Convex Q-Learning, Part 1: Deterministic Optimal Control
arXiv:2008.03559
Abstract
It is well known that the extension of Watkins' algorithm to general function approximation settings is challenging: does the projected Bellman equation have a solution? If so, is the solution useful in the sense of generating a good policy? And, if the preceding questions are answered in the affirmative, is the algorithm consistent? These questions are unanswered even in the special case of Q-function approximations that are linear in the parameter. The challenge seems paradoxical, given the long history of convex analytic approaches to dynamic programming. The paper begins with a brief survey of linear programming approaches to optimal control, leading to a particular over parameterization that lends itself to applications in reinforcement learning. The main conclusions are summarized as follows: (i) The new class of convex Q-learning algorithms is introduced based on the convex relaxation of the Bellman equation. Convergence is established under general conditions, including a linear function approximation for the Q-function. (ii) A batch implementation appears similar to the famed DQN algorithm (one engine behind AlphaZero). It is shown that in fact the algorithms are very different: while convex Q-learning solves a convex program that approximates the Bellman equation, theory for DQN is no stronger than for Watkins' algorithm with function approximation: (a) it is shown that both seek solutions to the same fixed point equation, and (b) the ODE approximations for the two algorithms coincide, and little is known about the stability of this ODE. These results are obtained for deterministic nonlinear systems with total cost criterion. Many extensions are proposed, including kernel implementation, and extension to MDP models.
This pre-print is written in a tutorial style so it is accessible to new-comers. It will be a part of a handout for upcoming short courses on RL. A more compact version suitable for journal submission is in preparation
References in corpus (9)
- Playing Atari with Deep Reinforcement Learning
- Averaged-DQN: Variance Reduction and Stabilization for Deep Reinforcement Learning
- Fastest Convergence for Q-learning
- A Kernel Loss for Solving the Bellman Equation
- On Infinite Linear Programming and the Moment Approach to Deterministic Infinite Horizon Discounted Optimal Control Problems
- A Unified Switching System Perspective and O.D.E. Analysis of Q-Learning Algorithms
- Kalman meets Bellman: Improving Policy Evaluation through Value Tracking
- Zap Q-Learning With Nonlinear Function Approximation
- Model-Free Primal-Dual Methods for Network Optimization with Application to Real-Time Optimal Power Flow