Finite-Sample Analysis of Nonlinear Stochastic Approximation with Applications in Reinforcement Learning
This provides theoretical guarantees for reinforcement learning algorithms under more realistic Markovian noise conditions, though it is incremental in extending existing stochastic approximation theory.
The paper tackles the finite-sample convergence analysis of nonlinear stochastic approximation under Markovian noise, showing exponential convergence to a neighborhood with constant stepsize and O(log(k)/k) rate with diminishing stepsizes. It applies these results to derive finite-sample bounds for Q-learning with linear function approximation without i.i.d. assumptions or projection steps.
Motivated by applications in reinforcement learning (RL), we study a nonlinear stochastic approximation (SA) algorithm under Markovian noise, and establish its finite-sample convergence bounds under various stepsizes. Specifically, we show that when using constant stepsize (i.e., $α_k\equiv α$), the algorithm achieves exponential fast convergence to a neighborhood (with radius $O(α\log(1/α))$) around the desired limit point. When using diminishing stepsizes with appropriate decay rate, the algorithm converges with rate $O(\log(k)/k)$. Our proof is based on Lyapunov drift arguments, and to handle the Markovian noise, we exploit the fast mixing of the underlying Markov chain. To demonstrate the generality of our theoretical results on Markovian SA, we use it to derive the finite-sample bounds of the popular $Q$-learning with linear function approximation algorithm, under a condition on the behavior policy. Importantly, we do not need to make the assumption that the samples are i.i.d., and do not require an artificial projection step in the algorithm to maintain the boundedness of the iterates. Numerical simulations corroborate our theoretical results.