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Umbrella Reinforcement Learning -- computationally efficient tool for hard non-linear problems

Egor E. Nuzhin, Nikolai V. Brilliantov

TL;DR

Umbrella RL addresses hard reinforcement learning problems characterized by sparse rewards, state traps, and absence of a single terminal state by introducing a continuous ensemble of interacting agents and an entropy-regularized objective. The method combines umbrella sampling concepts with policy gradient, and harnesses three neural networks to estimate the policy, value, and ensemble density, with PDE-based, neural-network solutions to compute the gradient efficiently. Empirical results on the Multi-Valley Mountain Car and StandUp problems show superior performance, robustness to time-step discretization, and reduced memory requirements compared with traditional baselines such as PPO, RND, iLQR, and VI. This approach offers a scalable, universal framework for solving hard RL problems in continuous spaces, with potential extensions to model-free settings and discrete-time formulations.

Abstract

We report a novel, computationally efficient approach for solving hard nonlinear problems of reinforcement learning (RL). Here we combine umbrella sampling, from computational physics/chemistry, with optimal control methods. The approach is realized on the basis of neural networks, with the use of policy gradient. It outperforms, by computational efficiency and implementation universality, all available state-of-the-art algorithms, in application to hard RL problems with sparse reward, state traps and lack of terminal states. The proposed approach uses an ensemble of simultaneously acting agents, with a modified reward which includes the ensemble entropy, yielding an optimal exploration-exploitation balance.

Umbrella Reinforcement Learning -- computationally efficient tool for hard non-linear problems

TL;DR

Umbrella RL addresses hard reinforcement learning problems characterized by sparse rewards, state traps, and absence of a single terminal state by introducing a continuous ensemble of interacting agents and an entropy-regularized objective. The method combines umbrella sampling concepts with policy gradient, and harnesses three neural networks to estimate the policy, value, and ensemble density, with PDE-based, neural-network solutions to compute the gradient efficiently. Empirical results on the Multi-Valley Mountain Car and StandUp problems show superior performance, robustness to time-step discretization, and reduced memory requirements compared with traditional baselines such as PPO, RND, iLQR, and VI. This approach offers a scalable, universal framework for solving hard RL problems in continuous spaces, with potential extensions to model-free settings and discrete-time formulations.

Abstract

We report a novel, computationally efficient approach for solving hard nonlinear problems of reinforcement learning (RL). Here we combine umbrella sampling, from computational physics/chemistry, with optimal control methods. The approach is realized on the basis of neural networks, with the use of policy gradient. It outperforms, by computational efficiency and implementation universality, all available state-of-the-art algorithms, in application to hard RL problems with sparse reward, state traps and lack of terminal states. The proposed approach uses an ensemble of simultaneously acting agents, with a modified reward which includes the ensemble entropy, yielding an optimal exploration-exploitation balance.

Paper Structure

This paper contains 26 sections, 111 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Definition of hard problems
  3. The main ideas of Umbrella RL
  4. Policy gradient for Umbrella RL
  5. Effective computation of the policy gradient
  6. Algorithmic implementation of Umbrella RL
  7. Application of the Umbrella RL
  8. Multi-Valleys Mountain Car problem
  9. StandUp problem
  10. Numerical experiments. Implementation detail
  11. Results for the hard problems
  12. Umbrella RL for Multi-Value Mountain Car problem
  13. Umbrella RL for StandUp problem
  14. Discussion and conclusion
  15. Acknowledgements
  16. ...and 11 more sections

Figures (4)

  • Figure 1: Illustration of typical hard RL problems. Mountain Car (a) and Acrobot (b) problems are well-known for their complexity. Multi-Valley Mountain Car (c) and StandUp problem (d) possess all main features of hard problems. The bell-shaped shadowed region in panel (a) illustrates the distribution of ensemble of agents for Umbrella RL.
  • Figure 2: Average expected return for Multi-Valley Mountain Car problem: For UR, UR-NE, PPO, RND, and iLQR the simulation time step is $dt = 0.05$ in each training run of 50 episodes (10 runs in total). For VI algorithm additional time steps of $0.03$ and $0.01$ are included. The total simulation time is 100. The "Others" algorithms on the figure -- UR-NE, RND, iLQR (dt = 0.05) and VI (dt = 0.01) obviously fail.
  • Figure 3: The color map of the final action policy for different algorithms for Multi-Valley Mountain Car problem. Left panel -- UR, right panel -- VI.
  • Figure 4: Average expected return for StandUp problem: For UR, UR-NE, PPO, RND, and iLQR the simulation time step is $dt = 0.05$ in each training run of 50 episodes (10 runs in total). For VI algorithm additional time steps of $0.075$, $0.05$ and $0.01$ are included. The total simulation time is 200. The others algorithms of the figure -- RND, iLQR (dt = 0.05) and VI (dt = 0.01) obviosly fail.