A Projection-free Algorithm for Constrained Stochastic Multi-level Composition Optimization
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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 ϵ-stationary solution, are of order 𝒪_T(ϵ^-2) and 𝒪_T(ϵ^-3) respectively, where 𝒪_T hides constants in T. Notably, the dependence of these complexity bounds on ϵ and T are separate in the sense that changing one does not impact the dependence of the bounds on the other. 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.
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