Inexact and Stochastic Generalized Conditional Gradient with Augmented Lagrangian and Proximal Step
This work addresses computationally intensive settings in machine learning, such as high-dimensional Hilbert spaces, by enabling practical application through inexact computations, but it is incremental as it builds on prior CGALP algorithm.
The paper tackles composite convex minimization problems with affine constraints by proposing inexact and stochastic versions of the CGALP algorithm, showing almost sure convergence of the Lagrangian to an optimum and asymptotic feasibility, with numerical experiments verifying predicted convergence rates.
In this paper we propose and analyze inexact and stochastic versions of the CGALP algorithm developed in the authors' previous paper, 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.