Semantic Apparatus – Robust Linear Regression: Optimal Rates in Polynomial Time

Cited by Lee Sonogan

Linear Regression in Python – Real Python

Abstract by Ainesh Bakshi , Adarsh Prasad

We obtain robust and computationally efficient estimators for learning several linear models that achieve statistically optimal convergence rate under minimal distributional assumptions. Concretely, we assume our data is drawn from a k-hypercontractive distribution and an є-fraction is adversarially corrupted. We then describe an estimator that converges to the optimal least-squares minimizer for the true distribution at a rate proportional to є2−2/k, when the noise is independent of the covariates. We note that no such estimator was known prior to our work, even with access to unbounded computation. The rate we achieve is information-theoretically optimal and thus we resolve the main open question in Klivans, Kothari and Meka [COLT’18].

Our key insight is to identify an analytic condition that serves as a polynomial relaxation of independence of random variables. In particular, we show that when the moments of the noise and covariates are negatively-correlated, we obtain the same rate as independent noise. Further, when the condition is not satisfied, we obtain a rate proportional to є2−4/k, and again match the information-theoretic lower bound. Our central technical contribution is to algorithmically exploit independence of random variables in the ”sum-of-squares” framework by formulating it as the aforementioned polynomial inequality.

Publication: 2021 International Joint Conference on Neural Networks (IJCNN) (Peer-Reviewed Journal)

Pub Date: 15 June, 2021 Doi: https://doi.org/10.1145/3406325.3451001

Keywords: Linear Regression, Optimal Rates, Polynomial Time, Robust Semantic

https://dl.acm.org/doi/abs/10.1145/3406325.3451001 (Plenty more sections and references in this research article)

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