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Alexandre Belloni (bellonimit.edu) Abstract: The classical perceptron algorithm is an elementary rowaction/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 rescaled 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 polynomialtime in the bitlength 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 rescaled perceptron algorithm based on the notion of a deepseparation oracle of a cone, which essentially computes a certificate of strong separation. We give a general condition under which the rescaled perceptron algorithm is itself theoretically efficient; this includes the cases when K is the crossproduct of halfspaces, secondorder cones, and the positive semidefinite 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 Modify/Update this entry  
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