A model of double descent for high-dimensional binary linear classification
Embargo End Date2022-04-03
Permanent link to this recordhttp://hdl.handle.net/10754/660756
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AbstractWe 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 on logistic loss. For this model, we investigate the dependence of the classification error on the ratio $\kappa =p/n$. First, building on known deterministic results on the implicit bias of gradient descent, we uncover a phase-transition phenomenon for the case of Gaussian features: the classification error of the gradient descent solution is the same as that of the maximum-likelihood solution when $\kappa <\kappa _\star $, and that of the support vector machine when $\kappa>\kappa _\star $, where $\kappa _\star $ is a phase-transition threshold. Next, using the convex Gaussian min–max theorem, we sharply characterize the performance of both the maximum-likelihood and the support vector machine solutions. Combining these results, we obtain curves that explicitly characterize the classification error for varying values of $\kappa $. The numerical results validate the theoretical predictions and unveil double-descent phenomena that complement similar recent findings in linear regression settings as well as empirical observations in more complex learning scenarios.
CitationDeng, Z., Kammoun, A., & Thrampoulidis, C. (2021). A model of double descent for high-dimensional binary linear classification. Information and Inference: A Journal of the IMA. https://doi.org/10.1093/imaiai/iaab002
SponsorsThis work was supported by the NSF under grant number CCF-2009030 and by a grant from the King Abdullah University of Science and Technology. The work was completed while C.T. was with the University of California, Santa Barbara, USA. The authors would also like to thank the Editor and the anonymous reviewers for their careful reading of our manuscript and for their suggestions that helped improve the paper’s organization and readability.
PublisherOxford University Press (OUP)