Single-Loop Stochastic Algorithms for Difference of Max-Structured Weakly Convex Functions
Quanqi Hu, Qi Qi, Zhaosong Lu, Tianbao Yang
TL;DR
A stochastic Moreau envelope approximate gradient method dubbed SMAG is proposed, the first single-loop algorithm for solving non-smooth non-convex problems, and the state-of-the-art non-asymptotic convergence rate is provided.
Abstract
In this paper, we study a class of non-smooth non-convex problems in the form of $\min_{x}[\max_{y\in Y}φ(x, y) - \max_{z\in Z}ψ(x, z)]$, where both $Φ(x) = \max_{y\in Y}φ(x, y)$ and $Ψ(x)=\max_{z\in Z}ψ(x, z)$ are weakly convex functions, and $φ(x, y), ψ(x, z)$ 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 $Φ, Ψ$ 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.