Asymmetric Valleys: Beyond Sharp and Flat Local Minima
This addresses the problem of understanding generalization in deep learning for researchers, by providing a new theoretical framework beyond sharp/flat minima, though it is incremental in building on prior work.
The paper identifies asymmetric valleys in deep neural network loss landscapes, where loss increases abruptly on one side and slowly on the opposite side, and proves that solutions biased toward the flat side generalize better than exact minimizers, with empirical evidence linking batch normalization to these valleys.
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.