A deterministic rescaled perceptron algorithm

The perceptron algorithm is a simple iterative procedure for finding a point in a convex cone F . At each iteration, the algorithm only involves a query to a separation oracle for F and a simple update on a trial solution. The perceptron algorithm is guaranteed to find a point in F after O(1/τ F ) iterations, where τF is the width of the cone F . We propose a version of the perceptron algorithm that includes a periodic rescaling of the ambient space. In contrast to the classical version, our rescaled version finds a point in F in O(m log(1/τF )) perceptron updates. This result is inspired by and strengthens the previous work on randomized rescaling of the perceptron algorithm by Dunagan and Vempala [Math. Program. 114 (2006), 101–114] and by Belloni, Freund, and Vempala [Math. Oper. Res. 34 (2009), 621–641]. In particular, our algorithm and its complexity analysis are simpler and shorter. Furthermore, our algorithm does not require randomization or deep separation oracles.