# Stochastic Approximation with Unbounded Markovian Noise: A General-Purpose Theorem

arXiv: https://arxiv.org/abs/2410.21704 · HTML: https://mlsanity.com/paper/2410.21704/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Placed in RL theory and sample complexity, one of 15 directions in Deep reinforcement learning methods.
- First posted on arXiv 29 Oct 2024.

## Placed in

- Deep reinforcement learning methods / [RL theory and sample complexity](https://mlsanity.com/deeprl/rl-theory-and-sample-complexity/)

## Built on by (most cited first, 10 of 10)

- [Finite-Time Bounds for Two-Time-Scale Stochastic Approximation with Arbitrary Norm Contractions and Markovian Noise](https://mlsanity.com/paper/2503.18391/) (2025)
- [Finite Sample Analysis of Linear Temporal Difference Learning with Arbitrary Features](https://mlsanity.com/paper/2505.21391/) (2025)
- [A Minimal-Assumption Analysis of Q-Learning with Time-Varying Policies](https://mlsanity.com/paper/2510.16132/) (2025)
- [Multiscale replay: A robust algorithm for stochastic variational inequalities with a Markovian buffer](https://mlsanity.com/paper/2601.01502/) (2026)
- [Parameter-Free Federated TD Learning with Markov Noise in Heterogeneous Environments](https://mlsanity.com/paper/2510.07436/) (2025)
- [Persistent-Transient Policy Evaluation for Markov Chains via Minimal Peripheral Quotients](https://mlsanity.com/paper/2602.00474/) (2026)
- [Periodic Regularized Q-Learning](https://mlsanity.com/paper/2602.03301/) (2026)
- [Non-Asymptotic Convergence of Stochastic Iterative Algorithms: A Lyapunov Framework](https://mlsanity.com/paper/2605.31309/) (2026)
- [A Banach-Space Theory of Markovian Halpern Iteration for Non-Expansive Maps](https://mlsanity.com/paper/2608.15966/) (2026)
- [Bridging the Gap Between Average and Discounted TD Learning](https://mlsanity.com/paper/2605.02103/) (2026)

## Abstract (arXiv)

Motivated by engineering applications such as resource allocation in networks and inventory systems, we consider average-reward Reinforcement Learning with unbounded state space and reward function. Recent works studied this problem in the actor-critic framework and established finite sample bounds assuming access to a critic with certain error guarantees. We complement their work by studying Temporal Difference (TD) learning with linear function approximation and establishing finite-time bounds with the optimal Oleft(1/ε2right) sample complexity. These results are obtained using the following general-purpose theorem for non-linear Stochastic Approximation (SA). Suppose that one constructs a Lyapunov function for a non-linear SA with certain drift condition. Then, our theorem establishes finite-time bounds when this SA is driven by unbounded Markovian noise under suitable conditions. It serves as a black box tool to generalize sample guarantees on SA from i.i.d. or martingale difference case to potentially unbounded Markovian noise. The generality and the mild assumption of the setup enables broad applicability of our theorem. We illustrate its power by studying two more systems: (i) We improve upon the finite-time bounds of Q-learning by tightening the error bounds and also allowing for a larger class of behavior policies. (ii) We establish the first ever finite-time bounds for distributed stochastic optimization of high-dimensional smooth strongly convex function using cyclic block coordinate descent.