The rapid development of machine learning and deep learning has introduced increasingly complex optimization challenges that must be addressed. Indeed, training modern, advanced models has become difficult to implement without leveraging multiple computing nodes in a distributed environment. Distributed optimization is also fundamental to emerging fields such as federated learning. Specifically, there is a need to organize the training process to minimize the time lost due to communication. A widely used and extensively researched technique to mitigate the communication bottleneck involves performing local training before communication. This approach is the focus of our paper. Concurrently, adaptive methods that incorporate scaling, notably led by Adam, have gained significant popularity in recent years. Therefore, this paper aims to merge the local training technique with the adaptive approach to develop efficient distributed learning methods. We consider the classical Local SGD method and enhance it with a scaling feature. A crucial aspect is that the scaling is described generically, allowing us to analyze various approaches, including Adam, RMSProp, and OASIS, in a unified manner. In addition to theoretical analysis, we validate the performance of our methods in practice by training a neural network.
We propose a novel analysis framework for non-descent-type optimization methodologies in nonconvex scenarios based on the Kurdyka-Lojasiewicz property. Our framework allows covering a broad class of algorithms, including those commonly employed in stochastic and distributed optimization. Specifically, it enables the analysis of first-order methods that lack a sufficient descent property and do not require access to full (deterministic) gradient information. We leverage this framework to establish, for the first time, iterate convergence and the corresponding rates for the decentralized gradient method and federated averaging under mild assumptions. Furthermore, based on the new analysis techniques, we show the convergence of the random reshuffling and stochastic gradient descent method without necessitating typical a priori bounded iterates assumptions.
Distributed model predictive control (DMPC) has attracted extensive attention as it can explicitly handle system constraints and achieve optimal control in a decentralized manner. However, the deployment of DMPC strategies generally requires the sharing of sensitive data among subsystems, which may violate the privacy of participating systems. In this paper, we propose a differentially-private DMPC algorithm for linear discrete-time systems subject to coupled global constraints. Specifically, we first show that a conventional distributed dual gradient algorithm can be used to address the considered DMPC problem but cannot provide strong privacy preservation. Then, to protect privacy against the eavesdropper, we incorporate a differential-privacy noise injection mechanism into the DMPC framework and prove that the resulting distributed optimization algorithm can ensure both provable convergence to a global optimal solution and rigorous $\epsilon$-differential privacy. In addition, an implementation strategy of the DMPC is designed such that the recursive feasibility and stability of the closed-loop system are guaranteed. Simulation results are provided to demonstrate the effectiveness of the developed approach.
We propose a new variant of the Adam optimizer called MicroAdam that specifically minimizes memory overheads, while maintaining theoretical convergence guarantees. We achieve this by compressing the gradient information before it is fed into the optimizer state, thereby reducing its memory footprint significantly. We control the resulting compression error via a novel instance of the classical \emph{error feedback} mechanism from distributed optimization in which *the error correction information is itself compressed* to allow for practical memory gains. We prove that the resulting approach maintains theoretical convergence guarantees competitive to those of AMSGrad, while providing good practical performance. Specifically, we show that MicroAdam can be implemented efficiently on GPUs: on both million-scale (BERT) and billion-scale (LLaMA) models, MicroAdam provides practical convergence competitive to that of the uncompressed Adam baseline, with lower memory usage and similar running time. Our code is available at https://github.com/IST-DASLab/MicroAdam.
We propose a novel algorithm for distributed stochastic gradient descent (SGD) with compressed gradient communication in the parameter-server framework. Our gradient compression technique, named flattened one-bit stochastic gradient descent (FO-SGD), relies on two simple algorithmic ideas: (i) a one-bit quantization procedure leveraging the technique of dithering, and (ii) a randomized fast Walsh-Hadamard transform to flatten the stochastic gradient before quantization. As a result, the approximation of the true gradient in this scheme is biased, but it prevents commonly encountered algorithmic problems, such as exploding variance in the one-bit compression regime, deterioration of performance in the case of sparse gradients, and restrictive assumptions on the distribution of the stochastic gradients. In fact, we show SGD-like convergence guarantees under mild conditions. The compression technique can be used in both directions of worker-server communication, therefore admitting distributed optimization with full communication compression.
