Deep neural networks algorithms for stochastic control problems on finite horizon: convergence analysis
This addresses computational challenges in stochastic control for applications like finance or robotics, offering a novel algorithmic approach with theoretical guarantees, though it is incremental in combining deep learning with dynamic programming.
The paper tackles high-dimensional stochastic control problems by developing deep learning algorithms that approximate the optimal policy with neural networks and the value function via Monte Carlo regression, providing theoretical convergence analysis in terms of approximation and statistical errors.
This paper develops algorithms for high-dimensional stochastic control problems based on deep learning and dynamic programming. Unlike classical approximate dynamic programming approaches, we first approximate the optimal policy by means of neural networks in the spirit of deep reinforcement learning, and then the value function by Monte Carlo regression. This is achieved in the dynamic programming recursion by performance or hybrid iteration, and regress now methods from numerical probabilities. We provide a theoretical justification of these algorithms. Consistency and rate of convergence for the control and value function estimates are analyzed and expressed in terms of the universal approximation error of the neural networks, and of the statistical error when estimating network function, leaving aside the optimization error. Numerical results on various applications are presented in a companion paper (arxiv.org/abs/1812.05916) and illustrate the performance of the proposed algorithms.