Fast and Accurate Least-Mean-Squares Solvers

Ibrahim Jubran, Alaa Maalouf, Dan Feldman

Abstract: Least-mean squares (LMS) solvers such as Linear / Ridge / Lasso-Regressions, SVD and Elastic-Nets not only solve fundamental machine learning problems, but are also the building blocks in a variety of other methods, such as decision trees and matrix factorizations. %The solutions are usually a function of the corresponding covariance matrix or its variants. We suggest an algorithm that gets a finite set of $d$-dimensional real vectors and returns a weighted subset of $d+1$ vectors whose sum is \emph{exactly} the same. The proof in Caratheodory's Theorem (1907) computes such a subset in $O(n^2d^2)$ time and thus not used in practice. Our algorithm computes this subset in $O(nd)$ time, using $O(\log n)$ calls to Caratheodory's construction on small but "smart" subsets. This is based on a novel paradigm of fusion between different data summarization techniques, known as sketches and coresets. As an example application, we show how it can be used to boost the performance of existing LMS solvers, such as those in scikit-learn library, up to x100. Generalization for streaming and distributed (big) data is trivial. Extensive experimental results and complete open source code are also provided.

Code Link: https://github.com/ibramjub/Fast-and-Accurate-Least-Mean-Squares-Solvers

CMT Num: 4511