Revisiting the softmax bellman operator: New benefits and new perspective

Zhao Song, Ronald E. Parr, Lawrence Carin

Research output: Chapter in Book/Report/Conference proceedingConference contribution

12 Scopus citations


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, wc 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 over-estimation 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.
Original languageEnglish (US)
Title of host publication36th International Conference on Machine Learning, ICML 2019
PublisherInternational Machine Learning Society (IMLS)
Number of pages16
ISBN (Print)9781510886988
StatePublished - Jan 1 2019
Externally publishedYes

Bibliographical note

Generated from Scopus record by KAUST IRTS on 2021-02-09


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