Title: Lower-level Duality Based Penalty Methods for Nonsmooth Bilevel Hyperparameter Optimization
Primary area: optimization
Keywords: Bilevel Optimization, Nonsmooth Optimization, Nonconvex Optimization, Hyperparameter Optimization

Abstract: Hyperparameter optimization (HO) is a critical task in machine learning and can be naturally formulated as bilevel optimization (BLO) with nonsmooth lower-level (LL) problems. However, many existing approaches rely on smoothing strategies or sequential subproblem solvers, both of which introduce significant computational overhead. To address these challenges, we develop a penalization framework that exploits strong duality of the LL problem and its dual. Building on this, we design first-order single-loop projection-based algorithms to solve the penalized problems efficiently. Our methods avoid smoothing and off-the-shelf solvers, thereby greatly reducing per-iteration complexity and overall runtime. We provide rigorous convergence guarantees and analyze the stationary conditions of BLO with nonsmooth LL problems under penalty perspective. Through extensive numerical experiments on a variety of benchmark and real-world tasks, we demonstrate the efficiency, scalability and superiority of our method over existing BLO algorithms.

TL;DR: We provide a penalization framework and first-order single-loop algorithms for nonsmooth bilevel hyperparameter optimization.