This paper introduces the first differentially private first-order algorithms for bilevel optimization, ensuring privacy with theoretical convergence guarantees for hypergradient norms in both empirical and population settings while avoiding Hessian computations.
Differential Privacy, Bilevel Optimization, Hyperparameter Tuning, First-Order Methods, Privacy-Preserving Machine Learning
Guy Kornowski
Weizmann Institute of Science, Apple
Generated by grok-3
Background Problem
Bilevel optimization, a hierarchical optimization framework where one problem is nested within another, is crucial in machine learning for tasks like hyperparameter tuning, meta-learning, and neural architecture search. However, its application to data-dependent objectives raises significant privacy concerns, as sensitive information can leak through the interaction between upper and lower levels, even when the upper level is data-independent. Prior to this work, no first-order (gradient-based) algorithms existed for bilevel optimization under standard differential privacy (DP) constraints, and existing methods relied on computationally expensive Hessian computations, limiting scalability in large-scale settings. This paper addresses these gaps by developing the first DP algorithms for bilevel optimization that use only gradient queries, ensuring privacy while maintaining practical applicability.
Method
The core idea is to design differentially private first-order algorithms for bilevel optimization by approximating the hypergradient without computing exact lower-level solutions, which could leak private information. The method employs a penalty-based approach to approximate the hyperobjective using a smooth function, avoiding Hessian inversions. Key steps include: (1) using a private localized gradient descent (DP-Loc-GD) to estimate lower-level solutions privately for both the original and penalized objectives; (2) constructing an inexact hypergradient oracle with added Gaussian noise to ensure (ǫ, δ)-DP; (3) updating the upper-level variables using a projected gradient mapping to handle constraints. A mini-batch variant is also proposed, sampling gradients for efficiency. The method crucially accounts for bias introduced by private approximations and controls privacy loss through advanced composition and amplification techniques.
Experiment
The paper does not present empirical experiments or datasets, focusing instead on theoretical guarantees. The experimental setup is purely analytical, deriving convergence rates for the hypergradient norm under (ǫ, δ)-DP constraints for both empirical risk minimization (ERM) and population loss settings. For ERM, the full-batch algorithm achieves a hypergradient norm of Õ((√d_x/ǫn)^(1/2) + (√d_y/ǫn)^(1/3)), where d_x and d_y are upper and lower level dimensions, and n is the dataset size. The mini-batch variant adds a 1/b_out term (b_out being the outer batch size), indicating dependency on batch size for convergence. For population loss, an additional term (d_x/n)^(1/2) accounts for generalization error. While the setup is comprehensive in covering constrained/unconstrained and full/mini-batch settings, the lack of empirical validation raises concerns about practical performance and whether the theoretical bounds are tight or overly pessimistic. The dependency on b_out in mini-batch settings also suggests potential scalability issues, not fully justified against single-level optimization results.
Further Thoughts
This work opens intriguing avenues for privacy-preserving machine learning in complex optimization settings, particularly in hyperparameter tuning where privacy is often overlooked. The penalty-based first-order approach is a significant step forward, but I wonder if integrating variance reduction techniques, as hinted in the discussion, could tighten the convergence rates, especially given recent advances in single-level DP optimization (e.g., Arora et al., 2023). Additionally, the strong convexity assumption on the lower-level problem, while necessary for current guarantees, limits applicability to non-strongly convex settings like certain neural network optimizations—exploring relaxations of this assumption could broaden impact. The application to hyperparameter tuning also suggests a connection to automated machine learning (AutoML), where privacy constraints are increasingly relevant; future work could explore integrating this with existing AutoML frameworks to test real-world efficacy. Lastly, the lack of empirical results is a critical gap—comparing this method against non-private baselines or prior second-order DP approaches on benchmark datasets could validate the trade-offs between privacy, accuracy, and computational cost.