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Stochastics and Statistics Seminar

Characterizing the Type 1-Type 2 Error Trade-off for SLOPE

November 12, 2021 @ 11:00 am - 12:00 pm

Cynthia Rush (Columbia University)


Abstract:  Sorted L1 regularization has been incorporated into many methods for solving high-dimensional statistical estimation problems, including the SLOPE estimator in linear regression. In this talk, we study how this relatively new regularization technique improves variable selection by characterizing the optimal SLOPE trade-off between the false discovery proportion (FDP) and true positive proportion (TPP) or, equivalently, between measures of type I and type II error. Additionally, we show that on any problem instance, SLOPE with a certain regularization sequence outperforms the Lasso, in the sense of having a smaller FDP, larger TPP and smaller L2 estimation risk simultaneously. Our proofs are based on a novel technique that reduces a variational calculus problem to a class of infinite-dimensional convex optimization problems and a very recent result from approximate message passing (AMP) theory. With SLOPE being a particular example, we discuss these results in the context of a general program for systematically deriving exact expressions for the asymptotic risk of estimators that are solutions to a broad class of convex optimization problems via AMP. Collaborators on this work include Zhiqi Bu, Jason Klusowski, and Weijie Su (https://arxiv.org/abs/1907.07502 and https://arxiv.org/abs/2105.13302) and Oliver Feng, Ramji Venkataramanan, and Richard Samworth (https://arxiv.org/abs/2105.02180.

Bio: Cynthia Rush is the Howard Levene Assistant Professor of Statistics in the Department of Statistics at Columbia University. In May, 2016, she received a Ph.D. in Statistics from Yale University under the supervision of Andrew Barron and she completed her undergraduate coursework at the University of North Carolina at Chapel Hill where she obtained a B.S. in Mathematics. She received an NSF CRIII award in 2019, was an NTT Research Fellow at the Simons Institute for the Theory of Computing for the program on Probability, Computation, and Geometry in High Dimensions in Fall 2020, and is currently a Google Research Fellow at the Simons Institute for the Theory of Computing for the program on Computational Complexity of Statistical Inference.

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