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Publication# Learning Proximal Operators with Gaussian Processes

Abstract

Several distributed-optimization setups involve a group of agents coordinated by a central entity (coordinator), altogether operating in a collaborative framework. In such environments, it is often common that the agents solve proximal minimization problems that are hidden from the central coordinator. We develop a scheme for reducing communication between the agents and the coordinator based on learning the agents' proximal operators with Gaussian Processes. The scheme learns a Gaussian Process model of the proximal operator associated with each agent from historical data collected at past query points. These models enable probabilistic predictions of the solutions to the local proximal minimization problems. Based on the predictive variance returned by a model, representative of its prediction confidence, an adaptive mechanism allows the coordinator to decide whether to communicate with the associated agent. The accuracy of the Gaussian Process models results in significant communication reduction, as demonstrated in simulations of a distributed optimal power dispatch application.

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Related concepts (27)

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Regularization (mathematics)

In mathematics, statistics, finance, computer science, particularly in machine learning and inverse problems, regularization is a process that changes the result answer to be "simpler". It is often used to obtain results for ill-posed problems or to prevent overfitting. Although regularization procedures can be divided in many ways, the following delineation is particularly helpful: Explicit regularization is regularization whenever one explicitly adds a term to the optimization problem.

Gaussian process

In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution, i.e. every finite linear combination of them is normally distributed. The distribution of a Gaussian process is the joint distribution of all those (infinitely many) random variables, and as such, it is a distribution over functions with a continuous domain, e.g.

Gaussian function

In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form and with parametric extension for arbitrary real constants a, b and non-zero c. It is named after the mathematician Carl Friedrich Gauss. The graph of a Gaussian is a characteristic symmetric "bell curve" shape. The parameter a is the height of the curve's peak, b is the position of the center of the peak, and c (the standard deviation, sometimes called the Gaussian RMS width) controls the width of the "bell".

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