# UCB Momentum Q-learning: Correcting the bias without forgetting

arXiv: https://arxiv.org/abs/2103.01312 · HTML: https://mlsanity.com/paper/2103.01312/

## Key facts

- Cited by 46 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 26 references that are in the map.
- First posted on arXiv 1 Mar 2021.

## Builds on (most cited first, 17 of 17)

- [Minimax Regret Bounds for Reinforcement Learning](https://mlsanity.com/paper/1703.05449/) (2017)
- [Is Q-learning Provably Efficient?](https://mlsanity.com/paper/1807.03765/) (2018)
- [Tighter Problem-Dependent Regret Bounds in Reinforcement Learning without Domain Knowledge using Value Function Bounds](https://mlsanity.com/paper/1901.00210/) (2019)
- [Unifying PAC and Regret: Uniform PAC Bounds for Episodic Reinforcement Learning](https://mlsanity.com/paper/1703.07710/) (2017)
- [Almost Optimal Model-Free Reinforcement Learning via Reference-Advantage Decomposition](https://mlsanity.com/paper/2004.10019/) (2020)
- [Episodic Reinforcement Learning in Finite MDPs: Minimax Lower Bounds Revisited](https://mlsanity.com/paper/2010.03531/) (2020)
- [Is Reinforcement Learning More Difficult Than Bandits? A Near-optimal Algorithm Escaping the Curse of Horizon](https://mlsanity.com/paper/2009.13503/) (2020)
- [Fast active learning for pure exploration in reinforcement learning](https://mlsanity.com/paper/2007.13442/) (2020)
- [Adaptive Reward-Free Exploration](https://mlsanity.com/paper/2006.06294/) (2020)
- [Efficient Bias-Span-Constrained Exploration-Exploitation in Reinforcement Learning](https://mlsanity.com/paper/1802.04020/) (2018)
- [Variance Reduced Value Iteration and Faster Algorithms for Solving Markov Decision Processes](https://mlsanity.com/paper/1710.09988/) (2017)
- [Variance-Aware Regret Bounds for Undiscounted Reinforcement Learning in MDPs](https://mlsanity.com/paper/1803.01626/) (2018)
- [Tight Regret Bounds for Model-Based Reinforcement Learning with Greedy Policies](https://mlsanity.com/paper/1905.11527/) (2019)
- [Is Long Horizon Reinforcement Learning More Difficult Than Short Horizon Reinforcement Learning?](https://mlsanity.com/paper/2005.00527/) (2020)
- [Near-Optimal Time and Sample Complexities for Solving Discounted Markov Decision Process with a Generative Model](https://mlsanity.com/paper/1806.01492/) (2018)
- [Adaptive Discretization for Model-Based Reinforcement Learning](https://mlsanity.com/paper/2007.00717/) (2020)
- [Momentum Q-learning with Finite-Sample Convergence Guarantee](https://mlsanity.com/paper/2007.15418/) (2020)

