Inexact and Stochastic Generalized Conditional Gradient with Augmented Lagrangian and Proximal Step

Abstract

In this paper we propose and analyze inexact and stochastic versions of the CGALP algorithm developed in [16], which we denote ICGALP , that allows for errors in the computation of several important quantities. In particular this allows one to compute some gradients, proximal terms, and/or linear minimization oracles in an inexact fashion that facilitates the practical application of the algorithm to computationally intensive settings, e.g. in high (or possibly infinite) dimensional Hilbert spaces commonly found in machine learning problems. The algorithm is able to solve composite minimization problems involving the sum of three convex proper lower-semicontinuous functions subject to an affine constraint of the form Ax = b for some bounded linear operator A. Only one of the functions in the objective is assumed to be differentiable, the other two are assumed to have an accessible prox operator and a linear minimization oracle. As main results, we show convergence of the Lagrangian to an optimum and asymptotic feasibility of the affine constraint as well as weak convergence of the dual variable to a solution of the dual problem, all in an almost sure sense. Almost sure convergence rates, both pointwise and ergodic, are given for the Lagrangian values and the feasibility gap. Numerical experiments verifying the predicted rates of convergence are shown as well.

Dynamical Programming for off-the-grid dynamic Inverse Problems

“…The proofs of convergence for inexact and stochastic Frank-Wolfe are given in Section 6 where we confirm that our proposed algorithm converges almost-surely to an optimal reconstruction of our variational model. The stochastic algorithm is similar to that considered by Silveti-Falls et al [2020], although we compute the step-size adaptively rather than fixing it a priori. Our novel reformulation to the shortest-path problem is explained in Section 7, the dynamical programming approach is standard but presented in the current context for clarity.…”