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

TL;DR

This paper introduces a projection-free stochastic algorithm for multi-level constrained optimization that achieves efficient oracle complexity bounds without increasing mini-batch sizes.

Contribution

It presents a novel parameter-free conditional gradient algorithm with separate complexity bounds for multi-level stochastic constrained problems.

Findings

Oracle call complexity is $ ilde{O}_T(rac{1}{\e^2})$ for stationarity.

Linear-minimization oracle calls are $ ilde{O}_T(rac{1}{\e^3})$.

Algorithm does not require increasing mini-batches for convergence.

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.