# Finite-Time Bounds for Two-Time-Scale Stochastic Approximation with Arbitrary Norm Contractions and Markovian Noise

arXiv: https://arxiv.org/abs/2503.18391 · HTML: https://mlsanity.com/paper/2503.18391/

## Key facts

- Cited by 9 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 19 of its 43 references that are in the map.
- First posted on arXiv 24 Mar 2025.

## 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, 19 of 19)

- [Finite-Time Error Bounds For Linear Stochastic Approximation and TD Learning](https://mlsanity.com/paper/1902.00923/) (2019)
- [Finite-Time Analysis of Asynchronous Stochastic Approximation and Q-Learning](https://mlsanity.com/paper/2002.00260/) (2020)
- [Finite Time Analysis of Linear Two-timescale Stochastic Approximation with Markovian Noise](https://mlsanity.com/paper/2002.01268/) (2020)
- [Nonlinear Two-Time-Scale Stochastic Approximation: Convergence and Finite-Time Performance](https://mlsanity.com/paper/2011.01868/) (2020)
- [A Tale of Two-Timescale Reinforcement Learning with the Tightest Finite-Time Bound](https://mlsanity.com/paper/1911.09157/) (2019)
- [Concentration of Contractive Stochastic Approximation: Additive and Multiplicative Noise](https://mlsanity.com/paper/2303.15740/) (2023)
- [A Two-Time-Scale Stochastic Optimization Framework with Applications in Control and Reinforcement Learning](https://mlsanity.com/paper/2109.14756/) (2021)
- [Concentration of Contractive Stochastic Approximation and Reinforcement Learning](https://mlsanity.com/paper/2106.14308/) (2021)
- [Constant Stepsize Q-learning: Distributional Convergence, Bias and Extrapolation](https://mlsanity.com/paper/2401.13884/) (2024)
- [Tight Finite Time Bounds of Two-Time-Scale Linear Stochastic Approximation with Markovian Noise](https://mlsanity.com/paper/2401.00364/) (2023)
- [Stochastic Approximation with Unbounded Markovian Noise: A General-Purpose Theorem](https://mlsanity.com/paper/2410.21704/) (2024)
- [First-Order Algorithms for Nonlinear Generalized Nash Equilibrium Problems](https://mlsanity.com/paper/2204.03132/) (2022)
- [Concentration bounds for two time scale stochastic approximation](https://mlsanity.com/paper/1806.10798/) (2018)
- [Fast Nonlinear Two-Time-Scale Stochastic Approximation: Achieving O(1/k) Finite-Sample Complexity](https://mlsanity.com/paper/2401.12764/) (2024)
- [Stochastic generalized Nash equilibrium seeking under partial-decision information](https://mlsanity.com/paper/2011.05357/) (2020)
- [Equilibrium Computation of Generalized Nash Games: A New Lagrangian-Based Approach](https://mlsanity.com/paper/2106.00109/) (2021)
- [Learning to Control Unknown Strongly Monotone Games](https://mlsanity.com/paper/2407.00575/) (2024)
- [Non-Expansive Mappings in Two-Time-Scale Stochastic Approximation: Finite-Time Analysis](https://mlsanity.com/paper/2501.10806/) (2025)
- [Randomized Lagrangian Stochastic Approximation for Large-Scale Constrained Stochastic Nash Games](https://mlsanity.com/paper/2304.07688/) (2023)

## Built on by (most cited first, 9 of 9)

- [O(1/k) Finite-Time Bound for Non-Linear Two-Time-Scale Stochastic Approximation](https://mlsanity.com/paper/2504.19375/) (2025)
- [A General-Purpose Theorem for High-Probability Bounds of Stochastic Approximation with Polyak Averaging](https://mlsanity.com/paper/2505.21796/) (2025)
- [Central Limit Theorems for Asynchronous Averaged Q-Learning](https://mlsanity.com/paper/2509.18964/) (2025)
- [Parameter-Free Federated TD Learning with Markov Noise in Heterogeneous Environments](https://mlsanity.com/paper/2510.07436/) (2025)
- [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)
- [High-Probability PL-SGD with Markovian Noise: Optimal Mixing and Tail Dependence](https://mlsanity.com/paper/2606.26316/) (2026)
- [Concentration and Mean-Square Bounds for Contractive Stochastic Approximation: A Unified Elementary Approach](https://mlsanity.com/paper/2607.17595/) (2026)

## Abstract (arXiv)

Two-time-scale Stochastic Approximation (SA) is an iterative algorithm with applications in reinforcement learning and optimization. Prior finite time analysis of such algorithms has focused on fixed point iterations with mappings contractive under Euclidean norm. Motivated by applications in reinforcement learning, we give the first mean square bound on non linear two-time-scale SA where the iterations have arbitrary norm contractive mappings and Markovian noise. We show that the mean square error decays at a rate of O(1/n2/3) in the general case, and at a rate of O(1/n) in a special case where the slower timescale is noiseless. Our analysis uses the generalized Moreau envelope to handle the arbitrary norm contractions and solutions of Poisson equation to deal with the Markovian noise. By analyzing the SSP Q-Learning algorithm, we give the first O(1/n) bound for an algorithm for asynchronous control of MDPs under the average reward criterion. We also obtain a rate of O(1/n) for Q-Learning with Polyak-averaging and provide an algorithm for learning Generalized Nash Equilibrium (GNE) for strongly monotone games which converges at a rate of O(1/n2/3).