Variance-Covariance Regularization Improves Representation Learning

## Abstract

Abstract Transfer learning plays a key role in advancing machine learning models, yet conventional supervised pretraining often undermines feature transferability by prioritizing features that minimize the pretraining loss. In this work, we adapt a self-supervised learning regularization technique from the VICReg method to supervised learning contexts, introducing Variance-Covariance Regularization (VCReg). This adaptation encourages the network to learn high-variance, low-covariance representations, promoting learning more diverse features. We outline best practices for an efficient implementation of our framework, including applying it to the intermediate representations. Through extensive empirical evaluation, we demonstrate that our method significantly enhances transfer learning for images and videos, achieving state-of-the-art performance across numerous tasks and datasets. VCReg also improves performance in scenarios like long-tail learning and hierarchical classification. Additionally, we show its effectiveness may stem from its success in addressing challenges like gradient starvation and neural collapse. In summary, VCReg offers a universally applicable regularization framework that significantly advances transfer learning and highlights the connection between gradient starvation, neural collapse, and feature transferability.

## 1 Introduction

Figure: Figure 1: VCReg regularizes the network by encouraging the intermediate representations to have high variance and low covariance. VCReg is applied to the output of each network block to make all the intermediate representations capture diverse features.
Refer to caption: extracted/5425843/images/vis.png

Transfer learning enables models to apply knowledge from one domain to enhance performance in another, particularly when data are scarce or costly to obtain (Pan & Yang, ; Weiss et al., ; Zhuang et al., ; Bommasani et al., ). One of the key challenges arises during the supervised pretraining phase. In this phase, models often lack detailed information about the downstream tasks to which they will be applied. Nevertheless, they must aim to capture a broad spectrum of features beneficial across various applications (Bengio, ; Caruana, ; Yosinski et al., ). Without proper regularization techniques, these supervised pretrained models tend to overly focus on features that minimize supervised loss, resulting in limited generalization capabilities and issues such as gradient starvation and neural collapse (Zhang et al., ; Neyshabur et al., ; Zhang et al., ; Pezeshki et al., ; Papyan et al., ; Shwartz-Ziv, ).

To tackle these challenges, we adapt the regularization techniques of the self-supervised VICReg method (Bardes et al., ) for the supervised learning paradigm. Our method, termed Variance-Covariance Regularization (VCReg), aims to encourage the learning of representations with high variance and low covariance, thus avoiding the overemphasis on features that merely minimize supervised loss. Instead of simply applying VCReg to the final representation of the network, we explore the most effective ways to incorporate it throughout the intermediate representations.

The structure of the paper is as follows: we begin with an introduction of our method, including an outline of a fast implementation strategy designed to minimize computational overhead. Following this, we present a series of experiments aimed at validating the method’s efficacy across a wide range of tasks, datasets, and architectures. Subsequently, we conduct analyses on the learned representations to demonstrate VCReg’s effectiveness in mitigating common issues in transfer learning, such as neural collapse and gradient starvation.

Our paper makes the following contributions:

- 1.
We introduce a robust strategy for applying VCReg to neural networks, including integrating it into the intermediate layers.
- 2.
We propose a computationally efficient implementation of VCReg. This implementation is optimized to ensure minimal additional computational overhead, allowing for seamless integration into existing workflows.
- 3.
Through extensive experiments on benchmark datasets both in images and videos, we demonstrate that VCReg suppresses the prior state-of-the-art results in transfer learning performance across various network architectures, including ResNet (He et al., ), ConvNeXt (Liu et al., ), and ViT (Dosovitskiy et al., ). Moreover, we also show that VCReg improves performance in diverse scenarios like long-tail learning and hierarchical classification.
- 4.
We investigate the representations learned by VCReg, revealing its effectiveness in combating challenges such as gradient starvation (Pezeshki et al., ), neural collapse (Papyan et al., ), information compression (Shwartz-Ziv, ), and sensitivity to noise.

Before delving into VCReg’s details in the following sections, it is key to note its divergence from VICReg, namely by omitting the invariance loss and focusing on variance and covariance loss for a wider application, especially in transfer learning. This approach tackles challenges like gradient starvation and neural collapse, advancing neural network training across various architectures. Our work further distinguishes itself by exploring optimal regularization strategies, moving beyond generic application to significantly enhance its
effectiveness.

## 2 Related Work

### 2.1 Variance-Invariance-Covariance Regularization (VICReg)

VICReg (Bardes et al., ) is a novel SSL method that encourages the learned representation to be invariant to data augmentation. However, focusing solely on this invariance criterion can result in the network producing a constant representation, making it invariant to both data augmentation and the input data itself.

VICReg primarily regularizes the network by combining variance loss and covariance loss. The variance loss encourages high variance in the learned representations, thereby promoting the learning of diverse features. The covariance loss, on the other hand, aims to minimize redundancy in the learned features by reducing the overlap in information captured by different dimensions of the representation. This dual-objective optimization framework effectively promotes diverse feature learning for SSL (Shwartz-Ziv et al., ). To improve the performance of supervised network training, we adapt the SSL feature collapse prevention mechanism from VICReg and propose a variance-covariance regularization method.

To calculate the loss function of VICReg with a batch of data $\{x_{1}\ldots x_{n}\}$, we first need to have a pair of inputs $(x^{\prime}_{i},x^{\prime\prime}_{i})$ such that $x^{\prime}_{i}$ and $x^{\prime\prime}_{i}$ are two augmented versions of the original input $x_{i}$. Given the neural network $f_{\theta}(\cdot)$ and the final representations $z^{\prime}_{i}=f_{\theta}(x^{\prime}_{i})$ and $z^{\prime\prime}_{i}=f_{\theta}(x^{\prime\prime}_{i})$ such that $z^{\prime}_{i},z^{\prime\prime}_{i}\in\mathbb{R}^{D}$, VICReg minimizes the following loss:

$$ $\displaystyle\ell_{\mathrm{VICReg}}(z^{\prime}_{1}\ldots z^{\prime}_{n},z^{ \prime\prime}_{1}\ldots z^{\prime\prime}_{n})$ (1) $\displaystyle=\alpha\ell_{\mathrm{var}}(z^{\prime}_{1},\ldots,z^{\prime}_{n})+ \alpha\ell_{\mathrm{var}}(z^{\prime\prime}_{1},\ldots,z^{\prime\prime}_{n})$ (2) $\displaystyle+\beta\ell_{\mathrm{cov}}(z^{\prime}_{1},\ldots,z^{\prime}_{n})+ \beta\ell_{\mathrm{cov}}(z^{\prime\prime}_{1},\ldots,z^{\prime\prime}_{n})$ $\displaystyle+\sum_{i=1}^{n}\ell_{\mathrm{inv}}(z^{\prime}_{i},z^{\prime\prime }_{i}).$ $$

The variance and covariance loss functions are defined as:

$$ $\displaystyle\ell_{\mathrm{var}}$ $\displaystyle=\frac{1}{D}\sum_{i=1}^{D}\max(0,1-\sqrt{C_{ii}})$ (3) $\displaystyle\ell_{\mathrm{cov}}$ $\displaystyle=\frac{1}{D(D-1)}\sum_{i\neq j}C_{ij}^{2}$ (4) $$

where $C=\frac{1}{N-1}\sum_{i=1}^{N}(z_{i}-\bar{z})(z_{i}-\bar{z})^{T}$ denotes the covariance matrix, and $\bar{z}$ represents the mean vector, given by $\bar{z}=\frac{1}{N}\sum_{i=1}^{N}z_{i}$.

