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

Abstract

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.

Appendices

Appendix

A More Background on Matrix Estimation

Matrix estimation is mainly divided into two types, matrix completion and matrix restoration. In this paper, we focus on the former. We use matrix estimation and matrix completion alternatively. Consider a low-rank matrix M, in which the values of some entries are missing or unknown. The goal of matrix completion problem is to recover the matrix by leveraging the low-rank structure. We use $\varOmega$ to denote the set of subscripts of known entries and the use $\hat{M}$ to denote the recovered (also called reconstructed in this paper) matrix. Formally, the matrix completion problem is formalized as follows:

$$\begin{aligned} \begin{aligned} \min \ \Vert \hat{M}\Vert _{*},&\\ \text {subject to} \ \hat{M}_{i,j}=M_{i,j}, \quad&\forall (i,j) \in \varOmega \end{aligned} \end{aligned}$$

(5)

Here $\Vert \cdot \Vert _{*}$ is nuclear norm. The purpose of the above optimization problem is to fill the original matrix as much as possible while keeping the rank of the filled matrix $\hat{M}$ low.

Some representative methods to solve this optimization problem are Fix Point Continuation (FPC) algorithm [14], Singular Value Thresholding (SVT) algorithm [2], OptSpace algorithm [9], Soft-Impute algorithm [15] and so on.

In this paper, we use Soft-Impute algorithm for matrix completion, i.e., Q-matrix reconstruction. A pseudocode for Soft-Impute is shown in Algorithm 2. Define function $P_{\varOmega }$ as below:

$$\begin{aligned} P_{\varOmega }(M) (i, j) = \left\{ \begin{array}{cc} &{} M_{ij}, \ if\ (i,j) \in \varOmega \\ &{} 0 , \ if \ (i,j) \not \in \varOmega \end{array} \right. \end{aligned}$$

(6)

Similar, we define function $P_{\bar{\varOmega }}$. Soft-Impute algorithm completes the missing entries in $\bar{\varOmega }$ in an iterative reconstruction fashion. During each iteration, the singular matrix S is obtained by SVD decomposition, and then a filtering is performed through subtracting the value $\lambda$ controlled by a filtering divider hyperparameter $\zeta$; next, the new matrix $M_{\text {new}}$ is reconstructed with the subtracted singular matrix $S_{\lambda }$. Afterwards, an iteration termination check is performed by comparing whether the difference between the matrices before and after the iteration exceeds the termination threshold $\epsilon$. When the termination threshold is met or the max iteration number is reached, the algorithm finally outputs the completed matrix $\hat{M}$.

B Additional Experimental Details

All regular hyperparametyers and implementation details are described in the main body of the paper.

For code-level details, our codes are implemented with Python 3.6.9 and Torch 1.3.1. All experiments were run on a single NVIDIA GeForce GTX 1660Ti GPU. Our Q-matrix reconstruction algorithm is implemented with reference to https://github.com/YyzHarry/SV-RL.

For matrix reconstruction algorithm, i.e., SoftImpute, we set the filtering divider $\zeta = 50$, termination threshold $\epsilon = 0.0001$ and Max Iteration Number to be 100.

C Complete Learning Curves of Table 2

Fig. 6.

For the reward curve, the first four are the results of reconstruction on the target Q-matrix, and the last four are the results of reconstruction on the current Q-matrix. Results are the means and stds over 3 trials.