# Extensions of Robbins-Siegmund Theorem with Applications in Reinforcement Learning

arXiv: https://arxiv.org/abs/2509.26442 · HTML: https://mlsanity.com/paper/2509.26442/

## Key facts

- Cited by 5 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 42 of its 83 references that are in the map.
- First posted on arXiv 30 Sep 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, 30 of 42)

- [A Finite Time Analysis of Temporal Difference Learning With Linear Function Approximation](https://mlsanity.com/paper/1806.02450/) (2018)
- [A Theoretical Analysis of Deep Q-Learning](https://mlsanity.com/paper/1901.00137/) (2019)
- [Finite-Time Error Bounds For Linear Stochastic Approximation and TD Learning](https://mlsanity.com/paper/1902.00923/) (2019)
- [Global Convergence of Policy Gradient Methods to (Almost) Locally Optimal Policies](https://mlsanity.com/paper/1906.08383/) (2019)
- [Finite-Time Analysis of Asynchronous Stochastic Approximation and Q-Learning](https://mlsanity.com/paper/2002.00260/) (2020)
- [Is Q-Learning Minimax Optimal? A Tight Sample Complexity Analysis](https://mlsanity.com/paper/2102.06548/) (2021)
- [Finite-Sample Analysis for SARSA with Linear Function Approximation](https://mlsanity.com/paper/1902.02234/) (2019)
- [On Linear Stochastic Approximation: Fine-grained Polyak-Ruppert and Non-Asymptotic Concentration](https://mlsanity.com/paper/2004.04719/) (2020)
- [An improved convergence analysis for decentralized online stochastic non-convex optimization](https://mlsanity.com/paper/2008.04195/) (2020)
- [Convergence and Dynamical Behavior of the ADAM Algorithm for Non-Convex Stochastic Optimization](https://mlsanity.com/paper/1810.02263/) (2018)
- [A Finite-Time Analysis of Q-Learning with Neural Network Function Approximation](https://mlsanity.com/paper/1912.04511/) (2019)
- [Breaking the Deadly Triad with a Target Network](https://mlsanity.com/paper/2101.08862/) (2021)
- [On Almost Sure Convergence Rates of Stochastic Gradient Methods](https://mlsanity.com/paper/2202.04295/) (2022)
- [Optimal and instance-dependent guarantees for Markovian linear stochastic approximation](https://mlsanity.com/paper/2112.12770/) (2021)
- [The ODE Method for Asymptotic Statistics in Stochastic Approximation and Reinforcement Learning](https://mlsanity.com/paper/2110.14427/) (2021)
- [Target Network and Truncation Overcome The Deadly Triad in Q-Learning](https://mlsanity.com/paper/2203.02628/) (2022)
- [Bias and Extrapolation in Markovian Linear Stochastic Approximation with Constant Stepsizes](https://mlsanity.com/paper/2210.00953/) (2022)
- [A Statistical Analysis of Polyak-Ruppert Averaged Q-learning](https://mlsanity.com/paper/2112.14582/) (2021)
- [Tight High Probability Bounds for Linear Stochastic Approximation with Fixed Stepsize](https://mlsanity.com/paper/2106.01257/) (2021)
- [Concentration of Contractive Stochastic Approximation: Additive and Multiplicative Noise](https://mlsanity.com/paper/2303.15740/) (2023)
- [The ODE Method for Stochastic Approximation and Reinforcement Learning with Markovian Noise](https://mlsanity.com/paper/2401.07844/) (2024)
- [Concentration of Contractive Stochastic Approximation and Reinforcement Learning](https://mlsanity.com/paper/2106.14308/) (2021)
- [On TD(0) with function approximation: Concentration bounds and a centered variant with exponential convergence](https://mlsanity.com/paper/1411.3224/) (2014)
- [Neural Temporal-Difference and Q-Learning Provably Converge to Global Optima](https://mlsanity.com/paper/1905.10027/) (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)
- [Global Optimality and Finite Sample Analysis of Softmax Off-Policy Actor Critic under State Distribution Mismatch](https://mlsanity.com/paper/2111.02997/) (2021)
- [Optimal variance-reduced stochastic approximation in Banach spaces](https://mlsanity.com/paper/2201.08518/) (2022)
- [On the Convergence and Sample Complexity Analysis of Deep Q-Networks with ε-Greedy Exploration](https://mlsanity.com/paper/2310.16173/) (2023)
- [Constant Stepsize Q-learning: Distributional Convergence, Bias and Extrapolation](https://mlsanity.com/paper/2401.13884/) (2024)
- [A Concentration Bound for Stochastic Approximation via Alekseev's Formula](https://mlsanity.com/paper/1506.08657/) (2015)

## Built on by (most cited first, 5 of 5)

- [Towards Formalizing Reinforcement Learning Theory: A Robbins-Siegmund Approach](https://mlsanity.com/paper/2511.03618/) (2025)
- [Almost Sure Convergence of Differential Temporal Difference Learning for Average Reward Markov Decision Processes](https://mlsanity.com/paper/2602.16629/) (2026)
- [Almost Sure Convergence Rates of Stochastic Approximation and Reinforcement Learning via a Poisson-Moreau Drift](https://mlsanity.com/paper/2605.07104/) (2026)
- [When Sample Selection Bias Precipitates Model Collapse](https://mlsanity.com/paper/2606.13732/) (2026)
- [On the Divergence of Differential Temporal Difference Learning without Local Clocks](https://mlsanity.com/paper/2605.06874/) (2026)

## Abstract (arXiv)

The Robbins-Siegmund theorem establishes the convergence of stochastic processes that are almost supermartingales and is one of the most commonly used approaches for analyzing stochastic iterative algorithms in stochastic approximation and reinforcement learning (RL). However, its original form has a significant limitation as it requires the zero-order term to be summable. In many important RL applications, this summable condition, however, cannot be met. This limitation motivates us to extend the Robbins-Siegmund theorem for almost supermartingales where the zero-order term is not summable, but only square-summable. In particular, we introduce a novel and mild assumption on the increments of the stochastic processes. This together with the square-summable condition enables an almost sure convergence to a bounded set. Additionally, we further provide almost sure convergence rates, high probability concentration bounds, and Lp convergence rates. We then apply the new results to stochastic approximation and RL. Notably, we obtain the first almost sure convergence rate, the first high probability concentration bound, and the first Lp convergence rate for Q-learning with linear function approximation.