High Probability Guarantees for Nonconvex Stochastic Gradient Descent with Heavy Tails

Stochastic gradient descent (SGD) is the workhorse in modern machine learning and datadriven optimization. Despite its popularity, existing theoretical guarantees for SGD are mainly derived in expectation and for convex learning problems. High probability guarantees of nonconvex SGD are scarce, and typically rely on "lighttail" noise assumptions and study the optimization and generalization performance separately. In this paper, we develop high probability bounds for nonconvex SGD with a joint perspective of optimization and generalization performance. Instead of the light tail assumption, we consider the gradient noise following a heavy-tailed sub-Weibull distribution, a novel class generalizing the sub-Gaussian and sub-Exponential families to potentially heavier-tailed distributions. Under these complicated settings, we first present high probability bounds with best-known rates in general nonconvex learning, then move to nonconvex learning with a gradient dominance curvature condition, for which we improve the learning guarantees to fast rates. We further obtain sharper learning guarantees by considering a mild Bernsteintype noise condition. Our analysis also reveals the effect of trade-offs between the optimization and generalization performance under different conditions. In the last, we show that gradient clipping can be employed to remove the bounded gradient-type assumptions. Additionally, in this case, the stepsize of SGD is completely oblivious to the knowledge of smoothness.