# Exploration versus exploitation in reinforcement learning: a stochastic control approach

arXiv: https://arxiv.org/abs/1812.01552 · HTML: https://mlsanity.com/paper/1812.01552/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Dec 2018.

## Built on by (most cited first, 18 of 18)

- [Recent Advances in Reinforcement Learning in Finance](https://mlsanity.com/paper/2112.04553/) (2021)
- [Reinforcement learning for linear-convex models with jumps via stability analysis of feedback controls](https://mlsanity.com/paper/2104.09311/) (2021)
- [Co-rewarding: Stable Self-supervised RL for Eliciting Reasoning in Large Language Models](https://mlsanity.com/paper/2508.00410/) (2025)
- [Fast Policy Learning for Linear Quadratic Control with Entropy Regularization](https://mlsanity.com/paper/2311.14168/) (2023)
- [Good Learners Think Their Thinking: Generative PRM Makes Large Reasoning Model More Efficient Math Learner](https://mlsanity.com/paper/2507.23317/) (2025)
- [Learn the Ropes, Then Trust the Wins: Self-imitation with Progressive Exploration for Agentic Reinforcement Learning](https://mlsanity.com/paper/2509.22601/) (2025)
- [Token Hidden Reward: Steering Exploration-Exploitation in Group Relative Deep Reinforcement Learning](https://mlsanity.com/paper/2510.03669/) (2025)
- [Constrained Policy Gradient Method for Safe and Fast Reinforcement Learning: a Neural Tangent Kernel Based Approach](https://mlsanity.com/paper/2107.09139/) (2021)
- [Choquet regularization for reinforcement learning](https://mlsanity.com/paper/2208.08497/) (2022)
- [Policy Optimization finds Nash Equilibrium in Regularized General-Sum LQ Games](https://mlsanity.com/paper/2404.00045/) (2024)
- [A Generative Framework for Predictive Modeling of Multiple Chronic Conditions Using Graph Variational Autoencoder and Bandit-Optimized Graph Neural Network](https://mlsanity.com/paper/2409.13671/) (2024)
- [Entropy-Regularized Reinforcement Learning for Linear-Quadratic Stackelberg Differential Games in Regime-Switching Diffusion Models](https://mlsanity.com/paper/2606.28671/) (2026)
- [A Method to Improve the Performance of Reinforcement Learning Based on the Y Operator for a Class of Stochastic Differential Equation-Based Child-Mother Systems](https://mlsanity.com/paper/2311.04014/) (2023)
- [On the Effect of Instability on Learning Continuous-Time Linear Control Systems](https://mlsanity.com/paper/2409.11327/) (2024)
- [Two Kinds of Learning Algorithms for Continuous-Time VWAP Targeting Execution](https://mlsanity.com/paper/2411.06645/) (2024)
- [An Overview and Discussion on Using Large Language Models for Implementation Generation of Solutions to Open-Ended Problems](https://mlsanity.com/paper/2501.00562/) (2024)
- [Policy Transfer for Continuous-Time Reinforcement Learning: A (Rough) Differential Equation Approach](https://mlsanity.com/paper/2510.15165/) (2025)
- [DecNefSimulator: A Modular, Interpretable Framework for Decoded Neurofeedback Simulation Using Generative Models](https://mlsanity.com/paper/2511.14555/) (2025)

## Abstract (arXiv)

We consider reinforcement learning (RL) in continuous time and study the problem of achieving the best trade-off between exploration of a black box environment and exploitation of current knowledge. We propose an entropy-regularized reward function involving the differential entropy of the distributions of actions, and motivate and devise an exploratory formulation for the feature dynamics that captures repetitive learning under exploration. The resulting optimization problem is a revitalization of the classical relaxed stochastic control. We carry out a complete analysis of the problem in the linear--quadratic (LQ) setting and deduce that the optimal feedback control distribution for balancing exploitation and exploration is Gaussian. This in turn interprets and justifies the widely adopted Gaussian exploration in RL, beyond its simplicity for sampling. Moreover, the exploitation and exploration are captured, respectively and mutual-exclusively, by the mean and variance of the Gaussian distribution. We also find that a more random environment contains more learning opportunities in the sense that less exploration is needed. We characterize the cost of exploration, which, for the LQ case, is shown to be proportional to the entropy regularization weight and inversely proportional to the discount rate. Finally, as the weight of exploration decays to zero, we prove the convergence of the solution of the entropy-regularized LQ problem to the one of the classical LQ problem.