We provide the first proof that gradient descent $\left({\color{green}\sf GD}\right)$ with greedy sparsification $\left({\color{green}\sf TopK}\right)$ and error feedback $\left({\color{green}\sf EF}\right)$ can obtain better communication complexity than vanilla ${\color{green}\sf GD}$ when solving the distributed optimization problem $\min_{x\in \mathbb{R}^d} {f(x)=\frac{1}{n}\sum_{i=1}^n f_i(x)}$, where $n$ = # of clients, $d$ = # of features, and $f_1,\dots,f_n$ are smooth nonconvex functions. Despite intensive research since 2014 when ${\color{green}\sf EF}$ was first proposed by Seide et al., this problem remained open until now. We show that ${\color{green}\sf EF}$ shines in the regime when features are rare, i.e., when each feature is present in the data owned by a small number of clients only. To illustrate our main result, we show that in order to find a random vector $\hat{x}$ such that $\lVert {\nabla f(\hat{x})} \rVert^2 \leq \varepsilon$ in expectation, ${\color{green}\sf GD}$ with the ${\color{green}\sf Top1}$ sparsifier and ${\color{green}\sf EF}$ requires ${\cal O} \left(\left( L+{\color{blue}r} \sqrt{ \frac{{\color{red}c}}{n} \min \left( \frac{{\color{red}c}}{n} \max_i L_i^2, \frac{1}{n}\sum_{i=1}^n L_i^2 \right) }\right) \frac{1}{\varepsilon} \right)$ bits to be communicated by each worker to the server only, where $L$ is the smoothness constant of $f$, $L_i$ is the smoothness constant of $f_i$, ${\color{red}c}$ is the maximal number of clients owning any feature ($1\leq {\color{red}c} \leq n$), and ${\color{blue}r}$ is the maximal number of features owned by any client ($1\leq {\color{blue}r} \leq d$). Clearly, the communication complexity improves as ${\color{red}c}$ decreases (i.e., as features become more rare), and can be much better than the ${\cal O}({\color{blue}r} L \frac{1}{\varepsilon})$ communication complexity of ${\color{green}\sf GD}$ in the same regime.
When large-scale uncertain centralized and distributed renewable energy sources are connected to a power system, separate dispatching of the transmission power system (TPS) and the active distribution network (ADN) will lower the network security and frequency security of the system. To address these problems, this paper proposes a coordinated frequency-constrained stochastic economic dispatch (CFC-SED) model for an integrated transmission and distribution (ITD) system. In this model, the dynamic frequency security constraints and network security constraints of the ITD system are constructed, and the joint chance constraints are adopted to handle the uncertainty. Then, the control parameters of inverter-based resources, the base point power, and the regulation reserve of all dispatchable resources in the ITD system are jointly optimized for the minimum operating cost. TPS and ADNs can deliver base point power bidirectionally and provide frequency regulation support bidirectionally, which extend the existing reserve assumption in ITD dispatch and enhance the operational security of the ITD system. Moreover, based on the alternating direction of multiplier algorithm, a two-layer distributed optimization framework is proposed to solve the CFC-SED model. Case studies show that the CFC-SED model can fully utilize the potential of multiple regulation resources to improve the security performance of the ITD system, and TPS and ADNs can be coordinated efficiently through the proposed distributed optimization framework.
Edge networks call for communication efficient (low overhead) and robust distributed optimization (DO) algorithms. These are, in fact, desirable qualities for DO frameworks, such as federated edge learning techniques, in the presence of data and system heterogeneity, and in scenarios where internode communication is the main bottleneck. Although computationally demanding, Newton-type (NT) methods have been recently advocated as enablers of robust convergence rates in challenging DO problems where edge devices have sufficient computational power. Along these lines, in this work we propose Q-SHED, an original NT algorithm for DO featuring a novel bit-allocation scheme based on incremental Hessian eigenvectors quantization. The proposed technique is integrated with the recent SHED algorithm, from which it inherits appealing features like the small number of required Hessian computations, while being bandwidth-versatile at a bit-resolution level. Our empirical evaluation against competing approaches shows that Q-SHED can reduce by up to 60% the number of communication rounds required for convergence.
