Online learning has been in the spotlight from the machine learning society for a long time. To handle massive data in Big Data era, one single learner could never efficiently finish this heavy task. Hence, in this paper, we propose a novel distributed online learning algorithm to solve the problem. Comparing to typical centralized online learner, the distributed learners optimize their own learning parameters based on local data sources and timely communicate with neighbors. However, communication may lead to a privacy breach. Thus, we use differential privacy to preserve the privacy of learners, and study the influence of guaranteeing differential privacy on the utility of the distributed online learning algorithm. Furthermore, by using the results from Kakade and Tewari (2009), we use the regret bounds of online learning to achieve fast convergence rates for offline learning algorithms in distributed scenarios, which provides tighter utility performance than the existing state-of-the-art results. In simulation, we demonstrate that the differentially private offline learning algorithm has high variance, but we can use mini-batch to improve the performance. Finally, the simulations show that the analytical results of our proposed theorems are right and our private distributed online learning algorithm is a general framework.
The method to solve distributed online learning was pioneered in distributed optimization. Hazan has studied online convex optimization in his book @cite_5 . They proposed that the framework of convex online learning is closely tied to statistical learning theory and convex optimization. @cite_11 developed an efficient algorithm for distributed optimization based on dual averaging of subgradients method. They demonstrated that their algorithm could work, even the communication matrix is random and not fixed. Nedic and Ozdaglar @cite_6 considered a subgradient method for distributed convex optimization, where the functions are convex but not necessarily smooth. They demonstrated that a time-variant communication could ensure the convergence of the distributed optimization algorithm. @cite_1 tried to analyze the influence of stochastic subgradient errors on distributed convex optimization based on a time-variant network topology. They studied the convergence rate of their distributed optimization algorithm. Our work extends the works of Nedic and Ozdaglar @cite_6 and @cite_1 . All these papers have made great contributions to distributed convex optimization, but they did not consider the privacy-preserving problem.
@cite_5 @cite_1 @cite_6 @cite_11
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We consider a convex unconstrained optimization problem that arises in a network of agents whose goal is to cooperatively optimize the sum of the individual agent objective functions through local computations and communications. For this problem, we use averaging algorithms to develop distributed subgradient methods that can operate over a time-varying topology. Our focus is on the convergence rate of these methods and the degradation in performance when only quantized information is available. Based on our recent results on the convergence time of distributed averaging algorithms, we derive improved upper bounds on the convergence rate of the unquantized subgradient method. We then propose a distributed subgradient method under the additional constraint that agents can only store and communicate quantized information, and we provide bounds on its convergence rate that highlight the dependence on the number of quantization levels. We study nonconvex distributed optimization in multiagent networks with time-varying (nonsymmetric) connectivity. We introduce the first algorithmic framework for the distributed minimization of the sum of a smooth (possibly nonconvex and nonseparable) function—the agents’ sum-utility—plus a convex (possibly nonsmooth and nonseparable) regularizer. The latter is usually employed to enforce some structure in the solution, typically sparsity. The proposed method hinges on successive convex approximation techniques while leveraging dynamic consensus as a mechanism to distribute the computation among the agents: each agent first solves (possibly inexactly) a local convex approximation of the nonconvex original problem, and then performs local averaging operations. Asymptotic convergence to (stationary) solutions of the nonconvex problem is established. Our algorithmic framework is then customized to a variety of convex and nonconvex problems in several fields, including signal processing, communications, networking, and machine learning. Numerical results show that the new method compares favorably to existing distributed algorithms on both convex and nonconvex problems. We consider distributed convex optimization problems originating from sample average approximation of stochastic optimization, or empirical risk minimization in machine learning. We assume that each machine in the distributed computing system has access to a local empirical loss function, constructed with i.i.d. data sampled from a common distribution. We propose a communication-efficient distributed algorithm to minimize the overall empirical loss, which is the average of the local empirical losses. The algorithm is based on an inexact damped Newton method, where the inexact Newton steps are computed by a distributed preconditioned conjugate gradient method. We analyze its iteration complexity and communication efficiency for minimizing self-concordant empirical loss functions, and discuss the results for ridge regression, logistic regression and binary classification with a smoothed hinge loss. In a standard setting for supervised learning where the condition number of the problem grows with square root of the sample size, the required number of communication rounds of the algorithm does not increase with the sample size, and only grows slowly with the number of machines. We consider the adversarial convex bandit problem and we build the first @math -time algorithm with @math -regret for this problem. To do so we introduce three new ideas in the derivative-free optimization literature: (i) kernel methods, (ii) a generalization of Bernoulli convolutions, and (iii) a new annealing schedule for exponential weights (with increasing learning rate). The basic version of our algorithm achieves @math -regret, and we show that a simple variant of this algorithm can be run in @math -time per step at the cost of an additional @math factor in the regret. These results improve upon the @math -regret and @math -time result of the first two authors, and the @math -regret and @math -time result of Hazan and Li. Furthermore we conjecture that another variant of the algorithm could achieve @math -regret, and moreover that this regret is unimprovable (the current best lower bound being @math and it is achieved with linear functions). For the simpler situation of zeroth order stochastic convex optimization this corresponds to the conjecture that the optimal query complexity is of order @math .