Differentially Private Bilevel Optimization

TL;DR

This paper introduces the first differentially private algorithms for bilevel optimization that avoid Hessian computations, applicable to large-scale machine learning problems with theoretical guarantees on hypergradient accuracy.

Contribution

The authors develop the first DP algorithms for bilevel optimization that do not require Hessian calculations, with analysis covering various problem settings and practical hyperparameter tuning.

Findings

Achieves hypergradient norm at most $ ilde{O}((rac{ oot{d_ ext{up}}}{ ext{epsilon} imes n})^{1/2}+(rac{ oot{d_ ext{low}}}{ ext{epsilon} imes n})^{1/3})$

Applicable to constrained and unconstrained problems, with mini-batch and empirical loss considerations

Provides a simple private rule for tuning regularization hyperparameters.

Abstract

We present differentially private (DP) algorithms for bilevel optimization, a problem class that received significant attention lately in various machine learning applications. These are the first algorithms for such problems under standard DP constraints, and are also the first to avoid Hessian computations which are prohibitive in large-scale settings. Under the well-studied setting in which the upper-level is not necessarily convex and the lower-level problem is strongly-convex, our proposed gradient-based $(\epsilon,\delta)$-DP algorithm returns a point with hypergradient norm at most $\widetilde{\mathcal{O}}\left((\sqrt{d_\mathrm{up}}/\epsilon n)^{1/2}+(\sqrt{d_\mathrm{low}}/\epsilon n)^{1/3}\right)$ where $n$ is the dataset size, and $d_\mathrm{up}/d_\mathrm{low}$ are the upper/lower level dimensions. Our analysis covers constrained and unconstrained problems alike, accounts for mini-batch gradients, and applies to both empirical and population losses. As an application, we specialize our analysis to derive a simple private rule for tuning a regularization hyperparameter.