Adaptive Online Value Function Approximation with Wavelets
This addresses a scalability issue in reinforcement learning for continuous and high-dimensional state spaces, though it is an incremental improvement over existing linear approximation methods.
The paper tackles the problem of exponential growth in basis functions for linear value function approximation in reinforcement learning by introducing wavelet bases, which enable adaptive refinement and achieve comparable or better performance to Fourier bases on Mountain Car and Acrobot.
Using function approximation to represent a value function is necessary for continuous and high-dimensional state spaces. Linear function approximation has desirable theoretical guarantees and often requires less compute and samples than neural networks, but most approaches suffer from an exponential growth in the number of functions as the dimensionality of the state space increases. In this work, we introduce the wavelet basis for reinforcement learning. Wavelets can effectively be used as a fixed basis and additionally provide the ability to adaptively refine the basis set as learning progresses, making it feasible to start with a minimal basis set. This adaptive method can either increase the granularity of the approximation at a point in state space, or add in interactions between different dimensions as necessary. We prove that wavelets are both necessary and sufficient if we wish to construct a function approximator that can be adaptively refined without loss of precision. We further demonstrate that a fixed wavelet basis set performs comparably against the high-performing Fourier basis on Mountain Car and Acrobot, and that the adaptive methods provide a convenient approach to addressing an oversized initial basis set, while demonstrating performance comparable to, or greater than, the fixed wavelet basis.