Building on insights from prior studies (Shwartz-Ziv, ; Shwartz-Ziv et al., ), it is understood that the invariance term does not play a pivotal role in diversifying features. Consequently, in adapting to the supervised regime, we exclude the invariance term from the regularization.

### 2.2 Representation Whitening and Feature Diversity Regularizers

Representation whitening is a technique for processing inputs before they enter a network layer. It transforms the input so that its components are uncorrelated with unit variance (Kessy et al., ). This transformation achieves enhanced model optimization and generalization. It uses a whitening matrix derived from the data’s covariance matrix and results in an identity covariance matrix, thereby aiding gradient flow during training and acting as a lightweight regularizer to reduce overfitting and encourage robust data representations (LeCun et al., ).

In addition to whitening as a processing step, additional regularization terms can be introduced to enforce decorrelation in the representations. Various prior works have explored these feature diversity regularization techniques to enhance neural network training (Cogswell et al., ; Ayinde et al., ; Laakom et al., ). These methods encourage diverse features in the representation by adding a regularization term. Recent methods like WLD-Reg (Laakom et al., ) and DeCov (Cogswell et al., ) also employ covariance-matrix-based regularization to promote feature diversity, similarly to our approach.

However, the studies above mainly focus on the benefits of optimization and generalization for the source task, often neglecting their implications for supervised transfer learning. VCReg distinguishes itself by explicitly targeting enhancements in transfer learning performance. Our results indicate that such regularization techniques yield only modest performance improvements in in-domain evaluations.

### 2.3 Gradient Starvation and Neural Collapse

Gradient starvation and neural collapse are two recently recognized phenomena that can significantly affect the quality of learned representations and a network’s generalization ability (Pezeshki et al., ; Papyan et al., ; Ben-Shaul et al., ).
Gradient starvation occurs when certain parameters in a deep learning model receive very small gradients during the training process, thereby leading to slower or non-existent learning for these parameters (Pezeshki et al., ).
Neural collapse, on the other hand, is a phenomenon observed during the late stages of training when the internal representations of the network tend to collapse towards each other, resulting in a loss of feature diversity (Papyan et al., ).
Both phenomena are particularly relevant in the context of transfer learning, where models are initially trained on a source task before being fine-tuned for a target task. Our work, through the use of VCReg, seeks to mitigate these issues, offering a pathway to more effective transfer learning.

## 3 Variance-Covariance Regularization

### 3.1 Vanilla VCReg

Consider a labeled dataset comprising $N$ samples, denoted as $\{(x_{1},y_{1})\ldots(x_{N},y_{N})\}$ and a neural network $f_{\theta}(\cdot)$, which takes these inputs $x_{i}$ and produces final predictions $\tilde{y}_{i}=f_{\theta}(x_{i})$. In standard supervised learning, the loss is defined as $L_{\mathrm{sup}}=\frac{1}{N}\sum_{i=1}^{N}\ell_{\mathrm{sup}}(\tilde{y}_{i},y_
{i})$.

The core objective of the Vanilla VCReg is to ensure that the $D$-dimensional input representations $\{h_{i}\}_{i=1}^{N}$ to the last layer of the network exhibit both high variance and low covariance. To achieve this, we employ variance and covariance regularization, same as mentioned in equation 2:

$$ $\displaystyle\ell_{\mathrm{vcreg}}(h_{1}\ldots h_{N})$ $\displaystyle=\alpha\ell_{\mathrm{var}}(h_{1}\ldots h_{N})+\beta\ell_{\mathrm{ cov}}(h_{1}\ldots h_{N})$ (5) $$

Intuitively speaking, the covariance matrix captures the interdependencies among the dimensions of the feature vectors $h_{i}$. Maximizing $\ell_{\mathrm{var}}$ encourages each feature dimension to contain unique, non-redundant information, while minimizing $\ell_{\mathrm{cov}}$ aims to reduce the correlation between different dimensions, thus promoting feature independence. The overall training loss, which includes also the supervised loss, then becomes:

$$ $\displaystyle L_{\mathrm{vanilla}}$ $\displaystyle=\alpha\ell_{\mathrm{var}}(h_{1}\ldots h_{N})+\beta\ell_{\mathrm{ cov}}(h_{1}\ldots h_{N})$ (6) $\displaystyle+\frac{1}{N}\sum_{i=1}^{N}\ell_{\mathrm{sup}}(\tilde{y}_{i},y_{i}).$ (7) $$

Here, $\alpha$ and $\beta$ serve as hyperparameters to control the strength of each regularization term.

### 3.2 Extending VCReg to Intermediate Representations

While regularizing the final layer in a neural network offers certain benefits, extending this approach to intermediate layers via VCReg provides additional advantages (for empirical evidence supporting this claim, please refer to Appendix A). Regularizing intermediate layers enables the model to capture more complex, higher-level abstractions. This strategy minimizes internal covariate shifts across layers, which in turn improves both the stability of training and the model’s generalization capabilities. Furthermore, it fosters the development of feature hierarchies and enriches the latent space, leading to enhanced model interpretability and improved transfer learning performance.

To implement this extension, VCReg is applied at $M$ strategically chosen layers throughout the neural network. For each intermediate layer $j$, we denote the feature representation for an input $x_{i}$ as $h_{i}^{(j)}\in\mathbb{R}^{D_{j}}$. This culminates in a composite loss function, expressed as follows:

$$ $\displaystyle L_{\text{VCReg}}$ $\displaystyle=\sum_{j=1}^{M}\left[\alpha\ell_{\text{var}}(h_{1}^{(j)}\ldots h_ {N}^{(j)})+\beta\ell_{\text{cov}}(h_{1}^{(j)}\ldots h_{N}^{(j)})\right]$ (8) $\displaystyle+\frac{1}{N}\sum_{i=1}^{N}\ell_{\text{sup}}(\tilde{y}_{i},y_{i}).$ (9) $$

Spatial Dimensions However, applying VCReg to intermediate layers of real-world neural networks presents challenges due to the spatial dimensions in these intermediate representations. Naively reshaping these representations into long vectors would lead to unmanageably large covariance matrices, thereby increasing computational costs and risking numerical instability.
To address this issue, we adapt VCReg to accommodate networks with spatial dimensions. Each vector at a different spatial location is treated as an individual sample when calculating the covariance matrix. Both the variance loss and the covariance loss are then calculated based on this modified covariance matrix.

In terms of practical implementation, a VCReg is usually applied subsequently to each block within the neural network architecture, often succeeding residual connections. This placement allows for seamless incorporation into current network architectures and training paradigms.

