Statistics seminar: Johannes Royset

Event Date

Location
Mathematical Sciences Building 1147 (Colloquium Room)

SPEAKER: Johannes O. Royset, Naval Postgraduate School, Dept of O.R.

TITLE: “Variational Analysis in Statistics”

ABSTRACT: Optimization theory and algorithms have long been employed in support of statistical applications. Modern challenges in machine learning and nonparametric statistics, however, strain the classical optimization machinery. The broader tools of variational analysis with their ability to handle nonconvexity, nonsmoothness, and nonuniqueness offer remedies for high generalization errors in machine learning as well as new models in shape-constrained estimation. The presentation illustrates results in two directions:  First, sup-projections in empirical risk minimization lead to generalization bounds that are independent of Lipschitz moduli for convex as well as nonconvex problems and produce quality solutions in chaotic landscapes from benchmark neural network classification problems with corrupted labels. Second, spaces of functions with variational metrics form the foundation of nonparametric M-estimators that account for a variety of information and assumptions about shape, pointwise bounds, location of modes, height at modes, location of level-sets, values of moments, Lipschitz moduli, multivariate total positivity, and any combination of the above. Variational analysis yields consistency of subsequent plug-in estimators of modes of densities, maximizers of regression functions, level-sets of classifiers, and related quantities, and also enables computation by means of approximating parametric classes.