# Tight Finite Time Bounds of Two-Time-Scale Linear Stochastic Approximation with Markovian Noise

arXiv: https://arxiv.org/abs/2401.00364 · HTML: https://mlsanity.com/paper/2401.00364/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 57 references that are in the map.
- First posted on arXiv 31 Dec 2023.

## Builds on (most cited first, 25 of 25)

- [Safe and Efficient Off-Policy Reinforcement Learning](https://mlsanity.com/paper/1606.02647/) (2016)
- [A Finite Time Analysis of Temporal Difference Learning With Linear Function Approximation](https://mlsanity.com/paper/1806.02450/) (2018)
- [Finite-Time Error Bounds For Linear Stochastic Approximation and TD Learning](https://mlsanity.com/paper/1902.00923/) (2019)
- [Non-strongly-convex smooth stochastic approximation with convergence rate O(1/n)](https://mlsanity.com/paper/1306.2119/) (2013)
- [Stochastic approximation with cone-contractive operators: Sharp ℓ_∞-bounds for Q-learning](https://mlsanity.com/paper/1905.06265/) (2019)
- [On Linear Stochastic Approximation: Fine-grained Polyak-Ruppert and Non-Asymptotic Concentration](https://mlsanity.com/paper/2004.04719/) (2020)
- [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)
- [Finite Sample Analysis of Two-Time-Scale Natural Actor-Critic Algorithm](https://mlsanity.com/paper/2101.10506/) (2021)
- [Nonlinear Two-Time-Scale Stochastic Approximation: Convergence and Finite-Time Performance](https://mlsanity.com/paper/2011.01868/) (2020)
- [Optimal and instance-dependent guarantees for Markovian linear stochastic approximation](https://mlsanity.com/paper/2112.12770/) (2021)
- [Adaptivity of averaged stochastic gradient descent to local strong convexity for logistic regression](https://mlsanity.com/paper/1303.6149/) (2013)
- [Finite-Sample Analysis of Proximal Gradient TD Algorithms](https://mlsanity.com/paper/2006.14364/) (2020)
- [Two Time-scale Off-Policy TD Learning: Non-asymptotic Analysis over Markovian Samples](https://mlsanity.com/paper/1909.11907/) (2019)
- [A Tale of Two-Timescale Reinforcement Learning with the Tightest Finite-Time Bound](https://mlsanity.com/paper/1911.09157/) (2019)
- [A Statistical Analysis of Polyak-Ruppert Averaged Q-learning](https://mlsanity.com/paper/2112.14582/) (2021)
- [Concentration of Contractive Stochastic Approximation: Additive and Multiplicative Noise](https://mlsanity.com/paper/2303.15740/) (2023)
- [Sample Complexity Bounds for Two Timescale Value-based Reinforcement Learning Algorithms](https://mlsanity.com/paper/2011.05053/) (2020)
- [Finite-sample Analysis of Greedy-GQ with Linear Function Approximation under Markovian Noise](https://mlsanity.com/paper/2005.10175/) (2020)
- [High-probability sample complexities for policy evaluation with linear function approximation](https://mlsanity.com/paper/2305.19001/) (2023)
- [Finite Sample Analysis of the GTD Policy Evaluation Algorithms in Markov Setting](https://mlsanity.com/paper/1809.08926/) (2018)
- [Finite-Time Convergence Rates of Nonlinear Two-Time-Scale Stochastic Approximation under Markovian Noise](https://mlsanity.com/paper/2104.01627/) (2021)
- [Linear Stochastic Approximation: Constant Step-Size and Iterate Averaging](https://mlsanity.com/paper/1709.04073/) (2017)
- [A Fixed-Point of View on Gradient Methods for Big Data](https://mlsanity.com/paper/1706.09880/) (2017)

## Built on by (most cited first, 15 of 15)

- [Rates of Convergence in the Central Limit Theorem for Markov Chains, with an Application to TD Learning](https://mlsanity.com/paper/2401.15719/) (2024)
- [Fast Two-Time-Scale Stochastic Gradient Method with Applications in Reinforcement Learning](https://mlsanity.com/paper/2405.09660/) (2024)
- [Finite-Time Bounds for Two-Time-Scale Stochastic Approximation with Arbitrary Norm Contractions and Markovian Noise](https://mlsanity.com/paper/2503.18391/) (2025)
- [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)
- [Nonasymptotic CLT and Error Bounds for Linear Two-Time-Scale Stochastic Approximation](https://mlsanity.com/paper/2502.09884/) (2025)
- [Two-Timescale Linear Stochastic Approximation: Constant Stepsizes Go a Long Way](https://mlsanity.com/paper/2410.13067/) (2024)
- [O(1/k) Finite-Time Bound for Non-Linear Two-Time-Scale Stochastic Approximation](https://mlsanity.com/paper/2504.19375/) (2025)
- [Non-Expansive Mappings in Two-Time-Scale Stochastic Approximation: Finite-Time Analysis](https://mlsanity.com/paper/2501.10806/) (2025)
- [Finite-Sample Wasserstein Error Bounds and Concentration Inequalities for Nonlinear Stochastic Approximation](https://mlsanity.com/paper/2602.02445/) (2026)
- [Markov Chain Variance Estimation: A Stochastic Approximation Approach](https://mlsanity.com/paper/2409.05733/) (2024)
- [Is Thompson Sampling Susceptible to Algorithmic Collusion?](https://mlsanity.com/paper/2405.17463/) (2024)
- [Instance-optimal stochastic convex optimization: Can we improve upon sample-average and robust stochastic approximation?](https://mlsanity.com/paper/2603.25657/) (2026)
- [Statistical analysis of Inverse Entropy-regularized Reinforcement Learning](https://mlsanity.com/paper/2512.06956/) (2025)
- [Nonlinear Two-Time-Scale Stochastic Approximation: A Sharp Phase Transition and How to Beat It](https://mlsanity.com/paper/2606.14488/) (2026)

## Abstract (arXiv)

Stochastic approximation (SA) is an iterative algorithm for finding the fixed point of an operator using noisy samples and widely used in optimization and Reinforcement Learning (RL). The noise in RL exhibits a Markovian structure, and in some cases, such as gradient temporal difference (GTD) methods, SA is employed in a two-time-scale framework. This combination introduces significant theoretical challenges for analysis. We derive an upper bound on the error for the iterations of linear two-time-scale SA with Markovian noise. We demonstrate that the mean squared error decreases as trace (Sigmay)/k + o(1/k) where k is the number of iterates, and Sigmay is an appropriately defined covariance matrix. A key feature of our bounds is that the leading term, Sigmay, exactly matches with the covariance in the Central Limit Theorem (CLT) for the two-time-scale SA, and we call them tight finite-time bounds. We illustrate their use in RL by establishing sample complexity for off-policy algorithms, TDC, GTD, and GTD2. A special case of linear two-time-scale SA that is extensively studied is linear SA with Polyak-Ruppert averaging. We present tight finite time bounds corresponding to the covariance matrix of the CLT. Such bounds can be used to study TD-learning with Polyak-Ruppert averaging.