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A new analytical approach to consistency and overfitting in regularized empirical risk minimization

Part of: Nonparametric inference Geometric probability and stochastic geometry Existence theories Discrete mathematics in relation to computer science

Published online by Cambridge University Press:  20 July 2017

NICOLÁS GARCÍA TRILLOS  and
RYAN MURRAY
NICOLÁS GARCÍA TRILLOS
Affiliation:
Division of Applied Mathematics, Brown University, Providence, RI, 02912, USA email: nicolas_garcia_trillos@brown.edu
RYAN MURRAY
Affiliation:
Mathematics Department, The Pennsylvania State University, University Park, PA 16802, USA email: rwm22@psu.edu
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Abstract

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This work considers the problem of binary classification: given training data x1, . . ., xn from a certain population, together with associated labels y1,. . ., yn ∈ {0,1}, determine the best label for an element x not among the training data. More specifically, this work considers a variant of the regularized empirical risk functional which is defined intrinsically to the observed data and does not depend on the underlying population. Tools from modern analysis are used to obtain a concise proof of asymptotic consistency as regularization parameters are taken to zero at rates related to the size of the sample. These analytical tools give a new framework for understanding overfitting and underfitting, and rigorously connect the notion of overfitting with a loss of compactness.

Keywords

Overfittingempirical risk minimizationgraph total variationdiscrete to continuumclassification

MSC classification

Primary: 49J55: Problems involving randomness
Secondary: 49J45: Methods involving semicontinuity and convergence; relaxation 60D05: Geometric probability and stochastic geometry 68R10: Graph theory (including graph drawing) 62G20: Asymptotic properties
Type
Papers
Information
European Journal of Applied Mathematics , Volume 28 , Special Issue 6: Big Data and Partial Differential Equations , December 2017 , pp. 886 - 921
Creative Commons
This is a work of the U.S. Government and is not subject to copyright protection in the United States.
Copyright
Copyright © Cambridge University Press 2017

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