A Projection-free Algorithm for Constrained Stochastic Multi-level Composition Optimization

We propose a projection-free conditional gradient-type algorithm for smooth stochastic multi-level composition optimization, where the objective function is a nested composition of T functions and the constraint set is a closed convex set. Our algorithm assumes access to noisy evaluations of the functions and their gradients, through a stochastic first-order oracle satisfying certain standard unbiasedness and second-moment assumptions. We show that the number of calls to the stochastic first-order oracle and the linear-minimization oracle required by the proposed algorithm, to obtain an epsilon-stationary solution, are of order O-T (epsilon(-2)) and O-T (epsilon(-3)) respectively, where OT hides constants in T. Notably, the dependence of these complexity bounds on epsilon and T are separate in the sense that changing one does not impact the dependence of the bounds on the other. For the case of T = 1, we also provide a high-probability convergence result that depends poly-logarithmically on the inverse confidence level. Moreover, our algorithm is parameter-free and does not require any (increasing) order of mini-batches to converge unlike the common practice in the analysis of stochastic conditional gradient-type algorithms.