This dissertation comprises three lines of work addressing societal challenges related to safe water access, fair incentive design, and secure data sharing, utilizing mathematical modeling to propose comprehensive solutions. These projects collectively advocate for universal resource and information access while prioritizing equity and fairness for marginalized communities. The first project centers on optimizing water storage decisions, considering factors such as cost, water quality, and wastage, with the core objective of promoting equitable access to safe drinking water, particularly for those facing severe water scarcity. In the second project, we address deficiencies in current incentive mechanisms for fairness, introducing quantitative definitions rooted in fairness principles to prevent harm to specific groups and encompass disadvantaged segments of society. Lastly, the third project delves into the intricacies of responsible scientific data sharing, presenting a novel cybersecurity insurance contract to incentivize healthcare providers while safeguarding privacy and livelihoods, especially for vulnerable populations. In sum, this research strives to develop solutions that facilitate equal and inclusive access while mitigating risks, with the goal of addressing cultural and societal issues concerning resource allocation and information sharing for underserved communities.

The recent increase in data and computing resource availability has made the use of data analytics more practical in practice than ever before. In particular, the ubiquity of technologies such as smartphones, wearable devices, and smart sensors has allowed for the collection of a large amount of individual level data. In contrast to traditional data analytics, which relies on wide sampling from a population to draw conclusions (i.e. cross-sectional sampling), this so called deep sampled data can be used through precision analytics to create customized experiences that better serve individuals and organizations. In this thesis, we explore this precision analytics framework that builds upon the fields of reinforcement learning and data driven decision making by extending their results to applications with individual level deep sampled data. One of the main applications of this framework is to systems where a decision maker is interested in applying an intervention (or policy) to affect the behavior of an individual agent or group of agents. Two key challenges that arise when analyzing such systems are that the decision maker may have scarce resources or high risk decisions that constrain how they apply their intervention, and that the decision maker may only have partial knowledge of how an agent will react to the intervention. The main focus of this thesis is to begin to analyze these challenges by providing predictive models that can accurately capture individual agent behavior, new estimation and machine learning techniques to efficiently estimate model parameters, and effective online and batch optimization methods to calculate these interventions. We will also discuss how these approaches can be implemented in practice, particularly in the precision healthcare setting.

Biological networks are comprised of thousands of interacting components, and these networks have complicated patterns of feedback and feed-forward motifs. It is practically impossible to use intuition to determine whether simultaneously modifying multiple pharmaceutical targets has a good therapeutic response. Even when a drug is discovered which is safe in humans and highly-effective against its target, the medical effect on the disease may be underwhelming. This provides a strong impetus for developing a systems theory for pharmaceutical drug discovery. This thesis discusses system theoretic tools which are useful for doing drug discovery. The first class of tools discussed is system identification tools, and case studies of parametric modeling are given. A new statistical system identification procedure which exploits the geometric and hierarchical structure of many biological (and engineering) systems is presented, and this new procedure is applied to engineering and biological systems. The second class of tools discussed is a new set of target selection tools. Given mathematical models of biological networks, these tools select a set of targets for pharmaceutical drugs. The targets are selected to achieve good medical outcomes for patients by reducing the effect of diseases on pathways and ensuring that the targets do not too adversely affect healthy cells. The ultimate goal of the work presented in this thesis is to create a framework which can be used to rationally select new drug targets and also be able to create personalized medicine treatments which are tailored to the particular phenotypic behavior of an individual's disease.

This thesis investigates the problem of fair statistical learning. We argue that critical notions of fairness can be represented by independence constraints on certain random variables, and take the approach of approximating independence by bounding moments. We propose a hierarchical Fair Optimization (FO) framework for generalized fair decision-making, prove desirable statistical properties, and extend the framework to a number of settings, ranging from supervised learning to unsupervised learning and hypothesis testing.

