Tracking the Best Expert Privately

2025-03-12

AI Summary

This paper studies dynamic regret minimization for expert prediction under differential privacy, against three adversary models: stochastic drift, oblivious, and adaptive. It establishes the first fundamental separation between oblivious and adaptive adversaries in terms of dynamic regret under privacy constraints. The authors propose the first ε-differentially private algorithm achieving sublinear dynamic regret against an S-drift stochastic adversary, with expected regret (Oig(sqrt{S T log(NT)} + S log(NT)/varepsilonig)). They further characterize the precise privacy–regret trade-off: for adaptive adversaries, dynamic regret is necessarily linear when (varepsilon leq sqrt{S/T}), whereas sublinear regret is attainable when (varepsilon gg sqrt{S/T}). These results lay the theoretical foundation for privately tracking optimal sequences in non-stationary environments and provide a practical algorithmic framework for privacy-preserving online learning under concept drift.

Technology Category

• Machine Learning: Privacy
• Game Theory and Economic Paradigms: Adversarial Learning
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

We design differentially private algorithms for the problem of prediction with expert advice under dynamic regret, also known as tracking the best expert. Our work addresses three natural types of adversaries, stochastic with shifting distributions, oblivious, and adaptive, and designs algorithms with sub-linear regret for all three cases. In particular, under a shifting stochastic adversary where the distribution may shift $S$ times, we provide an $epsilon$-differentially private algorithm whose expected dynamic regret is at most $Oleft( sqrt{S T log (NT)} + frac{S log (NT)}{epsilon} ight)$, where $T$ and $N$ are the epsilon horizon and number of experts, respectively. For oblivious adversaries, we give a reduction from dynamic regret minimization to static regret minimization, resulting in an upper bound of $Oleft(sqrt{S T log(NT)} + frac{S T^{1/3}log(T/delta) log(NT)}{epsilon^{2/3}} ight)$ on the expected dynamic regret, where $S$ now denotes the allowable number of switches of the best expert. Finally, similar to static regret, we establish a fundamental separation between oblivious and adaptive adversaries for the dynamic setting: while our algorithms show that sub-linear regret is achievable for oblivious adversaries in the high-privacy regime $epsilon le sqrt{S/T}$, we show that any $(epsilon, delta)$-differentially private algorithm must suffer linear dynamic regret under adaptive adversaries for $epsilon le sqrt{S/T}$. Finally, to complement this lower bound, we give an $epsilon$-differentially private algorithm that attains sub-linear dynamic regret under adaptive adversaries whenever $epsilon gg sqrt{S/T}$.

Problem

• Designs differentially private algorithms for expert advice prediction.
• Addresses dynamic regret under stochastic, oblivious, and adaptive adversaries.
• Provides sub-linear regret bounds for high-privacy regimes.

Innovation

• Differentially private algorithms for expert advice
• Sub-linear regret under shifting stochastic adversaries
• Dynamic regret minimization for oblivious adversaries