Distributed frameworks are widely used to handle massive data, where sample size $n$ is very large, and data are often stored in $k$ different machines. For a random vector $X\in \mathbb{R}^p$ with expectation $\mu$, testing the mean vector $H_0: \mu=\mu_0$ vs $H_1: \mu\ne \mu_0$ for a given vector $\mu_0$ is a basic problem in statistics. The centralized test statistics require heavy communication costs, which can be a burden when $p$ or $k$ is large. To reduce the communication cost, distributed test statistics are proposed in this paper for this problem based on the divide and conquer technique, a commonly used approach for distributed statistical inference. Specifically, we extend two commonly used centralized test statistics to the distributed ones, that apply to low and high dimensional cases, respectively. Comparing the power of centralized test statistics and the distributed ones, it is observed that there is a fundamental tradeoff between communication costs and the powers of the tests. This is quite different from the application of the divide and conquer technique in many other problems such as estimation, where the associated distributed statistics can be as good as the centralized ones. Numerical results confirm the theoretical findings.
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This paper studies a distributed policy gradient in collaborative multi-agent reinforcement learning (MARL), where agents over a communication network aim to find the optimal policy to maximize the average of all agents' local returns. Due to the non-concave performance function of policy gradient, the existing distributed stochastic optimization methods for convex problems cannot be directly used for policy gradient in MARL. This paper proposes a distributed policy gradient with variance reduction and gradient tracking to address the high variances of policy gradient, and utilizes importance weight to solve the non-stationary problem in the sampling process. We then provide an upper bound on the mean-squared stationary gap, which depends on the number of iterations, the mini-batch size, the epoch size, the problem parameters, and the network topology. We further establish the sample and communication complexity to obtain an $\epsilon$-approximate stationary point. Numerical experiments on the control problem in MARL are performed to validate the effectiveness of the proposed algorithm.
Large linear systems of saddle-point type have arisen in a wide variety of applications throughout computational science and engineering. The discretizations of distributed control problems have a saddle-point structure. The numerical solution of saddle-point problems has attracted considerable interest in recent years. In this work, we propose a novel Braess-Sarazin multigrid relaxation scheme for finite element discretizations of the distributed control problems, where we use the stiffness matrix obtained from the five-point finite difference method for the Laplacian to approximate the inverse of the mass matrix arising in the saddle-point system. We apply local Fourier analysis to examine the smoothing properties of the Braess-Sarazin multigrid relaxation. From our analysis, the optimal smoothing factor for Braess-Sarazin relaxation is derived. Numerical experiments validate our theoretical results. The relaxation scheme considered here shows its high efficiency and robustness with respect to the regularization parameter and grid size.
In mobile robotics, area exploration and coverage are critical capabilities. In most of the available research, a common assumption is global, long-range communication and centralised cooperation. This paper proposes a novel swarm-based coverage control algorithm that relaxes these assumptions. The algorithm combines two elements: swarm rules and frontier search algorithms. Inspired by natural systems in which large numbers of simple agents (e.g., schooling fish, flocking birds, swarming insects) perform complicated collective behaviors, the first element uses three simple rules to maintain a swarm formation in a distributed manner. The second element provides means to select promising regions to explore (and cover) using the minimization of a cost function involving the agent's relative position to the frontier cells and the frontier's size. We tested our approach's performance on both heterogeneous and homogeneous groups of mobile robots in different environments. We measure both coverage performance and swarm formation statistics that permit the group to maintain communication. Through a series of comparison experiments, we demonstrate the proposed strategy has superior performance over recently presented map coverage methodologies and the conventional artificial potential field based on a percentage of cell-coverage, turnaround, and safe paths while maintaining a formation that permits short-range communication.
In this paper we revisit the binary hypothesis testing problem with one-sided compression. Specifically we assume that the distribution in the null hypothesis is a mixture distribution of iid components. The distribution under the alternative hypothesis is a mixture of products of either iid distributions or finite order Markov distributions with stationary transition kernels. The problem is studied under the Neyman-Pearson framework in which our main interest is the maximum error exponent of the second type of error. We derive the optimal achievable error exponent and under a further sufficient condition establish the maximum $\epsilon$-achievable error exponent. It is shown that to obtain the latter, the study of the exponentially strong converse is needed. Using a simple code transfer argument we also establish new results for the Wyner-Ahlswede-K{\"o}rner problem in which the source distribution is a mixture of iid components.
