# Exponential Hardness of Reinforcement Learning with Linear Function Approximation

arXiv: https://arxiv.org/abs/2302.12940 · HTML: https://mlsanity.com/paper/2302.12940/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 41 references that are in the map.
- First posted on arXiv 25 Feb 2023.

## Builds on (most cited first, 26 of 26)

- [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)
- [Is a Good Representation Sufficient for Sample Efficient Reinforcement Learning?](https://mlsanity.com/paper/1910.03016/) (2019)
- [Learning with Good Feature Representations in Bandits and in RL with a Generative Model](https://mlsanity.com/paper/1911.07676/) (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)
- [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)
- [Learning in Observable POMDPs, without Computationally Intractable Oracles](https://mlsanity.com/paper/2206.03446/) (2022)
- [Planning in Observable POMDPs in Quasipolynomial Time](https://mlsanity.com/paper/2201.04735/) (2022)
- [Efficient Local Planning with Linear Function Approximation](https://mlsanity.com/paper/2108.05533/) (2021)
- [Provably Efficient Reinforcement Learning with Aggregated States](https://mlsanity.com/paper/1912.06366/) (2019)
- [Efficient Reinforcement Learning in Deterministic Systems with Value Function Generalization](https://mlsanity.com/paper/1307.4847/) (2013)
- [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)
- [Computationally Efficient PAC RL in POMDPs with Latent Determinism and Conditional Embeddings](https://mlsanity.com/paper/2206.12081/) (2022)
- [Confident Approximate Policy Iteration for Efficient Local Planning in qπ-realizable MDPs](https://mlsanity.com/paper/2210.15755/) (2022)
- [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)
- [Computational-Statistical Gaps in Reinforcement Learning](https://mlsanity.com/paper/2202.05444/) (2022)

## Built on by (most cited first, 4 of 4)

- [Linear Q-Learning Does Not Diverge in L2: Convergence Rates to a Bounded Set](https://mlsanity.com/paper/2501.19254/) (2025)
- [Exploration is Harder than Prediction: Cryptographically Separating Reinforcement Learning from Supervised Learning](https://mlsanity.com/paper/2404.03774/) (2024)
- [Linear Bellman Completeness Suffices for Efficient Online Reinforcement Learning with Few Actions](https://mlsanity.com/paper/2406.11640/) (2024)
- [Computational Hardness of Reinforcement Learning with Partial qπ-Realizability](https://mlsanity.com/paper/2510.21888/) (2025)

## Abstract (arXiv)

A fundamental question in reinforcement learning theory is: suppose the optimal value functions are linear in given features, can we learn them efficiently? This problem's counterpart in supervised learning, linear regression, can be solved both statistically and computationally efficiently. Therefore, it was quite surprising when a recent work citekane2022computational showed a computational-statistical gap for linear reinforcement learning: even though there are polynomial sample-complexity algorithms, unless NP = RP, there are no polynomial time algorithms for this setting. In this work, we build on their result to show a computational lower bound, which is exponential in feature dimension and horizon, for linear reinforcement learning under the Randomized Exponential Time Hypothesis. To prove this we build a round-based game where in each round the learner is searching for an unknown vector in a unit hypercube. The rewards in this game are chosen such that if the learner achieves large reward, then the learner's actions can be used to simulate solving a variant of 3-SAT, where (a) each variable shows up in a bounded number of clauses (b) if an instance has no solutions then it also has no solutions that satisfy more than (1-ε)-fraction of clauses. We use standard reductions to show this 3-SAT variant is approximately as hard as 3-SAT. Finally, we also show a lower bound optimized for horizon dependence that almost matches the best known upper bound of exp(√H).