Despite the non-convex nature of their loss functions, deep neural networks are known to generalize well when optimized with stochastic gradient descent (SGD). Recent work conjectures that SGD with proper configuration is able to find wide and flat local minima, which have been proposed to be associated with good generalization performance. In this paper, we observe that local minima of modern deep networks are more than being flat or sharp. Specifically, at a local minimum there exist many asymmetric directions such that the loss increases abruptly along one side, and slowly along the opposite side--we formally define such minima as asymmetric valleys. Under mild assumptions, we prove that for asymmetric valleys, a solution biased towards the flat side generalizes better than the exact minimizer. Further, we show that simply averaging the weights along the SGD trajectory gives rise to such biased solutions implicitly. This provides a theoretical explanation for the intriguing phenomenon observed by Izmailov et al. (2018). In addition, we empirically find that batch normalization (BN) appears to be a major cause for asymmetric valleys.

本研究观察到现代深度网络的局部最小值不仅是平坦或尖锐，而且存在许多不对称方向。我们形式化地将这样的最小值定义为不对称山谷，并证明在不对称山谷中，偏向平面的解决方案比确切的最小值更好地推广。此外，我们还发现，批量归一化（BN）似乎是不对称山谷产生的主要原因。

非对称山谷：突出和平坦局部极值之外