Computing the Bias of Constant-step Stochastic Approximation with Markovian Noise
This work addresses a theoretical bottleneck in optimization and control for researchers, providing incremental improvements in bias analysis for stochastic approximation methods.
The paper tackles the problem of analyzing the bias in stochastic approximation algorithms with Markovian noise and constant step-size, showing that the bias is of order O(α) and providing a precise characterization with a constant V from a Lyapunov equation, leading to an iterative scheme with bias O(α²) using extrapolation.
We study stochastic approximation algorithms with Markovian noise and constant step-size $α$. We develop a method based on infinitesimal generator comparisons to study the bias of the algorithm, which is the expected difference between $θ_n$ -- the value at iteration $n$ -- and $θ^*$ -- the unique equilibrium of the corresponding ODE. We show that, under some smoothness conditions, this bias is of order $O(α)$. Furthermore, we show that the time-averaged bias is equal to $αV + O(α^2)$, where $V$ is a constant characterized by a Lyapunov equation, showing that $\mathbb{E}[\barθ_n] \approx θ^*+Vα+ O(α^2)$, where $\barθ_n=(1/n)\sum_{k=1}^nθ_k$ is the Polyak-Ruppert average. We also show that $\barθ_n$ converges with high probability around $θ^*+αV$. We illustrate how to combine this with Richardson-Romberg extrapolation to derive an iterative scheme with a bias of order $O(α^2)$.