发表机构
School of Mathematical Sciences, Zhejiang Normal University（浙江师范大学数学与计算机科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究随机不动点迭代的锚定调度，揭示常数密度下的精确权衡，提出同时最优的稀疏调度，并给出压缩与非扩张算子的极小极大复杂度。
AI 中文摘要
我们研究希尔伯特空间上非扩张与压缩算子的随机不动点迭代 $x_{n+1}=\theta_nx_0+(1-\theta_n)\widehat T(x_n)$，其中使用有界方差的单点无偏预言机。在确定性情形下，每个密度 $c_n\in(0,1]$ 不在尺度间振荡的锚定调度 $\theta_n=c_n/(n+2)$ 要么在压缩算子上多项式次优，要么在旋转算子上超多项式缓慢；对于常数密度 $\theta_n=c/(n+2)$，权衡是精确的，在旋转角度 $\varphi$（$\varphi\to0$，$n\varphi\to\infty$）下 $\norm{x_n}\asymp\Gamma(c+1)(n\varphi)^{-c}$，而对于经典调度，$\norm{x_n}=\frac{2|\sin(n\varphi/2)|}{n\varphi}+O(\frac1n)$。这种二分法仅在稀疏锚定浓度下失效：一个显式的 $\gamma$-不知情调度（每个目标精度一个）同时实现压缩最优和旋转多项式指数 $1/\alpha_0$（$\alpha_0=\log_2(3/2)$），并且对于每个固定目标精度，匹配的下界对勒贝格几乎处处角度成立。对于随机压缩，在仿射映射上的极小极大速率为 $\Theta(\sigma^2\eps^{-2}(1-\gamma)^{-2})$，而在锚定类中，对于已知模量通过几何批处理为 $\Theta(\sigma^2\eps^{-2}(1-\gamma)^{-2}+(1-\gamma)^{-1}\ln(D/\eps))$；没有认证的模量界则不存在可靠的距离证书，而认证上限是精确边界。对于非扩张映射，我们在每个 $2$-一致光滑巴拿赫空间中证明 $O(K\sigma^2D^2\eps^{-4})$，并通过归约到单调包含得到希尔伯特速率 $\tilde\Theta(\sigma^2\eps^{-2}+D\eps^{-1})$。数值实验与所有缩放预测一致。
英文摘要
We study stochastic fixed-point iterations $x_{n+1}=θ_nx_0+(1-θ_n)\widehat T(x_n)$ for nonexpansive and contractive operators on Hilbert spaces, with a single-point unbiased oracle of bounded variance. Deterministically, every anchor schedule $θ_n=c_n/(n+2)$ whose density $c_n\in(0,1]$ does not oscillate between scales is either polynomially suboptimal on contractions or super-polynomially slow on rotations; for constant densities $θ_n=c/(n+2)$ the tradeoff is exact, with $\norm{x_n}\asympΓ(c+1)(nφ)^{-c}$ on rotations by angle $φ$ ($φ\to0$, $nφ\to\infty$) and, for the classical schedule, $\norm{x_n}=\frac{2|\sin(nφ/2)|}{nφ}+O(\frac1n)$. The dichotomy fails exactly under lacunary anchor concentration: an explicit $γ$-oblivious schedule, one per target accuracy, is simultaneously contraction-optimal and rotation-polynomial with exponent $1/α_0$, $α_0=\log_2(3/2)$, and, for each fixed target accuracy, the matching lower bound holds for Lebesgue-a.e.\ angle. For stochastic contractions, the minimax rate is $Θ(σ^2\eps^{-2}(1-γ)^{-2})$ on affine maps, and within the anchored class $Θ(σ^2\eps^{-2}(1-γ)^{-2}+(1-γ)^{-1}\ln(D/\eps))$ for known modulus via geometric batching; without a certified modulus bound no sound distance certificate exists, and a certified ceiling is the exact boundary. For nonexpansive maps we prove $O(Kσ^2D^2\eps^{-4})$ in every $2$-uniformly smooth Banach space and the Hilbert rate $\tildeΘ(σ^2\eps^{-2}+D\eps^{-1})$ by reduction to monotone inclusions. Numerical experiments are consistent with every scaling prediction.
Comments55 pages, 5 figures
