@inproceedings{
hu2024singleloop,
title={Single-Loop Stochastic Algorithms for Difference of Max-Structured Weakly Convex Functions},
author={Quanqi Hu and Qi Qi and Zhaosong Lu and Tianbao Yang},
booktitle={The Thirty-eighth Annual Conference on Neural Information Processing Systems},
year={2024},
url={https://openreview.net/forum?id=NhtBXSNXKA}
}
In this paper, we study a class of non-smooth non-convex problems in the form of $\min_{x}[\max_{y\in\mathcal Y}\phi(x, y) - \max_{z\in\mathcal Z}\psi(x, z)]$, where both $\Phi(x) = \max_{y\in\mathcal Y}\phi(x, y)$ and $\Psi(x)=\max_{z\in\mathcal Z}\psi(x, z)$ 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.
Single-Loop Stochastic Algorithms for Difference of Max-Structured Weakly Convex Functions
Quanqi Hu Qi Qi Zhaosong Lu Tianbao Yang