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CATEGORIES:Statistics
SUMMARY:Minimum L1-norm interpolators: Precise asymptotics
  and multiple descent - Yuting Wei (University of 
 Pennsylvania)
DTSTART;TZID=Europe/London:20220506T140000
DTEND;TZID=Europe/London:20220506T150000
UID:TALK173333AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/173333
DESCRIPTION:An evolving line of machine learning works observe
  empirical evidence that suggests interpolating es
 timators --- the ones that achieve zero training e
 rror --- may not necessarily be harmful. In this t
 alk\, we pursue theoretical understanding for an i
 mportant type of interpolators: the minimum L1-nor
 m interpolator\, which is motivated by the observa
 tion that several learning algorithms favor low L1
 -norm solutions in the over-parameterized regime. 
 Concretely\, we consider the noisy sparse regressi
 on model under Gaussian design\, focusing on linea
 r sparsity and high-dimensional asymptotics (so th
 at both the number of features and the sparsity le
 vel scale proportionally with the sample size).\n\
 nWe observe\, and provide rigorous theoretical jus
 tification for\, a curious multi-descent phenomeno
 n\; that is\, the generalization risk of the minim
 um L1-norm interpolator undergoes multiple (and po
 ssibly more than two) phases of descent and ascent
  as one increases the model capacity. This phenome
 non stems from the special structure of the minimu
 m L1-norm interpolator as well as the delicate int
 erplay between the over-parameterized ratio and th
 e sparsity\,  thus unveiling a fundamental distinc
 tion in geometry from the minimum L2-norm interpol
 ator. Our finding is built upon an exact character
 ization of the risk behavior\, which is governed b
 y a system of two non-linear equations with two un
 knowns.
LOCATION:MR12\, Centre for Mathematical Sciences
CONTACT:Qingyuan Zhao
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