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New directions in solving structured nonconvex problems in multivariate statistics

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Nonconvex problems arise frequently in modern applied statistics and machine learning, posing outstanding challenges from a computational and statistical viewpoint. Continuous especially convex optimization, has played a key role in our computational understanding of (relaxations or approximations of)  these problems. However, some other well-grounded techniques in mathematical optimization (for example, mixed integer optimization) have not been explored to their fullest potential. When the underlying statistical problem becomes difficult, simple convex relaxations and/or greedy methods have shortcomings. Fortunately, many of these can be ameliorated by using estimators that can be posed as solutions to structured discrete optimization problems. To this end, I will demonstrate how techniques in modern computational mathematical optimization (especially, discrete optimization) can be used to address the canonical problem of best-subset selection and cousins. I will describe how recent algorithms based on local combinatorial optimization can lead to high quality solutions in times comparable to (or even faster than) the fastest algorithms based on L1-regularization. I will also discuss the relatively less understood low Signal to Noise ratio regime, where usual subset selection performs unfavorably from a  statistical viewpoint; and propose simple alternatives that rely on nonconvex optimization. If time permits, I will outline problems arising in the context robust statistics (least median squares/least trimmed squares), low-rank factor analysis and nonparametric function estimation where, these techniques seem to be promising.

This talk is part of the Isaac Newton Institute Seminar Series series.

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