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Greedy Algorithms for Cone Constrained Optimization with Convergence Guarantees

Francesco Locatello,Michael Tschannen,G. Rätsch,Martin Jaggi

Published 2017 in Neural Information Processing Systems

ABSTRACT

Greedy optimization methods such as Matching Pursuit (MP) and Frank-Wolfe (FW) algorithms regained popularity in recent years due to their simplicity, effectiveness and theoretical guarantees. MP and FW address optimization over the linear span and the convex hull of a set of atoms, respectively. In this paper, we consider the intermediate case of optimization over the convex cone, parametrized as the conic hull of a generic atom set, leading to the first principled definitions of non-negative MP algorithms for which we give explicit convergence rates and demonstrate excellent empirical performance. In particular, we derive sublinear ($\mathcal{O}(1/t)$) convergence on general smooth and convex objectives, and linear convergence ($\mathcal{O}(e^{-t})$) on strongly convex objectives, in both cases for general sets of atoms. Furthermore, we establish a clear correspondence of our algorithms to known algorithms from the MP and FW literature. Our novel algorithms and analyses target general atom sets and general objective functions, and hence are directly applicable to a large variety of learning settings.

PUBLICATION RECORD

  • Publication year

    2017

  • Venue

    Neural Information Processing Systems

  • Publication date

    2017-05-01

  • Fields of study

    Mathematics, Computer Science

  • Identifiers
    arXiv 1705.11041
  • External record

    Open on Semantic Scholar

  • Source metadata

    Semantic Scholar

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