Inexact methods for model predictive control (MPC), such as real-time iterative schemes or time-distributed optimization, alleviate the computational burden of exact MPC by providing suboptimal solutions. While the asymptotic stability of such algorithms is well studied, their finite-time performance has not received much attention. In this work, we quantify the performance of suboptimal linear model predictive control in terms of the additional closed-loop cost incurred due to performing only a finite number of optimization iterations. Leveraging this novel analysis framework, we propose a novel suboptimal MPC algorithm with a diminishing horizon length and finite-time closed-loop performance guarantees. This analysis allows the designer to plan a limited computational power budget distribution to achieve a desired performance level. We provide numerical examples to illustrate the algorithm's transient behavior and computational complexity.
We propose a new training algorithm, named DualFL (Dualized Federated Learning), for solving distributed optimization problems in federated learning. DualFL achieves communication acceleration for very general convex cost functions, thereby providing a solution to an open theoretical problem in federated learning concerning cost functions that may not be smooth nor strongly convex. We provide a detailed analysis for the local iteration complexity of DualFL to ensure the overall computational efficiency of DualFL. Furthermore, we introduce a completely new approach for the convergence analysis of federated learning based on a dual formulation. This new technique enables concise and elegant analysis, which contrasts the complex calculations used in existing literature on convergence of federated learning algorithms.
This paper considers the problem of distributed multi-agent learning, where the global aim is to minimize a sum of local objective (empirical loss) functions through local optimization and information exchange between neighbouring nodes. We introduce a Newton-type fully distributed optimization algorithm, Network-GIANT, which is based on GIANT, a Federated learning algorithm that relies on a centralized parameter server. The Network-GIANT algorithm is designed via a combination of gradient-tracking and a Newton-type iterative algorithm at each node with consensus based averaging of local gradient and Newton updates. We prove that our algorithm guarantees semi-global and exponential convergence to the exact solution over the network assuming strongly convex and smooth loss functions. We provide empirical evidence of the superior convergence performance of Network-GIANT over other state-of-art distributed learning algorithms such as Network-DANE and Newton-Raphson Consensus.
Federated learning (FL) has recently gained much attention due to its effectiveness in speeding up supervised learning tasks under communication and privacy constraints. However, whether similar speedups can be established for reinforcement learning remains much less understood theoretically. Towards this direction, we study a federated policy evaluation problem where agents communicate via a central aggregator to expedite the evaluation of a common policy. To capture typical communication constraints in FL, we consider finite capacity up-link channels that can drop packets based on a Bernoulli erasure model. Given this setting, we propose and analyze QFedTD - a quantized federated temporal difference learning algorithm with linear function approximation. Our main technical contribution is to provide a finite-sample analysis of QFedTD that (i) highlights the effect of quantization and erasures on the convergence rate; and (ii) establishes a linear speedup w.r.t. the number of agents under Markovian sampling. Notably, while different quantization mechanisms and packet drop models have been extensively studied in the federated learning, distributed optimization, and networked control systems literature, our work is the first to provide a non-asymptotic analysis of their effects in multi-agent and federated reinforcement learning.
With the phenomenal growth in renewable energy generation, the conventional synchronous generator-based power plants are gradually getting replaced by renewable energy sources-based microgrids. Such transition gives rise to the challenges of procuring various ancillary services from microgrids. We propose a distributed optimization framework that coordinates multiple microgrids in an active distribution network for provisioning passive voltage support-based ancillary services while satisfying operational constraints. Specifically, we exploit the reactive power support capability of the inverters and the flexibility offered by storage systems available with microgrids for provisioning ancillary service support to the transmission grid. We develop novel mixed-integer inequalities to represent the set of feasible active and reactive power exchange with the transmission grid that ensures passive voltage support. The proposed alternating direction method of multipliers-based algorithm is fully distributed, and does not require the presence of a centralized entity to achieve coordination among the microgrids. We present detailed numerical results on the IEEE 33-bus distribution test system to demonstrate the effectiveness of the proposed approach and examine the scalability and convergence behavior of the distributed algorithm for different choice of hyperparameters and network sizes.
