In distributed optimization, a popular technique to reduce communication is quantization. In this paper, we provide a general analysis framework for inexact gradient descent that is applicable to quantization schemes. We also propose a quantization scheme Double Encoding and Error Diminishing (DEED). DEED can achieve small communication complexity in three settings: frequent-communication large-memory, frequent-communication small-memory, and infrequent-communication (e.g. federated learning). More specifically, in the frequent-communication large-memory setting, DEED can be easily combined with Nesterov's method, so that the total number of bits required is $\tilde{O}( \sqrt{\kappa} \log 1/\epsilon )$, where $\tilde{O}$ hides numerical constant and $\log \kappa$ factors. In the frequent-communication small-memory setting, DEED combined with SGD only requires $\tilde{O}( \kappa \log 1/\epsilon)$ number of bits in the interpolation regime. In the infrequent communication setting, DEED combined with Federated averaging requires a smaller total number of bits than Federated Averaging. All these algorithms converge at the same rate as their non-quantized versions, while using a smaller number of bits.
In distributed optimization problems, a technique called gradient coding, which involves replicating data points, has been used to mitigate the effect of straggling machines. Recent work has studied approximate gradient coding, which concerns coding schemes where the replication factor of the data is too low to recover the full gradient exactly. Our work is motivated by the challenge of creating approximate gradient coding schemes that simultaneously work well in both the adversarial and stochastic models. To that end, we introduce novel approximate gradient codes based on expander graphs, in which each machine receives exactly two blocks of data points. We analyze the decoding error both in the random and adversarial straggler setting, when optimal decoding coefficients are used. We show that in the random setting, our schemes achieve an error to the gradient that decays exponentially in the replication factor. In the adversarial setting, the error is nearly a factor of two smaller than any existing code with similar performance in the random setting. We show convergence bounds both in the random and adversarial setting for gradient descent under standard assumptions using our codes. In the random setting, our convergence rate improves upon block-box bounds. In the adversarial setting, we show that gradient descent can converge down to a noise floor that scales linearly with the adversarial error to the gradient. We demonstrate empirically that our schemes achieve near-optimal error in the random setting and converge faster than algorithms which do not use the optimal decoding coefficients.
We develop a distributed second order optimization algorithm that is communication-efficient as well as robust against Byzantine failures of the worker machines. We propose COMRADE (COMunication-efficient and Robust Approximate Distributed nEwton), an iterative second order algorithm, where the worker machines communicate only once per iteration with the center machine. This is in sharp contrast with the state-of-the-art distributed second order algorithms like GIANT [34] and DINGO[7], where the worker machines send (functions of) local gradient and Hessian sequentially; thus ending up communicating twice with the center machine per iteration. Moreover, we show that the worker machines can further compress the local information before sending it to the center. In addition, we employ a simple norm based thresholding rule to filter-out the Byzantine worker machines. We establish the linear-quadratic rate of convergence of COMRADE and establish that the communication savings and Byzantine resilience result in only a small statistical error rate for arbitrary convex loss functions. To the best of our knowledge, this is the first work that addresses the issue of Byzantine resilience in second order distributed optimization. Furthermore, we validate our theoretical results with extensive experiments on synthetic and benchmark LIBSVM [5] data-sets and demonstrate convergence guarantees.
We propose Federated Accelerated Stochastic Gradient Descent (FedAc), a principled acceleration of Federated Averaging (FedAvg, also known as Local SGD) for distributed optimization. FedAc is the first provable acceleration of FedAvg that improves convergence speed and communication efficiency on various types of convex functions. For example, for strongly convex and smooth functions, when using $M$ workers, the previous state-of-the-art FedAvg analysis can achieve a linear speedup in $M$ if given $M$ rounds of synchronization, whereas FedAc only requires $M^{\frac{1}{3}}$ rounds. Moreover, we prove stronger guarantees for FedAc when the objectives are third-order smooth. Our technique is based on a potential-based perturbed iterate analysis, a novel stability analysis of generalized accelerated SGD, and a strategic tradeoff between acceleration and stability.
In Byzantine robust distributed or federated learning, a central server wants to train a machine learning model over data distributed across multiple workers. However, a fraction of these workers may deviate from the prescribed algorithm and send arbitrary messages. While this problem has received significant attention recently, most current defenses assume that the workers have identical data. For realistic cases when the data across workers are heterogeneous (non-iid), we design new attacks which circumvent current defenses, leading to significant loss of performance. We then propose a simple bucketing scheme that adapts existing robust algorithms to heterogeneous datasets at a negligible computational cost. We also theoretically and experimentally validate our approach, showing that combining bucketing with existing robust algorithms is effective against challenging attacks. Our work is the first to establish guaranteed convergence for the non-iid Byzantine robust problem under realistic assumptions.
