A Control-Theoretic Perspective on Optimal High-Order Optimization

We provide a control-theoretic perspective on optimal tensor algorithms for minimizing a convex function in a finite-dimensional Euclidean space. Given a function $\Phi: \mathbb{R}^d \rightarrow \mathbb{R}$ that is convex and twice continuously differentiable, we study a closed-loop control system that is governed by the operators $\nabla \Phi$ and $\nabla^2 \Phi$ together with a feedback control law $\lambda(\cdot)$ satisfying the algebraic equation $(\lambda(t))^p\|\nabla\Phi(x(t))\|^{p-1} = \theta$ for some $\theta \in (0, 1)$. Our first contribution is to prove the existence and uniqueness of a local solution to this system via the Banach fixed-point theorem. We present a simple yet nontrivial Lyapunov function that allows us to establish the existence and uniqueness of a global solution under certain regularity conditions and analyze the convergence properties of trajectories. The rate of convergence is $O(1/t^{(3p+1)/2})$ in terms of objective function gap and $O(1/t^{3p})$ in terms of squared gradient norm. Our second contribution is to provide two algorithmic frameworks obtained from discretization of our continuous-time system, one of which generalizes the large-step A-HPE framework and the other of which leads to a new optimal $p$-th order tensor algorithm. While our discrete-time analysis can be seen as a simplification and generalization of~\citet{Monteiro-2013-Accelerated}, it is largely motivated by the aforementioned continuous-time analysis, demonstrating the fundamental role that the feedback control plays in optimal acceleration and the clear advantage that the continuous-time perspective brings to algorithmic design. A highlight of our analysis is that we show that all of the $p$-th order optimal tensor algorithms that we discuss minimize the squared gradient norm at a rate of $O(k^{-3p})$, which complements the recent analysis.Comment: Accepted by Mathematical Programming Series A; 45 page

Data-driven approximations of dynamical systems operators for control

The Koopman and Perron Frobenius transport operators are fundamentally changing how we approach dynamical systems, providing linear representations for even strongly nonlinear dynamics. Although there is tremendous potential benefit of such a linear representation for estimation and control, transport operators are infinite-dimensional, making them difficult to work with numerically. Obtaining low-dimensional matrix approximations of these operators is paramount for applications, and the dynamic mode decomposition has quickly become a standard numerical algorithm to approximate the Koopman operator. Related methods have seen rapid development, due to a combination of an increasing abundance of data and the extensibility of DMD based on its simple framing in terms of linear algebra. In this chapter, we review key innovations in the data-driven characterization of transport operators for control, providing a high-level and unified perspective. We emphasize important recent developments around sparsity and control, and discuss emerging methods in big data and machine learning.Comment: 37 pages, 4 figure

A Game Theoretic Perspective on Self-organizing Optimization for Cognitive Small Cells

In this article, we investigate self-organizing optimization for cognitive small cells (CSCs), which have the ability to sense the environment, learn from historical information, make intelligent decisions, and adjust their operational parameters. By exploring the inherent features, some fundamental challenges for self-organizing optimization in CSCs are presented and discussed. Specifically, the dense and random deployment of CSCs brings about some new challenges in terms of scalability and adaptation; furthermore, the uncertain, dynamic and incomplete information constraints also impose some new challenges in terms of convergence and robustness. For providing better service to the users and improving the resource utilization, four requirements for self-organizing optimization in CSCs are presented and discussed. Following the attractive fact that the decisions in game-theoretic models are exactly coincident with those in self-organizing optimization, i.e., distributed and autonomous, we establish a framework of game-theoretic solutions for self-organizing optimization in CSCs, and propose some featured game models. Specifically, their basic models are presented, some examples are discussed and future research directions are given.Comment: 8 Pages, 8 Figures, to appear in IEEE Communications Magazin

The Water-Filling Game in Fading Multiple Access Channels

We adopt a game theoretic approach for the design and analysis of distributed resource allocation algorithms in fading multiple access channels. The users are assumed to be selfish, rational, and limited by average power constraints. We show that the sum-rate optimal point on the boundary of the multipleaccess channel capacity region is the unique Nash Equilibrium of the corresponding water-filling game. This result sheds a new light on the opportunistic communication principle and argues for the fairness of the sum-rate optimal point, at least from a game theoretic perspective. The base-station is then introduced as a player interested in maximizing a weighted sum of the individual rates. We propose a Stackelberg formulation in which the base-station is the designated game leader. In this set-up, the base-station announces first its strategy defined as the decoding order of the different users, in the successive cancellation receiver, as a function of the channel state. In the second stage, the users compete conditioned on this particular decoding strategy. We show that this formulation allows for achieving all the corner points of the capacity region, in addition to the sum-rate optimal point. On the negative side, we prove the non-existence of a base-station strategy in this formulation that achieves the rest of the boundary points. To overcome this limitation, we present a repeated game approach which achieves the capacity region of the fading multiple access channel. Finally, we extend our study to vector channels highlighting interesting differences between this scenario and the scalar channel case.Comment: 26 pages, submitted to IEEE Transactions on Information Theor

Distributed Learning Algorithms for Spectrum Sharing in Spatial Random Access Wireless Networks

