This paper investigates new families of compositional optimization problems, called $\underline{\bf n}$on-$\underline{\bf s}$mooth $\underline{\bf w}$eakly-$\underline{\bf c}$onvex $\underline{\bf f}$inite-sum $\underline{\bf c}$oupled $\underline{\bf c}$ompositional $\underline{\bf o}$ptimization (NSWC FCCO). There has been a growing interest in FCCO due to its wide-ranging applications in machine learning and AI, as well as its ability to address the shortcomings of stochastic algorithms based on empirical risk minimization. However, current research on FCCO presumes that both the inner and outer functions are smooth, limiting their potential to tackle a more diverse set of problems. Our research expands on this area by examining non-smooth weakly-convex FCCO, where the outer function is weakly convex and non-decreasing, and the inner function is weakly-convex. We analyze a single-loop algorithm and establish its complexity for finding an $\epsilon$-stationary point of the Moreau envelop of the objective function. Additionally, we also extend the algorithm to solving novel non-smooth weakly-convex tri-level finite-sum coupled compositional optimization problems, which feature a nested arrangement of three functions. Lastly, we explore the applications of our algorithms in deep learning for two-way partial AUC maximization and multi-instance two-way partial AUC maximization, using empirical studies to showcase the effectiveness of the proposed algorithms.

翻译：本文研究了一类新的组合优化问题，称为非光滑弱凸有限和耦合组合优化（NSWC FCCO）。由于其在机器学习和人工智能中的广泛应用，以及能够解决基于经验风险最小化的随机算法的不足，FCCO日益受到关注。然而，当前关于FCCO的研究假设内函数和外函数都是光滑的，这限制了其在更广泛问题上的应用潜力。我们的研究通过考察非光滑弱凸FCCO扩展了这一领域，其中外函数是弱凸且非递减的，内函数是弱凸的。我们分析了一种单循环算法，并建立了其寻找目标函数的Moreau包络的ε-稳定点的复杂度。此外，我们还将该算法扩展到解决新颖的非光滑弱凸三层有限和耦合组合优化问题，该问题具有三个函数的嵌套结构。最后，我们探索了所提算法在深度学习中用于双向部分AUC最大化和多实例双向部分AUC最大化的应用，并通过实证研究展示了所提算法的有效性。