Coresets, sparse greedy approximation, and the Frank-Wolfe algorithm

The problem of maximizing a concave function f(x) in the unit simplex Δ can be solved approximately by a simple greedy algorithm. For given k , the algorithm can find a point x ( k ) on a k -dimensional face of Δ, such that f ( x ( k ) ≥ f(x * ) − O (1/ k ). Here f ( x * ) is the maximum value of f in Δ, and the constant factor depends on f . This algorithm and analysis were known before, and related to problems of statistics and machine learning, such as boosting, regression, and density mixture estimation. In other work, coming from computational geometry, the existence of ϵ-coresets was shown for the minimum enclosing ball problem by means of a simple greedy algorithm. Similar greedy algorithms, which are special cases of the Frank-Wolfe algorithm, were described for other enclosure problems. Here these results are tied together, stronger convergence results are reviewed, and several coreset bounds are generalized or strengthened.