Addressing Outliers with Smooth L1 Loss After treating spatial locations as independent samples for covariance computation, the resulting samples are no longer statistically independent. This can lead to outliers in the covariance matrix and unstable gradient updates. To address this, we introduce a smooth L1 penalty into the covariance loss term. Specifically, we replace the traditional squared covariance values $C_{ij}$ in $\ell_{\text{cov}}$ with a smooth L1 function:

$$ $\displaystyle\text{SmoothL1}(x)=\begin{cases}x^{2},&\text{if }|x|\leq\delta\\ 2\delta|x|-\delta^{2},&\text{otherwise}\end{cases}$ (10) $$

By implementing this modification, we ensure that the loss function increases in a more controlled manner with respect to large covariance values. Empirically, this minimizes the impact of outliers, thereby enhancing the stability of the training process.

### 3.3 Fast Implementation

To optimize VCRef speed, we use the fact that VCReg only affects the loss function and not the forward pass. This allows us to focus on directly modifying the backward function for improvements. Specifically, we sidestep the usual process of calculating the VCReg loss and subsequent backpropagation. Instead, we directly adjust the computed gradients, which is feasible since the VCReg loss calculation relies solely on the current representation. Further details of this speed-optimized technique are outlined in Appendix B. Our optimized VCReg implementation exhibits similar latency as batch normalization layers and is more than 5 times faster than the naive VCReg implementation. The results are presented in Table 8.

## 4 Experiments

In this section, we first outline the experimental framework and findings highlighting the effectiveness of our proposed regularization approach, VCReg, within the realm of transfer learning that utilizes supervised pretraining for both images and videos. Subsequently, we extend our experiments to three specialized learning scenarios: 1) class imbalance via long-tail learning, 2) synergizing with self-supervised learning frameworks, and 3) hierarchical classification problems. The objective is to assess the adaptability of VCReg across various data distributions and learning paradigms, thereby evaluating its broader utility in machine learning applications. For details on reproducing our experiments, please consult Appendix C.

### 4.1 Transfer Learning for Images

In this section, we adhere to evaluation protocols established by seminal works such as (Chen et al., ; Kornblith et al., ; Misra & Maaten, ) for our transfer learning experiments.

Initially, we pretrain models using three different architectures: ResNet-50 (He et al., ), ConvNeXt-Tiny (Liu et al., ), and ViT-Base-32 (Dosovitskiy et al., ), on the full ImageNet dataset. We follow the standard PyTorch recipes (Paszke et al., ) for all networks and do not modify any hyperparameters other than those related to VCReg to ensure a fair baseline comparison.
Subsequently, we perform a linear probing evaluation across 9 different benchmark to evaluate the transfer learning performance.

For ResNet-50, we include two other feature diversity regularizer methods for comparison: DeCov (Cogswell et al., ) and WLD-Reg (Laakom et al., ). We conduct experiments solely with ResNet-50 because it is the principal architecture used in the WLD-Reg paper. To ensure a fair comparison, we source hyperparameters from Laakom et al. () for both DeCov and WLD-Reg.

**Table 1: Transfer Learning Performance with ImageNet Supervised Pretraining The table shows performance metrics for different architectures. Each model is pretrained on the full ImageNet dataset and then tested on different downstream datasets using linear probing. Application of VCReg consistently improves performance and beats other feature diversity regularizer. Averages are calculated excluding ImageNet results.**
| Architecture | iNat18 | Places | Food | Cars | Aircraft | Pets | Flowers | DTD | Average |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| ResNet-50 | 42.8% | 50.6% | 69.1% | 43.6% | 54.8% | 91.9% | 77.1% | 68.7% | 62.33% |
| ResNet-50 (DeCov) | 43.1% | 50.4% | 69.0% | 45.7% | 55.5% | 90.6% | 79.2% | 69.1% | 62.83% |
| ResNet-50 (WLD-Reg) | 43.9% | 51.2% | 70.2% | 43.9% | 58.7% | 91.4% | 80.7% | 69.0% | 63.63% |
| ResNet-50 (VCReg) | 45.3% | 51.2% | 71.7% | 54.1% | 70.5% | 92.1% | 88.0% | 70.8% | 67.96% |
| ConvNeXt-T | 51.6% | 53.8% | 78.4% | 62.9% | 74.7% | 93.9% | 91.3% | 72.9% | 72.44% |
| ConvNeXt-T (VCReg) | 52.3% | 54.7% | 79.6% | 64.2% | 76.3% | 94.1% | 92.7% | 73.3% | 73.40% |
| ViT-Base-32 | 39.1% | 47.9% | 70.6% | 51.2% | 63.8% | 90.3% | 84.6% | 66.1% | 64.20% |
| ViT-Base-32 (VCReg) | 40.6% | 48.1% | 70.9% | 52.0% | 65.8% | 91.0% | 86.6% | 66.5% | 65.19% |

The results in Table 1 demonstrate that VCReg significantly enhances performance in transfer learning for images across almost all downstream datasets, achieving the highest performance for 9 out of 10 datasets, and for all three architectures. Clearly, VCReg acts as a versatile plug-in, effectively boosting transfer learning outcomes. Its effectiveness spans ConvNet and Transformer architectures, confirming its wide-ranging applicability.

### 4.2 Transfer Learning for Videos

To extend our evaluation of VCReg’s efficacy, we conduct additional experiments using networks pretrained on video datasets. Specifically, we utilized models pretrained on Kinetics-400 (Kay et al., ) and Kinetics-710 (Li et al., ), subsequently finetuning them for action recognition tasks on HMDB51 (Kuehne et al., ). This set of experiments encompassed models trained with self-supervised learning objectives, including VideoMAE (Tong et al., ) and VideoMAEv2 (Wang et al., ), as well as models trained with conventional supervised learning objectives, such as ViViT (Arnab et al., ).

We follow the finetuning protocols detailed by Tong et al. () and the conventional evaluation method used in the field, where the final performance is measured by the mean classification accuracy across three provided splits (Simonyan & Zisserman, ). To pinpoint the optimal VCReg coefficients, we conducted a grid search based on validation set accuracy. For simplicity, in this setup, VCReg regularization is exclusively applied to the final output of each network during finetuning, just before the classification head.

Table 2 illustrates that incorporating VCReg as a plugin regularizer enhances video classification performance across various methods (VideoMAE, VideoMAE2, and ViViT-B) and backbone architectures (ViT-B and ViT-S). The performance gains are evident in the improvements seen with VCReg across all models in the table. This consistent enhancement across a spectrum of models solidifies VCReg’s status as a practical and versatile regularizer, capable of substantially improving the performance of pretrained networks in transfer learning scenarios.

