Papers › Computing the Bias of Constant-step Stochastic Approximation with Markovian Noise
Computing the Bias of Constant-step Stochastic Approximation with Markovian Noise
Sebastian Allmeier, Nicolas Gast
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 θₙ -- 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(α²), where V is a constant characterized by a Lyapunov equation, showing that 𝔼[θ̅ₙ] ≈θ^*+Vα+ O(α²), where θ̅ₙ=(1/n)∑ₖ₌₁ⁿθₖ is the Polyak-Ruppert average. We also show that θ̅ₙ 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(α²).
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