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

• URL: http://arxiv.org/abs/2405.18577v3
• Date: Mon, 28 Oct 2024 20:04:57 GMT
• Title: Single-Loop Stochastic Algorithms for Difference of Max-Structured Weakly Convex Functions
• Authors: Quanqi Hu, Qi Qi, Zhaosong Lu, Tianbao Yang,
• Abstract summary: We show a class of non-smooth non-asymptotic fairness problems in the form of $min_x[yin Yphi(x, y) - max_zin Zpsix(x, z)]$. We propose an envelope approximate gradient SMAG, the first method for solving these problems, provide a state-of-the-art non-asymptotic convergence rate.
• Score: 41.43895948769255
• License:
• 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.

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