SGD Converges to Global Minimum in Deep Learning via Star-convex Path

Yi Zhou, Junjie Yang, Huishuai Zhang, Yingbin Liang, Vahid Tarokh

Abstract: Stochastic gradient descent (SGD) has been found to be surprisingly effective in training a variety of deep neural networks. However, there is still a lack of understanding on how and why SGD can train these complex networks towards a global minimum. In this study, we establish the convergence of SGD to a global minimum for nonconvex optimization problems that are commonly encountered in neural network training. Our argument exploits the following two important properties: 1) the training loss can achieve zero value (approximately), which has been widely observed in deep learning; 2) SGD follows a star-convex path, which is verified by various experiments in this paper. In such a context, our analysis shows that SGD, although has long been considered as a randomized algorithm, converges in an intrinsically deterministic manner to a global minimum.

Keywords: SGD, deep learning, global minimum, convergence

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