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

TL;DR

This paper introduces variance-reduced stochastic first-order methods for constrained nonconvex optimization, ensuring near-zero constraint violations with strong convergence guarantees and improved complexity bounds.

Contribution

It proposes single-loop variance-reduced methods with deterministic gradient computation for constrained stochastic nonconvex problems, achieving optimal complexity bounds for stronger stationarity.

Findings

Achieves $ ilde{O}( ext{epsilon}^{- ext{max}\{ heta+2, 2 heta ight ext{)}}$ complexity for constraint satisfaction.

Achieves $ ilde{O}( ext{epsilon}^{- ext{max}\{4, 2 heta ight ext{)}}$ complexity for stationarity.

Complexity bounds match best-known results for unconstrained problems when $ heta=1$.

Abstract

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 $\widetilde O(\epsilon^{-\max\{\theta+2, 2\theta\}})$ 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.