Continuous Tensor Train-Based Dynamic Programming for High-Dimensional Zero-Sum Differential Games

Abstract

© 2018 AACC. Zero-sum differential games constitute a prominent research topic in several fields ranging from economics to motion planning. Unfortunately, analytical techniques for differential games can address only simple, illustrative problem instances, and most existing computational methods suffer from the curse of dimensionality, i.e., the computational requirements grow exponentially with the dimensionality of the state space. In order to alleviate the curse of dimensionality for a certain class of two-player pursuit-evasion games, we propose a novel dynamic-programming-based algorithm that uses a continuous tensor-train approximation to represent the value function. In this way, the algorithm can represent high-dimensional tensors using computational resources that grow only polynomially with dimensionality of the state space and with the rank of the value function. The proposed algorithm is shown to converge to optimal solutions. It is demonstrated in several problem instances; in case of a seven-dimensional game, the value function representation was obtained with seven orders of magnitude savings in computational and memory cost, when compared to standard value iteration.