Regret minimization in Linear Bandits with offline data via extended D-optimal exploration

TL;DR

This paper introduces OOPE, an algorithm that leverages offline data with extended D-optimal design to minimize online regret in linear bandits, adapting to data quality and improving efficiency.

Contribution

The paper proposes a novel algorithm, OOPE, that effectively incorporates offline data into online regret minimization for linear bandits, with theoretical guarantees and design improvements.

Findings

Achieves regret bounds depending on offline data quality and quantity.

Provides minimax lower bounds that match the upper bounds in key regimes.

Improves design complexity from O(d^2) to O(d^2/δ_eff) in high dimensions.

Abstract

We consider the problem of online regret minimization in linear bandits with access to prior observations (offline data) from the underlying bandit model. There are numerous applications where extensive offline data is often available, such as in recommendation systems, online advertising. Consequently, this problem has been studied intensively in recent literature. Our algorithm, Offline-Online Phased Elimination (OOPE), effectively incorporates the offline data to substantially reduce the online regret compared to prior work. To leverage offline information prudently, OOPE uses an extended D-optimal design within each exploration phase. OOPE achieves an online regret is $\tilde{O}(\sqrt{\deff T \log \left(|\mathcal{A}|T\right)}+d^2)$. $\deff \leq d)$ is the effective problem dimension which measures the number of poorly explored directions in offline data and depends on the eigen-spectrum $(\lambda_k)_{k \in [d]}$ of the Gram matrix of the offline data. The eigen-spectrum $(\lambda_k)_{k \in [d]}$ is a quantitative measure of the \emph{quality} of offline data. If the offline data is poorly explored ($\deff \approx d$), we recover the established regret bounds for purely online setting while, when offline data is abundant ($\Toff >> T$) and well-explored ($\deff = o(1) $), the online regret reduces substantially. Additionally, we provide the first known minimax regret lower bounds in this setting that depend explicitly on the quality of the offline data. These lower bounds establish the optimality of our algorithm in regimes where offline data is either well-explored or poorly explored. Finally, by using a Frank-Wolfe approximation to the extended optimal design we further improve the $O(d^{2})$ term to $O\left(\frac{d^{2}}{\deff} \min \{ \deff,1\} \right)$, which can be substantial in high dimensions with moderate quality of offline data $\deff = \Omega(1)$.