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An Efficient Re-scaled Perceptron Algorithm for Conic Systems

Alexandre Belloni (belloni***at***mit.edu)
Robert M. Freund (rfreund***at***mit.edu)
Santosh Vempala (vempala***at***math.mit.edu)

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 \in 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.

Keywords: Convex Cones ; Perceptron ; Conic System ; Separation Oracle

Category 1: Linear, Cone and Semidefinite Programming

Citation: Technical Report RC24073, October/2006 IBM T.J. Watson Research Center 1101 Kitchawan Road, Yorktown Heights, NY 10598

Download: [PDF]

Entry Submitted: 10/10/2006
Entry Accepted: 10/10/2006
Entry Last Modified: 10/10/2006

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