Private optimization in the interpolation regime: faster rates and hardness results

TL;DR

This paper explores the limits of private stochastic convex optimization, showing that quadratic growth enables exponential improvements in sample complexity, but such benefits are limited in the general interpolation setting.

Contribution

It introduces an adaptive algorithm that significantly improves private sample complexity in quadratic growth scenarios and establishes tight lower bounds and necessity of polynomial terms.

Findings

Adaptive algorithm achieves exponential improvements in sample complexity.

Lower bounds show the tightness of the dimension-dependent term.

Demonstrates the necessity of polynomial terms for adaptive algorithms.

Abstract

In non-private stochastic convex optimization, stochastic gradient methods converge much faster on interpolation problems -- problems where there exists a solution that simultaneously minimizes all of the sample losses -- than on non-interpolating ones; we show that generally similar improvements are impossible in the private setting. However, when the functions exhibit quadratic growth around the optimum, we show (near) exponential improvements in the private sample complexity. In particular, we propose an adaptive algorithm that improves the sample complexity to achieve expected error $\alpha$ from $\frac{d}{\varepsilon \sqrt{\alpha}}$ to $\frac{1}{\alpha^\rho} + \frac{d}{\varepsilon} \log\left(\frac{1}{\alpha}\right)$ for any fixed $\rho >0$, while retaining the standard minimax-optimal sample complexity for non-interpolation problems. We prove a lower bound that shows the dimension-dependent term is tight. Furthermore, we provide a superefficiency result which demonstrates the necessity of the polynomial term for adaptive algorithms: any algorithm that has a polylogarithmic sample complexity for interpolation problems cannot achieve the minimax-optimal rates for the family of non-interpolation problems.