# Bellman Residual Orthogonalization for Offline Reinforcement Learning

arXiv: https://arxiv.org/abs/2203.12786 · HTML: https://mlsanity.com/paper/2203.12786/

## Key facts

- Cited by 10 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.
- Builds on 51 of its 89 references that are in the map.
- First posted on arXiv 24 Mar 2022.

## Placed in

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

## Builds on (most cited first, 30 of 51)

- [Stabilizing Off-Policy Q-Learning via Bootstrapping Error Reduction](https://mlsanity.com/paper/1906.00949/) (2019)
- [Introduction to Online Convex Optimization](https://mlsanity.com/paper/1909.05207/) (2019)
- [AWAC: Accelerating Online Reinforcement Learning with Offline Datasets](https://mlsanity.com/paper/2006.09359/) (2020)
- [Behavior Regularized Offline Reinforcement Learning](https://mlsanity.com/paper/1911.11361/) (2019)
- [MOPO: Model-based Offline Policy Optimization](https://mlsanity.com/paper/2005.13239/) (2020)
- [MOReL : Model-Based Offline Reinforcement Learning](https://mlsanity.com/paper/2005.05951/) (2020)
- [Is Pessimism Provably Efficient for Offline RL?](https://mlsanity.com/paper/2012.15085/) (2020)
- [Doubly Robust Off-policy Value Evaluation for Reinforcement Learning](https://mlsanity.com/paper/1511.03722/) (2015)
- [Contextual Decision Processes with Low Bellman Rank are PAC-Learnable](https://mlsanity.com/paper/1610.09512/) (2016)
- [Information-Theoretic Considerations in Batch Reinforcement Learning](https://mlsanity.com/paper/1905.00360/) (2019)
- [Data-Efficient Off-Policy Policy Evaluation for Reinforcement Learning](https://mlsanity.com/paper/1604.00923/) (2016)
- [Way Off-Policy Batch Deep Reinforcement Learning of Implicit Human Preferences in Dialog](https://mlsanity.com/paper/1907.00456/) (2019)
- [Bridging Offline Reinforcement Learning and Imitation Learning: A Tale of Pessimism](https://mlsanity.com/paper/2103.12021/) (2021)
- [Critic Regularized Regression](https://mlsanity.com/paper/2006.15134/) (2020)
- [Bellman-consistent Pessimism for Offline Reinforcement Learning](https://mlsanity.com/paper/2106.06926/) (2021)
- [Breaking the Curse of Horizon: Infinite-Horizon Off-Policy Estimation](https://mlsanity.com/paper/1810.12429/) (2018)
- [Keep Doing What Worked: Behavioral Modelling Priors for Offline Reinforcement Learning](https://mlsanity.com/paper/2002.08396/) (2020)
- [Tighter Problem-Dependent Regret Bounds in Reinforcement Learning without Domain Knowledge using Value Function Bounds](https://mlsanity.com/paper/1901.00210/) (2019)
- [AlgaeDICE: Policy Gradient from Arbitrary Experience](https://mlsanity.com/paper/1912.02074/) (2019)
- [Pessimistic Model-based Offline Reinforcement Learning under Partial Coverage](https://mlsanity.com/paper/2107.06226/) (2021)
- [Minimax Weight and Q-Function Learning for Off-Policy Evaluation](https://mlsanity.com/paper/1910.12809/) (2019)
- [Safe Policy Improvement with Baseline Bootstrapping](https://mlsanity.com/paper/1712.06924/) (2017)
- [More Robust Doubly Robust Off-policy Evaluation](https://mlsanity.com/paper/1802.03493/) (2018)
- [What are the Statistical Limits of Offline RL with Linear Function Approximation?](https://mlsanity.com/paper/2010.11895/) (2020)
- [Offline Reinforcement Learning with Realizability and Single-policy Concentrability](https://mlsanity.com/paper/2202.04634/) (2022)
- [GenDICE: Generalized Offline Estimation of Stationary Values](https://mlsanity.com/paper/2002.09072/) (2020)
- [Minimax-Optimal Off-Policy Evaluation with Linear Function Approximation](https://mlsanity.com/paper/2002.09516/) (2020)
- [Towards Optimal Off-Policy Evaluation for Reinforcement Learning with Marginalized Importance Sampling](https://mlsanity.com/paper/1906.03393/) (2019)
- [Provable Benefits of Actor-Critic Methods for Offline Reinforcement Learning](https://mlsanity.com/paper/2108.08812/) (2021)
- [Batch Value-function Approximation with Only Realizability](https://mlsanity.com/paper/2008.04990/) (2020)

## Built on by (most cited first, 10 of 10)

- [On the Statistical Efficiency of Reward-Free Exploration in Non-Linear RL](https://mlsanity.com/paper/2206.10770/) (2022)
- [Future-Dependent Value-Based Off-Policy Evaluation in POMDPs](https://mlsanity.com/paper/2207.13081/) (2022)
- [When is Realizability Sufficient for Off-Policy Reinforcement Learning?](https://mlsanity.com/paper/2211.05311/) (2022)
- [Policy Finetuning in Reinforcement Learning via Design of Experiments using Offline Data](https://mlsanity.com/paper/2307.04354/) (2023)
- [Offline Minimax Soft-Q-learning Under Realizability and Partial Coverage](https://mlsanity.com/paper/2302.02392/) (2023)
- [A Primal-Dual-Critic Algorithm for Offline Constrained Reinforcement Learning](https://mlsanity.com/paper/2306.07818/) (2023)
- [Policy evaluation from a single path: Multi-step methods, mixing and mis-specification](https://mlsanity.com/paper/2211.03899/) (2022)
- [Fitted Q-Evaluation without Bellman Completeness via Occupancy Weighting](https://mlsanity.com/paper/2512.23805/) (2025)
- [A Covering Framework for Offline POMDPs Learning using Belief Space Metric](https://mlsanity.com/paper/2603.03191/) (2026)
- [Continuous-time reinforcement learning: ellipticity enables model-free value function approximation](https://mlsanity.com/paper/2602.06930/) (2026)

## Abstract (arXiv)

We propose and analyze a reinforcement learning principle that approximates the Bellman equations by enforcing their validity only along an user-defined space of test functions. Focusing on applications to model-free offline RL with function approximation, we exploit this principle to derive confidence intervals for off-policy evaluation, as well as to optimize over policies within a prescribed policy class. We prove an oracle inequality on our policy optimization procedure in terms of a trade-off between the value and uncertainty of an arbitrary comparator policy. Different choices of test function spaces allow us to tackle different problems within a common framework. We characterize the loss of efficiency in moving from on-policy to off-policy data using our procedures, and establish connections to concentrability coefficients studied in past work. We examine in depth the implementation of our methods with linear function approximation, and provide theoretical guarantees with polynomial-time implementations even when Bellman closure does not hold.