Differentially Private Bilevel Optimization: Efficient Algorithms with Near-Optimal Rates
This work addresses privacy concerns in hierarchical machine learning tasks, offering near-optimal solutions that could impact fields like healthcare or finance where data sensitivity is critical, though it builds incrementally on existing private optimization methods.
The paper tackles the problem of ensuring differential privacy in bilevel optimization, which is common in sensitive applications like meta-learning, by providing nearly tight upper and lower bounds on excess empirical risk for convex settings and developing efficient algorithms with state-of-the-art rates for non-convex settings, achieving bounds that are dimension-independent for the inner problem.
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.