Active Sampling for Linear Regression Beyond the ℓ_2 Norm
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We study active sampling algorithms for linear regression, which aim to query only a small number of entries of a target vector b∈ℝ^n and output a near minimizer to min_x∈ℝ^dAx-b, where A∈ℝ^n × d is a design matrix and · is some loss function. For ℓ_p norm regression for any 0<p<∞, we give an algorithm based on Lewis weight sampling that outputs a (1+ϵ) approximate solution using just Õ(d^max(1,p/2)/poly(ϵ)) queries to b. We show that this dependence on d is optimal, up to logarithmic factors. Our result resolves a recent open question of Chen and Dereziński, who gave near optimal bounds for the ℓ_1 norm, and suboptimal bounds for ℓ_p regression with p∈(1,2). We also provide the first total sensitivity upper bound of O(d^max{1,p/2}log^2 n) for loss functions with at most degree p polynomial growth. This improves a recent result of Tukan, Maalouf, and Feldman. By combining this with our techniques for the ℓ_p regression result, we obtain an active regression algorithm making Õ(d^1+max{1,p/2}/poly(ϵ)) queries, answering another open question of Chen and Dereziński. For the important special case of the Huber loss, we further improve our bound to an active sample complexity of Õ(d^(1+√(2))/2/ϵ^c) and a non-active sample complexity of Õ(d^4-2√(2)/ϵ^c), improving a previous d^4 bound for Huber regression due to Clarkson and Woodruff. Our sensitivity bounds have further implications, improving a variety of previous results using sensitivity sampling, including Orlicz norm subspace embeddings and robust subspace approximation. Finally, our active sampling results give the first sublinear time algorithms for Kronecker product regression under every ℓ_p norm.
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