Haowei He, Gao Huang, Yang Yuan
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 are correlated with good generalization performance. In this paper, we observe that local minima of modern deep networks are more than being ﬂat or sharp. Instead, 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 ﬁrst prove that for asymmetric valleys, a solution biased towards the ﬂat side generalizes better than the exact empirical minimizer. Then, we show that performing weight averaging along the SGD trajectory implicitly induces such biased solutions. This provides theoretical explanations for a series of intriguing phenomena observed in recent work [25, 5, 51]. Finally, extensive empirical experiments on both modern deep networks and simple 2 layer networks are conducted to validate our assumptions and analyze the intriguing properties of asymmetric valleys.