有限样本Wasserstein误差界与非线性随机逼近的浓度不等式
Finite-Sample Wasserstein Error Bounds and Concentration Inequalities for Nonlinear Stochastic Approximation
AI总结：
本文研究了非线性随机逼近算法在有限样本下的Wasserstein误差界和浓度不等式，通过耦合论证和直接分析，推导了迭代和Polyak-Ruppert平均的收敛速率及分布特性。
摘要：
本文推导了非线性随机逼近算法在Wasserstein-p距离中的非渐近误差界。为了获得最后迭代的显式有限样本保证，我们开发了一种耦合论证，将离散时间过程与极限Ornstein-Uhlenbeck过程进行比较。我们的分析适用于由一般噪声条件驱动的算法，包括martingale差异和ergodic马尔可夫链的函数。补充这一结果，我们通过直接分析处理Polyak-Ruppert平均的收敛速度，该分析适用于相同的通用设置。假设驱动噪声满足非渐近的中心极限定理，我们证明归一化的最后迭代在p-Wasserstein距离中以阶γ_n^{1/6}收敛到正态分布，其中γ_n是步长。同样，Polyak-Ruppert平均在Wasserstein距离中以阶n^{-1/6}收敛。这些分布保证意味着高概率集中不等式，这些不等式优于由矩界和Markov不等式推导出的不等式。我们通过考虑两个应用来展示这种方法的实用性：(1)线性随机逼近，其中我们明确量化迭代从重尾到正态行为的转变，从而弥合最近的有限样本分析与渐近理论之间的差距；(2)随机梯度下降，其中我们建立了收敛到中心极限定理的速率。
英文摘要：
This paper derives non-asymptotic error bounds for nonlinear stochastic approximation algorithms in the Wasserstein-$p$ distance. To obtain explicit finite-sample guarantees for the last iterate, we develop a coupling argument that compares the discrete-time process to a limiting Ornstein-Uhlenbeck process. Our analysis applies to algorithms driven by general noise conditions, including martingale differences and functions of ergodic Markov chains. Complementing this result, we handle the convergence rate of the Polyak-Ruppert average through a direct analysis that applies under the same general setting. Assuming the driving noise satisfies a non-asymptotic central limit theorem, we show that the normalized last iterates converge to a Gaussian distribution in the $p$-Wasserstein distance at a rate of order $γ_n^{1/6}$, where $γ_n$ is the step size. Similarly, the Polyak-Ruppert average is shown to converge in the Wasserstein distance at a rate of order $n^{-1/6}$. These distributional guarantees imply high-probability concentration inequalities that improve upon those derived from moment bounds and Markov's inequality. We demonstrate the utility of this approach by considering two applications: (1) linear stochastic approximation, where we explicitly quantify the transition from heavy-tailed to Gaussian behavior of the iterates, thereby bridging the gap between recent finite-sample analyses and asymptotic theory and (2) stochastic gradient descent, where we establish rate of convergence to the central limit theorem.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
