Efficient Online Convex Optimization with Adaptively Minimax Optimal Dynamic Regret

We introduce an online convex optimization algorithm using projected sub-gradient descent with ideal adaptive learning rates, where each computation is efficiently done in a sequential manner. For the first time in the literature, this algorithm provides an adaptively minimax optimal dynamic regret guarantee for a sequence of convex functions without any restrictions -- such as strong convexity, smoothness or even Lipschitz continuity -- against a comparator decision sequence with bounded total successive changes. We show optimality by generating the worst-case dynamic regret adaptive lower bound, which constitutes of actual sub-gradient norms and matches with our guarantees. We discuss the advantages of our algorithm as opposed to adaptive projection with sub-gradient self outer products and also derive the extension for independent learning in each decision coordinate separately. Additionally, we demonstrate how to best preserve our guarantees when the bound on total successive changes in the dynamic comparator sequence grows as time goes, in a truly online manner.Comment: 10 pages, 1 figure, preprint, [v0] 201

Dynamic and Distributed Online Convex Optimization for Demand Response of Commercial Buildings

We extend the regret analysis of the online distributed weighted dual averaging (DWDA) algorithm [1] to the dynamic setting and provide the tightest dynamic regret bound known to date with respect to the time horizon for a distributed online convex optimization (OCO) algorithm. Our bound is linear in the cumulative difference between consecutive optima and does not depend explicitly on the time horizon. We use dynamic-online DWDA (D-ODWDA) and formulate a performance-guaranteed distributed online demand response approach for heating, ventilation, and air-conditioning (HVAC) systems of commercial buildings. We show the performance of our approach for fast timescale demand response in numerical simulations and obtain demand response decisions that closely reproduce the centralized optimal ones

Distributed Constrained Recursive Nonlinear Least-Squares Estimation: Algorithms and Asymptotics

This paper focuses on the problem of recursive nonlinear least squares parameter estimation in multi-agent networks, in which the individual agents observe sequentially over time an independent and identically distributed (i.i.d.) time-series consisting of a nonlinear function of the true but unknown parameter corrupted by noise. A distributed recursive estimator of the \emph{consensus} + \emph{innovations} type, namely $\mathcal{CIWNLS}$, is proposed, in which the agents update their parameter estimates at each observation sampling epoch in a collaborative way by simultaneously processing the latest locally sensed information~(\emph{innovations}) and the parameter estimates from other agents~(\emph{consensus}) in the local neighborhood conforming to a pre-specified inter-agent communication topology. Under rather weak conditions on the connectivity of the inter-agent communication and a \emph{global observability} criterion, it is shown that at every network agent, the proposed algorithm leads to consistent parameter estimates. Furthermore, under standard smoothness assumptions on the local observation functions, the distributed estimator is shown to yield order-optimal convergence rates, i.e., as far as the order of pathwise convergence is concerned, the local parameter estimates at each agent are as good as the optimal centralized nonlinear least squares estimator which would require access to all the observations across all the agents at all times. In order to benchmark the performance of the proposed distributed $\mathcal{CIWNLS}$ estimator with that of the centralized nonlinear least squares estimator, the asymptotic normality of the estimate sequence is established and the asymptotic covariance of the distributed estimator is evaluated. Finally, simulation results are presented which illustrate and verify the analytical findings.Comment: 28 pages. Initial Submission: Feb. 2016, Revised: July 2016, Accepted: September 2016, To appear in IEEE Transactions on Signal and Information Processing over Networks: Special Issue on Inference and Learning over Network

Distributed Online Convex Optimization with an Aggregative Variable

This paper investigates distributed online convex optimization in the presence of an aggregative variable without any global/central coordinators over a multi-agent network, where each individual agent is only able to access partial information of time-varying global loss functions, thus requiring local information exchanges between neighboring agents. Motivated by many applications in reality, the considered local loss functions depend not only on their own decision variables, but also on an aggregative variable, such as the average of all decision variables. To handle this problem, an Online Distributed Gradient Tracking algorithm (O-DGT) is proposed with exact gradient information and it is shown that the dynamic regret is upper bounded by three terms: a sublinear term, a path variation term, and a gradient variation term. Meanwhile, the O-DGT algorithm is also analyzed with stochastic/noisy gradients, showing that the expected dynamic regret has the same upper bound as the exact gradient case. To our best knowledge, this paper is the first to study online convex optimization in the presence of an aggregative variable, which enjoys new characteristics in comparison with the conventional scenario without the aggregative variable. Finally, a numerical experiment is provided to corroborate the obtained theoretical results