**Table 2: Transfer Learning Performance with Kinetics-400 and Kinetics-710 pretrained models: The table shows fine-tuning performance of Kinetics pre-trained models on HMDB51. VideoMAE-S, VideoMAE-B, and ViViT-B are pretrained on Kinetics-400 dataset while VideoMAEv2-S and VideoMAEv2-B are pre-trained on Kinetics-710. We apply VCReg only to the networks’ output preceding the classification head. The results show that VCReg can boost the transfer learning classification performance for networks pre-trained on video data.**
| Method | Backbone | HMDB51 |
| --- | --- | --- |
| VideoMAE-S | ViT-S | 79.9% |
| VideoMAE-S (VCReg) | ViT-S | 80.6% |
| VideoMAE-B | ViT-B | 82.2% |
| VideoMAE-B (VCReg) | ViT-B | 83.0% |
| VideoMAEv2-S | ViT-S | 83.6% |
| VideoMAEv2-S (VCReg) | ViT-S | 83.9% |
| VideoMAEv2-B | ViT-B | 86.5% |
| VideoMAEv2-B (VCReg) | ViT-B | 86.9% |
| ViViT-B | ViT-B | 70.9% |
| ViViT-B (VCReg) | ViT-B | 71.6% |

### 4.3 Class Imbalance with Long-Tail Learning

Class imbalance is a pervasive issue in many real-world datasets and poses a considerable challenge to standard neural network training algorithms. We conduct experiments to assess how well VCReg addresses this issue through long-tail learning.
We evaluate VCReg using the CIFAR10-LT and CIFAR100-LT (Krizhevsky et al., ) datasets, both engineered to have an imbalance ratio of 100. These experiments use a ResNet-32 backbone architecture. The per-class sample sizes ranges from 5,000 to 50 for CIFAR10-LT and from 500 to 5 for CIFAR100-LT.

**Table 3: Performance Comparison on Class-Imbalanced Datasets Using VCReg: This table shows the accuracy of standard ResNet-32 with and without VCReg when trained on class-imbalanced CIFAR10-LT and CIFAR100-LT datasets. The VCReg-enhanced models show improved performance, demonstrating the method’s effectiveness in addressing class imbalance.**
| Training Methods | CIFAR10-LT | CIFAR100-LT |
| --- | --- | --- |
| ResNet-32 | 69.6% | 37.4% |
| ResNet-32 (VCReg) | 71.2% | 40.4% |

Table 3 shows that models augmented with VCReg consistently outperform the standard ResNet-32 models on imbalanced datasets. These results are noteworthy because they demonstrate that VCReg effectively enhances the model’s ability to discriminate between classes in imbalanced settings. This establishes VCReg as a valuable tool for real-world applications where class imbalance is often a concern.

### 4.4 Self-Supervised Learning with VCReg

Our subsequent investigation focuses on examining the synergy between VCReg and existing self-supervised learning paradigms.
As mentioned in the previous sections, we apply VCReg not only to the final but also to intermediate representations.
So in all of the following experiments for self-supervised learning with VCReg, we apply the original loss function to the output of the network, and the VCReg loss to all the intermediate representations.

We employ a ResNet-50 architecture, training it for 100 epochs under four different configurations: using either SimCLR loss or VICReg loss, coupled with the ImageNet dataset. For evaluation, we conduct linear probing tests on multiple downstream task datasets, following the protocols prescribed by (Misra & Maaten, ; Zbontar et al., ).

**Table 4: Impact of VCReg on Self-Supervised Learning Methods: This table presents a comparative analysis of ResNet-50 models pretrained with SimCLR and VICReg losses on ImageNet, both with and without the VCReg applied. The models are evaluated using linear probing on various downstream task datasets. The VCReg models consistently outperform the non-VCReg models, showcasing the method’s broad utility in transfer learning for self-supervised learning scenarios. Averages are calculated excluding ImageNet results.**
| Pretraining Methods | ImageNet | iNat18 | Places | Food | Cars | Aircraft | Pets | Flowers | DTD | Average |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| SimCLR | 67.2% | 37.2% | 52.1% | 66.4% | 35.7% | 62.3% | 76.3% | 82.6% | 68.1% | 60.09% |
| SimCLR (VCReg) | 67.1% | 41.3% | 52.3% | 67.7% | 40.6% | 61.9% | 76.6% | 83.6% | 69.0% | 61.63% |
| VICReg | 65.2% | 41.7% | 48.2% | 61.0% | 27.3% | 51.2% | 79.1% | 74.3% | 65.4% | 56.03% |
| VICReg (VCReg) | 66.3% | 41.4% | 49.6% | 61.6% | 29.3% | 54.2% | 79.7% | 74.5% | 66.5% | 57.10% |

As indicated in Table 4, integrating VCReg into self-supervised learning paradigms such as SimCLR and VICReg results in consistent performance improvements for transfer learning. Specifically, the linear probing accuracies are enhanced across nearly all the evaluated datasets. These gains underscore the broad applicability and versatility of VCReg, demonstrating its potential to enhance various machine learning methodologies.

### 4.5 Hierarchical Classification

To evaluate the efficacy of the learned representations across multiple levels of class granularity, we conduct experiments on the CIFAR100 dataset as well as five distinct subsets of ImageNet (Engstrom et al., ). In each dataset, every data sample is tagged with both superclass and subclass labels, denoted as $(x_{i},y^{\mathrm{sup}}_{i},y^{\mathrm{sub}}_{i})$. Note that while samples sharing the same subclass label also share the same superclass label, the reverse does not necessarily hold true.
Initially, the model is trained using only the superclass labels, i.e., the $(x_{i},y^{\mathrm{sup}}_{i})$ pairs. Subsequently, linear probing is employed with the subclass labels $(x_{i},y^{\mathrm{sub}}_{i})$ to assess the quality of abstract features at the superclass level.

**Table 5: Impact of VCReg on Hierarchical Classification in ConvNeXt Models: This table summarizes the classification accuracies obtained with ConvNeXt models, both with and without the VCReg regularization, across multiple datasets featuring hierarchical class structures. The models were initially trained using superclass labels and subsequently probed using subclass labels. VCReg consistently boosts performance in subclass classification tasks.**
|  |  | Subsets of ImageNet |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- |
|  | CIFAR100 | living_9 | mixed_10 | mixed_13 | geirhos_16 | big_12 |
| Superclass Count | 20 | 9 | 10 | 13 | 16 | 12 |
| Subclass Count | 100 | 72 | 60 | 78 | 32 | 240 |
| ConvNeXt | 60.7% | 53.4% | 60.3% | 61.1% | 60.5% | 51.8% |
| ConvNeXt (VCReg) | 72.9% | 62.2% | 67.7% | 66.0% | 70.1% | 61.5% |

Table 5 presents key performance metrics, highlighting the substantial improvements VCReg brings to subclass classification. The improvements are consistent across all datasets, with the CIFAR100 dataset showing the most significant gain—an increase in accuracy from 60.7% to 72.9%.
These results underscore VCReg’s capability to assist neural networks in generating feature representations that are not only discriminative at the superclass level but are also well-suited for subclass distinctions. This attribute is particularly advantageous in real-world applications where class categorizations often exist within a hierarchical framework.

## 5 Exploring the Benefits of VCReg

This section aims to thoroughly unpack the multi-faceted benefits of VCReg in the context of supervised neural network training. Specifically, we discuss its capability to address challenges such as gradient starvation (Pezeshki et al., ), neural collapse (Papyan et al., ), noisy data, and the preservation of information richness during model training (Shwartz-Ziv, ).

