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An Efficient Re-Scaled Perceptron Algorithm for Conic Systems

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  • Belloni, Alexandre
  • Freund, Robert M
  • Vempala, Santosh
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    Abstract

    The classical perceptron algorithm is an elementary row-action/relaxation algorithm for solving a homogeneous linear inequality system Ax > 0. A natural condition measure associated with this algorithm is the Euclidean width T of the cone of feasible solutions, and the iteration complexity of the perceptron algorithm is bounded by 1/T^2, see Rosenblatt 1962. Dunagan and Vempala have developed a re-scaled version of the perceptron algorithm with an improved complexity of O(n ln(1/T)) iterations (with high probability), which is theoretically efficient in T, and in particular is polynomial-time in the bit-length model. We explore extensions of the concepts of these perceptron methods to the general homogeneous conic system Ax is an element of a set int K where K is a regular convex cone. We provide a conic extension of the re-scaled perceptron algorithm based on the notion of a deep-separation oracle of a cone, which essentially computes a certificate of strong separation. We give a general condition under which the re-scaled perceptron algorithm is itself theoretically efficient; this includes the cases when K is the cross-product of half-spaces, second-order cones, and the positive semi-definite cone.

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    File URL: http://hdl.handle.net/1721.1/37304
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    Bibliographic Info

    Paper provided by Massachusetts Institute of Technology (MIT), Sloan School of Management in its series Working papers with number 37304.

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    Date of creation: 27 Apr 2007
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    Handle: RePEc:mit:sloanp:37304

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    Postal: MASSACHUSETTS INSTITUTE OF TECHNOLOGY (MIT), SLOAN SCHOOL OF MANAGEMENT, 50 MEMORIAL DRIVE CAMBRIDGE MASSACHUSETTS 02142 USA

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    Keywords: convex cone; perceptron; conic system; separation oracle;

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