This thesis examines the critical relationship between data, complex models, and other methods to measure and analyze them. As models grow larger and more intricate, they require more data, making it vital to use that data effectively. The document starts with a deep dive into nonconvex functions, a fundamental element of modern complex systems, identifying key conditions that ensure these systems can be analyzed efficiently—a crucial consideration in an era of vast amounts of variables. Loss functions, traditionally seen as mere optimization tools, are analyzed and recast as measures of how accurately a model reflects reality. This redefined perspective permits the refinement of data-sourcing strategies for a better data economy. The aim of the investigation is the model itself, which is used to understand and harness the underlying patterns of complex systems. By incorporating structure both implicitly (through periodic patterns) and explicitly (using graphs), the model's ability to make sense of the data is enhanced. Moreover, online learning principles are applied to a crucial practical scenario: robotic resource monitoring. The results established in this thesis, backed by simulations and theoretical proofs, highlight the advantages of online learning methods over traditional ones commonly used in robotics. In sum, this thesis presents an integrated approach to measuring complex systems, providing new insights and methods that push forward the capabilities of machine learning.

When solving analysis, estimation, and control problems for Partial Differential Equations (PDEs) via computational methods, one must resolve three main challenges: (a) the lack of a universal parametric representation of PDEs; (b) handling unbounded differential operators that appear as parameters; and (c), enforcing auxiliary constraints such as Boundary conditions and continuity conditions. To address these challenges, an alternative representation of PDEs called the `Partial Integral Equation' (PIE) representation is proposed in this work. Primarily, the PIE representation alleviates the problem of the lack of a universal parametrization of PDEs since PIEs have, at most, $12$ Partial Integral (PI) operators as parameters. Naturally, this also resolves the challenges in handling unbounded operators because PI operators are bounded linear operators. Furthermore, for admissible PDEs, the PIE representation is unique and has no auxiliary constraints --- resolving the last of the $3$ main challenges. The PIE representation for a PDE is obtained by finding a unique unitary map from the states of the PIE to the states of the PDE. This map shows a PDE and its associated PIE have equivalent system properties, including well-posedness, internal stability, and I/O behavior. Furthermore, this unique map also allows us to construct a well-defined dual representation that can be used to solve optimal control problems for a PDE. Using the equivalent PIE representation of a PDE, mathematical and computational tools are developed to solve standard problems in Control theory for PDEs. In particular, problems such as a test for internal stability, Input-to-Output (I/O) $L_2$-gain, $\hinf$-optimal state observer design, and $\hinf$-optimal full state-feedback controller design are solved using convex-optimization and Lyapunov methods for linear PDEs in one spatial dimension. Once the PIE associated with a PDE is obtained, Lyapunov functions (or storage functions) are parametrized by positive PI operators to obtain a solvable convex formulation of the above-stated control problems. Lastly, the methods proposed here are applied to various PDE systems to demonstrate the application.

A distributed framework is proposed for addressing resource sharing problems in communications, micro-economics, and various other network systems. The approach uses a hierarchical multi-layer decomposition for network utility maximization. This methodology uses central management and distributed computations to allocate resources, and in dynamic environments, it aims to efficiently respond to network changes. The main contributions include a comprehensive description of an exemplary unifying optimization framework to share resources across different operators and platforms, and a detailed analysis of the generalized methods under the assumption that the network changes are on the same time-scale as the convergence time of the algorithms employed for local computations.Assuming strong concavity and smoothness of the objective functions, and under some stability conditions for each layer, convergence rates and optimality bounds are presented. The effectiveness of the framework is demonstrated through numerical examples. Furthermore, a novel Federated Edge Network Utility Maximization (FEdg-NUM) architecture is proposed for solving large-scale distributed network utility maximization problems in a fully decentralized way. In FEdg-NUM, clients with private utilities communicate with a peer-to-peer network of edge servers. Convergence properties are examined both through analysis and numerical simulations, and potential applications are highlighted. Finally, problems in a complex stochastic dynamic environment, specifically motivated by resource sharing during disasters occurring in multiple areas, are studied. In a hierarchical management scenario, a method of applying a primal-dual algorithm in higher-layer along with deep reinforcement learning algorithms in localities is presented. Analytical details as well as case studies such as pandemic and wildfire response are provided.

