Differentially Private Contextual Linear Bandits

TL;DR

This paper introduces a new approach to the contextual linear bandit problem that ensures privacy using joint differential privacy, modifies existing algorithms with noise addition, and establishes regret bounds and lower bounds for private algorithms.

Contribution

It proposes a joint differential privacy framework for contextual linear bandits, adapting linear-UCB with noise mechanisms, and provides regret bounds and fundamental lower bounds.

Findings

Joint differential privacy can be achieved with controlled regret.

Adding Gaussian or Wishart noise maintains privacy while bounding regret.

The paper establishes the first lower bound on regret for private bandit algorithms.

Abstract

We study the contextual linear bandit problem, a version of the standard stochastic multi-armed bandit (MAB) problem where a learner sequentially selects actions to maximize a reward which depends also on a user provided per-round context. Though the context is chosen arbitrarily or adversarially, the reward is assumed to be a stochastic function of a feature vector that encodes the context and selected action. Our goal is to devise private learners for the contextual linear bandit problem. We first show that using the standard definition of differential privacy results in linear regret. So instead, we adopt the notion of joint differential privacy, where we assume that the action chosen on day $t$ is only revealed to user $t$ and thus needn't be kept private that day, only on following days. We give a general scheme converting the classic linear-UCB algorithm into a joint differentially private algorithm using the tree-based algorithm. We then apply either Gaussian noise or Wishart noise to achieve joint-differentially private algorithms and bound the resulting algorithms' regrets. In addition, we give the first lower bound on the additional regret any private algorithms for the MAB problem must incur.