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    A Model of Double Descent for High-dimensional Binary Linear Classification

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    Type
    Preprint
    Authors
    Deng, Zeyu
    Kammoun, Abla cc
    Thrampoulidis, Christos
    KAUST Department
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    Date
    2019-11-13
    Permanent link to this record
    http://hdl.handle.net/10754/660756
    
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    Abstract
    We consider a model for logistic regression where only a subset of features of size $p$ is used for training a linear classifier over $n$ training samples. The classifier is obtained by running gradient-descent (GD) on the logistic-loss. For this model, we investigate the dependence of the generalization error on the overparameterization ratio $\kappa=p/n$. First, building on known deterministic results on convergence properties of the GD, we uncover a phase-transition phenomenon for the case of Gaussian regressors: the generalization error of GD is the same as that of the maximum-likelihood (ML) solution when $\kappa<\kappa_\star$, and that of the max-margin (SVM) solution when $\kappa>\kappa_\star$. Next, using the convex Gaussian min-max theorem (CGMT), we sharply characterize the performance of both the ML and SVM solutions. Combining these results, we obtain curves that explicitly characterize the generalization error of GD for varying values of $\kappa$. The numerical results validate the theoretical predictions and unveil double-descent phenomena that complement similar recent observations in linear regression settings.
    Publisher
    arXiv
    arXiv
    1911.05822
    Additional Links
    https://arxiv.org/pdf/1911.05822
    Collections
    Preprints; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

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