Modern life is full of challenging optimization problems that we unknowingly attempt to solve. For instance, a common dilemma often encountered is the decision of picking a parking spot while trying to minimize both the distance to the goal destination and time spent searching for parking; one strategy is to drive as close as possible to the goal destination but risk a penalty cost if no parking spaces can be found. Optimization problems of this class all have underlying time-varying processes that can be altered by a decision/input to minimize some cost. Such optimization problems are commonly solved by a class of methods called Dynamic Programming (DP) that breaks down a complex optimization problem into a simpler family of sub-problems. In the 1950s Richard Bellman introduced a class of DP methods that broke down Multi-Stage Optimization Problems (MSOP) into a nested sequence of ``tail problems”. Bellman showed that for any MSOP with a cost function that satisfies a condition called additive separability, the solution to the tail problem of the MSOP initialized at time-stage k>0 can be used to solve the tail problem initialized at time-stage k-1. Therefore, by recursively solving each tail problem of the MSOP, a solution to the original MSOP can be found. This dissertation extends Bellman`s theory to a broader class of MSOPs involving non-additively separable costs by introducing a new state augmentation solution method and generalizing the Bellman Equation. This dissertation also considers the analogous continuous-time counterpart to discrete-time MSOPs, called Optimal Control Problems (OCPs). OCPs can be solved by solving a nonlinear Partial Differential Equation (PDE) called the Hamilton-Jacobi-Bellman (HJB) PDE. Unfortunately, it is rarely possible to obtain an analytical solution to the HJB PDE. This dissertation proposes a method for approximately solving the HJB PDE based on Sum-Of-Squares (SOS) programming. This SOS algorithm can be used to synthesize controllers, hence solving the OCP, and also compute outer bounds of reachable sets of dynamical systems. This methodology is then extended to infinite time horizons, by proposing SOS algorithms that yield Lyapunov functions that can approximate regions of attraction and attractor sets of nonlinear dynamical systems arbitrarily well.

Coordination and control of Intelligent Agents as a team is considered in this thesis.

Intelligent agents learn from experiences, and in times of uncertainty use the knowl-

edge acquired to make decisions and accomplish their individual or team objectives.

Agent objectives are defined using cost functions designed uniquely for the collective

task being performed. Individual agent costs are coupled in such a way that group ob-

jective is attained while minimizing individual costs. Information Asymmetry refers

to situations where interacting agents have no knowledge or partial knowledge of cost

functions of other agents. By virtue of their intelligence, i.e., by learning from past

experiences agents learn cost functions of other agents, predict their responses and

act adaptively to accomplish the team’s goal.

Algorithms that agents use for learning others’ cost functions are called Learn-

ing Algorithms, and algorithms agents use for computing actuation (control) which

drives them towards their goal and minimize their cost functions are called Control

Algorithms. Typically knowledge acquired using learning algorithms is used in con-

trol algorithms for computing control signals. Learning and control algorithms are

designed in such a way that the multi-agent system as a whole remains stable during

learning and later at an equilibrium. An equilibrium is defined as the event/point

where cost functions of all agents are optimized simultaneously. Cost functions are

designed so that the equilibrium coincides with the goal state multi-agent system as

a whole is trying to reach.

In collective load transport, two or more agents (robots) carry a load from point

A to point B in space. Robots could have different control preferences, for example,

different actuation abilities, however, are still required to coordinate and perform

load transport. Control preferences for each robot are characterized using a scalar

parameter θ i unique to the robot being considered and unknown to other robots.

With the aid of state and control input observations, agents learn control preferences

of other agents, optimize individual costs and drive the multi-agent system to a goal

state.

Two learning and Control algorithms are presented. In the first algorithm(LCA-

1), an existing work, each agent optimizes a cost function similar to 1-step receding

horizon optimal control problem for control. LCA-1 uses recursive least squares as

the learning algorithm and guarantees complete learning in two time steps. LCA-1 is

experimentally verified as part of this thesis.

