L1 Regression with Lewis Weights Subsampling

• URL: http://arxiv.org/abs/2105.09433v1
• Date: Wed, 19 May 2021 23:15:00 GMT
• Title: L1 Regression with Lewis Weights Subsampling
• Authors: Aditya Parulekar, Advait Parulekar, Eric Price
• Abstract summary: We show that sampling from $X$ according to its Lewis weights and outputting the empirical minimizer succeeds with probability $1-delta$ for $m. We also give a corresponding lower bound of $Omega(fracdvarepsilon2 + (d + frac1varepsilon2) logfrac1delta)$.
• Score: 10.796388585158184
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We consider the problem of finding an approximate solution to $\ell_1$ regression while only observing a small number of labels. Given an $n \times d$ unlabeled data matrix $X$, we must choose a small set of $m \ll n$ rows to observe the labels of, then output an estimate $\widehat{\beta}$ whose error on the original problem is within a $1 + \varepsilon$ factor of optimal. We show that sampling from $X$ according to its Lewis weights and outputting the empirical minimizer succeeds with probability $1-\delta$ for $m > O(\frac{1}{\varepsilon^2} d \log \frac{d}{\varepsilon \delta})$. This is analogous to the performance of sampling according to leverage scores for $\ell_2$ regression, but with exponentially better dependence on $\delta$. We also give a corresponding lower bound of $\Omega(\frac{d}{\varepsilon^2} + (d + \frac{1}{\varepsilon^2}) \log\frac{1}{\delta})$.

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