Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A geometrical viewpoint on the benign overfitting property of the
  minimum $l_2$-norm interpolant estimator   Practitioners have observed that some deep learning models generalize well
even with a perfect fit to noisy training data [5,45,44]. Since then many
theoretical works have revealed some facets of this phenomenon [4,2,1,8] known
as benign overfitting. In particular, in the linear regression model, the
minimum $l_2$-norm interpolant estimator $\hat\beta$ has received a lot of
attention [1,39] since it was proved to be consistent even though it perfectly
fits noisy data under some condition on the covariance matrix $\Sigma$ of the
input vector. Motivated by this phenomenon, we study the generalization
property of this estimator from a geometrical viewpoint. Our main results
extend and improve the convergence rates as well as the deviation probability
from [39]. Our proof differs from the classical bias/variance analysis and is
based on the self-induced regularization property introduced in [2]:
$\hat\beta$ can be written as a sum of a ridge estimator $\hat\beta_{1:k}$ and
an overfitting component $\hat\beta_{k+1:p}$ which follows a decomposition of
the features space $\mathbb{R}^p=V_{1:k}\oplus^\perp V_{k+1:p}$ into the space
$V_{1:k}$ spanned by the top $k$ eigenvectors of $\Sigma$ and the ones
$V_{k+1:p}$ spanned by the $p-k$ last ones. We also prove a matching lower
bound for the expected prediction risk. The two geometrical properties of
random Gaussian matrices at the heart of our analysis are the Dvoretsky-Milman
theorem and isomorphic and restricted isomorphic properties. In particular, the
Dvoretsky dimension appearing naturally in our geometrical viewpoint coincides
with the effective rank from [1,39] and is the key tool to handle the behavior
of the design matrix restricted to the sub-space $V_{k+1:p}$ where overfitting
happens.

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The categories of this paper are: dg-ga
15
The categories of this paper are: nucl-ex
31
The categories of this paper are: ao-sci
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The categories of this paper are: nucl-th
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The categories of this paper are: q-bio
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The categories of this paper are: alg-geom
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The categories of this paper are: comp-gas
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The categories of this paper are: patt-sol
0
The categories of this paper are: hep-ph
34
The categories of this paper are: chem-ph
8
The categories of this paper are: hep-ex
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The categories of this paper are: plasm-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: gr-qc
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The categories of this paper are: chao-dyn