Hamilton-Jacobi Deep Q-Learning for Deterministic Continuous-Time Systems with Lipschitz Continuous Controls

• URL: http://arxiv.org/abs/2010.14087v1
• Date: Tue, 27 Oct 2020 06:11:04 GMT
• Title: Hamilton-Jacobi Deep Q-Learning for Deterministic Continuous-Time Systems with Lipschitz Continuous Controls
• Authors: Jeongho Kim, Jaeuk Shin, Insoon Yang
• Abstract summary: We propose Q-learning algorithms for continuous-time deterministic optimal control problems with Lipschitz continuous controls. A novel semi-discrete version of the HJB equation is proposed to design a Q-learning algorithm that uses data collected in discrete time without discretizing or approximating the system dynamics.
• Score: 2.922007656878633
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper, we propose Q-learning algorithms for continuous-time deterministic optimal control problems with Lipschitz continuous controls. Our method is based on a new class of Hamilton-Jacobi-Bellman (HJB) equations derived from applying the dynamic programming principle to continuous-time Q-functions. A novel semi-discrete version of the HJB equation is proposed to design a Q-learning algorithm that uses data collected in discrete time without discretizing or approximating the system dynamics. We identify the condition under which the Q-function estimated by this algorithm converges to the optimal Q-function. For practical implementation, we propose the Hamilton-Jacobi DQN, which extends the idea of deep Q-networks (DQN) to our continuous control setting. This approach does not require actor networks or numerical solutions to optimization problems for greedy actions since the HJB equation provides a simple characterization of optimal controls via ordinary differential equations. We empirically demonstrate the performance of our method through benchmark tasks and high-dimensional linear-quadratic problems.

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