Penalizing Gradient Norm for Efficiently Improving Generalization in Deep Learning (2202.03599v3)

Abstract: How to train deep neural networks (DNNs) to generalize well is a central concern in deep learning, especially for severely overparameterized networks nowadays. In this paper, we propose an effective method to improve the model generalization by additionally penalizing the gradient norm of loss function during optimization. We demonstrate that confining the gradient norm of loss function could help lead the optimizers towards finding flat minima. We leverage the first-order approximation to efficiently implement the corresponding gradient to fit well in the gradient descent framework. In our experiments, we confirm that when using our methods, generalization performance of various models could be improved on different datasets. Also, we show that the recent sharpness-aware minimization method (Foret et al., 2021) is a special, but not the best, case of our method, where the best case of our method could give new state-of-art performance on these tasks. Code is available at {https://github.com/zhaoyang-0204/gnp}.

• The paper introduces a gradient norm penalty term that guides optimization toward flat minima for enhanced generalization.
• It employs a first-order approximation, avoiding costly Hessian computations while effectively regulating the loss landscape.
• Experiments show up to 70% improvement, notably reducing WideResNet’s CIFAR-10 error from 2.78% to 2.52%.

Penalizing Gradient Norm to Enhance Generalization in Deep Learning

The paper presents a novel approach for enhancing the generalization performance of deep neural networks (DNNs) by penalizing the gradient norm of the loss function during optimization. This technique is particularly relevant for highly overparameterized networks, where the density of minima with varying generalization capabilities necessitates effective strategies for guiding optimization towards flat minima associated with good generalization.

Methodology

The core idea is to introduce an additional penalty term into the optimization objective, specifically targeting the gradient norm of the loss function. The rationale is that constraining the gradient norm can lead optimizers to identify flat minima, which have been empirically shown to enhance model generalization. The penalty is incorporated by modifying the loss function as follows:

$$L(\theta) = L_{\text{empirical}}(\theta) + \lambda \cdot || \nabla_\theta L_{\text{empirical}}(\theta) ||_p$$

where $L_{\text{empirical}}$ is the standard empirical risk, $\lambda$ is the penalty coefficient, and $|| \cdot ||_p$ denotes the $L^p$-norm.

To avoid the computational impracticalities of Hessian calculations, the authors propose a first-order approximation that eschews second-order derivatives. This approximation is akin to the Taylor series expansion, effectively enabling the term’s computation with only first-order derivatives.

The method bears resemblance to Sharpness-Aware Minimization (SAM), positing that SAM can be seen as a specific instance of their proposed scheme, where certain hyperparameters are fixed. However, the authors argue that their generalized approach can achieve superior generalization performance by allowing more flexibility in parameter tuning.

Experimental Validation

The authors conduct extensive experiments across various model architectures (including VGG, ResNet, and ViT) and datasets (Cifar-10, Cifar-100, and ImageNet) to assess the efficacy of their proposed scheme relative to both standard and SAM-based training regimes. The experiments demonstrate consistent improvements in test error rates when utilizing the gradient norm penalty scheme. Notably, the gains in generalization are reported to be as high as 70% over SAM's improvements under specific conditions.

For instance, on the Cifar-10 dataset with WideResNet architectures, their approach significantly outperformed SAM, reducing the testing error rate from 2.78% (SAM) to 2.52%. These results underscore the potential of gradient norm penalization to serve as a robust regularization mechanism for guiding large-scale networks towards better minima during training.

Implications and Future Directions

The approach has critical implications for optimizing deep learning models, particularly as model sizes continue to grow. By explicitly controlling the gradient norm, this method provides an additional tool for researchers to enhance the robustness and generalization of DNNs without the need for computationally demanding second-order methods.

Future research could explore the theoretical underpinnings of this approach, particularly its connections to Lipschitz continuity and the geometric properties of high-dimensional optimization landscapes. Additionally, extending this method to other domains within AI, such as reinforcement learning or generative modeling, could further elucidate its universal applicability.

In conclusion, the paper presents a compelling strategy for improving neural network training processes, contributing valuable insights into the ongoing discourse on model generalization and optimization in deep learning.