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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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**Published in:** LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

The online manipulation-resilient testing model, proposed by Kalemaj, Raskhodnikova and Varma (ITCS 2022 and Theory of Computing 2023), studies property testing in situations where access to the input degrades continuously and adversarially. Specifically, after each query made by the tester is answered, the adversary can intervene and either erase or corrupt t data points. In this work, we investigate a more nuanced version of the online model in order to overcome old and new impossibility results for the original model. We start by presenting an optimal tester for linearity and a lower bound for low-degree testing of Boolean functions in the original model. We overcome the lower bound by allowing batch queries, where the tester gets a group of queries answered between manipulations of the data. Our batch size is small enough so that function values for a single batch on their own give no information about whether the function is of low degree. Finally, to overcome the impossibility results of Kalemaj et al. for sortedness and the Lipschitz property of sequences, we extend the model to include t < 1, i.e., adversaries that make less than one erasure per query. For sortedness, we characterize the rate of erasures for which online testing can be performed, exhibiting a sharp transition from optimal query complexity to impossibility of testability (with any number of queries). Our online tester works for a general class of local properties of sequences. One feature of our results is that we get new (and in some cases, simpler) optimal algorithms for several properties in the standard property testing model.

Omri Ben-Eliezer, Esty Kelman, Uri Meir, and Sofya Raskhodnikova. Property Testing with Online Adversaries. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 11:1-11:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{beneliezer_et_al:LIPIcs.ITCS.2024.11, author = {Ben-Eliezer, Omri and Kelman, Esty and Meir, Uri and Raskhodnikova, Sofya}, title = {{Property Testing with Online Adversaries}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {11:1--11:25}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-309-6}, ISSN = {1868-8969}, year = {2024}, volume = {287}, editor = {Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.11}, URN = {urn:nbn:de:0030-drops-195395}, doi = {10.4230/LIPIcs.ITCS.2024.11}, annote = {Keywords: Linearity testing, low-degree testing, Reed-Muller codes, testing properties of sequences, erasure-resilience, corruption-resilience} }

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**Published in:** LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

We give alternate proofs for three related results in analysis of Boolean functions, namely the KKL Theorem, Friedgut’s Junta Theorem, and Talagrand’s strengthening of the KKL Theorem. We follow a new approach: looking at the first Fourier level of the function after a suitable random restriction and applying the Log-Sobolev inequality appropriately. In particular, we avoid using the hypercontractive inequality that is common to the original proofs. Our proofs might serve as an alternate, uniform exposition to these theorems and the techniques might benefit further research.

Esty Kelman, Subhash Khot, Guy Kindler, Dor Minzer, and Muli Safra. Theorems of KKL, Friedgut, and Talagrand via Random Restrictions and Log-Sobolev Inequality. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 26:1-26:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{kelman_et_al:LIPIcs.ITCS.2021.26, author = {Kelman, Esty and Khot, Subhash and Kindler, Guy and Minzer, Dor and Safra, Muli}, title = {{Theorems of KKL, Friedgut, and Talagrand via Random Restrictions and Log-Sobolev Inequality}}, booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)}, pages = {26:1--26:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-177-1}, ISSN = {1868-8969}, year = {2021}, volume = {185}, editor = {Lee, James R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.26}, URN = {urn:nbn:de:0030-drops-135657}, doi = {10.4230/LIPIcs.ITCS.2021.26}, annote = {Keywords: Fourier Analysis, Hypercontractivity, Log-Sobolev Inequality} }

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