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We develop a novel primal-dual algorithm to solve a class of nonsmooth andnonlinear compositional convex minimization problems, which covers manyexisting and brand-new models as special cases. Our approach relies on acombination of a new nonconvex potential function, Nesterov's acceleratedscheme, and an adaptive parameter updating strategy. Our algorithm issingle-loop and has low per-iteration complexity. Under only general convexityand mild assumptions, our algorithm achieves $\mathcal{O}(1/k)$ convergencerates through three different criteria: primal objective residual, dualobjective residual, and primal-dual gap, where $k$ is the iteration counter.Our rates are both ergodic (i.e., on an averaging sequence) and non-ergodic(i.e., on the last-iterate sequence). These convergence rates can beaccelerated up to $\mathcal{O}(1/k^2)$ if only one objective term is stronglyconvex (or equivalently, its conjugate is $L$-smooth). To the best of ourknowledge, this is the first algorithm achieving optimal rates on the primallast-iterate sequence for nonlinear compositional convex minimization. As aby-product, we specify our algorithm to solve a general convex cone constrainedprogram with both ergodic and non-ergodic rate guarantees. We test ouralgorithms and compare them with two recent methods on a binary classificationand a convex-concave game model.

A New Primal-Dual Algorithm for a Class of Nonlinear Compositional Convex Optimization Problems