### 5.1 Mitigating Gradient Starvation

Figure: Figure 2: Comparative evaluation between training with and without VCReg on a “Two-Moon” Synthetic Dataset. Decision boundaries are averaged over ten distinct runs with random data point sampling and model initialization. A single run’s data points are displayed for visual clarity. The contrast between VCReg and “No regularization” underscores the latter’s limitations in forming intricate decision boundaries, while highlighting VCReg’s effectiveness in generating meaningful ones.
Refer to caption: extracted/5425843/images/final.png

In line with the original study on gradient starvation (Pezeshki et al., ), we observe that most traditional regularization techniques fall short of capturing the vital features for the “two-moon” dataset experiment. To assess the effectiveness of VCReg, we replicate this setting with a three-layer network and apply our method during training. Our visualized results in Figure 2 make it apparent that VCReg has a marked advantage over traditional regularization techniques, particularly in the aspects of separation margins. Thus, it is reasonable to conclude that VCReg can help mitigate gradient starvation. Please check section E for the detailed information about experiments related to the “two-moon” dataset.

### 5.2 Preventing Neural Collapse and Information Compression

To deepen our understanding of VCReg and its training dynamics, we closely examine its learned representations. A recent study (Papyan et al., ) observed a peculiar trend in deep networks trained for classification tasks: the top-layer feature embeddings of training samples from the same class tend to cluster around their respective class means, which are as distant from each other as possible. However, this phenomenon could potentially result in a loss of diversity among the learned features (Papyan et al., ), thus curtailing the network’s capacity to grasp the complexity of the data and leading to suboptimal performance for transfer learning (Li et al., ).

Our neural collapse investigation includes two key metrics:

Class-Distance Normalized Variance (CDNV) For a feature map $f:\mathbb{R}^{d}\to\mathbb{R}^{p}$ and two unlabeled sets of samples $S_{1},S_{2}\subset\mathbb{R}^{d}$, the CDNV is defined as

$$ $\displaystyle V_{f}(S_{1},S_{2})=\frac{\sigma^{2}_{f}(S_{1})+\sigma^{2}_{f}(S_ {2})}{2\|\mu_{f}(S_{1})-\mu_{f}(S_{2})\|^{2}},$ (11) $$

where $\mu_{f}(S)$ and $\sigma^{2}_{f}(S)$ signify the mean and variance of the set $\{f(x)\mid x\in S\}$.
This metric measures the degree of clustering of the features extracted from $S_{1}$ and $S_{2}$, in relation to the distance between their respective features. A value approaching zero indicates perfect clustering.

Nearest Class-Center Classifier (NCC) This classifier is defined as

$$ $\displaystyle\hat{h}(x)=\operatorname*{arg\,min}_{c\in[C]}\|f(x)-\mu_{f}(S_{c})\|$ (12) $$

According to this measure, during training, collapsed feature embeddings in the penultimate layer become separable, and the classifier converges to the “nearest class-center classifier”.

Preventing Information Compression Although effective compression often yields superior representations, overly aggressive compression might cause the loss of crucial information about the target task (Shwartz-Ziv et al., ; Shwartz-Ziv & Alemi, ; Shwartz-Ziv & LeCun, ).
To investigate the compression during the learning, we use the mutual information neural estimation (MINE) (Belghazi et al., ), a method specifically designed to estimate the mutual information between the input and its corresponding embedded representation. This metric effectively gauges the complexity level of the representation, essentially indicating how much information (in terms of number of bits) it encodes.

We evaluate the learned representations of two ConvNeXt models (Liu et al., ), which are trained on ImageNet with supervised loss. One model is trained with VCReg, while the other is trained without VCReg.
As demonstrated in Table 6, both types of collapse, measured by CDNV and NCC, and the mutual information estimation reveal that VCReg representations have significantly more diverse features (lower neural collapse) and contain more information compared to regular training. This suggests that not only does VCReg achieve superior results, but it also yields representations which contain more information.

In summary, the VCReg method mitigates the neural collapse phenomenon and prevents excessive information compression, two crucial factors that often limit the effectiveness of deep learning models in transfer learning tasks. Our findings highlight the potential of VCReg as a valuable addition to the deep learning toolbox, significantly increasing the generalizability of learned representations.

**Table 6: VCReg learns richer representation and prevents neural collapse and information compression Metrics include Class-Distance Normalized Variance (CDNV), Nearest Class-Center Classifier (NCC), and Mutual Information (MI). Higher values in each metric for the VCReg model indicate reduced neural collapse and richer feature representations.**
| Network | CDNV | NCC | MI |
| --- | --- | --- | --- |
| ConvNeXt | 0.28 | 0.99 | 2.8 |
| ConvNeXt (VCReg) | 0.56 | 0.81 | 4.6 |

### 5.3 Providing Robustness to Noise

In real-world scenarios, encountering noise is a common challenge, making robustness against noise a crucial feature for any effective transfer learning algorithm. Recognizing the ubiquity of noise in practical applications, we aim to evaluate the capability of VCReg to bolster transfer learning performance in noisy environments.

For this purpose, we utilize video networks initially pretrained on Kinetics-400 and Kinetics-710, as mentioned in section 4.2. We then finetune these networks on the HMDB51 dataset, which is deliberately subjected to varying levels of Gaussian noise. The findings in Table 3 reveal a clear advantage: incorporating VCReg notably improves the resilience of VideoMAE-S and VideoMAEv2-S models to noisy data, a robustness not observed in models without VCReg. This trend of increased durability against noise is consistently seen in larger models, such as VideoMAE-B and VideoMAEv2-B. For a more granular analysis, Appendix D provides a thorough description of the results, complete with detailed figures.

This investigation highlights the necessity of achieving optimal performance in non-ideal settings. It emphasizes the critical need for maintaining robustness and reliability under the challenges commonly encountered in real-world settings, such as noise. This dual capability significantly boosts a model’s practical value and reliability.

Figure: Figure 3: Impact of VCReg amidst noisy data: This figure shows the top-1 accuracy of VideoMAE-S and VideoMAEv2-S when fine-tuned for action recognition using HMDB51 corrupted with synthetic noise. We corrupt the data with Gaussian noise with standard deviation $\sigma\in\{1,1.5,2\}$. Models with VCReg outperform their non-regularized counterparts in this setting.
Refer to caption: x1.png

## 6 Conclusion

In this work, we addressed prevalent challenges in supervised pretraining for transfer learning by introducing Variance-Covariance Regularization (VCReg). Building on the regularization technique of the self-supervised VICReg method, VCReg is designed to cultivate robust and generalizable features. Unlike conventional methods that attach regularization only to the final layer, we efficiently incorporate VCReg across intermediate layers to optimize its efficacy.

Our key contributions are threefold:

- 1.
We present a computationally efficient VCReg implementation that can be adapted to various network architectures.
- 2.
We provide empirical evidence through comprehensive evaluations on multiple benchmarks, demonstrating that using VCReg yields significant improvements in transfer learning performance across various network architectures and different learning paradigms, including video and image classification, long tail learning, and hierarchical classification.
- 3.
Our in-depth analyses confirm VCReg’s effectiveness in overcoming typical transfer learning hurdles such as neural collapse, gradient starvation, and noise.

To conclude, VCReg stands out as a potent and adaptable regularization technique that elevates the quality and applicability of learned representations. It enhances both the performance and reliability of models in transfer learning settings and paves the way for further research to achieve highly optimized and generalizable machine learning models.

## Acknowledgements

This material is partially based upon work supported by the National Science Foundation under NSF Award 1922658.

