  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.
