Single-loop Stochastic Algorithms for Difference of Max-Structured Weakly Convex Functions
<|reference_start|>Single-loop Stochastic Algorithms for Difference of Max-Structured Weakly Convex Functions: 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.<|reference_end|>
@article{hu2024single-loop,
    title={Single-Loop Stochastic Algorithms for Difference of Max-Structured
  Weakly Convex Functions},
    author={Quanqi Hu, Qi Qi, Zhaosong Lu, Tianbao Yang},
    journal={arXiv preprint arXiv:2405.18577},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.18577},
    primaryClass={math.OC cs.LG stat.ML}
}