Constrained Stochastic Recursive Momentum Successive Convex Approximation

Abstract

We consider stochastic optimization problems with functional constraints. If the objective and constraint functions are not convex, the classical stochastic approximation algorithms such as the proximal stochastic gradient descent do not lead to efficient algorithms. In this work, we put forth an accelerated SCA algorithm that utilizes the recursive momentum-based acceleration which is widely used in the unconstrained setting. Remarkably, the proposed algorithm also achieves the optimal SFO complexity, at par with that achieved by state-of-the-art (unconstrained) stochastic optimization algorithms and match the SFO-complexity lower bound for minimization of general smooth functions. At each iteration, the proposed algorithm entails constructing convex surrogates of the objective and the constraint functions, and solving the resulting convex optimization problem. A recursive update rule is employed to track the gradient of the objective function, and contributes to achieving faster convergence and improved SFO complexity. A key ingredient of the proof is a new parameterized version of the standard Mangasarian-Fromowitz Constraints Qualification, that allows us to bound the dual variables and hence establish that the iterates approach an \(\epsilon\)-stationary point. We also detail a obstacle-avoiding trajectory optimization problem that can be solved using the proposed algorithm, and show that its performance is superior to that of the existing algorithms. The performance of the proposed algorithm is also compared against that of a specialized sparse classification algorithm on a binary classification problem.