Algorithmic decision-making has steadily gained in prominence as more data is produced and computing resources become more abundant. However, it has been observed that these can often reflect, and perpetuate, biases apparent in the training data. To that end, we construct the FO framework as a general approach to statistical decision-making under fairness constraints. The framework revolves around bounding the moments between a ``score" function underlying the decision-making process and predefined ``protected" attributes. We prove that this framework is consistent and will thus asymptotically provide fair decision rules, and provide non-asymptotic bounds on how quickly the framework approaches truly fair decision-making rules. We also provide experimental results that show the efficacy of the FO hierarchy on a variety of datasets, and use it to construct fair, automated one-time and sequential dosage mechanisms for morphine and heparin.

Novel, adversarial notions of fairness are then defined for the problem of dimensionality reduction of data, and a Semidefinite Programming (SDP) relaxation of the FO hierarchy is defined that controls these notions. we provide experimental analysis, including a case study on insurance rate-setting that allows for mechanisms that are fair with respect to legally-motivated age restrictions. Similarly, we extend fairness to the problem of hypothesis testing, and make the connection between fairness and robustness in this realm. This is actuated in the form of a distributionally-robust dynamic watermarking scheme to detect attacks on dynamical systems. Finally, we extend the intuitions of data-dependent regularization underlying the FO hierarchy to design a data-dependent regularizer that promotes robustness in classifiers in the low-data regime when data lies in a low-dimensional manifold.

Cyber-Physical Systems (CPS) play an ubiquitous role in operation and control in many different domains: power systems, finance, robotics, and automation. The complex interplay between cyber components such as software, communication protocols, computer servers and physical components, such as sensors and pieces of dedicated hardware, requires advanced and sophisticated methods and algorithms that ensure safe and efficient operation. In this thesis we tackle both safety and efficiency: We develop novel detection algorithms that are able to identify malicious attacks, sensor corruption and faulty measurements. Our detection mechanisms have provable guarantees based on rigorous asymptotic and non-asymptotic statistical analysis and can be readily implemented in CPS, such as robotic systems and autonomous vehicles. In addition, we developed collusion detection mechanisms that can be used to identify whether two or more CPS are colluding or not. We also design a mechanism that is able to induce selfish systems/agents to behave cooperatively. We showcase the performance of our algorithms with several different case studies. In our analyses, we place emphasis on algorithms that can be implemented in real-time, that is can be used while the system is under operation in the real-world. On the efficiency side, we developed real-time non-linear Model Predictive Control (MPC) Methods that can provide optimal solutions to the Optimal Control problem faced by the CPS during operation. Our algorithm exploits the control structure and is tailored for implementation in embedded hardware and can operate both with memory and computation time constraints. We showcase the performance of our algorithm with a C/C++ implementation and we compare to several current state-of-the-art Optimal Control solvers. We also extend our methodology to be used together with Pseudo-spectral Methods and Hybrid Systems, developing an integrated Mixed-Integer MPC algorithm that can handle complex non-linear dynamics and both continuous and discrete variables. With this thesis, our goal is to provide real-time practical algorithms that have provable guarantees in performance both in the detection task and in the optimal control task. Our algorithms are based on rigorous theoretical analysis and display very good performance and can be readily implemented in practical Cyber-Physical Systems.

Personalized cancer therapy is an emerging treatment strategy based on the ability to predict which patients are more likely to respond well to specific treatments. It involves the systematic use of genetic or other information about an individual patient to optimally select a course of treatment. This dissertation presents mathematical models and algorithms to predict drug response on a personalized level and to understand the causes of different responses.

Recent large cancer studies have measured somatic mutations in an unprecedented number of tumours. These large datasets finally allow the identification of cancer-related sets of genetic mutations, in particular we are interested in identifying groups of genetic mutations that are associated with positive or negative drug response. We propose a combinatorial formulation for the problem, and prove that it's computationally hard. We design two optimization algorithms to solve the problem and implement them in our tool UNCOVER. We provide analytic evidence of the effectiveness of UNCOVER in finding high-quality solutions and show experimentally that UNCOVER finds sets of alterations significantly associated with functional targets in a variety of scenarios. In particular, we show that our algorithms find sets which are better than the ones obtained by the state-of-the-art method. In addition, our algorithms are much faster than the state-of-the-art, allowing the analysis of large datasets of thousands of target profiles from cancer cell lines. While we formulate this as a more general computational problem, we use UNCOVER to analyze drug response data, identifying sets of mutations associated with drug sensitivity.

