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B0371
Title: The role of robustness in modern statistical regression Authors:  Dimitris Bertsimas - MIT (United States)
Martin Copenhaver - MIT (United States) [presenting]
Abstract: Sparsity is a key driver in modern statistical problems, from linear regression via the Lasso to matrix regression with nuclear norm penalties in matrix completion and beyond. In stark contrast to sparsity motivations for such problems, it is known in the field of robust optimization that a variety of vector regression problems, such as Lasso which appears as a loss function plus a regularization penalty, can arise by simply immunizing a nominal problem (with only a loss function) to uncertainty in the data. Such a robustification offers an explanation for why some linear regression methods perform well in the face of noise, even when these methods do not reliably produce sparse solutions. We deepen and extend the understanding of the connection between robustification and regularization in regression problems. Specifically, (a) in the context of linear regression, we characterize under which conditions on the model of uncertainty used and on the loss function penalties robustification and regularization are equivalent; (b) we show how to tractably robustify median regression problems; and (c) we extend the characterization of robustification and regularization to matrix regression problems (matrix completion and Principal Component Analysis).