Basis Adaptation for Sparse Nonlinear Reinforcement Learning

This paper presents a new approach to representation discovery in reinforcement learning (RL) using basis adaptation. We introduce a general framework for basis adaptation as nonlinear separable least-squares value function approximation based on finding Fréchet gradients of an error function using variable projection functionals. We then present a scalable proximal gradientbased approach for basis adaptation using the recently proposed mirror-descent framework for RL. Unlike traditional temporal-difference (TD) methods for RL, mirror descent based RL methods undertake proximal gradient updates of weights in a dual space, which is linked together with the primal space using a Legendre transform involving the gradient of a strongly convex function. Mirror descent RL can be viewed as a proximal TD algorithm using Bregman divergence as the distance generating function. We present a new class of regularized proximal-gradient based TD methods, which combine feature selection through sparse L1 regularization and basis adaptation. Experimental results are provided to illustrate and validate the approach.  There has been rapidly growing interest in reinforcement learning in representation discovery (Mahadevan 2008). Basis construction algorithms (Mahadevan 2009) combine the learning of features, or basis functions, and control. Basis adaptation (Bertsekas and Yu 2009;  Menache, Shimkin, and Mannor 2005) enables tuning a given parametric basis, such as the Fourier basis (Konidaris, Osentoski, and Thomas 2011), radial basis functions (RBF) (Menache, Shimkin, and Mannor 2005), polynomial bases (Lagoudakis and Parr 2003), etc. to the geometry of a particular Markov decision process (MDP). Basis selection methods combine sparse feature selection through L1 regularization with traditional least-squares type RL methods (Kolter and Ng 2009; Johns, Painter-Wakefield, and Parr 2010), linear complementarity methods (Johns, Painter-Wakefield, and Parr 2010), approximate linear programming (Petrik et al. 2010), or convex-concave optimization methods for sparse off-policy TD-learning (Liu, Mahadevan, and Liu 2012).  In this paper, we present a new framework for basis adaptation as nonlinear separable least-squares approximation of value functions using variable projection functionals. This framework is adapted from a well-known classical method for nonlinear regression (Golub and Pereyra 1973), when the model parameters can be decomposed into a linear set, which is fit by classical least-squares, and a nonlinear set, which is fit by a Gauss-Newton method based on computing the gradient of an error function based on variable projection functionals. Mirror descent is a highly scalable online convex optimization framework (Nemirovksi and Yudin 1983). Online convex optimization (Zinkevich 2003) explores the use of first-order gradient methods for solving convex optimization problems. Mirror descent can be viewed as a first-order proximal gradient based method (Beck and Teboulle 2003) using a distance generating function that is a Bregman divergence (Bregman 1967). We combine basis adaptation with mirrordescent RL (Mahadevan and Liu 2012), a recently developed first-order approach to sparse RL. The proposed approach is also validated with some experiments showing improved performance compared to previous work. Reinforcement Learning Reinforcement learning can be viewed as a stochastic approximation framework (Borkar 2008) for solving MDPs, which are defined by a set of states S, a set of (possibly state-dependent) actions A (As), a dynamical system model comprised of the transition probabilities P a ss′ specifying the probability of transitioning to state s′ from state s under action a, and a reward model R specifying the payoffs received. A policy π : S → A is a deterministic mapping from states to actions. Associated with each policy π is a value function V π , which is a fixed point of the Bellman equation: V π = T(V ) = R + γPV π (1) where 0 ≤ γ < 1 is a discount factor, and T is the Bellman operator. An optimal policy π∗ is one whose associated value function dominates all others, and is defined by the following nonlinear system of equations: