Revisiting the Approximate Carath\'eodory Problem via the Frank-Wolfe Algorithm

TL;DR

This paper revisits the approximate Carathéodory problem using the Frank-Wolfe algorithm, providing simplified analysis, improved bounds, and practical methods for sparse projections in various $\, ext{l}_p$-norm scenarios.

Contribution

It introduces a Frank-Wolfe based approach for the approximate Carathéodory problem, offering simplified analysis, improved bounds, and methods for sparse projections across different norms.

Findings

Derived natural cardinality bounds using Frank-Wolfe convergence rates.

Provided bounds for cases when the point is interior or a convex combination with small diameter.

Extended bounds to nonsmooth variants for $p$ in [1,2) and infinity.

Abstract

The approximate Carath\'eodory theorem states that given a compact convex set $\mathcal{C}\subset\mathbb{R}^n$ and $p\in\left[2,+\infty\right[$, each point $x^*\in\mathcal{C}$ can be approximated to $\epsilon$-accuracy in the $\ell_p$-norm as the convex combination of $\mathcal{O}(pD_p^2/\epsilon^2)$ vertices of $\mathcal{C}$, where $D_p$ is the diameter of $\mathcal{C}$ in the $\ell_p$-norm. A solution satisfying these properties can be built using probabilistic arguments or by applying mirror descent to the dual problem. We revisit the approximate Carath\'eodory problem by solving the primal problem via the Frank-Wolfe algorithm, providing a simplified analysis and leading to an efficient practical method. Furthermore, improved cardinality bounds are derived naturally using existing convergence rates of the Frank-Wolfe algorithm in different scenarios, when $x^*$ is in the interior of $\mathcal{C}$, when $x^*$ is the convex combination of a subset of vertices with small diameter, or when $\mathcal{C}$ is uniformly convex. We also propose cardinality bounds when $p\in\left[1,2\right[\cup\{+\infty\}$ via a nonsmooth variant of the algorithm. Lastly, we address the problem of finding sparse approximate projections onto $\mathcal{C}$ in the $\ell_p$-norm, $p\in\left[1,+\infty\right]$.