# Nonasymptotic CLT and Error Bounds for Linear Two-Time-Scale Stochastic Approximation

arXiv: https://arxiv.org/abs/2502.09884 · HTML: https://mlsanity.com/paper/2502.09884/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 37 references that are in the map.
- First posted on arXiv 14 Feb 2025.

## Builds on (most cited first, 18 of 18)

- [Finite-Time Error Bounds For Linear Stochastic Approximation and TD Learning](https://mlsanity.com/paper/1902.00923/) (2019)
- [A Finite Time Analysis of Two Time-Scale Actor Critic Methods](https://mlsanity.com/paper/2005.01350/) (2020)
- [On Linear Stochastic Approximation: Fine-grained Polyak-Ruppert and Non-Asymptotic Concentration](https://mlsanity.com/paper/2004.04719/) (2020)
- [Finite-Sample Analysis of Nonlinear Stochastic Approximation with Applications in Reinforcement Learning](https://mlsanity.com/paper/1905.11425/) (2019)
- [Finite Time Analysis of Linear Two-timescale Stochastic Approximation with Markovian Noise](https://mlsanity.com/paper/2002.01268/) (2020)
- [Finite-time High-probability Bounds for Polyak-Ruppert Averaged Iterates of Linear Stochastic Approximation](https://mlsanity.com/paper/2207.04475/) (2022)
- [Finite-Time Performance Bounds and Adaptive Learning Rate Selection for Two Time-Scale Reinforcement Learning](https://mlsanity.com/paper/1907.06290/) (2019)
- [Two Time-scale Off-Policy TD Learning: Non-asymptotic Analysis over Markovian Samples](https://mlsanity.com/paper/1909.11907/) (2019)
- [Normal Approximation for Stochastic Gradient Descent via Non-Asymptotic Rates of Martingale CLT](https://mlsanity.com/paper/1904.02130/) (2019)
- [Gaussian Approximation and Multiplier Bootstrap for Polyak-Ruppert Averaged Linear Stochastic Approximation with Applications to TD Learning](https://mlsanity.com/paper/2405.16644/) (2024)
- [Regularized Gradient Descent Ascent for Two-Player Zero-Sum Markov Games](https://mlsanity.com/paper/2205.13746/) (2022)
- [Rates of Convergence in the Central Limit Theorem for Markov Chains, with an Application to TD Learning](https://mlsanity.com/paper/2401.15719/) (2024)
- [Tight Finite Time Bounds of Two-Time-Scale Linear Stochastic Approximation with Markovian Noise](https://mlsanity.com/paper/2401.00364/) (2023)
- [Fast Two-Time-Scale Stochastic Gradient Method with Applications in Reinforcement Learning](https://mlsanity.com/paper/2405.09660/) (2024)
- [Revisiting Step-Size Assumptions in Stochastic Approximation](https://mlsanity.com/paper/2405.17834/) (2024)
- [Central Limit Theorem for Two-Timescale Stochastic Approximation with Markovian Noise: Theory and Applications](https://mlsanity.com/paper/2401.09339/) (2024)
- [Fast Nonlinear Two-Time-Scale Stochastic Approximation: Achieving O(1/k) Finite-Sample Complexity](https://mlsanity.com/paper/2401.12764/) (2024)
- [Accelerated Multi-Time-Scale Stochastic Approximation: Optimal Complexity and Applications in Reinforcement Learning and Multi-Agent Games](https://mlsanity.com/paper/2409.07767/) (2024)

## Built on by (most cited first, 6 of 6)

- [Gaussian Approximation for Two-Timescale Linear Stochastic Approximation](https://mlsanity.com/paper/2508.07928/) (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)
- [Finite-Sample Wasserstein Error Bounds and Concentration Inequalities for Nonlinear Stochastic Approximation](https://mlsanity.com/paper/2602.02445/) (2026)
- [Gaussian Approximation for Asynchronous Q-learning](https://mlsanity.com/paper/2604.07323/) (2026)
- [Finite-Time Analysis of Projected Two-Time-Scale Stochastic Approximation](https://mlsanity.com/paper/2604.00179/) (2026)

## Abstract (arXiv)

We consider linear two-time-scale stochastic approximation algorithms driven by martingale noise. Recent applications in machine learning motivate the need to understand finite-time error rates, but conventional stochastic approximation analyses focus on either asymptotic convergence in distribution or finite-time bounds that are far from optimal. Prior work on asymptotic central limit theorems (CLTs) suggests that two-time-scale algorithms may be able to achieve 1/√K error in expectation, with a constant given by the expected norm of the limiting Gaussian vector. However, the best known finite-time rates are much slower. We derive the first nonasymptotic Wasserstein-1 CLT for linear two-time-scale stochastic approximation with Polyak-Ruppert averaging driven by martingale difference noise. As a corollary, we show that the expected error achieved by Polyak-Ruppert averaging decays at rate 1/√K, which significantly improves on the rates of convergence in prior works.