Empirical Risk Minimization for Stochastic Convex Optimization: O(1/n)- and O(1/n2)-type of Risk Bounds

Although there exist plentiful theories of empirical risk minimization (ERM) for supervised learning, current theoretical understandings of ERM for a related problem---stochastic convex optimization (SCO), are limited. In this work, we strengthen the realm of ERM for SCO by exploiting smoothness and strong convexity conditions to improve the risk bounds. First, we establish an widetildeO(d/n + √F_*/n) risk bound when the random function is nonnegative, convex and smooth, and the expected function is Lipschitz continuous, where d is the dimensionality of the problem, n is the number of samples, and F_* is the minimal risk. Thus, when F_* is small we obtain an widetildeO(d/n) risk bound, which is analogous to the widetildeO(1/n) optimistic rate of ERM for supervised learning. Second, if the objective function is also λ-strongly convex, we prove an widetildeO(d/n + kappa F_*/n ) risk bound where kappa is the condition number, and improve it to O(1/[λ n2] + kappa F_*/n) when n=widetildeOmega(kappa d). As a result, we obtain an O(kappa/n2) risk bound under the condition that n is large and F_* is small, which to the best of our knowledge, is the first O(1/n2)-type of risk bound of ERM. Third, we stress that the above results are established in a unified framework, which allows us to derive new risk bounds under weaker conditions, e.g., without convexity of the random function and Lipschitz continuity of the expected function. Finally, we demonstrate that to achieve an O(1/[λ n2] + kappa F_*/n) risk bound for supervised learning, the widetildeOmega(kappa d) requirement on n can be replaced with Omega(kappa2), which is dimensionality-independent.