Local SGD is a popular optimization method in distributed learning, often outperforming other algorithms in practice, including mini-batch SGD. Despite this success, theoretically proving the dominance of local SGD in settings with reasonable data heterogeneity has been difficult, creating a significant gap between theory and practice. In this paper, we provide new lower bounds for local SGD under existing first-order data heterogeneity assumptions, showing that these assumptions are insufficient to prove the effectiveness of local update steps. Furthermore, under these same assumptions, we demonstrate the min-max optimality of accelerated mini-batch SGD, which fully resolves our understanding of distributed optimization for several problem classes. Our results emphasize the need for better models of data heterogeneity to understand the effectiveness of local SGD in practice. Towards this end, we consider higher-order smoothness and heterogeneity assumptions, providing new upper bounds that imply the dominance of local SGD over mini-batch SGD when data heterogeneity is low.
This paper studies how a stochastic gradient algorithm (SG) can be controlled to hide the estimate of the local stationary point from an eavesdropper. Such problems are of significant interest in distributed optimization settings like federated learning and inventory management. A learner queries a stochastic oracle and incentivizes the oracle to obtain noisy gradient measurements and perform SG. The oracle probabilistically returns either a noisy gradient of the function} or a non-informative measurement, depending on the oracle state and incentive. The learner's query and incentive are visible to an eavesdropper who wishes to estimate the stationary point. This paper formulates the problem of the learner performing covert optimization by dynamically incentivizing the stochastic oracle and obfuscating the eavesdropper as a finite-horizon Markov decision process (MDP). Using conditions for interval-dominance on the cost and transition probability structure, we show that the optimal policy for the MDP has a monotone threshold structure. We propose searching for the optimal stationary policy with the threshold structure using a stochastic approximation algorithm and a multi-armed bandit approach. The effectiveness of our methods is numerically demonstrated on a covert federated learning hate-speech classification task.
This paper develops an adaptive traffic control policy inspired by Maximum Pressure (MP) while imposing coordination across intersections. The proposed Coordinated Maximum Pressure-plus-Penalty (CMPP) control policy features a local objective for each intersection that consists of the total pressure within the neighborhood and a penalty accounting for the queue capacities and continuous green time for certain movements. The corresponding control task is reformulated as a distributed optimization problem and solved via two customized algorithms: one based on the alternating direction method of multipliers (ADMM) and the other follows a greedy heuristic augmented with a majority vote. CMPP not only provides a theoretical guarantee of queuing network stability but also outperforms several benchmark controllers in simulations on a large-scale real traffic network with lower average travel and waiting time per vehicle, as well as less network congestion. Furthermore, CPMM with the greedy algorithm enjoys comparable computational efficiency as fully decentralized controllers without significantly compromising the control performance, which highlights its great potential for real-world deployment.
Distributed decision problems features a group of agents that can only communicate over a peer-to-peer network, without a central memory. In applications such as network control and data ranking, each agent is only affected by a small portion of the decision vector: this sparsity is typically ignored in distributed algorithms, while it could be leveraged to improve efficiency and scalability. To address this issue, our recent paper introduces Estimation Network Design (END), a graph theoretical language for the analysis and design of distributed iterations. END algorithms can be tuned to exploit the sparsity of specific problem instances, reducing communication overhead and minimizing redundancy, yet without requiring case-by-case convergence analysis. In this paper, we showcase the flexility of END in the context of distributed optimization. In particular, we study the sparsity-aware version of many established methods, including ADMM, AugDGM and Push-Sum DGD. Simulations on an estimation problem in sensor networks demonstrate that END algorithms can boost convergence speed and greatly reduce the communication and memory cost.
We consider an $n$ agents distributed optimization problem with imperfect information characterized in a parametric sense, where the unknown parameter can be solved by a distinct distributed parameter learning problem. Though each agent only has access to its local parameter learning and computational problem, they mean to collaboratively minimize the average of their local cost functions. To address the special optimization problem, we propose a coupled distributed stochastic approximation algorithm, in which every agent updates the current beliefs of its unknown parameter and decision variable by stochastic approximation method; and then averages the beliefs and decision variables of its neighbors over network in consensus protocol. Our interest lies in the convergence analysis of this algorithm. We quantitatively characterize the factors that affect the algorithm performance, and prove that the mean-squared error of the decision variable is bounded by $\mathcal{O}(\frac{1}{nk})+\mathcal{O}\left(\frac{1}{\sqrt{n}(1-\rho_w)}\right)\frac{1}{k^{1.5}}+\mathcal{O}\big(\frac{1}{(1-\rho_w)^2} \big)\frac{1}{k^2}$, where $k$ is the iteration count and $(1-\rho_w)$ is the spectral gap of the network weighted adjacency matrix. It reveals that the network connectivity characterized by $(1-\rho_w)$ only influences the high order of convergence rate, while the domain rate still acts the same as the centralized algorithm. In addition, we analyze that the transient iteration needed for reaching its dominant rate $\mathcal{O}(\frac{1}{nk})$ is $\mathcal{O}(\frac{n}{(1-\rho_w)^2})$. Numerical experiments are carried out to demonstrate the theoretical results by taking different CPUs as agents, which is more applicable to real-world distributed scenarios.
