Baseline
0
0
Seq-VCR VCR batch dim
0.03
0.97
0.99
0.97
Seq-VCR batch+length dim
0.87
0.99
0.98
1.0
Table 7: Accuracy of pre-trained and training from scratch by computing the covariance matrix across the batch length+batch dimensions, applying Seq-VCR 1st (LO) and last (L12) layers
Training models from scratch is more sensitive to hyperparameter choices, such as batch size, and performs poorly when Seq- VCR applied to the last layer. However, computing covariance across both the batch and length dimensions significantly improves performance. In contrast, fine-tuning GPT-2 proves to be more stable than training from scratch. Interestingly, as shown in Table 7 applying regularization to the 1st(LO) layer is effective for both fine-tuning and training from scratch. This likely because Seq- VCR at the 1st(LO) layer enhances collapse prevention, as shown in Figure 15
I EXPERIMENTS WITH LARGER MODEL
To test the effectiveness of our approach larger, recent models applied our Sec-Ve VCR regularization method during fine-tuning of a Llama3.2-18 model with LoRA(r = 64) adapters on the challenging 5x5 digit multiplication task.
We measure the representation collapse across layers for the finetuned model with and without our regularization and report results in Figure below.
LLama 3.2-1B
Accuracy
Vanilla
Seq- VCR
0.974
Table 8: Accuracy of finetuning Jlama3.2-11 B- model on 5x5 digit multiplication task
We find that the model trained with CCR has higher average entropy (reduced collapse) by 10-20% compared to the baseline llama model and achieving 97.4% (Table accuracy. This exper- iment highlights that collapse is not isolated case and is still issue in more recent models.
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Vanilla
Seq-VCR
Pretrained
0.7
0.6
Average Entropy
0.5
0.3
0.2
2
10
12
14
16
Layer
Figure 11: Representation collapse across llama3.2-1 layers on the 5 digit multiplication task.
J COLLAPSE AT SCALE
Llama-3.2-18
Llama-3.2-88
Meta-Llama-3-88
0.6
0.5
Average Entropy
0.4
0.3
0.2
10
25
Layer
Figure 12: Representation collapse across scale (1B, 3B and 8B) of LLama pretrained models
We tested pre-trained Llama models of varying scales on the 5x5 digit multiplication dataset and observed consistent representation collapse across all model sizes (Figure This highlights that the issue persists regardless of scale, emphasizing the need for effective regularization techniques like Seq-V VCR to mitigate collapse and improve intermediate reasoning capabilities.
K COLLAPSE MITIGATION IN OTHER BENCHMARKS
To test the effectiveness of our method to other domains, we finetuned Code-GPT-2 Small model on the SCGLLE-text-to-ooot benchmark") We measure the representation collapse across layers for the finetuned model with and without our regularization and report results in Figure [3 below.
We find that the model trained with Sec- VCR has a higher average entropy (reduced collapse) than the baseline pre-trained lama model by 3-4%. This experiment showcases that our method also reduces collapse other domains.
20

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Published as a conference paper at ICLR 2025
Vanilla
Seq-VCR
Pretrained
0.31
0.30
Average Entropy
0.29
0.28
0.27
0.26
10
12
Layer
Figure 13: Representation collapse across Code-GPT-2 Small layers on the CodeXGLUUE-txxttt
code task.
L COLLAPSE MITIGATION DURING PRE-TRAINING
To evaluate the effectiveness of our method in mitigating representation collapse during pre-training, trained GPT-2-small from scratch the C4 dataset. The regularizer applied either to the
last layer (L12) or the first layer (LO). As shown Figure [4] the validation perplexity achieved by method is similar to the baseline Vanilla validation perplexity. For LO, X 1.0 and X2 0.01
and for L12, A1 = 0.5 and A2 = 0.1
SeqVCR: LO
120
SeqVCRk L12
100
Vol Perpleaky
80
60
40
5000
10000
15000
20000
25000
30000
Steps
Figure 14: Validation Perplexity of GPT-2 Small trained on the C4 dataset.
However, Figure 15 demonstrates that Seq- VCR effectively mitigates collapse across layers. We find that the model trained with Sec- VCR has a higher average entropy (reduced collapse) than the vanila trained model by 3-5%. This experiment showcases that method also reduces collapse in other domains. Notably, applying the regularizer to the first layer resulted in a more stable prevention of collapse in intermediate layers.
21

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Published as a conference paper at ICLR 2025
Vanilla
Seq-VCR: L12
Seq-VCR:
Random
Average Entropy
0.8
0.6
2
8
10
12
Layer
Figure 15: Representation collapse across GPT-2 Small layers after the C4 pretraining.
M RESULTS ON GSM8K
We performed further experiments to fine-tune the GPT-2 Small model using an enhanced version of the GSM8K dataset[1], incorporating and excluding Seq- BCR, Pause (2), and COT. The findings are presented below.
Method
Accuracy
Vanilla
0.191
CoT
0.437
Pause(2)
0.197
Seq-VCR
0.198
Seq-VCR + Pause(2)
0,202
Table 9: Comparison of Various Methods by Fine-tuning GPT-2 Small on the GSM8K Dataset.
For this dataset, performance improves slightly compared Vanilla fine-tuning when using Seq- VCR without pauses, and even more when applying pauses with Seq.VCR.
22


Reviews:
[{"review_id": "sHhgqtuXiv", "reviewer": "Reviewer_STRW", "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.", "questions": "Same as the weaknesses section.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper presents a novel regularization technique that improves the model's performance in several reasoning tasks\n- 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.\n- 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.", "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": "SVrdhtK5HA", "reviewer": "Reviewer_DQGy", "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.", "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.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "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. \n\nThe 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.\n\nThe speedup vs CoT is intuitively reasonable but it would have been nice to see performance numbers as in the cite Deng 2024 paper.\n\nSimilarly, 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?\n\nFor completeness, it would be nice to see CoT in figures 7 and 8.", "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": "5a7QuhA5ZZ", "reviewer": "Reviewer_YSNL", "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.  \n\nThe 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.", "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.  \n\n(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?\n\n(3) Equation (3) introduces $\\lambda_1$ and $\\lambda_2$ as hyperparameters, but the paper fails to say what they are set to. \n\n(4) What batch size is used for computing the covariance matrix?\n\n(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?\n\n(6) How is the projection layer $f_{proj}$ trained?\n\n(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?", "limitations": "", "rating": 6, "confidence": 5, "soundness": 2, "presentation": 1, "contribution": 4, "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.\n\nOne 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.\n\nSimilarly, 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.\n\nFinally, 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.\n\nMore specific criticisms:\n\nEquation (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.   \n\n(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.)\n\nFor 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.", "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": "cn1kPcqaPu", "reviewer": "Reviewer_LszK", "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.", "questions": "see the weaknesses section.", "limitations": "", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. The identified representation collapse is quite interesting\n\n2. 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.\n\n2. 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.", "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)