Distributed optimization is vital in solving large-scale machine learning problems. A widely-shared feature of distributed optimization techniques is the requirement that all nodes complete their assigned tasks in each computational epoch before the system can proceed to the next epoch. In such settings, slow nodes, called stragglers, can greatly slow progress. To mitigate the impact of stragglers, we propose an online distributed optimization method called Anytime Minibatch. In this approach, all nodes are given a fixed time to compute the gradients of as many data samples as possible. The result is a variable per-node minibatch size. Workers then get a fixed communication time to average their minibatch gradients via several rounds of consensus, which are then used to update primal variables via dual averaging. Anytime Minibatch prevents stragglers from holding up the system without wasting the work that stragglers can complete. We present a convergence analysis and analyze the wall time performance. Our numerical results show that our approach is up to 1.5 times faster in Amazon EC2 and it is up to five times faster when there is greater variability in compute node performance.
Tuning hyperparameters for machine learning algorithms is a tedious task, one that is typically done manually. To enable automated hyperparameter tuning, recent works have started to use techniques based on Bayesian optimization. However, to practically enable automated tuning for large scale machine learning training pipelines, significant gaps remain in existing libraries, including lack of abstractions, fault tolerance, and flexibility to support scheduling on any distributed computing framework. To address these challenges, we present Mango, a Python library for parallel hyperparameter tuning. Mango enables the use of any distributed scheduling framework, implements intelligent parallel search strategies, and provides rich abstractions for defining complex hyperparameter search spaces that are compatible with scikit-learn. Mango is comparable in performance to Hyperopt, another widely used library. Mango is available open-source and is currently used in production at Arm Research to provide state-of-art hyperparameter tuning capabilities.
Motivated by federated learning, we consider the hub-and-spoke model of distributed optimization in which a central authority coordinates the computation of a solution among many agents while limiting communication. We first study some past procedures for federated optimization, and show that their fixed points need not correspond to stationary points of the original optimization problem, even in simple convex settings with deterministic updates. In order to remedy these issues, we introduce FedSplit, a class of algorithms based on operator splitting procedures for solving distributed convex minimization with additive structure. We prove that these procedures have the correct fixed points, corresponding to optima of the original optimization problem, and we characterize their convergence rates under different settings. Our theory shows that these methods are provably robust to inexact computation of intermediate local quantities. We complement our theory with some simple experiments that demonstrate the benefits of our methods in practice.
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 co-ordination, estimation in sensor networks, and large-scale optimization in machine learning. We develop and analyze distributed algorithms based on dual averaging of subgradients, and we provide sharp bounds on their convergence rates as a function of the network size and topology. Our method of analysis allows for a clear separation between the convergence of the optimization algorithm itself and the effects of communication constraints arising from the network structure. In particular, we show that the number of iterations required by our algorithm scales inversely in the spectral gap of the network. The sharpness of this prediction is confirmed both by theoretical lower bounds and simulations for various networks. Our approach includes both the cases of deterministic optimization and communication, as well as problems with stochastic optimization and/or communication.
In this paper, we study distributed algorithms for large-scale AUC maximization with a deep neural network as a predictive model. Although distributed learning techniques have been investigated extensively in deep learning, they are not directly applicable to stochastic AUC maximization with deep neural networks due to its striking differences from standard loss minimization problems (e.g., cross-entropy). Towards addressing this challenge, we propose and analyze a communication-efficient distributed optimization algorithm based on a {\it non-convex concave} reformulation of the AUC maximization, in which the communication of both the primal variable and the dual variable between each worker and the parameter server only occurs after multiple steps of gradient-based updates in each worker. Compared with the naive parallel version of an existing algorithm that computes stochastic gradients at individual machines and averages them for updating the model parameters, our algorithm requires a much less number of communication rounds and still achieves a linear speedup in theory. To the best of our knowledge, this is the \textbf{first} work that solves the {\it non-convex concave min-max} problem for AUC maximization with deep neural networks in a communication-efficient distributed manner while still maintaining the linear speedup property in theory. Our experiments on several benchmark datasets show the effectiveness of our algorithm and also confirm our theory.
