Dual-Anchor Acceleration Is Near-Optimal for Stochastic Monotone Root-Finding

Abstract

Among distinct optimal acceleration mechanisms for deterministic monotone root-finding problems and fixed-point problems, dual-anchoring has recently been shown to admit a more robust direct stochastic extension than standard anchor acceleration. However, without additional strong monotonicity, the existing stochastic dual-anchoring guarantee has two limitations: first, it requires cocoercivity in expectation, and second, it attains only $O(ε^{-3})$ oracle complexity, leaving a gap to the near-optimal $\tilde{O}(ε^{-2})$ complexity achieved by other methods. In this work, we address both of these limitations by combining dual-anchoring with stochastic resolvent approximation and optimized variance control. For unbiased stochastic oracles with variance bounded by $σ^2$, where sample operators are monotone and uniformly $L$-Lipschitz, our algorithm finds a point with $ε$-residual with a near-optimal oracle complexity of $O ( (LD / ε) \ell + (σ^2 / ε^2) \ell^2)$, where $\ell = \log (1 + LD / ε)$ and $D$ is the initial distance to a solution. This result improves the best known oracle complexity in the noise-dominated regime under these samplewise assumptions, reducing the poly-logarithmic factor from cubic to quadratic.