Distributed Pareto Optimization via Diffusion Strategies

We consider solving multi-objective optimization problems in a distributed manner by a network of cooperating and learning agents. The problem is equivalent to optimizing a global cost that is the sum of individual components. The optimizers of the individual components do not necessarily coincide and the network therefore needs to seek Pareto optimal solutions. We develop a distributed solution that relies on a general class of adaptive diffusion strategies. We show how the diffusion process can be represented as the cascade composition of three operators: two combination operators and a gradient descent operator. Using the Banach fixed-point theorem, we establish the existence of a unique fixed point for the composite cascade. We then study how close each agent converges towards this fixed point, and also examine how close the Pareto solution is to the fixed point. We perform a detailed mean-square error analysis and establish that all agents are able to converge to the same Pareto optimal solution within a sufficiently small mean-square-error (MSE) bound even for constant step-sizes. We illustrate one application of the theory to collaborative decision making in finance by a network of agents.Comment: 35 pages, 9 figures, submitted for publicatio

Robustness properties of estimators in generalized Pareto Models

We study global and local robustness properties of several estimators for shape and scale in a generalized Pareto model. The estimators considered in this paper cover maximum likelihood estimators, skipped maximum likelihood estimators, moment-based estimators, Cramér-von-Mises Minimum Distance estimators, and, as a special case of quantile-based estimators, Pickands Estimator as well as variants of the latter tuned for higher finite sample breakdown point (FSBP), and lower variance. We further consider an estimator matching population median and median of absolute deviations to the empirical ones (MedMad); again, in order to improve its FSBP, we propose a variant using a suitable asymmetric Mad as constituent, and which may be tuned to achieve an expected FSBP of 34%. These estimators are compared to one-step estimators distinguished as optimal in the shrinking neighborhood setting, i.e., the most bias-robust estimator minimizing the maximal (asymptotic) bias and the estimator minimizing the maximal (asymptotic) MSE. For each of these estimators, we determine the FSBP, the influence function, as well as statistical accuracy measured by asymptotic bias, variance, and mean squared error—all evaluated uniformly on shrinking convex contamination neighborhoods. Finally, we check these asymptotic theoretical findings against finite sample behavior by an extensive simulation study

Density functionals, with an option-pricing application

We present a method of estimating density-related functionals, without prior knowledge of the density’s functional form. The approach revolves around the specification of an explicit formula for a new class of distributions that encompasses many of the known cases in statistics, including the normal, gamma, inverse gamma, and mixtures thereof. The functionals are based on a couple of hypergeometric functions. Their parameters can be estimated, and the estimates then reveal both the functional form of the density and the parameters that determine centering, scaling, etc. The function to be estimated always leads to a valid density, by design, namely, one that is nonnegative everywhere and integrates to 1. Unlike fully nonparametric methods, our approach can be applied to small datasets. To illustrate our methodology, we apply it to finding risk-neutral densities associated with different types of financial options. We show how our approach fits the data uniformly very well. We also find that our estimated densities’ functional forms vary over the dataset, so that existing parametric methods will not do uniformly well

Multitask Diffusion Adaptation over Networks

Adaptive networks are suitable for decentralized inference tasks, e.g., to monitor complex natural phenomena. Recent research works have intensively studied distributed optimization problems in the case where the nodes have to estimate a single optimum parameter vector collaboratively. However, there are many important applications that are multitask-oriented in the sense that there are multiple optimum parameter vectors to be inferred simultaneously, in a collaborative manner, over the area covered by the network. In this paper, we employ diffusion strategies to develop distributed algorithms that address multitask problems by minimizing an appropriate mean-square error criterion with $\ell_2$-regularization. The stability and convergence of the algorithm in the mean and in the mean-square sense is analyzed. Simulations are conducted to verify the theoretical findings, and to illustrate how the distributed strategy can be used in several useful applications related to spectral sensing, target localization, and hyperspectral data unmixing.Comment: 29 pages, 11 figures, submitted for publicatio