An Efficient Re-scaled Perception Algorithm for Conic Systems

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 τ of the cone of feasible solutions, and the iteration complexity of the perceptron algorithm is bounded by 1/τ, see Rosenblatt 1962 [20]. Dunagan and Vempala [5] have developed a re-scaled version of the perceptron algorithm with an improved complexity of O(n ln(1/τ)) iterations (with high probability), which is theoretically efficient in τ , 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 ∈ 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 show that the re-scaled perceptron algorithm is theoretically efficient if an efficient deep-separation oracle is available for the feasible region. Furthermore, when K is the cross-product of basic cones that are either half-spaces or second-order cones, then a deep-separation oracle is available and hence the re-scaled perceptron algorithm is theoretically efficient. When the basic cones of K include semi-definite cones, then a probabilistic deep-separation oracle for K can be constructed that also yields a theoretically efficient version of the re-scaled perceptron algorithm.