Title: CROP: Certifying Robust Policies for Reinforcement Learning through Functional Smoothing. Abstract: As reinforcement learning (RL) has achieved great success and been even adopted in safety-critical domains such as autonomous vehicles, a range of empirical studies have been conducted to improve its robustness against adversarial attacks. However, how to certify its robustness with theoretical guarantees still remains challenging. In this paper, we present the ﬁrst uniﬁed framework CROP (Certifying Robust Policies for RL) to provide robustness certiﬁcation on both action and reward levels. In particular, we propose two robustness certiﬁcation criteria: robustness of per-state actions and lower bound of cumulative rewards. We then develop a local smoothing algorithm for policies derived from Q-functions to guarantee the robustness of actions taken along the trajectory; we also develop a global smoothing algorithm for certifying the lower bound of a ﬁnite-horizon cumulative reward, as well as a novel local smoothing algorithm to perform adaptive search in order to obtain tighter reward certiﬁcation. Empirically, we apply CROP to evaluate several existing empirically robust RL algorithms, including adversarial training and different robust regularization, in four environments (two representative Atari games, Highway, and CartPole). Furthermore, by evaluating these algorithms against adversarial attacks, we demonstrate that our certiﬁcations are often tight. All experiment results are available at website https://crop-leaderboard.github.io.