Stochastic Augmented Lagrangian Framework with Second-Order Convergence Guarantees for Nonconvex Expectation-Constrained Optimization

Abstract

In this paper, we propose and analyze an augmented Lagrangian framework for solving stochastic nonconvex optimization problems with expectation-based equality constraints over a closed and convex constraint set. The framework generates a sequence of nonconvex primal subproblems, which are solved inexactly using stochastic second-order methods. We establish iteration complexity results for obtaining approximate second-order stationary points, both in expectation and with prescribed probability, under corresponding conditions on the accuracy of the primal subproblem solutions. We further establish sample complexity results for the framework with general stochastic second-order subproblem solvers under reasonable assumptions on their theoretical guarantees. Moreover, we incorporate three existing stochastic second-order solvers and derive the corresponding deterministic and probabilistic sample complexity results for obtaining approximate second-order stationary points. To the best of our knowledge, such sample complexity guarantees for obtaining approximate second-order stationary points in nonconvex expectation-constrained optimization have not been established previously. Finally, we demonstrate the empirical performance of the proposed framework on two nonconvex machine learning problems.