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

## Citation

Preprint