Traffic flows in a distributed computing network require both transmission and processing, and can be interdicted by removing either communication or computation resources. We study the robustness of a distributed computing network under the failures of communication links and computation nodes. We define cut metrics that measure the connectivity, and show a non-zero gap between the maximum flow and the minimum cut. Moreover, we study a network flow interdiction problem that minimizes the maximum flow by removing communication and computation resources within a given budget. We develop mathematical programs to compute the optimal interdiction, and polynomial-time approximation algorithms that achieve near-optimal interdiction in simulation.
We propose a general method for distributed Bayesian model choice, using the marginal likelihood, where a data set is split in non-overlapping subsets. These subsets are only accessed locally by individual workers and no data is shared between the workers. We approximate the model evidence for the full data set through Monte Carlo sampling from the posterior on every subset generating a model evidence per subset. The results are combined using a novel approach which corrects for the splitting using summary statistics of the generated samples. Our divide-and-conquer approach enables Bayesian model choice in the large data setting, exploiting all available information but limiting communication between workers. We derive theoretical error bounds that quantify the resulting trade-off between computational gain and loss in precision. The embarrassingly parallel nature yields important speed-ups when used on massive data sets as illustrated by our real world experiments. In addition, we show how the suggested approach can be extended to model choice within a reversible jump setting that explores multiple feature combinations within one run.
The estimation of information measures of continuous distributions based on samples is a fundamental problem in statistics and machine learning. In this paper, we analyze estimates of differential entropy in $K$-dimensional Euclidean space, computed from a finite number of samples, when the probability density function belongs to a predetermined convex family $\mathcal{P}$. First, estimating differential entropy to any accuracy is shown to be infeasible if the differential entropy of densities in $\mathcal{P}$ is unbounded, clearly showing the necessity of additional assumptions. Subsequently, we investigate sufficient conditions that enable confidence bounds for the estimation of differential entropy. In particular, we provide confidence bounds for simple histogram based estimation of differential entropy from a fixed number of samples, assuming that the probability density function is Lipschitz continuous with known Lipschitz constant and known, bounded support. Our focus is on differential entropy, but we provide examples that show that similar results hold for mutual information and relative entropy as well.
This paper studies distributed binary test of statistical independence under communication (information bits) constraints. While testing independence is very relevant in various applications, distributed independence test is particularly useful for event detection in sensor networks where data correlation often occurs among observations of devices in the presence of a signal of interest. By focusing on the case of two devices because of their tractability, we begin by investigating conditions on Type I error probability restrictions under which the minimum Type II error admits an exponential behavior with the sample size. Then, we study the finite sample-size regime of this problem. We derive new upper and lower bounds for the gap between the minimum Type II error and its exponential approximation under different setups, including restrictions imposed on the vanishing Type I error probability. Our theoretical results shed light on the sample-size regimes at which approximations of the Type II error probability via error exponents became informative enough in the sense of predicting well the actual error probability. We finally discuss an application of our results where the gap is evaluated numerically, and we show that exponential approximations are not only tractable but also a valuable proxy for the Type II probability of error in the finite-length regime.
Alternating Direction Method of Multipliers (ADMM) is a widely used tool for machine learning in distributed settings, where a machine learning model is trained over distributed data sources through an interactive process of local computation and message passing. Such an iterative process could cause privacy concerns of data owners. The goal of this paper is to provide differential privacy for ADMM-based distributed machine learning. Prior approaches on differentially private ADMM exhibit low utility under high privacy guarantee and often assume the objective functions of the learning problems to be smooth and strongly convex. To address these concerns, we propose a novel differentially private ADMM-based distributed learning algorithm called DP-ADMM, which combines an approximate augmented Lagrangian function with time-varying Gaussian noise addition in the iterative process to achieve higher utility for general objective functions under the same differential privacy guarantee. We also apply the moments accountant method to bound the end-to-end privacy loss. The theoretical analysis shows that DP-ADMM can be applied to a wider class of distributed learning problems, is provably convergent, and offers an explicit utility-privacy tradeoff. To our knowledge, this is the first paper to provide explicit convergence and utility properties for differentially private ADMM-based distributed learning algorithms. The evaluation results demonstrate that our approach can achieve good convergence and model accuracy under high end-to-end differential privacy guarantee.
In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function $F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb)$ is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers.
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