Convex Q-Learning, Part 1: Deterministic Optimal Control
This addresses foundational issues in reinforcement learning for deterministic nonlinear systems, offering a novel convex approach that could improve algorithm stability and convergence, though it is incremental in extending existing convex analytic methods to RL.
The paper tackles the challenge of extending Watkins' Q-learning to general function approximation by introducing convex Q-learning algorithms based on a convex relaxation of the Bellman equation, establishing convergence under general conditions including linear function approximation. It shows that while batch implementations resemble DQN, they differ fundamentally as convex Q-learning solves a convex program, whereas DQN theory remains weak, with both sharing the same fixed point equation and ODE approximations but lacking stability guarantees.
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.