The Debiased Lasso

dc.contributor.author van de Geer, Sara
dc.contributor.corporatename Georgia Institute of Technology. Transdisciplinary Research Institute for Advancing Data Science en_US
dc.contributor.corporatename Georgia Institute of Technology. School of Mathematics en_US
dc.contributor.corporatename Eidgenössische Technische Hochschule Zürich en_US
dc.contributor.corporatename ETH Zürich en_US
dc.date.accessioned 2018-09-11T21:09:12Z
dc.date.available 2018-09-11T21:09:12Z
dc.date.issued 2018-09-06
dc.description Presented on September 6, 2018 from 3:05 p.m.-3:55 p.m. at the School of Mathematics, Skiles Room 006, Georgia Institute of Technology (Georgia Tech). en_US
dc.description Transdisciplinary Research Institute for Advancing Data Science (TRIAD) Distinguished Lecture Series - "Sparsity, Oracles and Inference in High Dimensional Statistics: Part 3". en_US
dc.description The seminar will be the third lecture of the TRIAD Distinguished Lecture Series by Prof. Sara van de Geer. en_US
dc.description Part 1: http://hdl.handle.net/1853/60424 Part 2: http://hdl.handle.net/1853/60426 en_US
dc.description Sara van de Geer has been Full Professor at the Seminar for Statistics at ETH Zurich since September 2005. Her main field of research is mathematical statistics, with special interest in high-dimensional problems. Focus points are: empirical processes, curve estimation, machine learning, model selection, and non- and semiparametric statistics. She is associate editor of Probability Theory and Related Fields, Journal of the European Mathematical Society, Scandinavian Journal of Statistics, Journal of Machine Learning Research, Statistical Surveys and Journal of Statistical Planning and Inference. She is a member of the Research Council of The Swiss National Science Foundation. She is a member of the International Statistical Institute and fellow of the Institute of Mathematical Statistics. She is correspondent of the Royal Dutch Academy of Sciences and member of Leopoldina German National Academy of Sciences. She is President of the Bernoulli Society. en_US
dc.description Runtime: 57:56 minutes
dc.description.abstract There will be three lectures, which in principle will be independent units. Their common theme is exploiting sparsity in high-dimensional statistics. Sparsity means that the statistical model is allowed to have quite a few parameters, but that it is believed that most of these parameters are actually not relevant. We let the data themselves decide which parameters to keep by applying a regularization method. The aim is then to derive so-called sparsity oracle inequalities. In the first lecture, we consider a statistical procedure called M-estimation. "M" stands here for "minimum": one tries to minimize a risk function, in order to obtain the best fit to the data. Lease squares is a prominent example. Regularization is done by adding a sparsity inducing penalty that discourages too good a fit to the data. An example is the l₁-penalty which together with least squares gives to an estimation procedure called the Lasso. We address the question: why does the l₁-penalty lead to sparsity oracle inequalities and how does this generalize to other norms? We will see in the first lecture that one needs conditions which relate the penalty to the risk function. They have in a certain sense to be “compatible”. We discuss these compatibility conditions in the second lecture in the context of the Lasso, where the l₁-penalty needs to be compatible with the least squares risk, i.e. with the l₂-norm. We give as example the total variation penalty. For D := {x1,…,xn} ⊂ R an increasing sequence, the total variation of a function f : D -> R is the sum of the absolute values of its jump sizes. We derive compatibility and as a consequence a sparsity oracle inequality which shows adaptation to the number of jumps. In the third lecture we use sparsity to establish confidence intervals for a parameter of interest. The idea is to use the penalized estimator as an initial estimator in a one-step Newton-Raphson procedure. Functionals of this new estimator that can under certain conditions be shown to be asymptotically normally distributed. We show that in the high-dimensional case, one may further profit from sparsity conditions if the inverse Hessian of the problem is not sparse. en_US
dc.format.extent 57:56 minutes
dc.identifier.uri http://hdl.handle.net/1853/60427
dc.identifier.uri http://hdl.handle.net/1853/60426
dc.identifier.uri http://hdl.handle.net/1853/60424
dc.language.iso en_US en_US
dc.publisher Georgia Institute of Technology en_US
dc.relation.ispartofseries TRIAD Distinguished Lecture Series en_US
dc.relation.ispartofseries Stochastics Seminar en_US
dc.subject M-estimation en_US
dc.subject Oracle inequalities en_US
dc.subject Sparsity en_US
dc.title The Debiased Lasso en_US
dc.type Moving Image
dc.type.genre Lecture
dspace.entity.type Publication
local.contributor.corporatename Transdisciplinary Research Institute for Advancing Data Science
local.relation.ispartofseries Transdisciplinary Research Institute for Advancing Data Science Lectures
relation.isOrgUnitOfPublication 09be376c-3b5f-4fa8-9e58-6a3595a8353b
relation.isSeriesOfPublication f402db73-162f-4a58-a9d2-bc56b6a0af52
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