The electric power distribution network (PDN) and the transportation network (TN) are generally operated/coordinated by different entities. However, they are coupled with each other due to electric vehicle charging stations (EVCSs). This paper proposes to coordinate the operation of the two systems via a fully decentralized framework where the PDN and TN operators solve their own operation problems by sharing only limited information. Nevertheless, the operation problems generally are in mixed-integer programming (MIP) form. To the best of our knowledge, the most existing decentralized/distributed optimization algorithms, such as the alternating direction method of multipliers (ADMM), are not always guaranteed to converge for such MIP problems. Therefore, a novel fully decentralized optimization algorithm is proposed, whose contributions include: 1) it is applicable to MIP problems with convergence and optimality guaranteed for mild assumptions, 2) it only requires limited information exchange between PDN and TN operators, which will help preserve the privacy of the two systems and reduce the investment in building communication channels, and 3) it is fully decentralized so that all the computations are carried out by PDN operator and TN coordinator only. Simulations on the test cases show the efficiency and efficacy of the proposed framework and algorithm.
We study finite-sum distributed optimization problems involving a master node and $n-1$ local nodes under the popular $\delta$-similarity and $\mu$-strong convexity conditions. We propose two new algorithms, SVRS and AccSVRS, motivated by previous works. The non-accelerated SVRS method combines the techniques of gradient sliding and variance reduction and achieves a better communication complexity of $\tilde{\mathcal{O}}(n {+} \sqrt{n}\delta/\mu)$ compared to existing non-accelerated algorithms. Applying the framework proposed in Katyusha X, we also develop a directly accelerated version named AccSVRS with the $\tilde{\mathcal{O}}(n {+} n^{3/4}\sqrt{\delta/\mu})$ communication complexity. In contrast to existing results, our complexity bounds are entirely smoothness-free and exhibit superiority in ill-conditioned cases. Furthermore, we establish a nearly matched lower bound to verify the tightness of our AccSVRS method.
This paper considers distributed optimization algorithms, with application in binary classification via distributed support-vector-machines (D-SVM) over multi-agent networks subject to some link nonlinearities. The agents solve a consensus-constraint distributed optimization cooperatively via continuous-time dynamics, while the links are subject to strongly sign-preserving odd nonlinear conditions. Logarithmic quantization and clipping (saturation) are two examples of such nonlinearities. In contrast to existing literature that mostly considers ideal links and perfect information exchange over linear channels, we show how general sector-bounded models affect the convergence to the optimizer (i.e., the SVM classifier) over dynamic balanced directed networks. In general, any odd sector-bounded nonlinear mapping can be applied to our dynamics. The main challenge is to show that the proposed system dynamics always have one zero eigenvalue (associated with the consensus) and the other eigenvalues all have negative real parts. This is done by recalling arguments from matrix perturbation theory. Then, the solution is shown to converge to the agreement state under certain conditions. For example, the gradient tracking (GT) step size is tighter than the linear case by factors related to the upper/lower sector bounds. To the best of our knowledge, no existing work in distributed optimization and learning literature considers non-ideal link conditions.
Crowdsourcing is an emerging computing paradigm that takes advantage of the intelligence of a crowd to solve complex problems effectively. Besides collecting and processing data, it is also a great demand for the crowd to conduct optimization. Inspired by this, this paper intends to introduce crowdsourcing into evolutionary computation (EC) to propose a crowdsourcing-based evolutionary computation (CEC) paradigm for distributed optimization. EC is helpful for optimization tasks of crowdsourcing and in turn, crowdsourcing can break the spatial limitation of EC for large-scale distributed optimization. Therefore, this paper firstly introduces the paradigm of crowdsourcing-based distributed optimization. Then, CEC is elaborated. CEC performs optimization based on a server and a group of workers, in which the server dispatches a large task to workers. Workers search for promising solutions through EC optimizers and cooperate with connected neighbors. To eliminate uncertainties brought by the heterogeneity of worker behaviors and devices, the server adopts the competitive ranking and uncertainty detection strategy to guide the cooperation of workers. To illustrate the satisfactory performance of CEC, a crowdsourcing-based swarm optimizer is implemented as an example for extensive experiments. Comparison results on benchmark functions and a distributed clustering optimization problem demonstrate the potential applications of CEC.