This paper considers the problem of multi-agent distributed optimization. In this problem, there are multiple agents in the system, and each agent only knows its local cost function. The objective for the agents is to collectively compute a common minimum of the aggregate of all their local cost functions. In principle, this problem is solvable using a distributed variant of the traditional gradient-descent method, which is an iterative method. However, the speed of convergence of the traditional gradient-descent method is highly influenced by the conditioning of the optimization problem being solved. Specifically, the method requires a large number of iterations to converge to a solution if the optimization problem is ill-conditioned. In this paper, we propose an iterative pre-conditioning approach that can significantly attenuate the influence of the problem's conditioning on the convergence-speed of the gradient-descent method. The proposed pre-conditioning approach can be easily implemented in distributed systems and has minimal computation and communication overhead. For now, we only consider a specific distributed optimization problem wherein the individual local cost functions of the agents are quadratic. Besides the theoretical guarantees, the improved convergence speed of our approach is demonstrated through experiments on a real data-set.
We consider the problem of communication efficient distributed optimization where multiple nodes exchange important algorithm information in every iteration to solve large problems. In particular, we focus on the stochastic variance-reduced gradient and propose a novel approach to make it communication-efficient. That is, we compress the communicated information to a few bits while preserving the linear convergence rate of the original uncompressed algorithm. Comprehensive theoretical and numerical analyses on real datasets reveal that our algorithm can significantly reduce the communication complexity, by as much as 95\%, with almost no noticeable penalty. Moreover, it is much more robust to quantization (in terms of maintaining the true minimizer and the convergence rate) than the state-of-the-art algorithms for solving distributed optimization problems. Our results have important implications for using machine learning over internet-of-things and mobile networks.
In this article, we study a Radio Resource Allocation (RRA) that was formulated as a non-convex optimization problem whose main aim is to maximize the spectral efficiency subject to satisfaction guarantees in multiservice wireless systems. This problem has already been previously investigated in the literature and efficient heuristics have been proposed. However, in order to assess the performance of Machine Learning (ML) algorithms when solving optimization problems in the context of RRA, we revisit that problem and propose a solution based on a Reinforcement Learning (RL) framework. Specifically, a distributed optimization method based on multi-agent deep RL is developed, where each agent makes its decisions to find a policy by interacting with the local environment, until reaching convergence. Thus, this article focuses on an application of RL and our main proposal consists in a new deep RL based approach to jointly deal with RRA, satisfaction guarantees and Quality of Service (QoS) constraints in multiservice celular networks. Lastly, through computational simulations we compare the state-of-art solutions of the literature with our proposal and we show a near optimal performance of the latter in terms of throughput and outage rate.
Consensus-based distributed optimization methods have recently been advocated as alternatives to parameter server and ring all-reduce paradigms for large scale training of machine learning models. In this case, each worker maintains a local estimate of the optimal parameter vector and iteratively updates it by averaging the estimates obtained from its neighbors, and applying a correction on the basis of its local dataset. While theoretical results suggest that worker communication topology should have strong impact on the number of epochs needed to converge, previous experiments have shown the opposite conclusion. This paper sheds lights on this apparent contradiction and show how sparse topologies can lead to faster convergence even in the absence of communication delays.
In this work, we investigate direction finding in the presence of sensor gain uncertainties and directional perturbations for sensor array processing in a multi-frequency scenario. Specifically, we adopt a distributed optimization scheme in which coherence models are incorporated and local agents exchange information only between connected nodes in the network, i.e., without a fusion center. Numerical simulations highlight the advantages of the proposed parallel iterative technique in terms of statistical and computational efficiency.
EXTRA is a popular method for dencentralized distributed optimization and has broad applications. This paper revisits EXTRA. First, we give a sharp complexity analysis for EXTRA with the improved $O\left(\left(\frac{L}{\mu}+\frac{1}{1-\sigma_2(W)}\right)\log\frac{1}{\epsilon(1-\sigma_2(W))}\right)$ communication and computation complexities for $\mu$-strongly convex and $L$-smooth problems, where $\sigma_2(W)$ is the second largest singular value of the weight matrix $W$. When the strong convexity is absent, we prove the $O\left(\left(\frac{L}{\epsilon}+\frac{1}{1-\sigma_2(W)}\right)\log\frac{1}{1-\sigma_2(W)}\right)$ complexities. Then, we use the Catalyst framework to accelerate EXTRA and obtain the $O\left(\sqrt{\frac{L}{\mu(1-\sigma_2(W))}}\log\frac{ L}{\mu(1-\sigma_2(W))}\log\frac{1}{\epsilon}\right)$ communication and computation complexities for strongly convex and smooth problems and the $O\left(\sqrt{\frac{L}{\epsilon(1-\sigma_2(W))}}\log\frac{1}{\epsilon(1-\sigma_2(W))}\right)$ complexities for non-strongly convex ones. Our communication complexities of the accelerated EXTRA are only worse by the factors of $\left(\log\frac{L}{\mu(1-\sigma_2(W))}\right)$ and $\left(\log\frac{1}{\epsilon(1-\sigma_2(W))}\right)$ from the lower complexity bounds for strongly convex and non-strongly convex problems, respectively.
