Differentially Private Bilevel Optimization: Efficient Algorithms with Near-Optimal Rates

TL;DR

This paper develops efficient differentially private algorithms for bilevel optimization, providing nearly optimal risk bounds in convex settings and state-of-the-art rates for non-convex cases, addressing privacy concerns in hierarchical machine learning tasks.

Contribution

It introduces the first nearly tight bounds for differentially private bilevel optimization and develops new algorithms with optimal rates, including a novel log-concave sampling method.

Findings

Bounds are nearly tight and match single-level DP ERM rates.

Algorithms achieve state-of-the-art rates in non-convex bilevel optimization.

Sampling method under inexact evaluations may be of independent interest.

Abstract

Bilevel optimization, in which one optimization problem is nested inside another, underlies many machine learning applications with a hierarchical structure -- such as meta-learning and hyperparameter optimization. Such applications often involve sensitive training data, raising pressing concerns about individual privacy. Motivated by this, we study differentially private bilevel optimization. We first focus on settings where the outer-level objective is convex, and provide novel upper and lower bounds on the excess empirical risk for both pure and approximate differential privacy. These bounds are nearly tight and essentially match the optimal rates for standard single-level differentially private ERM, up to additional terms that capture the intrinsic complexity of the nested bilevel structure. We also provide population loss bounds for bilevel stochastic optimization. The bounds are achieved in polynomial time via efficient implementations of the exponential and regularized exponential mechanisms. A key technical contribution is a new method and analysis of log-concave sampling under inexact function evaluations, which may be of independent interest. In the non-convex setting, we develop novel algorithms with state-of-the-art rates for privately finding approximate stationary points. Notably, our bounds do not depend on the dimension of the inner problem.