Distributed Online Convex Optimization with Adversarial Constraints: Reduced Cumulative Constraint Violation Bounds under Slater's Condition

This paper considers distributed online convex optimization with adversarial constraints. In this setting, a network of agents makes decisions at each round, and then only a portion of the loss function and a coordinate block of the constraint function are privately revealed to each agent. The loss and constraint functions are convex and can vary arbitrarily across rounds. The agents collaborate to minimize network regret and cumulative constraint violation. A novel distributed online algorithm is proposed and it achieves an $\mathcal{O}(T^{\max\{c,1-c\}})$ network regret bound and an $\mathcal{O}(T^{1-c/2})$ network cumulative constraint violation bound, where $T$ is the number of rounds and $c\in(0,1)$ is a user-defined trade-off parameter. When Slater's condition holds (i.e, there is a point that strictly satisfies the inequality constraints), the network cumulative constraint violation bound is reduced to $\mathcal{O}(T^{1-c})$. Moreover, if the loss functions are strongly convex, then the network regret bound is reduced to $\mathcal{O}(\log(T))$, and the network cumulative constraint violation bound is reduced to $\mathcal{O}(\sqrt{\log(T)T})$ and $\mathcal{O}(\log(T))$ without and with Slater's condition, respectively. To the best of our knowledge, this paper is the first to achieve reduced (network) cumulative constraint violation bounds for (distributed) online convex optimization with adversarial constraints under Slater's condition. Finally, the theoretical results are verified through numerical simulations

On data-selective learning

Adaptive filters are applied in several electronic and communication devices like smartphones, advanced headphones, DSP chips, smart antenna, and teleconference systems. Also, they have application in many areas such as system identification, channel equalization, noise reduction, echo cancellation, interference cancellation, signal prediction, and stock market. Therefore, reducing the energy consumption of the adaptive filtering algorithms has great importance, particularly in green technologies and in devices using battery. In this thesis, data-selective adaptive filters, in particular the set-membership (SM) adaptive filters, are the tools to reach the goal. There are well known SM adaptive filters in literature. This work introduces new algorithms based on the classical ones in order to improve their performances and reduce the number of required arithmetic operations at the same time. Therefore, firstly, we analyze the robustness of the classical SM adaptive filtering algorithms. Secondly, we extend the SM technique to trinion and quaternion systems. Thirdly, by combining SM filtering and partialupdating, we introduce a new improved set-membership affine projection algorithm with constrained step size to improve its stability behavior. Fourthly, we propose some new least-mean-square (LMS) based and recursive least-squares based adaptive filtering algorithms with low computational complexity for sparse systems. Finally, we derive some feature LMS algorithms to exploit the hidden sparsity in the parameters.Filtros adaptativos são aplicados em diversos aparelhos eletrônicos e de comunicação, como smartphones, fone de ouvido avançados, DSP chips, antenas inteligentes e sistemas de teleconferência. Eles também têm aplicação em várias áreas como identificação de sistemas, equalização de canal, cancelamento de eco, cancelamento de interferência, previsão de sinal e mercado de ações. Desse modo, reduzir o consumo de energia de algoritmos adaptativos tem importância significativa, especialmente em tecnologias verdes e aparelhos que usam bateria. Nesta tese, filtros adaptativos com seleção de dados, em particular filtros adaptativos da família set-membership (SM), são apresentados para cumprir essa missão. No presente trabalho objetivamos apresentar novos algoritmos, baseados nos clássicos, a fim de aperfeiçoar seus desempenhos e, ao mesmo tempo, reduzir o número de operações aritméticas exigidas. Dessa forma, primeiro analisamos a robustez dos filtros adaptativos SM clássicos. Segundo, estendemos o SM aos números trinions e quaternions. Terceiro, foram utilizadas também duas famílias de algoritmos, SM filtering e partial-updating, de uma maneira elegante, visando reduzir energia ao máximo possível e obter um desempenho competitivo em termos de estabilidade. Quarto, a tese propõe novos filtros adaptativos baseado em algoritmos least-mean-square (LMS) e mínimos quadrados recursivos com complexidade computacional baixa para espaços esparsos. Finalmente, derivamos alguns algoritmos feature LMS para explorar a esparsidade escondida nos parâmetros

Non-Convex Phase Retrieval Algorithms and Performance Analysis

University of Minnesota Ph.D. dissertation. April 2018. Major: Electrical Engineering. Advisor: Georgios Giannakis. 1 computer file (PDF); ix, 149 pages.High-dimensional signal estimation plays a fundamental role in various science and engineering applications, including optical and medical imaging, wireless communications, and power system monitoring. The ability to devise solution procedures that maintain high computational and statistical efficiency will facilitate increasing the resolution and speed of lensless imaging, identifying artifacts in products intended for military or national security, as well as protecting critical infrastructure including the smart power grid. This thesis contributes in both theory and methods to the fundamental problem of phase retrieval of high-dimensional (sparse) signals from magnitude-only measurements. Our vision is to leverage exciting advances in non-convex optimization and statistical learning to devise algorithmic tools that are simple, scalable, and easy-to-implement, while being computationally and statistically (near-)optimal. Phase retrieval is approached from a non-convex optimization perspective. To gain statistical and computational efficiency, the magnitude data (instead of the intensities) are fitted based on the least-squares or maximum likelihood criterion, which leads to optimization models that trade off smoothness for ‘low-order’ non-convexity. To solve the resultant challenging nonconvex and non-smooth optimization, the present thesis introduces a two-stage algorithmic framework, that is termed amplitude flow. The amplitude flows start with a careful initialization, which is subsequently refined by a sequence of regularized gradient-type iterations. Both stages are lightweight, and they scale well with problem dimensions. Due to the highly non-convex landscape, judicious gradient regularization techniques such as trimming (i.e., truncation) and iterative reweighting are devised to boost the exact phase recovery performance. It is shown that successive iterates of the amplitude flows provably converge to the global optimum at a geometric rate, corroborating their efficiency in terms of computational, storage, and data resources. The amplitude flows are also demonstrated to be stable vis-a-vis additive noise. Sparsity plays a instrumental role in many scientific fields - what has led to the upsurge of research referred to as compressive sampling. In diverse applications, the signal is naturally sparse or admits a sparse representation after some known and deterministic linear transformation. This thesis also accounts for phase retrieval of sparse signals, by putting forth sparsity-cognizant amplitude flow variants. Although analysis, comparisons, and corroborating tests focus on non-convex phase retrieval in this thesis, a succinct overview of other areas is provided to highlight the universality of the novel algorithmic framework to a number of intriguing future research directions