Variance-reduced first-order methods for deterministically constrained stochastic nonconvex optimization with strong convergence guarantees

In this paper, we study a class of deterministically constrained stochastic optimization problems. Existing methods typically aim to find an ε-stochastic stationary point, where the expected violations of both constraints and first-order stationarity are within a prescribed accuracy ε. However, in many practical applications, it is crucial that the constraints be nearly satisfied with certainty, making such an ε-stochastic stationary point potentially undesirable due to the risk of significant 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 θ ≥ 1 and other suitable assumptions, we establish that these methods respectively achieve a sample and first-order operation complexity of widetilde O(ε^-maxθ+2, 2θ) and widetilde O(ε^-max4, 2θ) for finding a stronger ε-stochastic stationary point, where the constraint violation is within ε with certainty, and the expected violation of first-order stationarity is within ε. For θ=1, these complexities reduce to widetilde O(ε-3) and widetilde O(ε-4) respectively, which match, up to a logarithmic factor, the best-known complexities achieved by existing methods for finding an ε-stochastic stationary point of unconstrained smooth stochastic optimization problems.