Grad Student Seminar: Tianxiao Sun & Jonathan Williams
Globally convergent Newton-type methods for convex optimization based
on smoothness structures
We study the smooth structure of convex functions by generalizing a powerful concept so-called self-concordance introduced by Nesterov and Nemirovskii to a broader class of convex functions. The proposed theory provides a mathematical tool to analyze both local and global convergence of Newton-type methods as long as the underlying functionals fall into our generalized self-concordant function class. First, we introduce the class of generalized self-concordant functions, which covers standard self-concordant functions as a special case. Next, we establish several properties and key estimates of this function class. Then, we apply this theory to develop Newton-type methods for solving a class of smooth convex optimization problems involving the generalized self-concordant functions. We provide an explicit step-size for the damped-step Newton-type scheme which can guarantee a global convergence without performing any globalization strategy. We also prove a local quadratic convergence of this method and its full-step variant without requiring the Lipschitz continuity of the objective Hessian. Then, we extend our result to develop proximal Newton-type methods for composite convex minimization. We also achieve both local and global convergence without additional assumptions. Finally, we verify our results via several numerical examples, and compare them with existing methods.
A Bayesian Approach to Multi-state Modeling: Application to
This is joint work with Curtis Storlie, and Terry Therneau
that is supported with funds from the Mayo Clinic.
A multi-state model is a useful way of describing the development of a disease. Here, a Hidden Markov Model (HMM) is specified to model dementia as it progresses from states associated with the buildup of amyloid plaque on the brain, and the loss of cortical thickness. Both of these processes are known in the medical community to be strongly associated with dementia. In previous studies, hard cutoff points were chosen to distinguish from states of high/low amyloid burden, and high/low cortical thickness loss burden. However, hard cutoff points for discretizing continuous measurements of biological processes are practically and philosophically problematic. Here, an approach is proposed which is cutoff point agnostic.
Data on 4742 individuals from the Mayo Clinic Study of Aging (MCSA) are analyzed. A hierarchical Bayesian approach is taken to estimate the parameters of the HMM. A notable feature of the analysis is that time is treated as continuous, and the infinitesimal generator matrix of the underlying Markov process is allowed to be time-inhomogeneous (as a function of an individual’s age). A novel contribution is that in addition to the affect of age, the affects of the covariates gender, number of years of education, and presence of an APOE4 allele on the infinitesimal transition rates are estimated. In this context, these have never been estimated in the literature.
In addition to the new insights these estimates bring to the medical community, a novel approach is illustrated for correcting a common bias in population-based studies. By ‘population-based’ study, it is meant that not all (or even none) of the study participants are observed at a given baseline age. This often introduces a strong downward bias on the death rates because dead individuals typically are not recruited into a study. Finally, standard R software is not capable of correcting for such biases, and often doesn’t include Bayesian estimation routines for a HMM.