A Walk with SGD: How SGD Explores Regions of Deep Network Loss?

Chen Xing, Devansh Arpit, Christos Tsirigotis, Yoshua Bengio

Abstract: The non-convex nature of the loss landscape of deep neural networks (DNN) lends them the intuition that over the course of training, stochastic optimization algorithms explore different regions of the loss surface by entering and escaping many local minima due to the noise induced by mini-batches. But is this really the case? This question couples the geometry of the DNN loss landscape with how stochastic optimization algorithms like SGD interact with it during training. Answering this question may help us qualitatively understand the dynamics of deep neural network optimization. We show evidence through qualitative and quantitative experiments that mini-batch SGD rarely crosses barriers during DNN optimization. As we show, the mini-batch induced noise helps SGD explore different regions of the loss surface using a seemingly different mechanism. To complement this finding, we also investigate the qualitative reason behind the slowing down of this exploration when using larger batch-sizes. We show this happens because gradients from larger batch-sizes align more with the top eigenvectors of the Hessian, which makes SGD oscillate in the proximity of the parameter initialization, thus preventing exploration.

Data: [CIFAR-10](https://paperswithcode.com/dataset/cifar-10), [CIFAR-100](https://paperswithcode.com/dataset/cifar-100)