On Coresets For Regularized Regression

06/09/2020
by   Rachit Chhaya, et al.
1

We study the effect of norm based regularization on the size of coresets for the regularized regression problems. Specifically, given a matrix A∈R^n × d with n≫ d and a vector b∈R ^ n and λ > 0, we analyze the size of coresets for regularized versions of regression of the form Ax-b_p^r + λx_q^s . It has been shown for the case of ridge regression (p,q,r,s=2) that we can obtain a coreset smaller than the coreset for its unregularized counterpart i.e. least squares regression (Avron et al.). We show that when r ≠ s, no coreset for some regularized regression can have size smaller than the optimal coreset of the unregularized version. The well known lasso problem falls under this category and hence does not allow a coreset smaller than the one for least squares regression. We propose a modified version of the lasso problem and obtain for it a coreset of size smaller than the least square regression. We empirically show that the modified version of lasso also induces sparsity in solution like the lasso. We also obtain smaller coresets for ℓ_p regression with ℓ_p regularization. We extend our methods to multi response regularized regression. Finally, we empirically demonstrate the coreset performance for the modified lasso and the ℓ_1 regression with ℓ_1 regularization.

READ FULL TEXT
research
06/08/2018

The Well Tempered Lasso

We study the complexity of the entire regularization path for least squa...
research
07/20/2022

Provably tuning the ElasticNet across instances

An important unresolved challenge in the theory of regularization is to ...
research
10/08/2011

Regularized Laplacian Estimation and Fast Eigenvector Approximation

Recently, Mahoney and Orecchia demonstrated that popular diffusion-based...
research
11/26/2014

A note relating ridge regression and OLS p-values to preconditioned sparse penalized regression

When the design matrix has orthonormal columns, "soft thresholding" the ...
research
02/16/2017

A new concentration inequality for the excess risk in least-squares regression with random design and heteroscedastic noise

We prove a new concentration inequality for the excess risk of a M-estim...
research
08/16/2007

Piecewise linear regularized solution paths

We consider the generic regularized optimization problem β̂(λ)=_βL(y,Xβ)...
research
03/22/2016

Localized Lasso for High-Dimensional Regression

We introduce the localized Lasso, which is suited for learning models th...

Please sign up or login with your details

Forgot password? Click here to reset