Implementing SLOPE++ for estimator selection

[usaito]

Overview

• Implement a method to tune hyperparameters of OPE estimators called SLOPE++ proposed by Tucker and Lee(2021)., which improves the vanilla SLOPE proposed by Su et al.(2020). This tuning method is usable by setting tuning_method="slope" when using estimators implemented in obp.ope.estimators_tuning.
• Implement some high probability lower bound estimators for the mean of independent random variables. The implemented methods include Hoeffding, empirical Bernstein, and the bound based on the Student's t distribution. Please see Thomas et al.(2015) for details.

Minor

• Fix docstrings and error messages

References

• George Tucker and Jonathan Lee .
  "Improved Estimator Selection for Off-Policy Evaluation.", 2021.
  https://lyang36.github.io/icml2021_rltheory/camera_ready/79.pdf

• Yi Su, Pavithra Srinath, and Akshay Krishnamurthy.
  "Adaptive Estimator Selection for Off-Policy Evaluation.", ICML2021.
  http://proceedings.mlr.press/v119/su20d.html

• Philip S. Thomas, Georgios Theocharous, and Mohammad Ghavamzadeh.
  "High Confidence Off-Policy Evaluation" AAAI2015.
  https://www.cs.utexas.edu/~sniekum/classes/RLFD-F17/papers/Thomas2015.pdf

• Philip S. Thomas, Georgios Theocharous, and Mohammad Ghavamzadeh.
  "High Confidence Off-Policy Improvement" AAAI2015.
  http://proceedings.mlr.press/v37/thomas15.html