Online Laplacian-Based Representation Learning in Reinforcement Learning

Representation learning plays a crucial role in reinforcement learning, especially in complex environments with high-dimensional and unstructured states. Effective representations can enhance the efficiency of learning algorithms by improving sample efficiency and generalization across tasks. This paper considers the Laplacian-based framework for representation learning, where the eigenvectors of the Laplacian matrix of the underlying transition graph are leveraged to encode meaningful features from raw sensory observations of the states. Despite the promising algorithmic advances in this framework, it remains an open question whether the Laplacian-based representations can be learned online and with theoretical guarantees along with policy learning. We address this by formulating an online optimization approach using the Asymmetric Graph Drawing Objective (AGDO) and analyzing its convergence via online projected gradient descent under mild assumptions. Our extensive simulation studies empirically validate the convergence guarantees to the true Laplacian representation. Furthermore, we provide insights into the compatibility of different reinforcement learning algorithms with online representation learning.