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### Extending Gossip Algorithms to Distributed Estimation of U-Statistics

Efficient and robust algorithms for decentralized estimation in networks are essential to many distributed systems. Whereas distributed estimation of sample mean statistics has been the subject of a good deal of attention, computation of $U$-statistics, relying on more expensive averaging over pairs of observations, is a less investigated area. Yet, such data functionals are essential to describe global properties of a statistical population, with important examples including Area Under the Curve, empirical variance, Gini mean difference and within-cluster point scatter. This paper proposes new synchronous and asynchronous randomized gossip algorithms which simultaneously propagate data across the network and maintain local estimates of the $U$-statistic of interest. We establish convergence rate bounds of $O(1/t)$ and $O(\log t / t)$ for the synchronous and asynchronous cases respectively, where $t$ is the number of iterations, with explicit data and network dependent terms. Beyond favorable comparisons in terms of rate analysis, numerical experiments provide empirical evidence the proposed algorithms surpasses the previously introduced approach.

### Decentralized Topic Modelling with Latent Dirichlet Allocation

Privacy preserving networks can be modelled as decentralized networks (e.g., sensors, connected objects, smartphones), where communication between nodes of the network is not controlled by an all-knowing, central node. For this type of networks, the main issue is to gather/learn global information on the network (e.g., by optimizing a global cost function) while keeping the (sensitive) information at each node. In this work, we focus on text information that agents do not want to share (e.g., text messages, emails, confidential reports). We use recent advances on decentralized optimization and topic models to infer topics from a graph with limited communication. We propose a method to adapt latent Dirichlet allocation (LDA) model to decentralized optimization and show on synthetic data that we still recover similar parameters and similar performance at each node than with stochastic methods accessing to the whole information in the graph.

### Extending Gossip Algorithms to Distributed Estimation of U-statistics

Efficient and robust algorithms for decentralized estimation in networks are essential to many distributed systems. Whereas distributed estimation of sample mean statistics has been the subject of a good deal of attention, computation of U-statistics, relying on more expensive averaging over pairs of observations, is a less investigated area. Yet, such data functionals are essential to describe global properties of a statistical population, with important examples including Area Under the Curve, empirical variance, Gini mean difference and within-cluster point scatter. This paper proposes new synchronous and asynchronous randomized gossip algorithms which simultaneously propagate data across the network and maintain local estimates of the U-statistic of interest. We establish convergence rate bounds of O(1 / t) and O(log t / t) for the synchronous and asynchronous cases respectively, where t is the number of iterations, with explicit data and network dependent terms.

### 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.

### Optimal algorithms for smooth and strongly convex distributed optimization in networks

In this paper, we determine the optimal convergence rates for strongly convex and smooth distributed optimization in two settings: centralized and decentralized communications over a network. For centralized (i.e. master/slave) algorithms, we show that distributing Nesterov's accelerated gradient descent is optimal and achieves a precision $\varepsilon > 0$ in time $O(\sqrt{\kappa_g}(1+\Delta\tau)\ln(1/\varepsilon))$, where $\kappa_g$ is the condition number of the (global) function to optimize, $\Delta$ is the diameter of the network, and $\tau$ (resp. $1$) is the time needed to communicate values between two neighbors (resp. perform local computations). For decentralized algorithms based on gossip, we provide the first optimal algorithm, called the multi-step dual accelerated (MSDA) method, that achieves a precision $\varepsilon > 0$ in time $O(\sqrt{\kappa_l}(1+\frac{\tau}{\sqrt{\gamma}})\ln(1/\varepsilon))$, where $\kappa_l$ is the condition number of the local functions and $\gamma$ is the (normalized) eigengap of the gossip matrix used for communication between nodes. We then verify the efficiency of MSDA against state-of-the-art methods for two problems: least-squares regression and classification by logistic regression.