# Q-learning with Uniformly Bounded Variance: Large Discounting is Not a Barrier to Fast Learning

arXiv: https://arxiv.org/abs/2002.10301 · HTML: https://mlsanity.com/paper/2002.10301/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Feb 2020.

## Built on by (most cited first, 13 of 13)

- [Is Q-Learning Minimax Optimal? A Tight Sample Complexity Analysis](https://mlsanity.com/paper/2102.06548/) (2021)
- [Reward Centering](https://mlsanity.com/paper/2405.09999/) (2024)
- [Linear Q-Learning Does Not Diverge in L2: Convergence Rates to a Bounded Set](https://mlsanity.com/paper/2501.19254/) (2025)
- [Stability of Q-Learning Through Design and Optimism](https://mlsanity.com/paper/2307.02632/) (2023)
- [Examining average and discounted reward optimality criteria in reinforcement learning](https://mlsanity.com/paper/2107.01348/) (2021)
- [Efficiency Ordering of Stochastic Gradient Descent](https://mlsanity.com/paper/2209.07446/) (2022)
- [Optimal Sample Complexity of Stable Discounted Markov Decision Processes](https://mlsanity.com/paper/2302.07477/) (2023)
- [Convex Q Learning in a Stochastic Environment: Extended Version](https://mlsanity.com/paper/2309.05105/) (2023)
- [Deflated Dynamics Value Iteration](https://mlsanity.com/paper/2407.10454/) (2024)
- [Accelerating Distributed Stochastic Optimization via Self-Repellent Random Walks](https://mlsanity.com/paper/2401.09665/) (2024)
- [Regularized Q-Learning with Linear Function Approximation](https://mlsanity.com/paper/2401.15196/) (2024)
- [An Empirical Study of Deep Reinforcement Learning in Continuing Tasks](https://mlsanity.com/paper/2501.06937/) (2025)
- [Stability and Sensitivity Analysis of Relative Temporal-Difference Learning: Extended Version](https://mlsanity.com/paper/2603.27874/) (2026)

## Abstract (arXiv)

Sample complexity bounds are a common performance metric in the Reinforcement Learning literature. In the discounted cost, infinite horizon setting, all of the known bounds have a factor that is a polynomial in 1/(1-γ), where γ < 1 is the discount factor. For a large discount factor, these bounds seem to imply that a very large number of samples is required to achieve an varepsilon-optimal policy. The objective of the present work is to introduce a new class of algorithms that have sample complexity uniformly bounded for all γ < 1. One may argue that this is impossible, due to a recent min-max lower bound. The explanation is that this previous lower bound is for a specific problem, which we modify, without compromising the ultimate objective of obtaining an varepsilon-optimal policy. Specifically, we show that the asymptotic covariance of the Q-learning algorithm with an optimized step-size sequence is a quadratic function of 1/(1-γ); an expected, and essentially known result. The new relative Q-learning algorithm proposed here is shown to have asymptotic covariance that is a quadratic in 1/(1- ρ^* γ), where 1 - ρ^* > 0 is an upper bound on the spectral gap of an optimal transition matrix.