Our next contribution is a computational method, named NETPHIX (NETwork-to-PHenotype assocIation with eXclusivity), which aims to identify subnetworks of genes whose genetic alterations are associated with a continuous cancer phenotype. Leveraging the properties of cancer mutations and the interactions among genes, we formulate the problem as an integer linear program and solve it optimally to obtain a set of associated genes. Note that this algorithm solves a related but different mathematical problem than the one considered by UNCOVER since it also takes into account functional relationship among genes, which can be captured as an input network. Additionally NETPHIX, unlike UNCOVER, allows to pick up mixed sensitivity modules. Applied to a large-scale drug screening dataset, NETPHIX uncovered gene modules significantly associated with drug response, and many of the modules are also validated in another independent dataset. Utilizing interaction information, NETPHIX modules are functionally coherent, and can thus provide important insights into drug action.

We also include a case study that provides novel biological insight obtained from NETPHIX and expression correlation analysis to investigate the genetic mutations associated with mutational signatures. Specifically, our analysis aims to answer the following two complementary questions: (i) what are functional pathways whose gene expression activities correlate with the strengths of mutational signatures, and (ii) are there pathways whose genetic alterations might have led to specific mutational signatures. Analyzing a breast cancer dataset, we identified pathways associated with mutational signatures on both expression and mutation levels, elucidating differences between related signatures .

UNCOVER and NETPHIX can be used directly for personalized cancer treatment by looking at the genomic alterations of patients and checking if these are correlated with drug sensitivity for any candidate treatment.

While these methods can lead to useful insight towards personalized drug treatment our last contribution attempt to solve the more practical problem of predicting drug response based on all the information oncologists have available about the patient, this can include genetic information but is more often based on demographics, histology report, baseline labs and medical history recorded in the patient's Electronic Health Record. We propose a framework to simultaneously predict multiple outcomes for each treatment i.e. we are not only concerned with the expected survival time of the patient, other relevant factors such as quality of life and side effects are also considered as important quantifiable outcomes. These outcomes are heavily correlated to each other and one can leverage this property to improve prediction performance over predicting each outcome separately. Furthermore to the authors best knowledge there is no current published work or package that is able to handle a mix of survival, continuous and categorical outcomes. We provide a unified framework for prediction of heterogeneous outcomes in a clinical setting, leveraging an ensemble learning method known as random forests. We propose an updated node splitting rule that captures the heterogeneity of clinical outcomes.

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This dissertation work combines two lines of work related to stochastic optimization, one focused on theoretical contributions to the field, and the other focused on modelling and philosophical contributions to the field. In the first line of work, we contribute to the theoretical underpinnings of two models that draw from data analysis, mathematical and statistical modeling, and optimization. In particular, there are several classes of machine learning problems which interface with stochastics and optimization, and, with the use of stochastic process theory and associated notions of complexity, we can recharacterize some of the theoretical expectations of these problems, which importantly model uncertainty. The improved results are gained from problem reformulations, improved analytical techniques, and a concerted effort to use the most recent advances in complexity analysis and their associated generalizations, and has implications for the construction and analysis of societal scale models. In the second line of work, we demonstrate why the techniques in stochastic optimization help form a mathematical and statistical basis that can solve large-scale societal problems over time, with precise engineering intervention focus. The main model researched is the Principal Agent Model (PAM). Considering sectors of the social economy as the principal and communities of people as the agents, we utilize development engineering and the associated economics of global poverty reduction to unveil the concept of Optimal Intervention Theory as a means to explore, adapt, and activate tractable solutions in specially-constrained environments. By tractable here, we mean interventions that last long beyond the intervention period, according to the education-, resource-, access-, and network-bases interventions afforded the impacted populations, especially those concerned with international development and global poverty alleviation.

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