Private Online Prediction from Experts: Separations and Faster Rates

TL;DR

This paper introduces new differentially private algorithms for online expert prediction that outperform existing methods, especially in high-dimensional and adaptive adversary settings, revealing fundamental separations in regret bounds.

Contribution

The paper presents the first algorithms achieving sub-linear regret under pure differential privacy for high-dimensional, oblivious adversaries, and establishes new lower bounds highlighting separations between adaptive and non-adaptive adversaries.

Findings

Achieves regret bounds of  O(\u221A{T D7 DAlog d} + DAlog d/D5) for approximate DP.

First to obtain sub-linear regret with pure DP in high-dimensional regimes.

Proves lower bounds demonstrating a separation between adaptive and non-adaptive adversaries.

Abstract

Online prediction from experts is a fundamental problem in machine learning and several works have studied this problem under privacy constraints. We propose and analyze new algorithms for this problem that improve over the regret bounds of the best existing algorithms for non-adaptive adversaries. For approximate differential privacy, our algorithms achieve regret bounds of $\tilde{O}(\sqrt{T \log d} + \log d/\varepsilon)$ for the stochastic setting and $\tilde{O}(\sqrt{T \log d} + T^{1/3} \log d/\varepsilon)$ for oblivious adversaries (where $d$ is the number of experts). For pure DP, our algorithms are the first to obtain sub-linear regret for oblivious adversaries in the high-dimensional regime $d \ge T$. Moreover, we prove new lower bounds for adaptive adversaries. Our results imply that unlike the non-private setting, there is a strong separation between the optimal regret for adaptive and non-adaptive adversaries for this problem. Our lower bounds also show a separation between pure and approximate differential privacy for adaptive adversaries where the latter is necessary to achieve the non-private $O(\sqrt{T})$ regret.