Active Linear Regression for ℓp Norms and Beyond

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 d Axb , 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 which outputs a (1 + ǫ)-approximate solution using just Õ(d/ǫ 2 ) queries to b for p ∈ (0, 1), Õ(d/ǫ) queries for p ∈ (1, 2), and Õ(d p/2 /ǫ p ) queries for p ∈ (2, ∞). For p ∈ (0, 2), our bounds are optimal up to logarithmic factors, thus settling the query complexity for this range of p. For p ∈ (2, ∞), our dependence on d is optimal, while our dependence on ǫ is off by at most a single ǫ factor, up to logarithmic factors. Our result resolves an open question of Chen and Dereziński, who gave near optimal bounds for the ℓ 1 norm, but required at least d 2 /ǫ 2 samples for ℓ p regression with p ∈ (1, 2), and gave no bounds for p ∈ (2, ∞) or p ∈ (0, 1).

We also provide the first total sensitivity upper bound of O(d max1,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 ℓ p regression, we obtain an active regression algorithm making Õ(d 1+max1,p/2 / poly(ǫ)) queries for such loss functions, including the important cases of the Tukey and Huber losses. This answers another question of Chen and Dereziński. For the Huber loss, we further improve our bound to a sample complexity of Õ(d 4-2 √ 2 / poly(ǫ)) where 4 -2 √ 2 ≈ 1.17157. Our sensitivity bounds also give improvements to a variety of previous results using sensitivity sampling, including Orlicz norm subspace embeddings, robust subspace approximation, and dimension reduction for smoothed p-norms.

Finally, our active sampling results give the first sublinear time algorithms for Kronecker product regression under every ℓ p norm. Previous results required reading the entire b vector in the kernel feature space. 1 Our work will also extend to other loss functions of the form n i=1 M ([Axb]i) that are not necessarily norms. 2 In principal, entries of b can be read adaptively -i.e., we can select indices to query based on the results of other queries. However, the benefits of adaptivity appear limited. Most methods for solving Problem 1.1 and those studied in this paper are non-adaptive.

3 All query complexity bounds in this section are stated for solving Problem 1.1 with high constant probabilitye.g., probability 99/100. In later sections we will include an explicit dependence on a failure probability δ.