## Appendix A Experimental Investigation on Effective Application of VCReg to Standard Networks

To determine the optimal manner of integrating the VCReg into a standard network, we conducted several experiments utilizing the ConvNeXt-Atto architecture, trained on ImageNet following the torchvision (Paszke et al., ) training recipe. To reduce the training time, we limited the network training to 90 epochs with a batch size of 4096.
The complete configuration comprised 90 epochs, a batch size of 4096, two learning rate of $\{0.016,0.008\}$ with a 5 epochs linear warmup followed by a cosine annealing decay. The weight decay was set at $0.05$ and the norm layers were excluded from the weight decay. we experimented with $\alpha\in\{1.28,0.64,0.32,0.16\}$ and $\beta\in\{0.16,0.08,0.04,0.02,0.01\}$.

We experimented with incorporating the VCReg layers in four different locations:

- 1.
Applying the VCReg exclusively to the second last representation (the input of the classification layer).
- 2.
Applying VCReg to the output of each ConvNeXt block.
- 3.
Applying VCReg to the output of each downsample layer.
- 4.
Applying VCReg to the output of both, each ConvNeXt block and each downsample layer.

The VCReg layer was implemented as detailed in 1, with the addition of a mean removal layer along the batch preceding the VCReg layer to ensure that the VCReg input exhibited a zero mean.

**Table 7: Transfer Learning Experiments with Different VCReg Configurations**
| Architecture | Food | Cars | Aircraft | Pets | Flowers | DTD |
| --- | --- | --- | --- | --- | --- | --- |
| ConvNeXt-Atto (VCReg1) | 63.2% | 39.6% | 55.9% | 89.1% | 85.3% | 65.1% |
| ConvNeXt-Atto (VCReg2) | 66.8% | 48.1% | 60.4% | 91.1% | 86.4% | 66.4% |
| ConvNeXt-Atto (VCReg3) | 64.0% | 40.9% | 56.5% | 89.4% | 85.9% | 65.1% |
| ConvNeXt-Atto (VCReg4) | 66.7% | 48.3% | 59.6% | 90.6% | 85.6% | 66.1% |

The results in Table 7 indicate superior performance when the VCReg layer is applied to the output of each block (second setup)
or applied to the output of blocks and downsample layers (fourth setup) compared to the other setups. Considering architectures like ViT lack downsample layers, for consistency across different architectures, we decided to use the second configuration for further experiments.

## Appendix B The Fast Implementation of the VCReg

The VCRegeg does not affect the forward pass in any way, allowing us to substantially speed up the implementation by modifying the backward function directly. Instead of computing the VCReg loss and backpropagating it, we can directly alter the calculated gradient. This is possible since the VCReg loss calculation only requires the current representation. The specifics of this speed-optimized implementation are outlined in Algorithm 1.

Figure: Algorithm 1 PyTorch-Style Pseudocode for Fast VCReg Implementation

We quantify the computational overhead by measuring the average time required for one NVIDIA A100 GPU to execute both the forward and backward passes on the entire network for a batch size of 128 using the ImageNet dataset. These results are summarized in Table 8. For the sake of comparison, we also include the latencies associated with adding Batch Normalization (BN) layers, revealing that our optimized VCReg implementation exhibits similar latencies to BN layers and is almost 5 times faster than the naive implementation.

**Table 8: Average Time Required for One Forward and Backward Pass with Various Layers Inserted Comparison of computational latencies across different configurations of ViT and ConvNeXt networks. The table demonstrates the efficacy of the optimized VCReg layer in terms of computational time, compared to both naive VCReg and Batch Normalization (BN) layers.**
| Network | Number of Inserted Layers | Identity | VCReg (Naive) | VCReg (Fast) | BN |
| --- | --- | --- | --- | --- | --- |
| ViT-Base-32 | 12 | 0.223s | 1.427s | 0.245s | 0.247s |
| ConvNeXt-T | 18 | 0.442s | 2.951s | 0.471s | 0.468s |

## Appendix C Implementation Details

### C.1 Transfer Learning Experiments with ImageNet Pretraining

In conducting the transfer learning experiments, we adhered primarily to the training recipe specified by PyTorch (Paszke et al., ) for each respective architecture during the supervised pretraining phase. We abstained from pretraining any of the baseline models, instead opting to directly download the weights from PyTorch’s own repository.
The only modifications applied were to the parameters associated with VCReg loss, and we experimented with $\alpha\in\{1.28,0.64,0.32,0.16\}$ and $\beta\in\{0.16,0.08,0.04,0.02,0.01\}$.

For iNaturalist 18 (Van Horn et al., ) and Place205 (Zhou et al., ), we relied on the experimental settings detailed in (Zbontar et al., ) for the linear probe evaluation.

Regarding Food-101 (Bossard et al., ), Stanford Cars (Krause et al., ), FGVC Aircraft (Maji et al., ), Oxford-IIIT Pets (Parkhi et al., ), Oxford 102 Flowers (Nilsback & Zisserman, ), and the Describable Textures Dataset (DTD) (Cimpoi et al., ), we complied with the evaluation protocol provided by (Chen et al., ; Kornblith et al., ). An $L2$-regularized multinomial logistic regression classifier was trained on features extracted from the frozen pretrained network. Optimization of the softmax cross-entropy objective was conducted using L-BFGS, without the application of data augmentation. All images were resized to 224 pixels along the shorter side through bicubic resampling, followed by a 224 x 224 center crop. The $L2$-regularization parameter was selected from a range of 45 logarithmically spaced values between $0.00001$ and $100000$.

All experiments were run three times, with the average results presented in Table 1.

### C.2 Transfer Learning Experiments with Kinetics pre-trained Models

In conducting experiments with video-pretrained models, we utilize the publicly available code bases and model checkpoints provided for VideoMAE and VideoMAEv2 ([https://github.com/MCG-NJU/VideoMAE](https://github.com/MCG-NJU/VideoMAE) and [https://github.com/OpenGVLab/VideoMAEv2](https://github.com/OpenGVLab/VideoMAEv2)). For both VideoMAE and VideoMAEv2 we use ViT-Small and ViT-Base checkpoints. VideoMAE models are pre-trained on Kinetics-400 while VideoMAEv2 on Kinetics-710. We use the pre-trained checkpoint for ViViT-B (ViT-Base backbone) pre-trained on Kinetics-400 from HuggingFace. For evaluation, we adopt the inference protocol of 10 clips $\times$ 3 crops. For VCReg hyperparameters experiments with values for $\alpha\in{1,3,5}$ and $\beta\in\{0.1,0..3,0.5\}$. For the rest of the finetuning hyperparameters as well as the data pre-processing and evaluation protocol, we use the configuration for HMDB51 available in VideoMAE (Tong et al., ) and its corresponding code base (linked above).

### C.3 Subclass Linear Probing Result with Network Pretrained on Superclass Label

For our subclass linear probing experiments, we employed a ConvNeXt-Atto network. Each model was pretrained for 200 epochs using the superclasses, adhering to the same procedure detailed in the Appendix A. Subsequent to this pretraining phase, we initiated a linear probing process using the subclass labels. This linear classifier was trained for 100 epochs, using a base learning rate of $0.016$ in conjunction with a cosine learning rate schedule. The optimizer used was AdamW, which worked to minimize cross-entropy loss with a weight decay set at $0.05$. We processed our training data in batches of 256.

