On the Exponential Convergence for Offline RLHF with Pairwise Comparisons

TL;DR

This paper introduces RL-LOW, an algorithm for offline reinforcement learning from human feedback with pairwise comparisons, achieving exponential convergence in simple regret and matching lower bounds, with extensions to privacy-preserving settings.

Contribution

The paper proposes RL-LOW, the first algorithm with exponential convergence guarantees for offline RLHF with pairwise comparisons, and establishes matching instance-dependent lower bounds.

Findings

RL-LOW achieves exponential simple regret decay of exp(-Ω(n/H)).

Lower bounds match the upper bounds order-wise, proving optimality.

The hardness parameter H remains unchanged under differential privacy constraints.

Abstract

We consider the problem of offline reinforcement learning from human feedback (RLHF) with pairwise comparisons proposed by Zhu et al. (2023), where the implicit reward is a linear function of an unknown parameter. Given an offline dataset, our objective consists in ascertaining the optimal action for each state, with the ultimate goal of minimizing the {\em simple regret}. We propose an algorithm, \underline{RL} with \underline{L}ocally \underline{O}ptimal \underline{W}eights or {\sc RL-LOW}, which yields an exponential form of simple regret of $\exp ( - \Omega(n/H) )$ where $n$ is the number of data samples and $H$ denotes an instance-dependent hardness quantity that depends explicitly on the suboptimality gap of each action. Furthermore, we derive a first-of-its-kind instance-dependent lower bound in offline RLHF with pairwise comparisons. Interestingly, we observe that the lower and upper bounds on the simple regret match order-wise in the exponent, demonstrating order-wise optimality of our {\sc RL-LOW}. In view of privacy considerations in practical applications, we also extend {\sc RL-LOW} to the setting of $(\varepsilon,\delta)$-differential privacy and show, somewhat surprisingly, that the hardness parameter $H$ is unchanged in the asymptotic regime as $n$ tends to infinity; this underscores the inherent efficiency of {\sc RL-LOW} in terms of preserving the privacy of the observed rewards. Given our focus on establishing instance-dependent bounds of exponential convergence, our research fills the research gap in existing studies that concentrate on establishing worst-case regrets of {\em inverse polynomial convergence} (e.g., $\widetilde{O}(\frac{1}{\sqrt{n}})$) for offline RLHF with pairwise comparisons.