# EMaQ: Expected-Max Q-Learning Operator for Simple Yet Effective Offline and Online RL

arXiv: https://arxiv.org/abs/2007.11091 · HTML: https://mlsanity.com/paper/2007.11091/

## Key facts

- Cited by 115 papers in the map (data through 14 Oct 2026).
- Placed in Offline RL algorithms, one of 13 directions in Offline RL and imitation learning from data.
- First posted on arXiv 21 Jul 2020.

## Placed in

- Offline RL and imitation learning from data / [Offline RL algorithms](https://mlsanity.com/offlinerl/offline-rl-algorithms/)

## Built on by (most cited first, 30 of 115)

- [Offline Reinforcement Learning with Implicit Q-Learning](https://mlsanity.com/paper/2110.06169/) (2021)
- [A Minimalist Approach to Offline Reinforcement Learning](https://mlsanity.com/paper/2106.06860/) (2021)
- [What Matters in Learning from Offline Human Demonstrations for Robot Manipulation](https://mlsanity.com/paper/2108.03298/) (2021)
- [Offline Reinforcement Learning as One Big Sequence Modeling Problem](https://mlsanity.com/paper/2106.02039/) (2021)
- [COMBO: Conservative Offline Model-Based Policy Optimization](https://mlsanity.com/paper/2102.08363/) (2021)
- [IDQL: Implicit Q-Learning as an Actor-Critic Method with Diffusion Policies](https://mlsanity.com/paper/2304.10573/) (2023)
- [Bridging Offline Reinforcement Learning and Imitation Learning: A Tale of Pessimism](https://mlsanity.com/paper/2103.12021/) (2021)
- [Offline Reinforcement Learning with Fisher Divergence Critic Regularization](https://mlsanity.com/paper/2103.08050/) (2021)
- [Cal-QL: Calibrated Offline RL Pre-Training for Efficient Online Fine-Tuning](https://mlsanity.com/paper/2303.05479/) (2023)
- [Offline-to-Online Reinforcement Learning via Balanced Replay and Pessimistic Q-Ensemble](https://mlsanity.com/paper/2107.00591/) (2021)
- [Offline Reinforcement Learning via High-Fidelity Generative Behavior Modeling](https://mlsanity.com/paper/2209.14548/) (2022)
- [OPAL: Offline Primitive Discovery for Accelerating Offline Reinforcement Learning](https://mlsanity.com/paper/2010.13611/) (2020)
- [Pessimistic Model-based Offline Reinforcement Learning under Partial Coverage](https://mlsanity.com/paper/2107.06226/) (2021)
- [HIQL: Offline Goal-Conditioned RL with Latent States as Actions](https://mlsanity.com/paper/2307.11949/) (2023)
- [Offline RL for Natural Language Generation with Implicit Language Q Learning](https://mlsanity.com/paper/2206.11871/) (2022)
- [Pessimistic Bootstrapping for Uncertainty-Driven Offline Reinforcement Learning](https://mlsanity.com/paper/2202.11566/) (2022)
- [Mildly Conservative Q-Learning for Offline Reinforcement Learning](https://mlsanity.com/paper/2206.04745/) (2022)
- [Reinforcement Learning with Action Chunking](https://mlsanity.com/paper/2507.07969/) (2025)
- [Generalized Decision Transformer for Offline Hindsight Information Matching](https://mlsanity.com/paper/2111.10364/) (2021)
- [Human-centric Dialog Training via Offline Reinforcement Learning](https://mlsanity.com/paper/2010.05848/) (2020)
- [Is Value Learning Really the Main Bottleneck in Offline RL?](https://mlsanity.com/paper/2406.09329/) (2024)
- [Policy Agnostic RL: Offline RL and Online RL Fine-Tuning of Any Class and Backbone](https://mlsanity.com/paper/2412.06685/) (2024)
- [Can Wikipedia Help Offline Reinforcement Learning?](https://mlsanity.com/paper/2201.12122/) (2022)
- [Why So Pessimistic? Estimating Uncertainties for Offline RL through Ensembles, and Why Their Independence Matters](https://mlsanity.com/paper/2205.13703/) (2022)
- [Supported Policy Optimization for Offline Reinforcement Learning](https://mlsanity.com/paper/2202.06239/) (2022)
- [How to Leverage Unlabeled Data in Offline Reinforcement Learning](https://mlsanity.com/paper/2202.01741/) (2022)
- [VRL3: A Data-Driven Framework for Visual Deep Reinforcement Learning](https://mlsanity.com/paper/2202.10324/) (2022)
- [Offline RL Policies Should be Trained to be Adaptive](https://mlsanity.com/paper/2207.02200/) (2022)
- [Offline Reinforcement Learning as Anti-Exploration](https://mlsanity.com/paper/2106.06431/) (2021)
- [Diffusion Policies creating a Trust Region for Offline Reinforcement Learning](https://mlsanity.com/paper/2405.19690/) (2024)

## Abstract (arXiv)

Off-policy reinforcement learning holds the promise of sample-efficient learning of decision-making policies by leveraging past experience. However, in the offline RL setting -- where a fixed collection of interactions are provided and no further interactions are allowed -- it has been shown that standard off-policy RL methods can significantly underperform. Recently proposed methods often aim to address this shortcoming by constraining learned policies to remain close to the given dataset of interactions. In this work, we closely investigate an important simplification of BCQ -- a prior approach for offline RL -- which removes a heuristic design choice and naturally restricts extracted policies to remain exactly within the support of a given behavior policy. Importantly, in contrast to their original theoretical considerations, we derive this simplified algorithm through the introduction of a novel backup operator, Expected-Max Q-Learning (EMaQ), which is more closely related to the resulting practical algorithm. Specifically, in addition to the distribution support, EMaQ explicitly considers the number of samples and the proposal distribution, allowing us to derive new sub-optimality bounds which can serve as a novel measure of complexity for offline RL problems. In the offline RL setting -- the main focus of this work -- EMaQ matches and outperforms prior state-of-the-art in the D4RL benchmarks. In the online RL setting, we demonstrate that EMaQ is competitive with Soft Actor Critic. The key contributions of our empirical findings are demonstrating the importance of careful generative model design for estimating behavior policies, and an intuitive notion of complexity for offline RL problems. With its simple interpretation and fewer moving parts, such as no explicit function approximator representing the policy, EMaQ serves as a strong yet easy to implement baseline for future work.