Composite Optimization with Indicator Functions: Stationary Duality and a Semismooth Newton Method

TL;DR

This paper develops a dual optimization framework for indicator functions in composite models, introducing a dual Newton method with proven convergence properties, and demonstrates its effectiveness in machine learning applications.

Contribution

It constructs the dual problem for composite indicator functions, extending conjugate properties, and proposes a dual Newton method with global and local convergence guarantees.

Findings

The dual problem is a sparse optimization with an $\,l_0$ regularizer.

The proposed dual Newton method achieves fast convergence.

Applications show improved speed and accuracy in AUC maximization and multi-label classification.

Abstract

Indicator functions of taking values of zero or one are essential to numerous applications in machine learning and statistics. The corresponding primal optimization model has been researched in several recent works. However, its dual problem is a more challenging topic that has not been well addressed. One possible reason is that the Fenchel conjugate of any indicator function is finite only at the origin. This work aims to explore the dual optimization for the sum of a strongly convex function and a composite term with indicator functions on positive intervals. For the first time, a dual problem is constructed by extending the classic conjugate subgradient property to the indicator function. This extension further helps us establish the equivalence between the primal and dual solutions. The dual problem turns out to be a sparse optimization with a $\ell_0$ regularizer and a nonnegative constraint. The proximal operator of the sparse regularizer is used to identify a dual subspace to implement gradient and/or semismooth Newton iteration with low computational complexity. This gives rise to a dual Newton-type method with both global convergence and local superlinear (or quadratic) convergence rate under mild conditions. Finally, when applied to AUC maximization and sparse multi-label classification, our dual Newton method demonstrates satisfactory performance on computational speed and accuracy.