Finite-Time Convergent Algorithms for Time-Varying Distributed Optimization

This paper focuses on finite-time (FT) convergent distributed algorithms for solving time-varying distributed optimization (TVDO). The objective is to minimize the sum of local time-varying cost functions subject to the possible time-varying constraints by the coordination of multiple agents in finite time. We first provide a unified approach for designing finite/fixed-time convergent algorithms to solve centralized time-varying optimization, where an auxiliary dynamics is introduced to achieve prescribed performance. Then, two classes of TVDO are investigated included unconstrained distributed consensus optimization and distributed optimal resource allocation problems (DORAP) with both time-varying cost functions and coupled equation constraints. For the previous one, based on nonsmooth analysis, a continuous-time distributed discontinuous dynamics with FT convergence is proposed based on an extended zero-gradient-sum method with a local auxiliary subsystem. Different from the existing methods, the proposed algorithm does not require the initial state of each agent to be the optimizer of the local cost function. Moreover, the provided algorithm has a simpler structure without estimating the global information and can be used for TVDO with nonidentical Hessians. Then, an FT convergent distributed dynamics is further obtained for time-varying DORAP by dual transformation. Particularly, the inverse of Hessians is not required from a dual perspective, which reduces the computation complexity significantly. Finally, two numerical examples are conducted to verify the proposed algorithms

Finite-Time Distributed Optimization with Quantized Gradient Descent

In this paper, we consider the unconstrained distributed optimization problem, in which the exchange of information in the network is captured by a directed graph topology, and thus nodes can send information to their out-neighbors only. Additionally, the communication channels among the nodes have limited bandwidth, to alleviate the limitation, quantized messages should be exchanged among the nodes. For solving the distributed optimization problem, we combine a distributed quantized consensus algorithm (which requires the nodes to exchange quantized messages and converges in a finite number of steps) with a gradient descent method. Specifically, at every optimization step, each node performs a gradient descent step (i.e., subtracts the scaled gradient from its current estimate), and then performs a finite-time calculation of the quantized average of every node's estimate in the network. As a consequence, this algorithm approximately mimics the centralized gradient descent algorithm. The performance of the proposed algorithm is demonstrated via simple illustrative examples

A duality-based approach for distributed min-max optimization with application to demand side management

In this paper we consider a distributed optimization scenario in which a set of processors aims at minimizing the maximum of a collection of "separable convex functions" subject to local constraints. This set-up is motivated by peak-demand minimization problems in smart grids. Here, the goal is to minimize the peak value over a finite horizon with: (i) the demand at each time instant being the sum of contributions from different devices, and (ii) the local states at different time instants being coupled through local dynamics. The min-max structure and the double coupling (through the devices and over the time horizon) makes this problem challenging in a distributed set-up (e.g., well-known distributed dual decomposition approaches cannot be applied). We propose a distributed algorithm based on the combination of duality methods and properties from min-max optimization. Specifically, we derive a series of equivalent problems by introducing ad-hoc slack variables and by going back and forth from primal and dual formulations. On the resulting problem we apply a dual subgradient method, which turns out to be a distributed algorithm. We prove the correctness of the proposed algorithm and show its effectiveness via numerical computations.Comment: arXiv admin note: substantial text overlap with arXiv:1611.0916

Differentially Private Distributed Stochastic Optimization with Time-Varying Sample Sizes

Differentially private distributed stochastic optimization has become a hot topic due to the urgent need of privacy protection in distributed stochastic optimization. In this paper, two-time scale stochastic approximation-type algorithms for differentially private distributed stochastic optimization with time-varying sample sizes are proposed using gradient- and output-perturbation methods. For both gradient- and output-perturbation cases, the convergence of the algorithm and differential privacy with a finite cumulative privacy budget $\varepsilon$ for an infinite number of iterations are simultaneously established, which is substantially different from the existing works. By a time-varying sample sizes method, the privacy level is enhanced, and differential privacy with a finite cumulative privacy budget $\varepsilon$ for an infinite number of iterations is established. By properly choosing a Lyapunov function, the algorithm achieves almost-sure and mean-square convergence even when the added privacy noises have an increasing variance. Furthermore, we rigorously provide the mean-square convergence rates of the algorithm and show how the added privacy noise affects the convergence rate of the algorithm. Finally, numerical examples including distributed training on a benchmark machine learning dataset are presented to demonstrate the efficiency and advantages of the algorithms