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This paper revisits a class of convex Finite-Sum Coupled CompositionalStochastic Optimization (cFCCO) problems with many applications, includinggroup distributionally robust optimization (GDRO), learning with imbalanceddata, reinforcement learning, and learning to rank. To better solve theseproblems, we introduce an efficient single-loop primal-dual block-coordinateproximal algorithm, dubbed ALEXR. This algorithm leverages block-coordinatestochastic mirror ascent updates for the dual variable and stochastic proximalgradient descent updates for the primal variable. We establish the convergencerates of ALEXR in both convex and strongly convex cases under smoothness andnon-smoothness conditions of involved functions, which not only improve thebest rates in previous works on smooth cFCCO problems but also expand the realmof cFCCO for solving more challenging non-smooth problems such as the dual formof GDRO. Finally, we present lower complexity bounds to demonstrate that theconvergence rates of ALEXR are optimal among first-order block-coordinatestochastic algorithms for the considered class of cFCCO problems.

ALEXR: An Optimal Single-Loop Algorithm for Convex Finite-Sum Coupled Compositional Stochastic Optimization