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 the proposed methods achieve a sample complexity and first-order operation complexity of $\widetilde O(\epsilon^{-\max\{4, 2\theta\}})$ 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 $\epsilon$. To the best of our knowledge, this is the first work to develop methods with provable complexity guarantees for finding an approximate stochastic stationary point of such problems that nearly satisfies all constraints with certainty.

本研究解决了确定性约束的随机优化问题，现有方法往往只寻求满足一定准确度的随机平衡点，但在实际应用中，确保约束的确定性满足更为重要。我们提出了一种单循环方差减少的随机一阶方法，实验证明，该方法在满足约束的可能性上具有显著改进，且实现了样本复杂度和操作复杂度的优化。

具有强收敛保证的确定性约束随机非凸优化的一阶方差减少方法