Momentum based Projection Free Stochastic Optimization Under Affine Constraints

Abstract

This paper considers stochastic convex optimization problems with affine constraints, in addition to other deterministic convex constraints on the domain of the variables. Usually, to solve the stochastic optimization problems, the gradient based approaches make use of stochastic gradients of the objective arriving sequentially over iterations and then project back the iterate on the feasible domain. However, there are scenarios where carrying out projection step onto the convex domain constraints at every iteration is not viable. Traditionally, the projection operation can be avoided by considering the Frank-Wolfe (FW) a.k.a. conditional gradient variant of the algorithm. Existing approaches towards solving the problems having costly projections along with affine constraints combine homotopy and smoothing techniques under the conditional gradient framework. The state-of-the-art stochastic FW variants to solve such problems achieve a convergence rate of O(k <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1/3</sup> ) on the optimality gap and O(k <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-5/1</sup> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) on the feasibility gap. In this work, we propose the Momentum based Stochastic Homotopy Frank-Wolfe (MSHFW) algorithm that tracks gradient using momentum technique and guarantees an optimal convergence rate of O(k <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1/2</sup> ) in terms of both optimality and feasibility gap. The efficacy of the proposed algorithm is tested on relevant problems of K-means clustering and sparse matrix estimation.