Non-Smooth Weakly-Convex Finite-sum Coupled Compositional Optimization

TL;DR

This paper introduces new algorithms for non-smooth weakly-convex finite-sum coupled compositional optimization problems, broadening the scope of FCCO applications in machine learning and deep learning tasks.

Contribution

It extends FCCO to non-smooth weakly-convex functions, analyzes a single-loop algorithm's complexity, and applies it to complex deep learning problems with empirical validation.

Findings

Proposed algorithms effectively solve non-smooth weakly-convex FCCO problems.

Established complexity bounds for finding stationary points.

Demonstrated improved performance in deep learning AUC maximization tasks.

Abstract

This paper investigates new families of compositional optimization problems, called $\underline{\bf n}$on-$\underline{\bf s}$mooth $\underline{\bf w}$eakly-$\underline{\bf c}$onvex $\underline{\bf f}$inite-sum $\underline{\bf c}$oupled $\underline{\bf c}$ompositional $\underline{\bf o}$ptimization (NSWC FCCO). There has been a growing interest in FCCO due to its wide-ranging applications in machine learning and AI, as well as its ability to address the shortcomings of stochastic algorithms based on empirical risk minimization. However, current research on FCCO presumes that both the inner and outer functions are smooth, limiting their potential to tackle a more diverse set of problems. Our research expands on this area by examining non-smooth weakly-convex FCCO, where the outer function is weakly convex and non-decreasing, and the inner function is weakly-convex. We analyze a single-loop algorithm and establish its complexity for finding an $\epsilon$-stationary point of the Moreau envelop of the objective function. Additionally, we also extend the algorithm to solving novel non-smooth weakly-convex tri-level finite-sum coupled compositional optimization problems, which feature a nested arrangement of three functions. Lastly, we explore the applications of our algorithms in deep learning for two-way partial AUC maximization and multi-instance two-way partial AUC maximization, using empirical studies to showcase the effectiveness of the proposed algorithms.