DP-Dueling: Learning from Preference Feedback without Compromising User Privacy

TL;DR

A general class of utility-based preference matrices for large (potentially unbounded) decision spaces for large (potentially unbounded) decision spaces is considered and the first differentially private dueling bandit algorithm for active learning with user preferences is given.

Abstract

We consider the well-studied dueling bandit problem, where a learner aims to identify near-optimal actions using pairwise comparisons, under the constraint of differential privacy. We consider a general class of utility-based preference matrices for large (potentially unbounded) decision spaces and give the first differentially private dueling bandit algorithm for active learning with user preferences. Our proposed algorithms are computationally efficient with near-optimal performance, both in terms of the private and non-private regret bound. More precisely, we show that when the decision space is of finite size $K$, our proposed algorithm yields order optimal $O\Big(\sum_{i = 2}^K\log\frac{KT}{Δ_i} + \frac{K}ε\Big)$ regret bound for pure $ε$-DP, where $Δ_i$ denotes the suboptimality gap of the $i$-th arm. We also present a matching lower bound analysis which proves the optimality of our algorithms. Finally, we extend our results to any general decision space in $d$-dimensions with potentially infinite arms and design an $ε$-DP algorithm with regret $\tilde{O} \left( \frac{d^6}{κε} + \frac{ d\sqrt{T }}κ \right)$, providing privacy for free when $T \gg d$.

Theorems & Definitions (46)

• Definition 1: ($\epsilon, \delta)$-differentially private dueling bandit
• Definition 2: $\varepsilon$-net for any set ${\mathcal{S}}$ matouvsek1989constructionvershynin2018high
• Remark 1
• Definition 3: G-Optimal Design
• Lemma 1: DworkNaPiRo10, Theorem 4.1
• Theorem 1: Regret Analysis of
• proof : Proof sketch of \ref{['thm:fin']}
• Lemma 2
• proof
• Lemma 3