With the proliferation of training data, distributed machine learning (DML) is becoming more competent for large-scale learning tasks. However, privacy concerns have to be given priority in DML, since training data may contain sensitive information of users. In this paper, we propose a privacy-preserving ADMM-based DML framework with two novel features: First, we remove the assumption commonly made in the literature that the users trust the server collecting their data. Second, the framework provides heterogeneous privacy for users depending on data's sensitive levels and servers' trust degrees. The challenging issue is to keep the accumulation of privacy losses over ADMM iterations minimal. In the proposed framework, a local randomization approach, which is differentially private, is adopted to provide users with self-controlled privacy guarantee for the most sensitive information. Further, the ADMM algorithm is perturbed through a combined noise-adding method, which simultaneously preserves privacy for users' less sensitive information and strengthens the privacy protection of the most sensitive information. We provide detailed analyses on the performance of the trained model according to its generalization error. Finally, we conduct extensive experiments using real-world datasets to validate the theoretical results and evaluate the classification performance of the proposed framework.
@cite_4: In decentralized consensus optimization, a connected network of agents collaboratively minimize the sum of their local objective functions over a common decision variable, where their information exchange is restricted between the neighbors. To this end, one can first obtain a problem reformulation and then apply the alternating direction method of multipliers (ADMM). The method applies iterative computation at the individual agents and information exchange between the neighbors. This approach has been observed to converge quickly and deemed powerful. This paper establishes its linear convergence rate for the decentralized consensus optimization problem with strongly convex local objective functions. The theoretical convergence rate is explicitly given in terms of the network topology, the properties of local objective functions, and the algorithm parameter. This result is not only a performance guarantee but also a guideline toward accelerating the ADMM convergence.
@cite_27: The goal of decentralized optimization over a network is to optimize a global objective formed by a sum of local (possibly nonsmooth) convex functions using only local computation and communication. It arises in various application domains, including distributed tracking and localization, multi-agent coordination, estimation in sensor networks, and large-scale machine learning. We develop and analyze distributed algorithms based on dual subgradient averaging, and we provide sharp bounds on their convergence rates as a function of the network size and topology. Our analysis allows us to clearly separate the convergence of the optimization algorithm itself and the effects of communication dependent on the network structure. We show that the number of iterations required by our algorithm scales inversely in the spectral gap of the network, and confirm this prediction's sharpness both by theoretical lower bounds and simulations for various networks. Our approach includes the cases of deterministic optimization and communication, as well as problems with stochastic optimization and or communication.
@cite_15: We study a distributed computation model for optimizing a sum of convex objective functions corresponding to multiple agents. For solving this (not necessarily smooth) optimization problem, we consider a subgradient method that is distributed among the agents. The method involves every agent minimizing his her own objective function while exchanging information locally with other agents in the network over a time-varying topology. We provide convergence results and convergence rate estimates for the subgradient method. Our convergence rate results explicitly characterize the tradeoff between a desired accuracy of the generated approximate optimal solutions and the number of iterations needed to achieve the accuracy.
@cite_16: We propose an adaptive diffusion mechanism to optimize global cost functions in a distributed manner over a network of nodes. The cost function is assumed to consist of a collection of individual components. Diffusion adaptation allows the nodes to cooperate and diffuse information in real-time; it also helps alleviate the effects of stochastic gradient noise and measurement noise through a continuous learning process. We analyze the mean-square-error performance of the algorithm in some detail, including its transient and steady-state behavior. We also apply the diffusion algorithm to two problems: distributed estimation with sparse parameters and distributed localization. Compared to well-studied incremental methods, diffusion methods do not require the use of a cyclic path over the nodes and are robust to node and link failure. Diffusion methods also endow networks with adaptation abilities that enable the individual nodes to continue learning even when the cost function changes with time. Examples involving such dynamic cost functions with moving targets are common in the context of biological networks.
@cite_20: In this paper, an algorithm is developed for collaboratively training networks of kernel-linear least-squares regression estimators. The algorithm is shown to distributively solve a relaxation of the classical centralized least-squares regression problem. A statistical analysis shows that the generalization error afforded agents by the collaborative training algorithm can be bounded in terms of the relationship between the network topology and the representational capacity of the relevant reproducing kernel Hilbert space. Numerical experiments suggest that the algorithm is effective at reducing noise. The algorithm is relevant to the problem of distributed learning in wireless sensor networks by virtue of its exploitation of local communication. Several new questions for statistical learning theory are proposed.
As one of the important applications of distributed optimization, DML has received widespread attentions from researchers. Besides the ADMM scheme, many distributed approaches have been proposed in the existing literature, e.g., subgradient descent methods @cite_15 , local message-passing algorithms @cite_20 , adaptive diffusion mechanisms @cite_16 , and dual averaging approaches @cite_27 . Compared with these approaches, ADMM schemes achieve faster empirical convergence @cite_4 , making it more suitable for large-scale DML tasks.