Basis Function Adaptation in Temporal Difference Reinforcement Learning

Abstract

Reinforcement Learning (RL) is an approach for solving complex multi-stage decision problems that fall under the general framework of Markov Decision Problems (MDPs), with possibly unknown parameters. Function approximation is essential for problems with a large state space, as it facilitates compact representation and enables generalization. Linear approximation architectures (where the adjustable parameters are the weights of pre-fixed basis functions) have recently gained prominence due to efficient algorithms and convergence guarantees. Nonetheless, an appropriate choice of basis function is important for the success of the algorithm. In the present paper we examine methods for adapting the basis function during the learning process in the context of evaluating the value function under a fixed control policy. Using the Bellman approximation error as an optimization criterion, we optimize the weights of the basis function while simultaneously adapting the (non-linear) basis function parameters. We present two algorithms for this problem. The first uses a gradient-based approach and the second applies the Cross Entropy method. The performance of the proposed algorithms is evaluated and compared in simulations.

A Tutorial on the Cross-Entropy Method

“…The CE method has been successfully applied to a diverse range of estimation and optimization problems, including buffer allocation [1], queueing models of telecommunication systems [14,16], optimal control of HIV/AIDS spread [48,49], signal detection [30], combinatorial auctions [9], DNA sequence alignment [24,38], scheduling and vehicle routing [3,8,11,20,23,53], neural and reinforcement learning [31,32,34,52,54], project management [12], rare-event simulation with light-and heavy-tail distributions [2,10,21,28], clustering analysis [4,5,29]. Applications to classical combinatorial optimization problems including the max-cut, traveling salesman, and Hamiltonian cycle 1…”

Algorithms for Reinforcement Learning

“…One class of methods aims at constructing a parsimonious set of features (basis functions). These include tuning the parameter of Gaussian RBF either using a gradient-or the cross-entropymethod in the context of LSTD (Menache et al, 2005), deriving new basis functions with nonparametric techniques (Keller et al, 2006;Parr et al, 2007) or using a combination of numerical analysis and nonparametric techniques (Mahadevan, 2009). These methods, however, do not attempt to control the tradeoff between the approximation and estimation errors.…”

A Nonlinear Approach to Robust Routing Based on Reinforcement Learning with State Space Compression and Adaptive Basis Construction

“…Various methods have been developed for adaptively constructing a basis function, most of which use the radial basis function (RBF) [4] and adjust the parameters of RBF [11]. However, orthonormal bases are superior to non-orthogonal bases such as RBF from the viewpoint of the trade-off between N and the approximation error [12].…”