A novel learning and control algorithm (LCA-2) is proposed and verified in sim-

ulations and on hardware. In LCA-2, each agent solves an infinite horizon linear

quadratic regulator (LQR) problem for computing control. LCA-2 uses a learning al-

gorithm similar to line search methods, and guarantees learning convergence to true

values asymptotically.

Simulations and hardware implementation show that the LCA-2 is stable for a

variety of systems. Load transport is demonstrated using both the algorithms. Ex-

periments running algorithm LCA-2 are able to resist disturbances and balance the

assumed load better compared to LCA-1.

In the past few decades, there has been a remarkable shift in the boundary between public and private information. The application of information technology and electronic communications allow service providers (businesses) to collect a large amount of data. However, this ``data collection" process can put the privacy of users at risk and also lead to user reluctance in accepting services or sharing data. This dissertation first investigates privacy sensitive consumer-retailers/service providers interactions under different scenarios, and then focuses on a unified framework for various information-theoretic privacy and privacy mechanisms that can be learned directly from data.

Existing approaches such as differential privacy or information-theoretic privacy try to quantify privacy risk but do not capture the subjective experience and heterogeneous expression of privacy-sensitivity. The first part of this dissertation introduces models to study consumer-retailer interaction problems and to better understand how retailers/service providers can balance their revenue objectives while being sensitive to user privacy concerns. This dissertation considers the following three scenarios: (i) the consumer-retailer interaction via personalized advertisements; (ii) incentive mechanisms that electrical utility providers need to offer for privacy sensitive consumers with alternative energy sources; (iii) the market viability of offering privacy guaranteed free online services. We use game-theoretic models to capture the behaviors of both consumers and retailers, and provide insights for retailers to maximize their profits when interacting with privacy sensitive consumers.

Preserving the utility of published datasets while simultaneously providing provable privacy guarantees is a well-known challenge. In the second part, a novel context-aware privacy framework called generative adversarial privacy (GAP) is introduced. Inspired by recent advancements in generative adversarial networks, GAP allows the data holder to learn the privatization mechanism directly from the data. Under GAP, finding the optimal privacy mechanism is formulated as a constrained minimax game between a privatizer and an adversary. For appropriately chosen adversarial loss functions, GAP provides privacy guarantees against strong information-theoretic adversaries. Both synthetic and real-world datasets are used to show that GAP can greatly reduce the adversary's capability of inferring private information at a small cost of distorting the data.

Our ability to understand networks is important to many applications, from the analysis and modeling of biological networks to analyzing social networks. Unveiling network dynamics allows us to make predictions and decisions. Moreover, network dynamics models have inspired new ideas for computational methods involving multi-agent cooperation, offering effective solutions for optimization tasks. This dissertation presents new theoretical results on network inference and multi-agent optimization, split into two parts -

The first part deals with modeling and identification of network dynamics. I study two types of network dynamics arising from social and gene networks. Based on the network dynamics, the proposed network identification method works like a `network RADAR', meaning that interaction strengths between agents are inferred by injecting `signal' into the network and observing the resultant reverberation. In social networks, this is accomplished by stubborn agents whose opinions do not change throughout a discussion. In gene networks, genes are suppressed to create desired perturbations. The steady-states under these perturbations are characterized. In contrast to the common assumption of full rank input, I take a laxer assumption where low-rank input is used, to better model the empirical network data. Importantly, a network is proven to be identifiable from low rank data of rank that grows proportional to the network's sparsity. The proposed method is applied to synthetic and empirical data, and is shown to offer superior performance compared to prior work. The second part is concerned with algorithms on networks. I develop three consensus-based algorithms for multi-agent optimization. The first method is a decentralized Frank-Wolfe (DeFW) algorithm. The main advantage of DeFW lies on its projection-free nature, where we can replace the costly projection step in traditional algorithms by a low-cost linear optimization step. I prove the convergence rates of DeFW for convex and non-convex problems. I also develop two consensus-based alternating optimization algorithms --- one for least square problems and one for non-convex problems. These algorithms exploit the problem structure for faster convergence and their efficacy is demonstrated by numerical simulations.

I conclude this dissertation by describing future research directions.