  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.
