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This talk surveys applications of space dilation in the perceptron-like algorithms for solving systems of linear inequalities in polynomial time: from the ellipsoid algorithm to the modified perceptron algorithm and the perceptron rescaling algorithm. It also suggests a more general version of the latter algorithm and poses an open problem about the lowest complexity of this approach. Space dilation is a linear non-orthogonal space transformation for subgradient descent algorithms to handle the case when the subgradient is almost orthogonal to the direction towards the optimum [5]. It aims to reduce the angle between the anti-subgradient and that direction by applying a linear operator that changes the metric of the space. It was exploited in the ellipsoid algorithm to make its running time polynomial [3]. We consider a problem of solving a system of linear inequalities by the perceptron algorithm (a simple incremental subgradient descent method) [4], which may have an exponential complexity in the worst case. The rescaling procedure, which is a special case of the space dilation, is designed for the perceptron rescaling algorithm [2]. It uses the modified perceptro

Year: 2013

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