The rapid development of parallel and distributed computing paradigms has brought about great revolution in computing. Thanks to the intrinsic parallelism of evolutionary computation (EC), it is natural to implement EC on parallel and distributed computing systems. On the one hand, the computing power provided by parallel computing systems can significantly improve the efficiency and scalability of EC. On the other hand, data are collected and processed in a distributed manner, which brings a novel development direction and new challenges to EC. In this paper, we intend to give a systematic review on distributed EC (DEC). First, a new taxonomy for DEC is proposed from top design mechanism to bottom implementation mechanism. Based on this taxonomy, existing studies on DEC are reviewed in terms of purpose, parallel structure of the algorithm, parallel model for implementation, and the implementation environment. Second, we clarify two major purposes of DEC, i.e., improving efficiency through parallel processing for centralized optimization and cooperating distributed individuals/sub-populations with partial information to perform distributed optimization. Third, noting that the latter purpose of DEC is an emerging and attractive trend for EC with the booming of spatially distributed paradigms, this paper gives a systematic definition of the distributed optimization and classifies it into dimension distributed-, data distributed-, and objective distributed-optimization problems. Formal formulations for these problems are provided and various DEC studies on these problems are reviewed. We also discuss challenges and potential research directions, aiming to enlighten the design of DEC and pave the way for future developments.
Recently, evolutionary computation (EC) has been promoted by machine learning, distributed computing, and big data technologies, resulting in new research directions of EC like distributed EC and surrogate-assisted EC. These advances have significantly improved the performance and the application scope of EC, but also trigger privacy leakages, such as the leakage of optimal results and surrogate model. Accordingly, evolutionary computation combined with privacy protection is becoming an emerging topic. However, privacy concerns in evolutionary computation lack a systematic exploration, especially for the object, motivation, position, and method of privacy protection. To this end, in this paper, we discuss three typical optimization paradigms (i.e., \textit{centralized optimization, distributed optimization, and data-driven optimization}) to characterize optimization modes of evolutionary computation and propose BOOM to sort out privacy concerns in evolutionary computation. Specifically, the centralized optimization paradigm allows clients to outsource optimization problems to the centralized server and obtain optimization solutions from the server. While the distributed optimization paradigm exploits the storage and computational power of distributed devices to solve optimization problems. Also, the data-driven optimization paradigm utilizes data collected in history to tackle optimization problems lacking explicit objective functions. Particularly, this paper adopts BOOM to characterize the object and motivation of privacy protection in three typical optimization paradigms and discusses potential privacy-preserving technologies balancing optimization performance and privacy guarantees in three typical optimization paradigms. Furthermore, this paper attempts to foresee some new research directions of privacy-preserving evolutionary computation.
We propose two distributed iterative algorithms that can be used to solve, in finite time, the distributed optimization problem over quadratic local cost functions in large-scale networks. The first algorithm exhibits synchronous operation whereas the second one exhibits asynchronous operation. Both algorithms share salient features. Specifically, the algorithms operate exclusively with quantized values, which means that the information stored, processed and exchanged between neighboring nodes is subject to deterministic uniform quantization. The algorithms rely on event-driven updates in order to reduce energy consumption, communication bandwidth, network congestion, and/or processor usage. Finally, once the algorithms converge, nodes distributively terminate their operation. We prove that our algorithms converge in a finite number of iterations to the exact optimal solution depending on the quantization level, and we present applications of our algorithms to (i) optimal task scheduling for data centers, and (ii) global model aggregation for distributed federated learning. We provide simulations of these applications to illustrate the operation, performance, and advantages of the proposed algorithms. Additionally, it is shown that our proposed algorithms compare favorably to algorithms in the current literature. Quantized communication and asynchronous updates increase the required time to completion, but finite-time operation is maintained.
Improving renewable energy resource utilization efficiency is crucial to reducing carbon emissions, and multi-parametric programming has provided a systematic perspective in conducting analysis and optimization toward this goal in smart grid operations. This paper focuses on two aspects of interest related to multi-parametric linear/quadratic programming (mpLP/QP). First, we study degeneracy issues of mpLP/QP. A novel approach to deal with degeneracies is proposed to find all critical regions containing the given parameter. Our method leverages properties of the multi-parametric linear complementary problem, vertex searching technique, and complementary basis enumeration. Second, an improved critical region exploration (CRE) method to solve distributed LP/QP is proposed under a general mpLP/QP-based formulation. The improved CRE incorporates the proposed approach to handle degeneracies. A cutting plane update and an adaptive stepsize scheme are also integrated to accelerate convergence under different problem settings. The computational efficiency is verified on multi-area tie-line scheduling problems with various testing benchmarks and initial states.
