Papers › On Coresets For Regularized Regression

On Coresets For Regularized Regression

9 Jun 2020ICML 2020 1arXiv:2006.05440archive 2025-07-28

Rachit Chhaya, Anirban Dasgupta, Supratim Shit

We study the effect of norm based regularization on the size of coresets for 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-𝐛ₚʳ + λ𝐱_qˢ . Prior work has shown that for ridge regression (where p,q,r,s=2) we can obtain a coreset that is smaller than the coreset for the unregularized counterpart i.e. least squares regression (Avron et al). We show that when r ≠s, no coreset for 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, similar to the original lasso. We also obtain smaller coresets for ℓₚ regression with ℓₚ regularization. We extend our methods to multi response regularized regression. Finally, we empirically demonstrate the coreset performance for the modified lasso and the ℓ₁ regression with ℓ₁ regularization.

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