Abstract: Residual connections significantly boost the performance of deep neural networks. However, there are few theoretical results that address the influence of residuals on the hypothesis complexity and the generalization ability of deep neural networks. This paper studies the influence of residual connections on the hypothesis complexity of the neural network in terms of the covering number of its hypothesis space. We prove that the upper bound of the covering number is the same as chain-like neural networks, if the total numbers of the weight matrices and nonlinearities are fixed, no matter whether they are in the residuals or not. This result demonstrates that residual connections may not increase the hypothesis complexity of the neural network compared with the chain-like counterpart. Based on the upper bound of the covering number, we then obtain an $\mathcal O(1 / \sqrt{N})$ margin-based multi-class generalization bound for ResNet, as an exemplary case of any deep neural network with residual connections. Generalization guarantees for similar state-of-the-art neural network architectures, such as DenseNet and ResNeXt, are straight-forward. From our generalization bound, a practical implementation is summarized: to approach a good generalization ability, we need to use regularization terms to control the magnitude of the norms of weight matrices not to increase too much, which justifies the standard technique of weight decay.
The paper demonstrates that introducing residual connections does not inherently increase hypothesis complexity when weight matrices and activations remain fixed.
It establishes a margin-based multiclass generalization bound of O(1/√N) for ResNet, extending theoretical guarantees to architectures like DenseNet and ResNeXt.
The study reveals a negative correlation between generalization and the product of weight norms, justifying the widespread use of weight decay as regularization.
Understanding the Generalization Capabilities of ResNet Through Theoretical Insights
The paper "Why ResNet Works? Residuals Generalize" by Fengxiang He, Tongliang Liu, and Dacheng Tao offers a much-needed theoretical exploration of the generalization capabilities of neural networks with residual connections, specifically focusing on ResNet. Despite the empirical success of residual networks, a clear theoretical understanding of their functioning, particularly their generalization ability, remained elusive until this research.
The primary focus of this paper is to theoretically examine how residual connections affect the hypothesis complexity and generalization ability of deep neural networks. The research assesses hypothesis complexity through the lens of the covering number of its hypothesis space, leading to a significant theoretical assertion: the introduction of residual connections doesn't inherently increase hypothesis complexity, provided the total number of weight matrices and nonlinearities remain unchanged. This finding challenges the intuitive expectation that residual connections, by introducing non-chain-like structures and potential loops in the networks, would contribute to increased complexity and, consequently, reduce generalization effectiveness. Instead, the authors prove that the upper bound of the covering number for residual networks aligns with that of chain-like neural networks under these conditions.
Building upon this foundation, the research introduces a margin-based multiclass generalization bound for ResNet, among other neural networks utilizing residual connections. The bounding approach, which takes the form O(1/N) where N is the training sample size, provides not only theoretical guarantees for ResNet but can also be generalized to architectures like DenseNet and ResNeXt. The results bolster the claim that residual connections do not compromise generalization, supporting their broader adoption in diverse, state-of-the-art neural network architectures.
An integral part of this analysis is the relationship between generalization ability and the product of the norms of all weight matrices, establishing a negative correlation. This insight leads to a practical recommendation that resonates with standard training techniques: the regulation of these norms—justifying the canonical use of weight decay as a regularization method in training deep neural networks.
This work carries substantial implications for both the theoretical understanding and practical deployment of advanced neural network architectures. By clarifying that residual connections do not detrimentally increase hypothesis complexity or affect generalization negatively, the paper encourages the expanded application of residual networks across numerous domains. Furthermore, the evidence supporting the effectiveness of weight decay enhances the methodological framework utilized to improve generalization capabilities in practical settings.
Looking ahead, this research opens avenues for further inquiry into the mechanisms of generalization in deep neural networks, particularly focusing on the potential influence of localized hypothesis space exploration driven by optimization methods like stochastic gradient descent. The authors suggest that incorporating localization properties into theoretical models might lead to even tighter bounds on generalization error, which remains a fertile ground for future work.
Overall, this paper bridges critical gaps in the theoretical understanding of Residual Networks, presenting a rigorous analysis that is both academically illuminating and practically applicable in the field of machine learning.