This paper considers a distributed adaptive optimization problem, where all agents only have access to their local cost functions with a common unknown parameter, whereas they mean to collaboratively estimate the true parameter and find the optimal solution over a connected network. A general mathematical framework for such a problem has not been studied yet. We aim to provide valuable insights for addressing parameter uncertainty in distributed optimization problems and simultaneously find the optimal solution. Thus, we propose a novel Prediction while Optimization scheme, which utilizes distributed fractional Bayesian learning through weighted averaging on the log-beliefs to update the beliefs of unknown parameters, and distributed gradient descent for renewing the estimation of the optimal solution. Then under suitable assumptions, we prove that all agents' beliefs and decision variables converge almost surely to the true parameter and the optimal solution under the true parameter, respectively. We further establish a sublinear convergence rate for the belief sequence. Finally, numerical experiments are implemented to corroborate the theoretical analysis.
Federated learning is a distributed optimization paradigm that allows training machine learning models across decentralized devices while keeping the data localized. The standard method, FedAvg, suffers from client drift which can hamper performance and increase communication costs over centralized methods. Previous works proposed various strategies to mitigate drift, yet none have shown uniformly improved communication-computation trade-offs over vanilla gradient descent. In this work, we revisit DANE, an established method in distributed optimization. We show that (i) DANE can achieve the desired communication reduction under Hessian similarity constraints. Furthermore, (ii) we present an extension, DANE+, which supports arbitrary inexact local solvers and has more freedom to choose how to aggregate the local updates. We propose (iii) a novel method, FedRed, which has improved local computational complexity and retains the same communication complexity compared to DANE/DANE+. This is achieved by using doubly regularized drift correction.
Modern Machine Learning (ML) training on large-scale datasets is a very time-consuming workload. It relies on the optimization algorithm Stochastic Gradient Descent (SGD) due to its effectiveness, simplicity, and generalization performance. Processor-centric architectures (e.g., CPUs, GPUs) commonly used for modern ML training workloads based on SGD are bottlenecked by data movement between the processor and memory units due to the poor data locality in accessing large datasets. As a result, processor-centric architectures suffer from low performance and high energy consumption while executing ML training workloads. Processing-In-Memory (PIM) is a promising solution to alleviate the data movement bottleneck by placing the computation mechanisms inside or near memory. Our goal is to understand the capabilities of popular distributed SGD algorithms on real-world PIM systems to accelerate data-intensive ML training workloads. To this end, we 1) implement several representative centralized parallel SGD algorithms on the real-world UPMEM PIM system, 2) rigorously evaluate these algorithms for ML training on large-scale datasets in terms of performance, accuracy, and scalability, 3) compare to conventional CPU and GPU baselines, and 4) discuss implications for future PIM hardware and highlight the need for a shift to an algorithm-hardware codesign. Our results demonstrate three major findings: 1) The UPMEM PIM system can be a viable alternative to state-of-the-art CPUs and GPUs for many memory-bound ML training workloads, especially when operations and datatypes are natively supported by PIM hardware, 2) it is important to carefully choose the optimization algorithms that best fit PIM, and 3) the UPMEM PIM system does not scale approximately linearly with the number of nodes for many data-intensive ML training workloads. We open source all our code to facilitate future research.
Cartesian reverse derivative categories (CRDCs) provide an axiomatic generalization of the reverse derivative, which allows generalized analogues of classic optimization algorithms such as gradient descent to be applied to a broad class of problems. In this paper, we show that generalized gradient descent with respect to a given CRDC induces a hypergraph functor from a hypergraph category of optimization problems to a hypergraph category of dynamical systems. The domain of this functor consists of objective functions that are 1) general in the sense that they are defined with respect to an arbitrary CRDC, and 2) open in that they are decorated spans that can be composed with other such objective functions via variable sharing. The codomain is specified analogously as a category of general and open dynamical systems for the underlying CRDC. We describe how the hypergraph functor induces a distributed optimization algorithm for arbitrary composite problems specified in the domain. To illustrate the kinds of problems our framework can model, we show that parameter sharing models in multitask learning, a prevalent machine learning paradigm, yield a composite optimization problem for a given choice of CRDC. We then apply the gradient descent functor to this composite problem and describe the resulting distributed gradient descent algorithm for training parameter sharing models.