### C.4 Long-Tail Learning Result

For our long-tail learning experiments, we use ResNet-32 as a backbone for experiments on the CIFAR10-LT and CIFAR100-LT datasets.
We trained 100 epochs with batch size 256, Adam optimizer with two learning rate of $\{0.016,0.008\}$ with a 10-epoch linear warm-up followed by a cosine annealing decay. The weight decay was set at $0.05$ and the norm layers were excluded from the weight decay. we experimented with $\alpha\in\{1.28,0.64,0.32,0.16\}$ and $\beta\in\{0.16,0.08,0.04,0.02,0.01\}$.

### C.5 VCReg with Self-Supervised Learning Methods

We train a ResNet-50 model in four different setups, using either the SimCLR loss or the VICReg loss with the ImageNet dataset.
The application of the VCReg is the same as described in Appendix A.

We closely follow the original setting in (Chen et al., ) for SimCLR pretraining and (Bardes et al., ) for VICReg pretraining.

Augmentation For both methods, we use the same augmentation methods. Each augmented view is generated from a random set of augmentations of the same input image.
We apply a series of standard augmentations for each view, including random cropping, resizing to 224x224, random horizontal flipping, random color-jittering, randomly converting to grayscale, and a random Gaussian blur. These augmentations are applied symmetrically on two branches (Geiping et al., )

Architecture
For SimCLR, the encoder is a ResNet-50 network without the final classification layer followed by a projector. The projector is a two-layer MLP with input dimension 2048, hidden dimension 2048, and output dimension 256. The projector has ReLU between the two layers and batch normalization after every layer. This 256-dimensional embedding is fed to the infoNCE loss.

For VICReg, the online encoder is a ResNet-50 network without the final classification layer. The online projector is a two-layer MLP with input dimension 2048, hidden dimension 8192, and output dimension 8192.
The projector has ReLU between the two layers and batch normalization after every layer. This 8192-dimensional embedding is fed to the infoNCE loss.

For VCReg, we just applied the VCReg layers to the ResNet-50 network as described in the Appendix A.

Optimization
We follow the training protocol in (Zbontar et al., ).
For SimCLR experiments, we used a LARS optimizer and a base learning rate 0.3 with cosine learning rate decay schedule. We pretrain the model for 100 epochs with 5 epochs warm-up with batch size 4096.

For VICReg, we use a LARS optimizer and a base learning rate 0.2 using cosine learning rate decay schedule. We pretrain the model for 100 epochs with 5 epochs warm-up with batch size 4096.

Evaluation We follow the standard evaluation protocol as prescribed by (Misra & Maaten, ; Zbontar et al., ), performing linear probing evaluations, on iNaturalist 18 (Van Horn et al., ) and Place205 (Zhou et al., ) datasets.

## Appendix D Robustness to noise

This section provides additional results on measuring VCReg’s ability to enhance transfer learning performance in the presence of noise. In these experiments we start with VideoMAE-B and VideoMAEv2-B networks (from section 4.2) pre-trained on Kinetics-400 and Kinetics-710, respectively, then fine-tune them on HMDB51 corrupted with varying levels of Gaussian noise. Figure 4 shows that VCReg models outperform their non-regularized counterparts in this setting.

Figure: Figure 4: Impact of VCReg amidst noisy data: This figure shows the top-1 accuracy of VideoMAE-B and VideoMAEv2-B when fine-tuned for action recognition using HMDB51 with synthetic noise. We corrupt the data with Gaussian noise with standard deviation $\sigma\in\{1,1.5,2\}$. Models with VCReg outperform their non-regularized counterparts in this setting.

## Appendix E Two-Moon Dataset

Figure: Figure 5: The effect of conventional regularization methods and the VCReg on a simple task of two-moon classification. Shown decision boundaries are the average over 10 runs in which data points and the model initialization parameters are sampled randomly. Here, only the data points of one particular seed are plotted for visual clarity. It can be seen that conventional regularizations of deep learning seem not to help with learning a curved decision boundary.
Refer to caption: extracted/5425843/images/alltwomoon.png

In alignment with the original gradient starvation study (Pezeshki et al., ), we notice that most regular routine regularization techniques do not sufficiently capture the necessary features for the “two-moon” dataset experiment. To evaluate our approach, we mirrored this setting and applied the VCReg during the training.

The synthetic “two-moon” dataset comprises two classes of points, each forming a moon-like shape. The gradient starvation study highlighted an issue where if the gap between the two moons is wide enough for a straight line to separate the two classes, the network stops learning additional features and focuses solely on a single feature. We duplicated this situation using a three-layer network and applied all the initially tested methods in the original study. The resulting decision boundary after training with the “two-moon” dataset is visualized in Figure 5.

From the visualization, it becomes apparent that not only does VCReg outperform other conventional regularization techniques in separation margins, but also it shows superior performance compared to spectral decoupling, a method specifically designed for this task. VCReg is effective in maximizing the variance while minimizing the covariance in the feature space, an achievement that is not obtained by other techniques such as L2, dropout (Hinton et al., ), and batch normalization (Ioffe & Szegedy, ). Consequently, these other techniques yield features that are less discriminative and informative.

## Appendix F Miscellaneous

### F.1 Compute Resources

The majority of our experiments were run using AMD MI50 GPUs.
The longest pretraining for ConvNeXt-Tiny takes about 48 hours on 2 nodes, where each node has 8 MI50 GPUs attached.
We estimate that the total amount of compute resources used for all the experiments can be roughly approximated by $60\text{ (days)}\times 24\text{ (hours per day)}\times 8\text{ (nodes)}\times 8
\text{ (GPUs per nodes)}=92,160\text{ (GPU hours)}$.

We are aware of potential environmental impact of consuming a lot of compute resources needed for this work, such as atmospheric $\text{CO}_{2}$ emissions due to the electricity used by the servers. However, we also believe that advancements in representation learning and transfer learning can potentially help mitigate these effects by reducing the need for data and compute resources in the future.

### F.2 Limitations

Due to a lack of compute resources, we were unable to conduct a large number of experiments with the goal of tuning hyperparameters and searching for the best configurations.
Therefore, the majority of hyperparameters and network configurations used in this work are the same as provided by PyTorch (Paszke et al., ).
The only hyperparameters that were tuned were $\alpha$ and $\beta$, the coefficients for VCR.
All the other hyperparameters may not be optimal.

In addition, all models were pretrained on the ImageNet (Deng et al., ) and (Krizhevsky et al., ) dataset, so their performances might differ if pretrained with other datasets containing different data distributions or different types of images (e.g., x-rays).
We encourage further exploration in this direction for current and future self-supervised learning frameworks.

## Appendix G Impact Statement

This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.

