Single-Loop Stochastic Algorithms for Difference of Max-Structured Weakly Convex Functions

TL;DR

This paper introduces SMAG, a novel single-loop stochastic algorithm for non-smooth, non-convex difference of weakly convex functions, achieving state-of-the-art convergence rates and validated through experiments in machine learning tasks.

Contribution

We propose SMAG, the first single-loop stochastic algorithm for difference of weakly convex functions, with a non-asymptotic convergence rate and practical effectiveness.

Findings

SMAG achieves improved convergence rates for complex non-convex problems.

Empirical results validate SMAG's effectiveness in PU learning and pAUC optimization.

The algorithm simplifies computation by using one stochastic gradient step for approximate gradients.

Abstract

In this paper, we study a class of non-smooth non-convex problems in the form of $\min_{x}[\max_{y\in Y}\phi(x, y) - \max_{z\in Z}\psi(x, z)]$, where both $\Phi(x) = \max_{y\in Y}\phi(x, y)$ and $\Psi(x)=\max_{z\in Z}\psi(x, z)$ are weakly convex functions, and $\phi(x, y), \psi(x, z)$ are strongly concave functions in terms of $y$ and $z$, respectively. It covers two families of problems that have been studied but are missing single-loop stochastic algorithms, i.e., difference of weakly convex functions and weakly convex strongly-concave min-max problems. We propose a stochastic Moreau envelope approximate gradient method dubbed SMAG, the first single-loop algorithm for solving these problems, and provide a state-of-the-art non-asymptotic convergence rate. The key idea of the design is to compute an approximate gradient of the Moreau envelopes of $\Phi, \Psi$ using only one step of stochastic gradient update of the primal and dual variables. Empirically, we conduct experiments on positive-unlabeled (PU) learning and partial area under ROC curve (pAUC) optimization with an adversarial fairness regularizer to validate the effectiveness of our proposed algorithms.