Decentralized learning and optimization is a central problem in control that encompasses several existing and emerging applications, such as federated learning. While there exists a vast literature on this topic and most methods centered around the celebrated average-consensus paradigm, less attention has been devoted to scenarios where the communication between the agents may be imperfect. To this end, this paper presents three different algorithms of Decentralized Federated Learning (DFL) in the presence of imperfect information sharing modeled as noisy communication channels. The first algorithm, Federated Noisy Decentralized Learning (FedNDL1), comes from the literature, where the noise is added to their parameters to simulate the scenario of the presence of noisy communication channels. This algorithm shares parameters to form a consensus with the clients based on a communication graph topology through a noisy communication channel. The proposed second algorithm (FedNDL2) is similar to the first algorithm but with added noise to the parameters, and it performs the gossip averaging before the gradient optimization. The proposed third algorithm (FedNDL3), on the other hand, shares the gradients through noisy communication channels instead of the parameters. Theoretical and experimental results demonstrate that under imperfect information sharing, the third scheme that mixes gradients is more robust in the presence of a noisy channel compared with the algorithms from the literature that mix the parameters.
Distributed optimization with open collaboration is a popular field since it provides an opportunity for small groups/companies/universities, and individuals to jointly solve huge-scale problems. However, standard optimization algorithms are fragile in such settings due to the possible presence of so-called Byzantine workers -- participants that can send (intentionally or not) incorrect information instead of the one prescribed by the protocol (e.g., send anti-gradient instead of stochastic gradients). Thus, the problem of designing distributed methods with provable robustness to Byzantine workers has been receiving a lot of attention recently. In particular, several works consider a very promising way to achieve Byzantine tolerance via exploiting variance reduction and robust aggregation. The existing approaches use SAGA- and SARAH-type variance-reduced estimators, while another popular estimator -- SVRG -- is not studied in the context of Byzantine-robustness. In this work, we close this gap in the literature and propose a new method -- Byzantine-Robust Loopless Stochastic Variance Reduced Gradient (BR-LSVRG). We derive non-asymptotic convergence guarantees for the new method in the strongly convex case and compare its performance with existing approaches in numerical experiments.
This paper considers distributed optimization (DO) where multiple agents cooperate to minimize a global objective function, expressed as a sum of local objectives, subject to some constraints. In DO, each agent iteratively solves a local optimization model constructed by its own data and communicates some information (e.g., a local solution) with its neighbors until a global solution is obtained. Even though locally stored data are not shared with other agents, it is still possible to reconstruct the data from the information communicated among agents, which could limit the practical usage of DO in applications with sensitive data. To address this issue, we propose a privacy-preserving DO algorithm for constrained convex optimization models, which provides a statistical guarantee of data privacy, known as differential privacy, and a sequence of iterates that converges to an optimal solution in expectation. The proposed algorithm generalizes a linearized alternating direction method of multipliers by introducing a multiple local updates technique to reduce communication costs and incorporating an objective perturbation method in the local optimization models to compute and communicate randomized feasible local solutions that cannot be utilized to reconstruct the local data, thus preserving data privacy. Under the existence of convex constraints, we show that, while both algorithms provide the same level of data privacy, the objective perturbation used in the proposed algorithm can provide better solutions than does the widely adopted output perturbation method that randomizes the local solutions by adding some noise. We present the details of privacy and convergence analyses and numerically demonstrate the effectiveness of the proposed algorithm by applying it in two different applications, namely, distributed control of power flow and federated learning, where data privacy is of concern.
We study differentially private distributed optimization under communication constraints. A server using SGD for optimization aggregates the client-side local gradients for model updates using distributed mean estimation (DME). We develop a communication-efficient private DME, using the recently developed multi-message shuffled (MMS) privacy framework. We analyze our proposed DME scheme to show that it achieves the order-optimal privacy-communication-performance tradeoff resolving an open question in [1], whether the shuffled models can improve the tradeoff obtained in Secure Aggregation. This also resolves an open question on the optimal trade-off for private vector sum in the MMS model. We achieve it through a novel privacy mechanism that non-uniformly allocates privacy at different resolutions of the local gradient vectors. These results are directly applied to give guarantees on private distributed learning algorithms using this for private gradient aggregation iteratively. We also numerically evaluate the private DME algorithms.