eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-09-23
12:1
12:17
10.4230/LIPIcs.ESA.2024.12
article
Outlier Robust Multivariate Polynomial Regression
Arora, Vipul
1
https://orcid.org/0000-0002-5522-9086
Bhattacharyya, Arnab
1
https://orcid.org/0000-0002-3332-0700
Boban, Mathews
1
https://orcid.org/0009-0002-9706-060X
Guruswami, Venkatesan
2
https://orcid.org/0000-0001-7926-3396
Kelman, Esty
3
4
https://orcid.org/0009-0007-4962-848X
School of Computing, National University of Singapore, Singapore
Department of EECS, and Department of Mathematics, University of California, Berkeley, CA, USA
CSAIL, Massachusetts Institute of Technology, Cambridge, MA, USA
Department of Computer Science, and Faculty of Computing & Data Sciences, Boston University, MA, USA
We study the problem of robust multivariate polynomial regression: let p: ℝⁿ → ℝ be an unknown n-variate polynomial of degree at most d in each variable. We are given as input a set of random samples (𝐱_i,y_i) ∈ [-1,1]ⁿ × ℝ that are noisy versions of (𝐱_i,p(𝐱_i)). More precisely, each 𝐱_i is sampled independently from some distribution χ on [-1,1]ⁿ, and for each i independently, y_i is arbitrary (i.e., an outlier) with probability at most ρ < 1/2, and otherwise satisfies |y_i-p(𝐱_i)| ≤ σ. The goal is to output a polynomial p̂, of degree at most d in each variable, within an 𝓁_∞-distance of at most O(σ) from p.
Kane, Karmalkar, and Price [FOCS'17] solved this problem for n = 1. We generalize their results to the n-variate setting, showing an algorithm that achieves a sample complexity of O_n(dⁿlog d), where the hidden constant depends on n, if χ is the n-dimensional Chebyshev distribution. The sample complexity is O_n(d^{2n}log d), if the samples are drawn from the uniform distribution instead. The approximation error is guaranteed to be at most O(σ), and the run-time depends on log(1/σ). In the setting where each 𝐱_i and y_i are known up to N bits of precision, the run-time’s dependence on N is linear. We also show that our sample complexities are optimal in terms of dⁿ. Furthermore, we show that it is possible to have the run-time be independent of 1/σ, at the cost of a higher sample complexity.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol308-esa2024/LIPIcs.ESA.2024.12/LIPIcs.ESA.2024.12.pdf
Robust Statistics
Polynomial Regression
Sample Efficient Learning