Revisiting the Softmax Bellman Operator: New Benefits and New Perspective

Zhao Song, Ron Parr, Lawrence Carin
Proceedings of the 36th International Conference on Machine Learning, PMLR 97:5916-5925, 2019.

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v97-song19c, title = {Revisiting the Softmax {B}ellman Operator: New Benefits and New Perspective}, author = {Song, Zhao and Parr, Ron and Carin, Lawrence}, booktitle = {Proceedings of the 36th International Conference on Machine Learning}, pages = {5916--5925}, year = {2019}, editor = {Chaudhuri, Kamalika and Salakhutdinov, Ruslan}, volume = {97}, series = {Proceedings of Machine Learning Research}, month = {09--15 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v97/song19c/song19c.pdf}, url = {https://proceedings.mlr.press/v97/song19c.html}, 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.} }
Endnote
%0 Conference Paper %T Revisiting the Softmax Bellman Operator: New Benefits and New Perspective %A Zhao Song %A Ron Parr %A Lawrence Carin %B Proceedings of the 36th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2019 %E Kamalika Chaudhuri %E Ruslan Salakhutdinov %F pmlr-v97-song19c %I PMLR %P 5916--5925 %U https://proceedings.mlr.press/v97/song19c.html %V 97 %X 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.
APA
Song, Z., Parr, R. & Carin, L.. (2019). Revisiting the Softmax Bellman Operator: New Benefits and New Perspective. Proceedings of the 36th International Conference on Machine Learning, in Proceedings of Machine Learning Research 97:5916-5925 Available from https://proceedings.mlr.press/v97/song19c.html.

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