We study reinforcement learning in the presence of an unknown reward perturbation. Existing methodologies for this problem make strong assumptions including reward smoothness, known perturbations, and/or perturbations that do not modify the optimal policy. We study the case of unknown arbitrary perturbations that discretize and shuffle reward space, but have the property that the true reward belongs to the most frequently observed class after perturbation. This class of perturbations generalizes existing classes (and, in the limit, all continuous bounded perturbations) and defeats existing methods. We introduce an adaptive distributional reward critic and show theoretically that it can recover the true rewards under technical conditions. Under the targeted perturbation in discrete and continuous control tasks, we win/tie the highest return in 40/57 settings (compared to 16/57 for the best baseline). Even under the untargeted perturbation, we still win an edge over the baseline designed especially for that setting. The use of reward as an objective is a central feature of reinforcement learning (RL) that has been hypothesized to constitute a path to general intelligence Silver et al. (2021). The reward is also the cause of a substantial amount of human effort associated with RL, from engineering to reduce difficulties caused by sparse, delayed, or misspecified rewards Ng et al. (1999); Hadfield-Menell et al. (2017); Qian et al. (2023) to gathering large volumes of human-labeled rewards used for tuning large language models (LLMs) Ouyang et al. (2022); Bai et al. (2022).

With the rapid development of neural network applications in NLP, model robustness problem is gaining more attention. Different from computer vision, the discrete nature of texts makes it more challenging to explore robustness in NLP. Therefore, in this paper, we aim to connect discrete perturbations with continuous perturbations, therefore we can use such connections as a bridge to help understand discrete perturbations in NLP models. Specifically, we first explore how to connect and measure the correlation between discrete perturbations and continuous perturbations. Then we design a regression task as a PerturbScore to learn the correlation automatically. Through experimental results, we find that we can build a connection between discrete and continuous perturbations and use the proposed PerturbScore to learn such correlation, surpassing previous methods used in discrete perturbation measuring. Further, the proposed PerturbScore can be well generalized to different datasets, perturbation methods, indicating that we can use it as a powerful tool to study model robustness in NLP.