Consistent Estimation for PCA and Sparse Regression with Oblivious Outliers
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Date
2021
Publication Type
Conference Paper
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yes
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Abstract
We develop machinery to design efficiently computable and consistent estimators, achieving estimation error approaching zero as the number of observations grows, when facing an oblivious adversary that may corrupt responses in all but an α fraction of the samples. As concrete examples, we investigate two problems: sparse regression and principal component analysis (PCA). For sparse regression, we achieve consistency for optimal sample size n≳(klogd)/α^2 and optimal error rate O([(klogd)/(n⋅α^2)]^{1/2} where n is the number of observations, d is the number of dimensions and k is the sparsity of the parameter vector, allowing the fraction of inliers to be inverse-polynomial in the number of samples. Prior to this work, no estimator was known to be consistent when the fraction of inliers α is o(1/loglogn), even for (non-spherical) Gaussian design matrices. Results holding under weak design assumptions and in the presence of such general noise have only been shown in dense setting (i.e., general linear regression) very recently by d'Orsi et al. [dNS21]. In the context of PCA, we attain optimal error guarantees under broad spikiness assumptions on the parameter matrix (usually used in matrix completion). Previous works could obtain non-trivial guarantees only under the assumptions that the measurement noise corresponding to the inliers is polynomially small in n (e.g., Gaussian with variance 1/n^2).
To devise our estimators, we equip the Huber loss with non-smooth regularizers such as the ℓ1 norm or the nuclear norm, and extend d'Orsi et al.'s approach [dNS21] in a novel way to analyze the loss function. Our machinery appears to be easily applicable to a wide range of estimation problems.
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Publication status
published
Book title
Advances in Neural Information Processing Systems 34
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Volume
Pages / Article No.
25427 - 25438
Publisher
Curran
Event
35th Annual Conference on Neural Information Processing Systems (NeurIPS 2021)
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Software
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Organisational unit
09622 - Steurer, David / Steurer, David
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Funding
815464 - Unified Theory of Efficient Optimization and Estimation (EC)