Uncertainty-aware Low-Rank Q-Matrix Estimation for Deep Reinforcement Learning

Value estimation is one key problem in Reinforcement Learning. Albeit many successes have been achieved by Deep Reinforcement Learning (DRL) in different fields, the underlying structure and learning dynamics of value function, especially with complex function approximation, are not fully understood. In this paper, we report that decreasing rank of $Q$-matrix widely exists during learning process across a series of continuous control tasks for different popular algorithms. We hypothesize that the low-rank phenomenon indicates the common learning dynamics of $Q$-matrix from stochastic high dimensional space to smooth low dimensional space. Moreover, we reveal a positive correlation between value matrix rank and value estimation uncertainty. Inspired by above evidence, we propose a novel Uncertainty-Aware Low-rank Q-matrix Estimation (UA-LQE) algorithm as a general framework to facilitate the learning of value function. Through quantifying the uncertainty of state-action value estimation, we selectively erase the entries of highly uncertain values in state-action value matrix and conduct low-rank matrix reconstruction for them to recover their values. Such a reconstruction exploits the underlying structure of value matrix to improve the value approximation, thus leading to a more efficient learning process of value function. In the experiments, we evaluate the efficacy of UA-LQE in several representative OpenAI MuJoCo continuous control tasks.