Online Stochastic Optimization with Multiple Objectives

In this paper we propose a general framework to characterize and solve the stochastic optimization problems with multiple objectives underlying many real world learning applications. We first propose a projection based algorithm which attains an O(T−1/3) convergence rate. Then, by leveraging on the theory of Lagrangian in constrained optimization, we devise a novel primal-dual stochastic approximation algorithm which attains the optimal convergence rate ofO(T−1/2) for general Lipschitz continuous objectives.