Stochastic Gradient Methods: Bias, Stability and Generalization

Recent developments of stochastic optimization often suggest biased gradient estimators to improve either the robustness, communication efficiency or computational speed. Representative biased stochastic gradient methods (BSGMs) include Zeroth-order stochastic gradient descent (SGD), Clipped-SGD and SGD with delayed gradients. The practical success of BSGMs motivates a lot of convergence analysis to explain their impressive training behaviour. As a comparison, there is far less work on their generalization analysis, which is a central topic in modern machine learning. In this paper, we present the first framework to study the stability and generalization of BSGMs for convex and smooth problems. We introduce a generalized Lipschitz-type condition on gradient estimators and bias, under which we develop a rather general stability bound to show how the bias and the gradient estimators affect the stability. We apply our general result to develop the first stability bound for Zeroth-order SGD with reasonable step size sequences, and the first stability bound for Clipped-SGD. While our stability analysis is developed for general BSGMs, the resulting stability bounds for both Zeroth-order SGD and Clipped-SGD match those of SGD under appropriate smoothing/clipping parameters. We combine the stability and convergence analysis together, and derive excess risk bounds of order $O(1/\sqrt{n})$ for both Zeroth-order SGD and Clipped-SGD, where $n$ is the sample size.