We consider distributed optimization over orthogonal collision channels in spatial random access networks. Users are spatially distributed and each user is in the interference range of a few other users. Each user is allowed to transmit over a subset of the shared channels with a certain attempt probability. We study both the non-cooperative and cooperative settings. In the former, the goal of each user is to maximize its own rate irrespective of the utilities of other users. In the latter, the goal is to achieve proportionally fair rates among users. Simple distributed learning algorithms are developed to solve these problems. The efficiencies of the proposed algorithms are demonstrated via both theoretical analysis and simulation results.Comment: 40 pages, 6 figures, accepted for publication in the IEEE Transactions on Automatic Control, part of this work was presented at the 13th International Symposium on Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks (WiOpt), 201

Transmitter and Precoding Order Optimization for Nonlinear Downlink Beamforming

The downlink of a multiple-input multiple output (MIMO) broadcast channel (BC) is considered, where each receiver is equipped with a single antenna and the transmitter performs nonlinear Dirty-Paper Coding (DPC). We present an efficient algorithm that finds the optimum transmit filters and power allocation as well as the optimum precoding order(s) possibly affording time-sharing between individual DPC orders. Subsequently necessary and sufficient conditions for the optimality of an arbitrary precoding order are derived. Based on these we propose a suboptimal algorithm showing excellent performance and having low complexity.Comment: Submitted to IEEE Int. Symposium on Inf. Theory (ISIT) 200

Differential Privacy Techniques for Cyber Physical Systems: A Survey

Modern cyber physical systems (CPSs) has widely being used in our daily lives because of development of information and communication technologies (ICT).With the provision of CPSs, the security and privacy threats associated to these systems are also increasing. Passive attacks are being used by intruders to get access to private information of CPSs. In order to make CPSs data more secure, certain privacy preservation strategies such as encryption, and k-anonymity have been presented in the past. However, with the advances in CPSs architecture, these techniques also needs certain modifications. Meanwhile, differential privacy emerged as an efficient technique to protect CPSs data privacy. In this paper, we present a comprehensive survey of differential privacy techniques for CPSs. In particular, we survey the application and implementation of differential privacy in four major applications of CPSs named as energy systems, transportation systems, healthcare and medical systems, and industrial Internet of things (IIoT). Furthermore, we present open issues, challenges, and future research direction for differential privacy techniques for CPSs. This survey can serve as basis for the development of modern differential privacy techniques to address various problems and data privacy scenarios of CPSs.Comment: 46 pages, 12 figure

Game-Theoretic Approaches for Wireless Communications with Unmanned Aerial Vehicles

Wireless communications with unmanned aerial vehicles (UAVs) offer a promising solution to provide cost-effective wireless connectivity and extend coverage. In recent years, the area of wireless communications for UAV system design and optimization has been receiving enormous attention from the research community. However, there are still existing challenges that are far from solved. To cope with those challenges, researchers have been exploring the applicability of game-theoretic approaches. This paper surveys the existing game-theoretic solutions and presents a number of novel solutions which are designed to optimize energy consumption, enhance network coverage, and improve connectivity in wireless communications with UAVs. We present main game components and the elements they represent in wireless communications with UAVs first and then give a classification of the current used game-theoretic approaches. We identify main problems in wireless communications with UAVs in which game theory has been used to find solutions. We provide a case to show the merits of applying game theory in wireless communication with UAVs. Finally, we discuss shortcomings of the traditional game-theoretic approaches and propose mean field game (MFG) as an appropriate method for solving novel technical problems in massive UAVs networks.Comment: 9 pages, 6 figures, This paper was accepted for publication in the IEEE Wireless Communications Magazine on Feb-201

Fully Decentralized Policies for Multi-Agent Systems: An Information Theoretic Approach

Learning cooperative policies for multi-agent systems is often challenged by partial observability and a lack of coordination. In some settings, the structure of a problem allows a distributed solution with limited communication. Here, we consider a scenario where no communication is available, and instead we learn local policies for all agents that collectively mimic the solution to a centralized multi-agent static optimization problem. Our main contribution is an information theoretic framework based on rate distortion theory which facilitates analysis of how well the resulting fully decentralized policies are able to reconstruct the optimal solution. Moreover, this framework provides a natural extension that addresses which nodes an agent should communicate with to improve the performance of its individual policy

Quadratic Privacy-Signaling Games and the MMSE Gaussian Information Bottleneck Problem

We introduce a privacy-signaling game problem in which a transmitter with privacy concerns observes a pair of correlated random vectors which are modeled as jointly Gaussian. The transmitter aims to hide one of these random vectors and convey the other one whereas the objective of the receiver is to accurately estimate both of the random vectors. We analyze these conflicting objectives in a game theoretic framework where depending on the commitment conditions (of the sender), we consider Nash or Stackelberg (Bayesian persuasion) equilibria. We show that a payoff dominant Nash equilibrium among all admissible policies is attained by a set of explicitly characterized linear policies. We also show that a payoff dominant Nash equilibrium coincides with a Stackelberg equilibrium. We formulate the information bottleneck problem within our Stackelberg framework under the mean squared error distortion criterion where the information bottleneck setup has a further restriction that only one of the parameters is observed at the sender. We show that this MMSE Gaussian Information Bottleneck Problem admits a linear solution which is explicitly characterized in the paper. We provide explicit conditions on when the optimal solutions, or equilibrium solutions in the Nash setup, are informative or noninformative.Comment: 14 pages, 4 figure