Abstract

In this paper, we are interested in the development of efficient algorithms for convex optimization problems in the simultaneous presence of multiple objectives and stochasticity in the first-order information. We cast the stochastic multiple objective optimization problem into a constrained optimization problem by choosing one function as the objective and try to bound other objectives by appropriate thresholds. We first examine a two stages exploration-exploitation based algorithm which first approximates the stochastic objectives by sampling and then solves a constrained stochastic optimization problem by projected gradient method. This method attains a suboptimal convergence rate even under strong assumption on the objectives. Our second approach is an efficient primal-dual stochastic algorithm. It leverages on the theory of Lagrangian method  in constrained optimization and attains the optimal convergence rate of in high probability for general Lipschitz continuous objectives.