# Computational-Statistical Gaps in Reinforcement Learning

arXiv: https://arxiv.org/abs/2202.05444 · HTML: https://mlsanity.com/paper/2202.05444/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 55 references that are in the map.
- First posted on arXiv 11 Feb 2022.

## Builds on (most cited first, 21 of 21)

- [Playing Atari with Deep Reinforcement Learning](https://mlsanity.com/paper/1312.5602/) (2013)
- [Dota 2 with Large Scale Deep Reinforcement Learning](https://mlsanity.com/paper/1912.06680/) (2019)
- [Provably Efficient Reinforcement Learning with Linear Function Approximation](https://mlsanity.com/paper/1907.05388/) (2019)
- [Contextual Decision Processes with Low Bellman Rank are PAC-Learnable](https://mlsanity.com/paper/1610.09512/) (2016)
- [Model-Based Reinforcement Learning with Value-Targeted Regression](https://mlsanity.com/paper/2006.01107/) (2020)
- [FLAMBE: Structural Complexity and Representation Learning of Low Rank MDPs](https://mlsanity.com/paper/2006.10814/) (2020)
- [Bellman Eluder Dimension: New Rich Classes of RL Problems, and Sample-Efficient Algorithms](https://mlsanity.com/paper/2102.00815/) (2021)
- [The Statistical Complexity of Interactive Decision Making](https://mlsanity.com/paper/2112.13487/) (2021)
- [Learning Near Optimal Policies with Low Inherent Bellman Error](https://mlsanity.com/paper/2003.00153/) (2020)
- [Bilinear Classes: A Structural Framework for Provable Generalization in RL](https://mlsanity.com/paper/2103.10897/) (2021)
- [Provably efficient RL with Rich Observations via Latent State Decoding](https://mlsanity.com/paper/1901.09018/) (2019)
- [Provably Efficient Reinforcement Learning for Discounted MDPs with Feature Mapping](https://mlsanity.com/paper/2006.13165/) (2020)
- [Sample Complexity of Reinforcement Learning using Linearly Combined Model Ensembles](https://mlsanity.com/paper/1910.10597/) (2019)
- [On Oracle-Efficient PAC RL with Rich Observations](https://mlsanity.com/paper/1803.00606/) (2018)
- [Exponential Lower Bounds for Planning in MDPs With Linearly-Realizable Optimal Action-Value Functions](https://mlsanity.com/paper/2010.01374/) (2020)
- [An Exponential Lower Bound for Linearly-Realizable MDPs with Constant Suboptimality Gap](https://mlsanity.com/paper/2103.12690/) (2021)
- [Provably Efficient Reinforcement Learning with Aggregated States](https://mlsanity.com/paper/1912.06366/) (2019)
- [Agnostic Q-learning with Function Approximation in Deterministic Systems: Tight Bounds on Approximation Error and Sample Complexity](https://mlsanity.com/paper/2002.07125/) (2020)
- [On Query-efficient Planning in MDPs under Linear Realizability of the Optimal State-value Function](https://mlsanity.com/paper/2102.02049/) (2021)
- [TensorPlan and the Few Actions Lower Bound for Planning in MDPs under Linear Realizability of Optimal Value Functions](https://mlsanity.com/paper/2110.02195/) (2021)
- [Detection in the stochastic block model with multiple clusters: proof of the achievability conjectures, acyclic BP, and the information-computation gap](https://mlsanity.com/paper/1512.09080/) (2015)

## Built on by (most cited first, 3 of 3)

- [Hybrid RL: Using Both Offline and Online Data Can Make RL Efficient](https://mlsanity.com/paper/2210.06718/) (2022)
- [Learning in Observable POMDPs, without Computationally Intractable Oracles](https://mlsanity.com/paper/2206.03446/) (2022)
- [Exponential Hardness of Reinforcement Learning with Linear Function Approximation](https://mlsanity.com/paper/2302.12940/) (2023)

## Abstract (arXiv)

Reinforcement learning with function approximation has recently achieved tremendous results in applications with large state spaces. This empirical success has motivated a growing body of theoretical work proposing necessary and sufficient conditions under which efficient reinforcement learning is possible. From this line of work, a remarkably simple minimal sufficient condition has emerged for sample efficient reinforcement learning: MDPs with optimal value function V^* and Q^* linear in some known low-dimensional features. In this setting, recent works have designed sample efficient algorithms which require a number of samples polynomial in the feature dimension and independent of the size of state space. They however leave finding computationally efficient algorithms as future work and this is considered a major open problem in the community. In this work, we make progress on this open problem by presenting the first computational lower bound for RL with linear function approximation: unless NP=RP, no randomized polynomial time algorithm exists for deterministic transition MDPs with a constant number of actions and linear optimal value functions. To prove this, we show a reduction from Unique-Sat, where we convert a CNF formula into an MDP with deterministic transitions, constant number of actions and low dimensional linear optimal value functions. This result also exhibits the first computational-statistical gap in reinforcement learning with linear function approximation, as the underlying statistical problem is information-theoretically solvable with a polynomial number of queries, but no computationally efficient algorithm exists unless NP=RP. Finally, we also prove a quasi-polynomial time lower bound under the Randomized Exponential Time Hypothesis.