Non-asymptotic Confidence Intervals of Off-policy Evaluation:  Primal and Dual Bounds 
Authors: Yihao Feng,Ziyang Tang,na zhang,qiang liu

Abstract: Off-policy evaluation (OPE) is the task of estimating the expected reward of a given policy based on offline data previously collected under different policies. Therefore, OPE is a key step in applying reinforcement learning to real-world domains such as medical treatment, where interactive data collection is expensive or even unsafe. As the observed data tends to be noisy and limited, it is essential to provide rigorous  uncertainty quantification, not just a point estimation, when applying OPE to make high stakes decisions. This work considers the problem of constructing non-asymptotic confidence intervals in infinite-horizon  off-policy evaluation, which remains a challenging open question. We develop a practical algorithm through a primal-dual optimization-based approach, which leverages the kernel Bellman loss (KBL) of Feng et al. 2019 and a new  martingale concentration inequality of KBL applicable to time-dependent data with unknown mixing conditions. Our algorithm makes  minimum assumptions on the data and the function class of the Q-function,  and works for the behavior-agnostic settings where the data is collected under a mix of arbitrary unknown behavior policies.  We present empirical results that clearly demonstrate the advantages of our approach over existing methods.