This paper studies the convergence rate of the Polyak-Ruppert average estimator in one-dimensional linear stochastic approximation recursion driven by Markov chains, allowing the Markov chain to start from any initial distribution. Under the conditions of positive Harris recurrence, non-periodicity, and one-step domination, combined with the fourth-moment condition of the coefficient function with respect to the stationary distribution, this paper uses the exchangeable pair technique to prove that after deterministic bias correction for the average estimator, a Berry--Esseen bound of order $n^{-\alpha/2}$ can be obtained, where the step size exponent is $\alpha \in (1/2, 1)$. Moreover, for the statistics without bias correction and after variance normalization, the asymptotic normal approximation error is asymptotically equivalent to a normal constant multiplied by $n^{1/2 - \alpha}$.
In this paper, we establish two nonasymptotic Berry--Esseen bounds over convex sets for the Gaussian approximation of multivariate nonlinear statistics. The statistics of interest can be written as a sum of independent centered random vectors plus a remainder that may depend on all observations. The first bound retains...
Locally stationary Markov chains have attracted a lot of attention in statistics in the past three decades. In this paper we prove a variety of limit theorems for partial sums generated by such chains. We first provide explicit formulas for the asymptotic mean and variance (and also higher moments), and obtain optimal...
We analyze stochastic mirror descent for minimizing a convex objective expressed as an expectation under the stationary distribution of a Markov chain, using noisy subgradient observations along a single sample path. Through Poisson's equation, we decompose the Markovian noise and control its contribution under regular...
Sharp lower bounds are established for Gaussian approximation of Polyak-Ruppert averaged stochastic gradient descent with polynomially decreasing step sizes for Polyak-Ruppert averaged stochastic gradient descent with polynomially decreasing step sizes.
D. Belomestny, Alexey Naumov, Sergey Samsonov· 0 citations
Stochastic approximation provides a general framework for online estimation and optimization. Statistical inference based on the resulting estimates requires understanding their fluctuations around the target. For Polyak--Ruppert averaging, functional central limit theorems describe the normalized cumulative estimation...
The analysis separates the contributions of initial conditions, fast-timescale tracking error, Markovian dependence, and timescale coupling, thereby clarifying the origin of the $\beta^2/\alpha^2$ term.
Djamel Rassem Lamouri, Dorian Baudry, Nicolas Gast· arXiv.org· 0 citations
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