We consider the problem of distributed mean estimation (DME), in which $n$ machines are each given a local $d$-dimensional vector $x_v \in \mathbb{R}^d$, and must cooperate to estimate the mean of their inputs $\mu = \frac 1n\sum_{v = 1}^n x_v$, while minimizing total communication cost. DME is a fundamental construct in distributed machine learning, and there has been considerable work on variants of this problem, especially in the context of distributed variance reduction for stochastic gradients in parallel SGD. Previous work typically assumes an upper bound on the norm of the input vectors, and achieves an error bound in terms of this norm. However, in many real applications, the input vectors are concentrated around the correct output $\mu$, but $\mu$ itself has large norm. In such cases, previous output error bounds perform poorly. In this paper, we show that output error bounds need not depend on input norm. We provide a method of quantization which allows distributed mean estimation to be performed with solution quality dependent only on the distance between inputs, not on input norm, and show an analogous result for distributed variance reduction. The technique is based on a new connection with lattice theory. We also provide lower bounds showing that the communication to error trade-off of our algorithms is asymptotically optimal. As the lattices achieving optimal bounds under $\ell_2$-norm can be computationally impractical, we also present an extension which leverages easy-to-use cubic lattices, and is loose only up to a logarithmic factor in $d$. We show experimentally that our method yields practical improvements for common applications, relative to prior approaches.
We study distributed optimization algorithms for minimizing the average of \emph{heterogeneous} functions distributed across several machines with a focus on communication efficiency. In such settings, naively using the classical stochastic gradient descent (SGD) or its variants (e.g., SVRG) with a uniform sampling of machines typically yields poor performance. It often leads to the dependence of convergence rate on maximum Lipschitz constant of gradients across the devices. In this paper, we propose a novel \emph{adaptive} sampling of machines specially catered to these settings. Our method relies on an adaptive estimate of local Lipschitz constants base on the information of past gradients. We show that the new way improves the dependence of convergence rate from maximum Lipschitz constant to \emph{average} Lipschitz constant across machines, thereby, significantly accelerating the convergence. Our experiments demonstrate that our method indeed speeds up the convergence of the standard SVRG algorithm in heterogeneous environments.
Sign-based optimization methods have become popular in machine learning due to their favorable communication cost in distributed optimization and their surprisingly good performance in neural network training. Furthermore, they are closely connected to so-called adaptive gradient methods like Adam. Recent works on signSGD have used a non-standard "separable smoothness" assumption, whereas some older works study sign gradient descent as steepest descent with respect to the $\ell_\infty$-norm. In this work, we unify these existing results by showing a close connection between separable smoothness and $\ell_\infty$-smoothness and argue that the latter is the weaker and more natural assumption. We then proceed to study the smoothness constant with respect to the $\ell_\infty$-norm and thereby isolate geometric properties of the objective function which affect the performance of sign-based methods. In short, we find sign-based methods to be preferable over gradient descent if (i) the Hessian is to some degree concentrated on its diagonal, and (ii) its maximal eigenvalue is much larger than the average eigenvalue. Both properties are common in deep networks.
We study local SGD (also known as parallel SGD and federated averaging), a natural and frequently used stochastic distributed optimization method. Its theoretical foundations are currently lacking and we highlight how all existing error guarantees in the convex setting are dominated by a simple baseline, minibatch SGD. (1) For quadratic objectives we prove that local SGD strictly dominates minibatch SGD and that accelerated local SGD is minimax optimal for quadratics; (2) For general convex objectives we provide the first guarantee that at least sometimes improves over minibatch SGD; (3) We show that indeed local SGD does not dominate minibatch SGD by presenting a lower bound on the performance of local SGD that is worse than the minibatch SGD guarantee.