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

• URL: http://arxiv.org/abs/2202.04296v1
• Date: Wed, 9 Feb 2022 06:05:38 GMT
• Title: A Projection-free Algorithm for Constrained Stochastic Multi-level Composition Optimization
• Authors: Tesi Xiao, Krishnakumar Balasubramanian, Saeed Ghadimi
• Abstract summary: We propose a projection-free conditional gradient-type algorithm for composition optimization. We show that the number of oracles and the linear-minimization oracle required by the proposed algorithm, are of order $mathcalO_T(epsilon-2)$ and $mathcalO_T(epsilon-3)$ respectively.
• Score: 12.096252285460814
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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 $\mathcal{O}_T(\epsilon^{-2})$ and $\mathcal{O}_T(\epsilon^{-3})$ respectively, where $\mathcal{O}_T$ 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. 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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