## Built on by (most cited first, 30 of 46)

- [Unpacking Reward Shaping: Understanding the Benefits of Reward Engineering on Sample Complexity](https://mlsanity.com/paper/2210.09579/) (2022)
- [Breaking the Sample Complexity Barrier to Regret-Optimal Model-Free Reinforcement Learning](https://mlsanity.com/paper/2110.04645/) (2021)
- [DrM: Mastering Visual Reinforcement Learning through Dormant Ratio Minimization](https://mlsanity.com/paper/2310.19668/) (2023)
- [Settling the Sample Complexity of Online Reinforcement Learning](https://mlsanity.com/paper/2307.13586/) (2023)
- [The Efficacy of Pessimism in Asynchronous Q-Learning](https://mlsanity.com/paper/2203.07368/) (2022)
- [MADE: Exploration via Maximizing Deviation from Explored Regions](https://mlsanity.com/paper/2106.10268/) (2021)
- [On Improving Model-Free Algorithms for Decentralized Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2110.05707/) (2021)
- [Stochastic Shortest Path: Minimax, Parameter-Free and Towards Horizon-Free Regret](https://mlsanity.com/paper/2104.11186/) (2021)
- [Horizon-Free Reinforcement Learning in Polynomial Time: the Power of Stationary Policies](https://mlsanity.com/paper/2203.12922/) (2022)
- [Distributionally Robust Reinforcement Learning with Interactive Data Collection: Fundamental Hardness and Near-Optimal Algorithms](https://mlsanity.com/paper/2404.03578/) (2024)
- [Regret Bounds for Information-Directed Reinforcement Learning](https://mlsanity.com/paper/2206.04640/) (2022)
- [From Dirichlet to Rubin: Optimistic Exploration in RL without Bonuses](https://mlsanity.com/paper/2205.07704/) (2022)
- [Settling the Horizon-Dependence of Sample Complexity in Reinforcement Learning](https://mlsanity.com/paper/2111.00633/) (2021)
- [Posterior Sampling for Deep Reinforcement Learning](https://mlsanity.com/paper/2305.00477/) (2023)
- [Reward-agnostic Fine-tuning: Provable Statistical Benefits of Hybrid Reinforcement Learning](https://mlsanity.com/paper/2305.10282/) (2023)
- [ACE : Off-Policy Actor-Critic with Causality-Aware Entropy Regularization](https://mlsanity.com/paper/2402.14528/) (2024)
- [Minimax-Optimal Reward-Agnostic Exploration in Reinforcement Learning](https://mlsanity.com/paper/2304.07278/) (2023)
- [Nearly Optimal Policy Optimization with Stable at Any Time Guarantee](https://mlsanity.com/paper/2112.10935/) (2021)
- [Optimistic Posterior Sampling for Reinforcement Learning with Few Samples and Tight Guarantees](https://mlsanity.com/paper/2209.14414/) (2022)
- [Probabilistic Inference in Reinforcement Learning Done Right](https://mlsanity.com/paper/2311.13294/) (2023)
- [A Reduction-Based Framework for Conservative Bandits and Reinforcement Learning](https://mlsanity.com/paper/2106.11692/) (2021)
- [The Sample-Communication Complexity Trade-off in Federated Q-Learning](https://mlsanity.com/paper/2408.16981/) (2024)
- [Regret-Optimal Model-Free Reinforcement Learning for Discounted MDPs with Short Burn-In Time](https://mlsanity.com/paper/2305.15546/) (2023)
- [Gap-Dependent Bounds for Q-Learning using Reference-Advantage Decomposition](https://mlsanity.com/paper/2410.07574/) (2024)
- [Hybrid Transfer Reinforcement Learning: Provable Sample Efficiency from Shifted-Dynamics Data](https://mlsanity.com/paper/2411.03810/) (2024)
- [Combinatorial Multivariant Multi-Armed Bandits with Applications to Episodic Reinforcement Learning and Beyond](https://mlsanity.com/paper/2406.01386/) (2024)
- [ORVIT: Near-Optimal Online Distributionally Robust Reinforcement Learning](https://mlsanity.com/paper/2508.03768/) (2025)
- [Adaptive Multi-Goal Exploration](https://mlsanity.com/paper/2111.12045/) (2021)
- [Gap-Dependent Bounds for Federated Q-learning](https://mlsanity.com/paper/2502.02859/) (2025)
- [Sharp Gap-Dependent Variance-Aware Regret Bounds for Tabular MDPs](https://mlsanity.com/paper/2506.06521/) (2025)

## Abstract (arXiv)

We propose UCBMQ, Upper Confidence Bound Momentum Q-learning, a new algorithm for reinforcement learning in tabular and possibly stage-dependent, episodic Markov decision process. UCBMQ is based on Q-learning where we add a momentum term and rely on the principle of optimism in face of uncertainty to deal with exploration. Our new technical ingredient of UCBMQ is the use of momentum to correct the bias that Q-learning suffers while, at the same time, limiting the impact it has on the second-order term of the regret. For UCBMQ, we are able to guarantee a regret of at most O(√H3SAT+ H4 S A ) where H is the length of an episode, S the number of states, A the number of actions, T the number of episodes and ignoring terms in poly-log(SAHT). Notably, UCBMQ is the first algorithm that simultaneously matches the lower bound of Omega(√H3SAT) for large enough T and has a second-order term (with respect to the horizon T) that scales only linearly with the number of states S.