为带马尔可夫噪声的线性随机逼近提供非渐近统计推断方法
Statistical inference for Linear Stochastic Approximation with Markovian Noise
- 基于Polyak-Ruppert平均迭代，分析马尔可夫噪声下的收敛速率
- 在Kolmogorov距离下达到O(n^{-1/4})的高斯逼近速度
- 首次给出基于自助法置信区间的非渐近有效性保证
本文推导了由马尔可夫噪声驱动的线性随机逼近（LSA）算法中Polyak-Ruppert平均迭代的非渐近Berry-Esseen界。分析表明，在Kolmogorov距离下收敛至高斯极限的速率为$/mathcal{O}(n^{-1/4})$。我们进一步建立了乘子块自助法构造置信区间的非渐近有效性，确保在马尔可夫采样下推断的一致性。本工作首次提供了带马尔可夫噪声的随机逼近中基于自助法置信区间的非渐近收敛速率保证。此外，我们恢复了经典渐近方差估计的速率$/mathcal{O}(n^{-1/8})$，仅相差对数因子。
原文摘要 · Abstract (English)
In this paper we derive non-asymptotic Berry-Esseen bounds for Polyak-Ruppert averaged iterates of the Linear Stochastic Approximation (LSA) algorithm driven by the Markovian noise. Our analysis yields $\mathcal{O}(n^{-1/4})$ convergence rates to the Gaussian limit in the Kolmogorov distance. We further establish the non-asymptotic validity of a multiplier block bootstrap procedure for constructing the confidence intervals, guaranteeing consistent inference under Markovian sampling. Our work provides the first non-asymptotic guarantees on the rate of convergence of bootstrap-based confidence intervals for stochastic approximation with Markov noise. Moreover, we recover the classical rate of order $\mathcal{O}(n^{-1/8})$ up to logarithmic factors for estimating the asymptotic variance of the iterates of the LSA algorithm.
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