高概率求解随机不动点方程

中文摘要 AI 辅助

我们研究赋范空间\((\mathcal{E}, \|\cdot\|)\)上的随机不动点方程\(\mathbf{T}(\mathbf{x}) = \mathbf{x}\)，其中算子\(\mathbf{T}\)是非扩张或压缩的，且仅通过具有有界二阶中心矩的无偏随机评估来访问。给定\(\epsilon > 0\)，\(\delta \in (0, 1)\)，目标是输出\(\mathbf{x} \in \mathcal{E}\)，使得\(\|\mathbf{T}(\mathbf{x}) - \mathbf{x}\| \leq \epsilon\)的概率至少为\(1 - \delta\)。我们引入VR - GHAL，一种用于二次可平滑Banach空间的方差减少渐进哈尔彭方法。关键算法要素是基于预言机评估的截断差的递归随机估计器。主要定理给出了随时高概率残差界，在概率至少为\(1 - \delta\)的单个事件上，残差在各轮中近似几何下降。还给出了不同条件下的预言机复杂度。

英文摘要

We study stochastic fixed-point equations $\mathbf{T}(\mathbf{x}) = \mathbf{x}$ over normed spaces $(\mathcal{E}, \|\cdot\|)$, where the operator $\mathbf{T}$ is nonexpansive or contractive and is accessed only through unbiased stochastic evaluations with bounded second central moment. Given $ε> 0, δ\in (0, 1)$, the goal is to output $\mathbf{x} \in \mathcal{E}$ such that $\|\mathbf{T}(\mathbf{x}) - \mathbf{x}\| \leq ε$ with probability at least $1-δ$. We introduce VR-GHAL, a variance-reduced gradual Halpern method for quadratically smoothable Banach spaces. The key algorithmic ingredient is a recursive stochastic estimator based on clipped differences of oracle evaluations: instead of clipping $τ(\mathbf{x}; ξ)$ itself, we clip stochastic differences at the Lipschitz scale $γ\|\mathbf{x} - \mathbf{y}\|$. This makes the estimator pathwise Lipschitz along the algorithmic trajectory while permitting martingale concentration under finite second moments in the native norm. Our main theorem gives an anytime high-probability residual bound: on a single event of probability at least $1 - δ$, the residual decreases nearly geometrically across epochs, up to lower-order logarithmic factors. Under only bounded variance, displaying only the dependence on the target error $ε$ and Lipschitz constant $γ\in (0, 1]$ of $\mathbf{T}$, the resulting oracle complexity is $\min\{ε^{-5}, (1-γ)^{-3}ε^{-2}\}$. Under a Lipschitz-in-expectation oracle, the dependence improves to the corresponding $ε^{-3}$ nonexpansive rate (i.e., for $γ= 1$), and under samplewise nonexpansiveness to $ε^{-2}$.

发表机构

• Department of Computer Sciences University of Wisconsin-Madison（计算机科学系明尼苏达大学）

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