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**Published in:** LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

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.

Vipul Arora, Arnab Bhattacharyya, Mathews Boban, Venkatesan Guruswami, and Esty Kelman. Outlier Robust Multivariate Polynomial Regression. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 12:1-12:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{arora_et_al:LIPIcs.ESA.2024.12, author = {Arora, Vipul and Bhattacharyya, Arnab and Boban, Mathews and Guruswami, Venkatesan and Kelman, Esty}, title = {{Outlier Robust Multivariate Polynomial Regression}}, booktitle = {32nd Annual European Symposium on Algorithms (ESA 2024)}, pages = {12:1--12:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-338-6}, ISSN = {1868-8969}, year = {2024}, volume = {308}, editor = {Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.12}, URN = {urn:nbn:de:0030-drops-210830}, doi = {10.4230/LIPIcs.ESA.2024.12}, annote = {Keywords: Robust Statistics, Polynomial Regression, Sample Efficient Learning} }

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