Gossip Dual Averaging for Decentralized Optimization of Pairwise Functions

In decentralized networks (of sensors, connected objects, etc.), there is an important need for efficient algorithms to optimize a global cost function, for instance to learn a global model from the local data collected by each computing unit. In this paper, we address the problem of decentralized minimization of pairwise functions of the data points, where these points are distributed over the nodes of a graph defining the communication topology of the network. This general problem finds applications in ranking, distance metric learning and graph inference, among others. We propose new gossip algorithms based on dual averaging which aims at solving such problems both in synchronous and asynchronous settings. The proposed framework is flexible enough to deal with constrained and regularized variants of the optimization problem. Our theoretical analysis reveals that the proposed algorithms preserve the convergence rate of centralized dual averaging up to an additive bias term. We present numerical simulations on Area Under the ROC Curve (AUC) maximization and metric learning problems which illustrate the practical interest of our approach

Polynomial-Time Under-Approximation of Winning Regions in Parity Games

We propose a pattern for designing algorithms that run in polynomial time by construction and under-approximate the winning regions of both players in parity games. This approximation is achieved by the interaction of finitely many aspects governed by a common ranking function, where the choice of aspects and ranking function instantiates the design pattern. Each aspect attempts to improve the under-approximation of winning regions or decrease the rank function by simplifying the structure of the parity game. Our design pattern is incremental as aspects may operate on the residual game of yet undecided nodes. We present several aspects and one higher-order transformation of our algorithms - based on efficient, static analyses - and illustrate the benefit of their interaction as well as their relative precision within pattern instantiations. Instantiations of our design pattern can be applied for local model checking and as preprocessors for algorithms whose worst-case running time is exponential. This design pattern and its aspects have already been implemented in [H. Wang. Framework for Under-Approximating Solutions of Parity Games in Polynomial Time. MEng Thesis, Department of Computing, Imperial College London, 78 pages, June 2007]. © 2008 Elsevier B.V. All rights reserved

Distributed top-k aggregation queries at large

Top-k query processing is a fundamental building block for efficient ranking in a large number of applications. Efficiency is a central issue, especially for distributed settings, when the data is spread across different nodes in a network. This paper introduces novel optimization methods for top-k aggregation queries in such distributed environments. The optimizations can be applied to all algorithms that fall into the frameworks of the prior TPUT and KLEE methods. The optimizations address three degrees of freedom: 1) hierarchically grouping input lists into top-k operator trees and optimizing the tree structure, 2) computing data-adaptive scan depths for different input sources, and 3) data-adaptive sampling of a small subset of input sources in scenarios with hundreds or thousands of query-relevant network nodes. All optimizations are based on a statistical cost model that utilizes local synopses, e.g., in the form of histograms, efficiently computed convolutions, and estimators based on order statistics. The paper presents comprehensive experiments, with three different real-life datasets and using the ns-2 network simulator for a packet-level simulation of a large Internet-style network

A Harmonic Extension Approach for Collaborative Ranking

We present a new perspective on graph-based methods for collaborative ranking for recommender systems. Unlike user-based or item-based methods that compute a weighted average of ratings given by the nearest neighbors, or low-rank approximation methods using convex optimization and the nuclear norm, we formulate matrix completion as a series of semi-supervised learning problems, and propagate the known ratings to the missing ones on the user-user or item-item graph globally. The semi-supervised learning problems are expressed as Laplace-Beltrami equations on a manifold, or namely, harmonic extension, and can be discretized by a point integral method. We show that our approach does not impose a low-rank Euclidean subspace on the data points, but instead minimizes the dimension of the underlying manifold. Our method, named LDM (low dimensional manifold), turns out to be particularly effective in generating rankings of items, showing decent computational efficiency and robust ranking quality compared to state-of-the-art methods