发表机构
Johns Hopkins University（约翰斯·霍普金斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对随机单调求根问题，本文结合双锚定与随机预解近似及优化方差控制，在无偏随机预言机下实现近乎最优的预言机复杂度，改进噪声主导区域的对数因子。
AI 中文摘要
在确定性单调求根问题和不动点问题的不同最优加速机制中，双锚定（dual-anchoring）最近被证明比标准锚定加速具有更稳健的直接随机扩展。然而，在没有额外强单调性的情况下，现有的随机双锚定保证存在两个局限性：首先，它要求期望意义上的余强单调性（cocoercivity）；其次，它仅达到$O(\epsilon^{-3})$的预言机复杂度，与其他方法实现的近乎最优的$\tilde{O}(\epsilon^{-2})$复杂度存在差距。在本工作中，我们通过将双锚定与随机预解近似（stochastic resolvent approximation）以及优化的方差控制相结合，解决了这两个局限性。对于方差以$\sigma^2$为界的无偏随机预言机，其中样本算子单调且一致$L$-Lipschitz，我们的算法找到一个具有$\epsilon$-残差的点，其近乎最优的预言机复杂度为$O((LD/\epsilon)\ell + (\sigma^2/\epsilon^2)\ell^2)$，其中$\ell = \log(1 + LD/\epsilon)$，$D$是到解的初始距离。这一结果在噪声主导区域下，在这些样本级假设下改进了已知的最佳预言机复杂度，将对数因子从三次降至二次。
英文摘要
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.
