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Some Results in High Dimensional Sta...

Verchand, Kabir Aladin,

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Some Results in High Dimensional Statistics: Iterative Algorithms and Regression with Missing Data /

Record Type:	Electronic resources : Monograph/item
Title/Author:	Some Results in High Dimensional Statistics: Iterative Algorithms and Regression with Missing Data // Kabir Aladin Verchand.
Author:	Verchand, Kabir Aladin,
Description:	1 electronic resource (386 pages)
Notes:	Source: Dissertations Abstracts International, Volume: 85-04, Section: B.
Contained By:	Dissertations Abstracts International85-04B.
Subject:	Sparsity. -
Online resource:	https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=30615141
ISBN:	9798380470094

Some Results in High Dimensional Statistics: Iterative Algorithms and Regression with Missing Data /
Verchand, Kabir Aladin,

Some Results in High Dimensional Statistics: Iterative Algorithms and Regression with Missing Data /Kabir Aladin Verchand. - 1 electronic resource (386 pages)

Source: Dissertations Abstracts International, Volume: 85-04, Section: B.

We study two problems at the interface between statistics and computation. In the first part of this thesis, we consider the problem of fitting a model to data by applying an iterative algorithm to perform empirical risk minimization. While the choice of iterative algorithm determines both the speed with which a fitted model is produced as well as the statistical quality of the produced model, the process of algorithm selection is often performed in a heuristic manner: via either expensive trial-and-error or appeal to (potentially) conservative worst case efficiency estimates. Motivated by this, we develop a framework-based on Gaussian comparison inequalities-to rigorously compare and contrast different iterative procedures to perform empirical risk minimization in an average case setting. In turn, we use this framework to demonstrate concrete separations in the convergence behavior of several iterative algorithms as well as to reveal some nonstandard convergence phenomena.In the second part of this thesis, we turn to studying parametric models with missing data in high dimensions. While this problem is well understood in the classical fixed dimension and infinite sample scaling, the situation changes drastically when the data is high dimensional. First, the high dimensional nature of the problem-which ensures that most samples contain missing entries- removes a complete case analysis from the statistician's toolbox. Second, the log-likelihood may be nonconcave in which case maximum likelihood estimation may prove computationally infeasible. Motivated by these issues, we study imputation-based methodology. In particular, we show that conditional mean imputation followed by a complete data method yields minimax optimal estimators in the linear model. We then show that the situation is quite different in the logistic model: in a stylized setting, conditional mean imputation can yield inconsistent estimators which nonetheless match the (Bayes) optimal prediction error.

English

ISBN: 9798380470094Subjects--Topical Terms:

3680690
Sparsity.

Some Results in High Dimensional Statistics: Iterative Algorithms and Regression with Missing Data /
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245 1 0 $a Some Results in High Dimensional Statistics: Iterative Algorithms and Regression with Missing Data / $c Kabir Aladin Verchand.
264 1 $a Ann Arbor : $b ProQuest Dissertations & Theses, $c 2023
300 $a 1 electronic resource (386 pages)
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500 $a Source: Dissertations Abstracts International, Volume: 85-04, Section: B.
500 $a Advisors: Duchi, John; Wootters, Mary; Montanari, Andrea Committee members: Bent, Stacey F.
502 $b Ph.D. $c Stanford University $d 2023.
520 $a We study two problems at the interface between statistics and computation. In the first part of this thesis, we consider the problem of fitting a model to data by applying an iterative algorithm to perform empirical risk minimization. While the choice of iterative algorithm determines both the speed with which a fitted model is produced as well as the statistical quality of the produced model, the process of algorithm selection is often performed in a heuristic manner: via either expensive trial-and-error or appeal to (potentially) conservative worst case efficiency estimates. Motivated by this, we develop a framework-based on Gaussian comparison inequalities-to rigorously compare and contrast different iterative procedures to perform empirical risk minimization in an average case setting. In turn, we use this framework to demonstrate concrete separations in the convergence behavior of several iterative algorithms as well as to reveal some nonstandard convergence phenomena.In the second part of this thesis, we turn to studying parametric models with missing data in high dimensions. While this problem is well understood in the classical fixed dimension and infinite sample scaling, the situation changes drastically when the data is high dimensional. First, the high dimensional nature of the problem-which ensures that most samples contain missing entries- removes a complete case analysis from the statistician's toolbox. Second, the log-likelihood may be nonconcave in which case maximum likelihood estimation may prove computationally infeasible. Motivated by these issues, we study imputation-based methodology. In particular, we show that conditional mean imputation followed by a complete data method yields minimax optimal estimators in the linear model. We then show that the situation is quite different in the logistic model: in a stylized setting, conditional mean imputation can yield inconsistent estimators which nonetheless match the (Bayes) optimal prediction error.
546 $a English
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650 4 $a Sparsity. $3 3680690
650 4 $a Missing data. $3 3697670
650 4 $a Recipes. $3 1009569
650 4 $a Theorems. $3 3686073
650 4 $a Neighborhoods. $3 993475
650 4 $a Mathematics. $3 515831
690 $a 0405
710 2 $a Stanford University. $e degree granting institution. $3 3765820
720 1 $a Duchi, John $e degree supervisor.
720 1 $a Wootters, Mary $e degree supervisor.
720 1 $a Montanari, Andrea $e degree supervisor.
773 0 $t Dissertations Abstracts International $g 85-04B.
790 $a 0212
791 $a Ph.D.
792 $a 2023
856 4 0 $u https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=30615141