Unified Smoothing Approach for Best Hyperparameter Selection Problem Using a Bilevel Optimization Strategy

Abstract

Strongly motivated from use in various fields including machine learning, the methodology of sparse optimization has been developed intensively so far. Especially, the recent advance of algorithms for solving problems with nonsmooth regularizers is remarkable. However, those algorithms suppose that weight parameters of regularizers, called hyperparameters hereafter, are pre-fixed, and it is a crucial matter how the best hyperparameter should be selected. In this paper, we focus on the hyperparameter selection of regularizers related to the $\ell_p$ function with $0<p\le 1$ and apply a bilevel programming strategy, wherein we need to solve a bilevel problem, whose lower-level problem is nonsmooth, possibly nonconvex and non-Lipschitz. Recently, for solving a bilevel problem for hyperparameter selection of the pure $\ell_p\ (0<p \le 1)$ regularizer Okuno et al. discovered new necessary optimality conditions, called SB(scaled bilevel)-KKT conditions, and further proposed a smoothing-type algorithm using a certain smoothing function. However, this optimality measure is loose in the sense that there could be many points that satisfy the SB-KKT conditions. In this work, we propose new bilevel KKT conditions, which are new necessary optimality conditions tighter than the ones proposed by Okuno et al. Moreover, we propose a unified smoothing approach using smoothing functions that belong to the Chen-Mangasarian class, and then prove that generated iteration points accumulate at \alert{bilevel KKT points under milder constraint qualifications. Another contribution is that our approach and analysis are applicable to a wider class of regularizers. Numerical comparisons demonstrate which smoothing functions work well for hyperparameter optimization via bilevel optimization approach.