# Improved Regret for Efficient Online Reinforcement Learning with Linear Function Approximation

arXiv: https://arxiv.org/abs/2301.13087 · HTML: https://mlsanity.com/paper/2301.13087/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 45 of its 79 references that are in the map.
- First posted on arXiv 30 Jan 2023.

## Builds on (most cited first, 30 of 45)

- [Continuous control with deep reinforcement learning](https://mlsanity.com/paper/1509.02971/) (2015)
- [Playing Atari with Deep Reinforcement Learning](https://mlsanity.com/paper/1312.5602/) (2013)
- [High-Dimensional Continuous Control Using Generalized Advantage Estimation](https://mlsanity.com/paper/1506.02438/) (2015)
- [Introduction to Online Convex Optimization](https://mlsanity.com/paper/1909.05207/) (2019)
- [Solving Rubik's Cube with a Robot Hand](https://mlsanity.com/paper/1910.07113/) (2019)
- [Deep Reinforcement Learning for Autonomous Driving: A Survey](https://mlsanity.com/paper/2002.00444/) (2020)
- [Provably Efficient Reinforcement Learning with Linear Function Approximation](https://mlsanity.com/paper/1907.05388/) (2019)
- [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)
- [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)
- [Model-Based Reinforcement Learning with Value-Targeted Regression](https://mlsanity.com/paper/2006.01107/) (2020)
- [Bellman Eluder Dimension: New Rich Classes of RL Problems, and Sample-Efficient Algorithms](https://mlsanity.com/paper/2102.00815/) (2021)
- [Provably Efficient Exploration in Policy Optimization](https://mlsanity.com/paper/1912.05830/) (2019)
- [Reinforcement Learning in Feature Space: Matrix Bandit, Kernels, and Regret Bound](https://mlsanity.com/paper/1905.10389/) (2019)
- [Nearly Minimax Optimal Reinforcement Learning for Linear Mixture Markov Decision Processes](https://mlsanity.com/paper/2012.08507/) (2020)
- [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)
- [Unifying PAC and Regret: Uniform PAC Bounds for Episodic Reinforcement Learning](https://mlsanity.com/paper/1703.07710/) (2017)
- [Policy Mirror Descent for Reinforcement Learning: Linear Convergence, New Sampling Complexity, and Generalized Problem Classes](https://mlsanity.com/paper/2102.00135/) (2021)
- [Global Optimality Guarantees For Policy Gradient Methods](https://mlsanity.com/paper/1906.01786/) (2019)
- [Sample Complexity of Episodic Fixed-Horizon Reinforcement Learning](https://mlsanity.com/paper/1510.08906/) (2015)
- [Provably Efficient Reinforcement Learning for Discounted MDPs with Feature Mapping](https://mlsanity.com/paper/2006.13165/) (2020)
- [Frequentist Regret Bounds for Randomized Least-Squares Value Iteration](https://mlsanity.com/paper/1911.00567/) (2019)
- [Sample Complexity of Reinforcement Learning using Linearly Combined Model Ensembles](https://mlsanity.com/paper/1910.10597/) (2019)
- [PC-PG: Policy Cover Directed Exploration for Provable Policy Gradient Learning](https://mlsanity.com/paper/2007.08459/) (2020)
- [Learning Adversarial MDPs with Bandit Feedback and Unknown Transition](https://mlsanity.com/paper/1912.01192/) (2019)
- [Policy Mirror Descent for Regularized Reinforcement Learning: A Generalized Framework with Linear Convergence](https://mlsanity.com/paper/2105.11066/) (2021)
- [Online Convex Optimization in Adversarial Markov Decision Processes](https://mlsanity.com/paper/1905.07773/) (2019)
- [Optimistic Policy Optimization with Bandit Feedback](https://mlsanity.com/paper/2002.08243/) (2020)

## Built on by (most cited first, 17 of 17)

- [Is RLHF More Difficult than Standard RL?](https://mlsanity.com/paper/2306.14111/) (2023)
- [A Theoretical Analysis of Optimistic Proximal Policy Optimization in Linear Markov Decision Processes](https://mlsanity.com/paper/2305.08841/) (2023)
- [Bypassing the Simulator: Near-Optimal Adversarial Linear Contextual Bandits](https://mlsanity.com/paper/2309.00814/) (2023)
- [Imitation Learning in Discounted Linear MDPs without exploration assumptions](https://mlsanity.com/paper/2405.02181/) (2024)
- [Rate-Optimal Policy Optimization for Linear Markov Decision Processes](https://mlsanity.com/paper/2308.14642/) (2023)
- [Towards Optimal Regret in Adversarial Linear MDPs with Bandit Feedback](https://mlsanity.com/paper/2310.11550/) (2023)
- [Delay-Adapted Policy Optimization and Improved Regret for Adversarial MDP with Delayed Bandit Feedback](https://mlsanity.com/paper/2305.07911/) (2023)
- [Improved Algorithm for Adversarial Linear Mixture MDPs with Bandit Feedback and Unknown Transition](https://mlsanity.com/paper/2403.04568/) (2024)
- [Learning Adversarial Low-rank Markov Decision Processes with Unknown Transition and Full-information Feedback](https://mlsanity.com/paper/2311.07876/) (2023)
- [Enhancing PPO with Trajectory-Aware Hybrid Policies](https://mlsanity.com/paper/2502.15968/) (2025)
- [An Improved Algorithm for Adversarial Linear Contextual Bandits via Reduction](https://mlsanity.com/paper/2508.11931/) (2025)
- [Beating Adversarial Low-Rank MDPs with Unknown Transition and Bandit Feedback](https://mlsanity.com/paper/2411.06739/) (2024)
- [An Improved Model-Free Decision-Estimation Coefficient with Applications in Adversarial MDPs](https://mlsanity.com/paper/2510.08882/) (2025)
- [Optimal Regret for Policy Optimization in Contextual Bandits](https://mlsanity.com/paper/2602.13700/) (2026)
- [Policy Regret for Embedding Model Routing: Contextual Bandits with Low-Rank Experts](https://mlsanity.com/paper/2606.14929/) (2026)
- [Primal-Dual Policy Optimization for Linear CMDPs with Adversarial Losses](https://mlsanity.com/paper/2605.11535/) (2026)
- [Follow the Perturbed Leader for Linear Bandits: √T Regret over General Convex Sets](https://mlsanity.com/paper/2610.17818/) (2026)

## Abstract (arXiv)

We study reinforcement learning with linear function approximation and adversarially changing cost functions, a setup that has mostly been considered under simplifying assumptions such as full information feedback or exploratory conditions.We present a computationally efficient policy optimization algorithm for the challenging general setting of unknown dynamics and bandit feedback, featuring a combination of mirror-descent and least squares policy evaluation in an auxiliary MDP used to compute exploration bonuses.Our algorithm obtains an widetilde O(K6/7) regret bound, improving significantly over previous state-of-the-art of widetilde O (K14/15) in this setting. In addition, we present a version of the same algorithm under the assumption a simulator of the environment is available to the learner (but otherwise no exploratory assumptions are made), and prove it obtains state-of-the-art regret of widetilde O (K2/3).