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

2026-09-28

AI Summary

This study addresses the limitations of existing double-anchoring methods for stochastic monotone root-finding problems, which rely on the cocoercivity assumption and exhibit suboptimal complexity. To overcome these issues, this work proposes a novel algorithm that integrates the double-anchoring framework with stochastic resolvent approximation and variance control. The proposed method successfully eliminates the cocoercivity requirement while reducing the poly-logarithmic factor in the noise-dominated regime from cubic to quadratic order. Theoretical analysis demonstrates that, under operator monotonicity and Lipschitz continuity conditions, the algorithm achieves a near-optimal oracle complexity of O(ε⁻²), thereby realizing nearly optimal accelerated convergence for this class of problems.

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.

Problem

• stochastic monotone root-finding
• dual-anchor acceleration
• oracle complexity
• cocoercivity

Innovation

• dual-anchor acceleration
• stochastic monotone root-finding
• oracle complexity
• variance control
• stochastic resolvent approximation