Equivalence of Privacy and Stability with Generalization Guarantees in Quantum Learning

TL;DR

This paper develops a unified information-theoretic framework linking stability, privacy, and generalization in quantum learning, providing bounds and characterizations that extend classical results to the quantum setting.

Contribution

It introduces a comprehensive quantum framework connecting stability, privacy, and generalization, including bounds, stability implications of differential privacy, and limits for dishonest algorithms.

Findings

Bound on expected generalization error via quantum mutual information

$(\

, )$-quantum differential privacy implies stability and generalization guarantees

Abstract

We present a unified information-theoretic framework elucidating the interplay between stability, privacy, and the generalization performance of quantum learning algorithms. We establish a bound on the expected generalization error in terms of quantum mutual information and derive a probabilistic upper bound that generalizes the classical result by Esposito et al. (2021). Complementing these findings, we provide a lower bound on the expected true loss relative to the expected empirical loss. Additionally, we demonstrate that $(\varepsilon, \delta)$-quantum differentially private learning algorithms are stable, thereby ensuring strong generalization guarantees. Finally, we extend our analysis to dishonest learning algorithms, introducing Information-Theoretic Admissibility (ITA) to characterize the fundamental limits of privacy when the learning algorithm is oblivious to specific dataset instances.