An Improved Deterministic Rescaling for Linear Programming Algorithms

TL;DR

This paper introduces an improved deterministic rescaling technique for the perceptron algorithm in linear programming, enabling earlier rescaling and faster convergence, with implications for polynomial-time algorithms via multiplicative weights.

Contribution

It presents a novel deterministic rescaling method that enhances the perceptron algorithm's efficiency and connects it to multiplicative weights update methods.

Findings

Faster running time for the rescaled perceptron algorithm

Enables earlier rescaling in the perceptron method

Establishes a connection to polynomial-time algorithms using multiplicative weights

Abstract

The perceptron algorithm for linear programming, arising from machine learning, has been around since the 1950s. While not a polynomial-time algorithm, it is useful in practice due to its simplicity and robustness. In 2004, Dunagan and Vempala showed that a randomized rescaling turns the perceptron method into a polynomial time algorithm, and later Pe\~{n}a and Soheili gave a deterministic rescaling. In this paper, we give a deterministic rescaling for the perceptron algorithm that improves upon the previous rescaling methods by making it possible to rescale much earlier. This results in a faster running time for the rescaled perceptron algorithm. We will also demonstrate that the same rescaling methods yield a polynomial time algorithm based on the multiplicative weights update method. This draws a connection to an area that has received a lot of recent attention in theoretical computer science.