# Almost Sure Convergence Rates and Concentration of Stochastic Approximation and Reinforcement Learning with Markovian Noise

arXiv: https://arxiv.org/abs/2411.13711 · HTML: https://mlsanity.com/paper/2411.13711/

## Key facts

- Cited by 12 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.
- Builds on 11 of its 84 references that are in the map.
- First posted on arXiv 20 Nov 2024.

## Placed in

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

## Builds on (most cited first, 11 of 11)

- [Stochastic approximation with cone-contractive operators: Sharp ℓ_∞-bounds for Q-learning](https://mlsanity.com/paper/1905.06265/) (2019)
- [A Lyapunov Theory for Finite-Sample Guarantees of Asynchronous Q-Learning and TD-Learning Variants](https://mlsanity.com/paper/2102.01567/) (2021)
- [The ODE Method for Asymptotic Statistics in Stochastic Approximation and Reinforcement Learning](https://mlsanity.com/paper/2110.14427/) (2021)
- [Simple and optimal high-probability bounds for strongly-convex stochastic gradient descent](https://mlsanity.com/paper/1909.00843/) (2019)
- [Concentration of Contractive Stochastic Approximation: Additive and Multiplicative Noise](https://mlsanity.com/paper/2303.15740/) (2023)
- [Finite-Sample Analysis of Stochastic Approximation Using Smooth Convex Envelopes](https://mlsanity.com/paper/2002.00874/) (2020)
- [Optimal variance-reduced stochastic approximation in Banach spaces](https://mlsanity.com/paper/2201.08518/) (2022)
- [A Concentration Bound for Stochastic Approximation via Alekseev's Formula](https://mlsanity.com/paper/1506.08657/) (2015)
- [A Unified Switching System Perspective and O.D.E. Analysis of Q-Learning Algorithms](https://mlsanity.com/paper/1912.02270/) (2019)
- [A Concentration Bound for TD(0) with Function Approximation](https://mlsanity.com/paper/2312.10424/) (2023)
- [Revisiting Step-Size Assumptions in Stochastic Approximation](https://mlsanity.com/paper/2405.17834/) (2024)

## Built on by (most cited first, 12 of 12)

- [Linear Q-Learning Does Not Diverge in L2: Convergence Rates to a Bounded Set](https://mlsanity.com/paper/2501.19254/) (2025)
- [Towards Formalizing Reinforcement Learning Theory: A Robbins-Siegmund Approach](https://mlsanity.com/paper/2511.03618/) (2025)
- [Finite Sample Analysis of Linear Temporal Difference Learning with Arbitrary Features](https://mlsanity.com/paper/2505.21391/) (2025)
- [Extensions of Robbins-Siegmund Theorem with Applications in Reinforcement Learning](https://mlsanity.com/paper/2509.26442/) (2025)
- [Concentration of General Stochastic Approximation Under Heavy-Tailed Markovian Noise](https://mlsanity.com/paper/2605.20999/) (2026)
- [Almost Sure Convergence of Differential Temporal Difference Learning for Average Reward Markov Decision Processes](https://mlsanity.com/paper/2602.16629/) (2026)
- [Regret and Sample Complexity of Online Q-Learning via Concentration of Stochastic Approximation with Time-Inhomogeneous Markov Chains](https://mlsanity.com/paper/2602.16274/) (2026)
- [Almost Sure Convergence Rates of Stochastic Approximation and Reinforcement Learning via a Poisson-Moreau Drift](https://mlsanity.com/paper/2605.07104/) (2026)
- [Non-Asymptotic Convergence of Stochastic Iterative Algorithms: A Lyapunov Framework](https://mlsanity.com/paper/2605.31309/) (2026)
- [Large-scale Uncertainty Quantification for Latent Variable Models Using Subsampling Markov Chain Monte Carlo](https://mlsanity.com/paper/2606.00309/) (2026)
- [Concentration and Mean-Square Bounds for Contractive Stochastic Approximation: A Unified Elementary Approach](https://mlsanity.com/paper/2607.17595/) (2026)
- [Shrinking-Tube Concentration for Adaptive Markovian Stochastic Approximation](https://mlsanity.com/paper/2609.29833/) (2026)

## Abstract (arXiv)

This paper establishes the first almost sure convergence rate and the first maximal concentration bound with exponential tails for general contractive stochastic approximation algorithms with Markovian noise. As a corollary, we also obtain convergence rates in Lp. Key to our successes is a novel discretization of the mean ODE of stochastic approximation algorithms using intervals with diminishing (instead of constant) length. As applications, we provide the first almost sure convergence rate for Q-learning with Markovian samples without count-based learning rates. We also provide the first concentration bound for off-policy temporal difference learning with Markovian samples.