We introduce a distributed algorithm, termed noise-robust distributed maximum consensus (RD-MC), for estimating the maximum value within a multi-agent network in the presence of noisy communication links. Our approach entails redefining the maximum consensus problem as a distributed optimization problem, allowing a solution using the alternating direction method of multipliers. Unlike existing algorithms that rely on multiple sets of noise-corrupted estimates, RD-MC employs a single set, enhancing both robustness and efficiency. To further mitigate the effects of link noise and improve robustness, we apply moving averaging to the local estimates. Through extensive simulations, we demonstrate that RD-MC is significantly more robust to communication link noise compared to existing maximum-consensus algorithms.
Community battery systems have been widely deployed to provide services to the grid. Unlike a single battery storage system in the community, coordinating multiple community batteries can further unlock their value, enhancing the viability of community battery solutions. However, the centralized control of community batteries relies on the full information of the system, which is less practical and may even lead to privacy leakage. In this paper, we formulate a value-stacking optimization problem for community batteries to interact with local solar, buildings, and the grid, within distribution network constraints. We then propose a distributed algorithm using asynchronous distributed alternate direction method of multipliers (ADMM) to solve the problem. Our algorithm is robust to communication latency between community batteries and the grid while preserving the operational privacy. The simulation results demonstrate the convergence of our proposed asynchronous distributed ADMM algorithm. We also evaluate the electricity cost and the contribution of each value stream in the value-stacking problem for community batteries using real-world data.
Distributed nonconvex optimization underpins key functionalities of numerous distributed systems, ranging from power systems, smart buildings, cooperative robots, vehicle networks to sensor networks. Recently, it has also merged as a promising solution to handle the enormous growth in data and model sizes in deep learning. A fundamental problem in distributed nonconvex optimization is avoiding convergence to saddle points, which significantly degrade optimization accuracy. We discover that the process of quantization, which is necessary for all digital communications, can be exploited to enable saddle-point avoidance. More specifically, we propose a stochastic quantization scheme and prove that it can effectively escape saddle points and ensure convergence to a second-order stationary point in distributed nonconvex optimization. With an easily adjustable quantization granularity, the approach allows a user to control the number of bits sent per iteration and, hence, to aggressively reduce the communication overhead. Numerical experimental results using distributed optimization and learning problems on benchmark datasets confirm the effectiveness of the approach.
In this study, we investigate stochastic optimization on Riemannian manifolds, focusing on the crucial variance reduction mechanism used in both Euclidean and Riemannian settings. Riemannian variance-reduced methods usually involve a double-loop structure, computing a full gradient at the start of each loop. Determining the optimal inner loop length is challenging in practice, as it depends on strong convexity or smoothness constants, which are often unknown or hard to estimate. Motivated by Euclidean methods, we introduce the Riemannian Loopless SVRG (R-LSVRG) and PAGE (R-PAGE) methods. These methods replace the outer loop with probabilistic gradient computation triggered by a coin flip in each iteration, ensuring simpler proofs, efficient hyperparameter selection, and sharp convergence guarantees. Using R-PAGE as a framework for non-convex Riemannian optimization, we demonstrate its applicability to various important settings. For example, we derive Riemannian MARINA (R-MARINA) for distributed settings with communication compression, providing the best theoretical communication complexity guarantees for non-convex distributed optimization over Riemannian manifolds. Experimental results support our theoretical findings.
In Distributed optimization and Learning, and even more in the modern framework of federated learning, communication, which is slow and costly, is critical. We introduce LoCoDL, a communication-efficient algorithm that leverages the two popular and effective techniques of Local training, which reduces the communication frequency, and Compression, in which short bitstreams are sent instead of full-dimensional vectors of floats. LoCoDL works with a large class of unbiased compressors that includes widely-used sparsification and quantization methods. LoCoDL provably benefits from local training and compression and enjoys a doubly-accelerated communication complexity, with respect to the condition number of the functions and the model dimension, in the general heterogenous regime with strongly convex functions. This is confirmed in practice, with LoCoDL outperforming existing algorithms.
Gradient-type distributed optimization methods have blossomed into one of the most important tools for solving a minimization learning task over a networked agent system. However, only one gradient update per iteration is difficult to achieve a substantive acceleration of convergence. In this paper, we propose an accelerated framework named as MUSIC allowing each agent to perform multiple local updates and a single combination in each iteration. More importantly, we equip inexact and exact distributed optimization methods into this framework, thereby developing two new algorithms that exhibit accelerated linear convergence and high communication efficiency. Our rigorous convergence analysis reveals the sources of steady-state errors arising from inexact policies and offers effective solutions. Numerical results based on synthetic and real datasets demonstrate both our theoretical motivations and analysis, as well as performance advantages.