Flat Minima

Understanding the loss-landscape geometry near a minimum is key to explaining the implicit bias of gradient-based methods in non-convex optimization problems such as deep neural network training and deep matrix factorization. A central quantity for characterizing this geometry is the maximum eigenvalue of the Hessian of the loss. Currently, its precise role has been obfuscated because no exact expressions for this sharpness measure were known in general settings. In this paper, we present the first exact expression for the maximum eigenvalue of the Hessian of the squared-error loss at any minimizer in deep matrix factorization problems, resolving an open question posed by Mulayoff and Michaeli [2020]. This expression reveals a fundamental property of the loss landscape in deep matrix factorization: It is sufficient, but not necessary, for flatness that, across all layers, the product of the spectral norms of the left and right intermediate factors equals a constant determined by the depth and the spectral norm of the target matrix. Most notably, in both depth-2 matrix factorization and deep overparameterized scalar factorization, we show that this condition is both necessary and sufficient for flatness, which implies that flat minima are spectral-norm balanced and not necessarily balanced in depth-2 matrix factorization. To complement our theory, we provide empirical verification, relying on our exact expression, of an escape phenomenon during gradient-based training near a minimizer of a deep matrix factorization problem.

Adversarial training has been widely studied in recent years due to its role in improving model robustness against adversarial attacks. This paper focuses on comparing different distributed adversarial training algorithms--including centralized and decentralized strategies--within multi-agent learning environments. Previous studies have highlighted the importance of model flatness in determining robustness. To this end, we develop a general theoretical framework to study the escaping efficiency of these algorithms from local minima, which is closely related to the flatness of the resulting models. We show that when the perturbation bound is sufficiently small (i.e., when the attack strength is relatively mild) and a large batch size is used, decentralized adversarial training algorithms--including consensus and diffusion--are guaranteed to escape faster from local minima than the centralized strategy, thereby favoring flatter minima. However, as the perturbation bound increases, this trend may no longer hold. In the simulation results, we illustrate our theoretical findings and systematically compare the performance of models obtained through decentralized and centralized adversarial training algorithms. The results highlight the potential of decentralized strategies to enhance the robustness of models in distributed settings.

Sharpness-Aware Minimization (SAM) enhances generalization by reducing a Max-Sharpness (MaxS). Despite the practical success, we empirically found that the MAxS behind SAM's generalization enhancements face the "Flatness Indicator Problem" (FIP), where SAM only considers the flatness in the direction of gradient ascent, resulting in a next minimization region that is not sufficiently flat. A better Flatness Indicator (FI) would bring a better generalization of neural networks. Because SAM is a greedy search method in nature. In this paper, we propose to utilize the difference between the training loss and the minimum loss over the neighborhood surrounding the current weight, which we denote as Min-Sharpness (MinS). By merging MaxS and MinS, we created a better FI that indicates a flatter direction during the optimization. Specially, we combine this FI with SAM into the proposed Bilateral SAM (BSAM) which finds a more flatter minimum than that of SAM. The theoretical analysis proves that BSAM converges to local minima. Extensive experiments demonstrate that BSAM offers superior generalization performance and robustness compared to vanilla SAM across various tasks, i.e., classification, transfer learning, human pose estimation, and network quantization.

The sharpness-aware minimization (SAM) algorithm and its variants, including gap guided SAM (GSAM), have been successful at improving the generalization capability of deep neural network models by finding flat local minima of the empirical loss in training. Meanwhile, it has been shown theoretically and practically that increasing the batch size or decaying the learning rate avoids sharp local minima of the empirical loss. In this paper, we consider the GSAM algorithm with increasing batch sizes or decaying learning rates, such as cosine annealing or linear learning rate, and theoretically show its convergence. Moreover, we numerically compare SAM (GSAM) with and without an increasing batch size and conclude that using an increasing batch size { achieves a lower worst-case $\ell_\infty$ adaptive sharpness} than compared with using a constant batch size and learning rate.