Variance-Reduced First-Order Methods for Deterministically Constrained Stochastic Nonconvex Optimization with Strong Convergence Guarantees

Abstract. In this paper, we study a class of deterministically constrained stochastic nonconvex optimization problems. Existing methods typically aim to find an [Formula: see text]- expectedly feasible stochastic stationary point, where the expected violations of both constraints and first-order stationarity are within a prescribed tolerance [Formula: see text]. However, in many practical applications, it is crucial that the constraints be nearly satisfied with certainty, making such an [Formula: see text]-stochastic stationary point potentially undesirable due to the risk of substantial constraint violations. To address this issue, we propose single-loop variance-reduced stochastic first-order methods, where the stochastic gradient of the stochastic component is computed using either a truncated recursive momentum scheme or a truncated Polyak momentum scheme for variance reduction, while the gradient of the deterministic component is computed exactly. Under the error bound condition with a parameter [Formula: see text] and other suitable assumptions, we establish that these methods respectively achieve sample complexity and first-order oracle complexity of [Formula: see text] and [Formula: see text] for finding an [Formula: see text]- surely feasible stochastic stationary point ([Formula: see text] represents [Formula: see text] with logarithmic factors hidden), where the constraint violation is within [Formula: see text] with certainty, and the expected violation of first-order stationarity is within [Formula: see text]. For [Formula: see text], these complexities reduce to [Formula: see text] and [Formula: see text], respectively, which match, up to a logarithmic factor, the best-known complexities achieved by existing methods for finding an [Formula: see text]-stochastic stationary point of unconstrained smooth stochastic nonconvex optimization problems.