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Aaditya RAMDAS (Carnegie Mellon University) – " Dimension-agnostic inference "

November 16, 2020 @ 2:00 pm - 3:15 pm
The Statistical Seminar: Every Monday at 2:00 pm.
Time: 2:00 pm – 3:15 pm
Date: 16th of November 2020
Place: Visio
Aaditya RAMDAS (Carnegie Mellon University)Dimension-agnostic inference

Abstract: Classical asymptotic theory for statistical hypothesis testing, for example Wilks’ theorem for likelihood ratios, usually involves calibrating the test statistic by fixing the dimension d while letting the sample size n increase to infinity. In the last few decades, a great deal of effort has been dedicated towards understanding how these methods behave in high-dimensional settings, where d_n and n both increase to infinity together at some prescribed relative rate. This often leads to different tests in the two settings, depending on the assumptions about the dimensionality. This leaves the practitioner in a bind: given a dataset with 100 samples in 20 dimensions, should they calibrate by assuming n >> d, or d_n/n \approx 0.2? This talk considers the goal of dimension-agnostic inference—developing methods whose validity does not depend on any assumption on d_n. I will summarize results from 3 papers that achieve this goal in parametric and nonparametric settings.

(A) Universal inference (PNAS’20) https://arxiv.org/abs/1912.11436 — we describe the split/crossfit likelihood ratio test whose validity holds nonasymptotically without any regularity conditions.
(B) Classification accuracy as a proxy for two-sample testing (Annals of Statistics’21) https://arxiv.org/abs/1602.02210.
(C) Dimension-agnostic inference (http://arxiv.org/abs/2011.05068)— We describe a generic approach that uses variational representations of existing test statistics along with sample splitting and self-normalization (studentization) to produce a Gaussian limiting null distribution. We exemplify this technique for a handful of classical problems, such as one-sample mean testing, testing if a covariance matrix equals the identity, and kernel methods for testing equality of distributions using degenerate U-statistics like the maximum mean discrepancy. Without explicitly targeting the high-dimensional setting, our tests are shown to be minimax rate-optimal, meaning that the power of our tests cannot be improved further up to a constant factor of \sqrt{2}.
This is primarily joint work with several excellent coauthors including Ilmun Kim (postdoc, Cambridge), Larry Wasserman, Sivaraman Balakrishnan and Aarti Singh.
Aaditya Ramdas (http://www.stat.cmu.edu/~aramdas/)