Complexity penalized methods for structured and unstructured data

A fundamental goal of statisticians is to make inferences from the sample about characteristics of the underlying population. This is an inverse problem, since we are trying to recover a feature of the input with the availability of observations on an output. Towards this end, we consider complexity penalized methods, because they balance goodness of fit and generalizability of the solution. The data from the underlying population may come in diverse formats - structured or unstructured - such as probability distributions, text tokens, or graph characteristics. Depending on the defining features of the problem we can chose the appropriate complexity penalized approach, and assess the quality of the estimate produced by it. Favorable characteristics are strong theoretical guarantees of closeness to the true value and interpretability. Our work fits within this framework and spans the areas of simulation optimization, text mining and network inference. The first problem we consider is model calibration under the assumption that given a hypothesized input model, we can use stochastic simulation to obtain its corresponding output observations. We formulate it as a stochastic program by maximizing the entropy of the input distribution subject to moment matching. We then propose an iterative scheme via simulation to approximately solve it. We prove convergence of the proposed algorithm under appropriate conditions and demonstrate the performance via numerical studies. The second problem we consider is summarizing text documents through an inferred set of topics. We propose a frequentist reformulation of a Bayesian regularization scheme. Through our complexity-penalized perspective we lend further insight into the nature of the loss function and the regularization achieved through the priors in the Bayesian formulation. The third problem is concerned with the impact of sampling on the degree distribution of a network. Under many sampling designs, we have a linear inverse problem characterized by an ill-conditioned matrix. We investigate the theoretical properties of an approximate solution for the degree distribution found by regularizing the solution of the ill-conditioned least squares objective. Particularly, we study the rate at which the penalized solution tends to the true value as a function of network size and sampling rate

Operational Decision Making under Uncertainty: Inferential, Sequential, and Adversarial Approaches

Modern security threats are characterized by a stochastic, dynamic, partially observable, and ambiguous operational environment. This dissertation addresses such complex security threats using operations research techniques for decision making under uncertainty in operations planning, analysis, and assessment. First, this research develops a new method for robust queue inference with partially observable, stochastic arrival and departure times, motivated by cybersecurity and terrorism applications. In the dynamic setting, this work develops a new variant of Markov decision processes and an algorithm for robust information collection in dynamic, partially observable and ambiguous environments, with an application to a cybersecurity detection problem. In the adversarial setting, this work presents a new application of counterfactual regret minimization and robust optimization to a multi-domain cyber and air defense problem in a partially observable environment