## Peer Reviews

confidence: 4
contribution: 2
initial_rating: 5
presentation: 3
questions: Please refer weakness section and I am waiting to get response on it.
soundness: 2
strengths: 1. VCReg repurpose the variance and covariance components of VICReg for supervised settings, intent to make transfer learning without dependance on invariance
2. Improvements are shown when using supervised pretraining on ImageNet and then transferring to other datasets so empirical evidence of presented task/scenarios is largely convincing.
3. presentation of this paper is well-structured with clarity.
summary: The paper introduces Variance-Covariance Regularization (VCReg), adaptation of VICReg designed for supervised learning. By focusing on high-variance, low-covariance representations, VCReg targets improved feature transferability and robustness, specifically addressing issues like gradient starvation and neural collapse which are prevalent in supervised pretraining for transfer learning. The authors apply VCReg not only to final representations but also to intermediate layers, achieving performance improvements across image and video-based transfer learning tasks, class-imbalanced datasets, and hierarchical classification.
weaknesses: 1. VICReg's regularization easily integrates with other SSL methods like SimCLR since it operates on dimensions across the batch without interfering with contrastive losses so it should ideally can be integration with SSL and supervised losses also. So Why I see VCReg as an special case of VICReg applied on supervised setting rather than novel approach.
2. Invariance component is removed to streamline VCReg for supervised tasks however this could reduce robustness to data variations. Without the invariance regularization, I am curious about VCReg generalization capabilities in scenarios like distribution shift. To counter the question, empirical evidence should justify that.
3. Application of VCReg across intermediate layers, along with its specialized backward-pass gradient adjustments, adds substantial computational overhead and results in Appendix A does not justify it entirely.  
4. Results on combining the VCReg with SSL methods in Table 4 is interesting. Focusing on ImageNet results- How to justify that with SimCLR + VCReg doesn't help however It improves performance when VCReg added with VICReg (VICReg having invariance component). Need to understand reason.
5. Could the authors elaborate on how VCReg specifically addresses generalization in context of connection between gradient starvation and transfer learning?

---

confidence: 4
contribution: 3
initial_rating: 5
presentation: 3
questions: Please refer to the weakness.
soundness: 2
strengths: 1.The authors perform extensive experiments across multiple tasks, showcasing the effectiveness of VCReg in diverse settings, including transfer learning for images and videos, long-tailed learning, self-supervised learning, and hierarchical classification.
2.The benefits of VCReg are explored thoroughly in Section 5 empirically, which is both interesting and convincing.
summary: This paper proposes VarianceCovariance Regularization (VCReg), a method designed to encourage learning representations with high variance and low covariance. Rather than applying VCReg only to the network’s final representation, the authors integrate it into intermediate layers, leveraging an efficient implementation to ensure minimal computational overhead and enable easy integration into existing workflows. The paper presents extensive experiments on multiple tasks, demonstrating VCReg’s effectiveness across various network architectures.
weaknesses: 1.The paper lacks a theoretical explanation for how VCReg improves generalization. For instance, can the authors provide a theoretical analysis of VCReg’s impact on the decision boundary or expected risk? A theoretical grounding would clarify VCReg’s influence on generalization and strengthen the methodology. Relevant references for further grounding could include:
- Empirical Bernstein Bounds and Sample Variance Penalization by Maurer et al., 2009
- Variance-based Regularization with Convex Objectives by Namkoong et al., 2017
- Feature Variance Regularization: A Simple Way to Improve the Generalizability of Neural Networks by Ranran et al., 2020
- PAC-Bayes-Empirical-Bernstein Inequality by Tolstikhin et al., 2013

2. The description in Section 3.2 could be clearer, particularly regarding spatial dimensions and covariance calculations. Is covariance calculated for each feature dimension, and if so, why do spatial dimensions influence the results? Providing explicit equations would enhance clarity and improve the understanding.

2. Minor typos: in line 076, suppresses -> surpasses.

---

confidence: 4
contribution: 2
initial_rating: 3
presentation: 2
questions: See weaknesses.
soundness: 2
strengths: 1. Writing and motivation are generally clear.
2. The proposed VCReg makes sense as whitening technique has been proved to enhance feature diversity in SL.
3. Extensive experiments are conducted to verify VCReg’s performance in transfer Learning.
summary: This paper proposes to adapt regularization components (variance, covariance) from the VICReg to supervised learning contexts, which aims to encourages the network to learn high-variance, low covariance representations, promoting learning more diverse features. The authors focus on an efficient implementation of VCReg and conduct experiments to show its effectiveness in transfer learning.
weaknesses: 1. **Lack of novelty and practicality.**
- Firstly, most SSL methods, such as VICReg, DINO and so on, have been verified to significantly outperform supervised learning on downstream transfer learning tasks. No matter what components are added, SL consistently focuses on matching label information. Therefore, in contemporary machine learning, SL is more commonly used for fine-tuning specific tasks rather than serving as a pretraining method for obtaining general-purpose representations. If the proposed SL+VCReg can outperform current self-supervised learning methods on downstream tasks, in my view, that would truly represent both novelty and practicality. I suggest that the authors conduct direct comparisons between their SL+VCReg method and leading SSL approaches on downstream transfer learning tasks. This would provide a clearer picture of where the proposed method stands in relation to current best practices.

- Secondly, the author proposes adding regularization components to multiple intermediate layers of the neural network. Although the runtime of each component is similar to that of batch normalization (BN), it still introduces additional computational overhead. Moreover, based on the experimental setup, selecting the coefficients $\alpha$ and $\beta$ for the two regularization components across different network architectures is challenging and requires significant cost. However, despite these additional burdens, the performance improvement is not particularly remarkable. These issues make it difficult for this method to have strong prospects in practical applications. The authors should discuss potential strategies for efficiently selecting the $\alpha$ and $\beta$ coefficients, such as automated hyperparameter optimization techniques. This would help readers better understand the practical implications of implementing VCReg.

2. **Some aspects of the method's setup are unclear and lack intuitive analysis.** Specifically, how VCReg uses spatial dimensions in its calculations is not well explained—I suggest that the authors provide explicit formulas. Additionally, it is unclear how the method determines which intermediate layers to add the regularization components. The lack of clarification on these critical settings raises concerns about the reproducibility of the method. The authors should provide a step-by-step algorithm or pseudocode for how VCReg is applied, including details on handling spatial dimensions and selecting intermediate layers. This would significantly enhance the reproducibility of the method and clarify its implementation.

3. **Some of the experimental results raise concerns.** CDNV measures the degree of clustering of the features, but the authors applied it to "two unlabeled sets of samples." It is unclear how the authors obtained and processed this data, and the significance of using unlabeled data in a supervised learning context is questionable. In Table 6, the results show that ConvNeXt (VCReg) demonstrates significantly weaker clustering ability compared to the standard ConvNeXt, which does not support the claim that this method prevents neural collapse. Furthermore, in Table 4, the reproduced performance of VICReg is even worse than SimCLR, which contradicts the prior knowledge of self-supervised learning. As a result, I find it difficult to be convinced by the experimental outcomes presented in the paper.

Meta-review: Inspired by self-supervised learning, this work introduces Variance-Covariance Regularization (VCReg) from VICReg to supervised learning. Reviewers were concerned about novelty, computational overhead, and theoretical analysis. After rebuttal, one reviewer decreased the score. AC agrees that the work needs further refinement and authors can improve the work according to the comments from all reviewers.

Decision: Reject