The multi-category support vector machine (MC-SVM) is one of the most popular learning algorithms. There are numerous MC-SVM variants, although different optimization algorithms were developed for diverse machines. In this study, we a new algorithm that can be applied to several variants. based on Frank-Wolfe framework requires two subproblems, direction-finding and line search, in each iterati...

The approximate Carathéodory theorem states that given a compact convex set $${\mathcal {C}}\subset {\mathbb {R}}^n$$ and $$p\in [2,+\infty [$$ , each point $$x^*\in {\mathcal {C}}$$ can be approximated to $$\epsilon $$ -accuracy in the $$\ell _p$$ -norm as combination of {O}}(pD_p^2/\epsilon ^2)$$ vertices where $$D_p$$ is diameter -norm. A solution satisfying these properties built using prob...

In this paper, we propose a new algorithm combining the Douglas-Rachford (DR) and Frank-Wolfe algorithm, also known as conditional gradient (CondG) method, for solving classic convex feasibility problem. Within which will be named {\it Approximate (ApDR) algorithm}, CondG method is used subroutine to compute feasible inexact projections on sets under consideration, ApDR iteration defined based ...

We establish novel generalization bounds for learning algorithms that converge to global minima. We do so by deriving black-box stability results that only depend on the convergence of a learning algorithm and the geometry around the minimizers of the loss function. The results are shown for nonconvex loss functions satisfying the Polyak-Łojasiewicz (PL) and the quadratic growth (QG) conditions...

The Frank-Wolfe (FW) optimization algorithm has lately re-gained popularity thanks in particular to its ability to nicely handle the structured constraints appearing in machine learning applications. However, its convergence rate is known to be slow (sublinear) when the solution lies at the boundary. A simple lessknown fix is to add the possibility to take ‘away steps’ during optimization, an o...

We present a new Frank-Wolfe (FW) type algorithm that is applicable to minimization problems with a nonsmooth convex objective. We provide convergence bounds and show that the scheme yields so-called coreset results for various Machine Learning problems including 1-median, Balanced Development, Sparse PCA, Graph Cuts, and the `1-norm-regularized Support Vector Machine (SVM) among others. This m...

Learning sparse combinations is a frequent theme in machine learning. In this paper, we study its associated optimization problem in the distributed setting where the elements to be combined are not centrally located but spread over a network. We address the key challenges of balancing communication costs and optimization errors. To this end, we propose a distributed Frank-Wolfe (dFW) algorithm...

We show that k-means clustering is a matrix factorization problem. Seen from this point of view, k-means clustering can be computed using alternating least squares techniques and we show how the constrained optimization steps involved in this procedure can be solved efficiently using the Frank-Wolfe algorithm.