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.

翻译：本文研究一类非光滑非凸问题，其形式为$\min_{x}[\max_{y\in Y}\phi(x, y) - \max_{z\in Z}\psi(x, z)]$，其中$\Phi(x) = \max_{y\in Y}\phi(x, y)$与$\Psi(x)=\max_{z\in Z}\psi(x, z)$均为弱凸函数，且$\phi(x, y)$和$\psi(x, z)$分别关于$y$和$z$为强凹函数。该形式涵盖了两类已被研究但尚缺单循环随机算法的问题族：弱凸函数之差问题与弱凸强凹极小极大问题。我们提出了一种名为SMAG的随机莫罗包络近似梯度法，这是首个用于求解此类问题的单循环算法，并提供了当前最优的非渐近收敛速率。该设计的关键思想是仅通过一次原始变量与对偶变量的随机梯度更新来计算$\Phi$和$\Psi$的莫罗包络之近似梯度。在实证研究中，我们通过在正例-未标记（PU）学习以及带有对抗公平正则化项的部分ROC曲线下面积（pAUC）优化任务上进行实验，验证了所提算法的有效性。