Revisiting the Softmax Bellman Operator: New Benefits and New Perspective

TL;DR

This paper revisits the softmax Bellman operator in reinforcement learning, revealing its convergence properties, potential to reduce overestimation errors, and its surprising practical benefits over traditional methods.

Contribution

It provides new theoretical insights into the softmax Bellman operator, including convergence bounds and error reduction, explaining its empirical success in deep Q-learning.

Findings

Softmax Bellman operator converges exponentially fast to the standard Bellman operator.

It can bound the distance of its Q-function from the optimal Q-function.

The operator reduces overestimation errors, improving policy performance.

Abstract

The impact of softmax on the value function itself in reinforcement learning (RL) is often viewed as problematic because it leads to sub-optimal value (or Q) functions and interferes with the contraction properties of the Bellman operator. Surprisingly, despite these concerns, and independent of its effect on exploration, the softmax Bellman operator when combined with Deep Q-learning, leads to Q-functions with superior policies in practice, even outperforming its double Q-learning counterpart. To better understand how and why this occurs, we revisit theoretical properties of the softmax Bellman operator, and prove that $(i)$ it converges to the standard Bellman operator exponentially fast in the inverse temperature parameter, and $(ii)$ the distance of its Q function from the optimal one can be bounded. These alone do not explain its superior performance, so we also show that the softmax operator can reduce the overestimation error, which may give some insight into why a sub-optimal operator leads to better performance in the presence of value function approximation. A comparison among different Bellman operators is then presented, showing the trade-offs when selecting them.