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Register a new userStatistical Estimation and Inference via Local SGD in Federated Learning
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Federated Learning (FL) makes a large amount of edge computing devices (e.g., mobile phones) jointly learn a global model without data sharing. In FL, data are generated in a decentralized manner with high heterogeneity. This paper studies how to perform statistical estimation and inference in the federated setting. We analyze the so-called Local SGD, a multi-round estimation procedure that uses intermittent communication to improve communication efficiency. We first establish a {it functional central limit theorem} that shows the averaged iterates of Local SGD weakly converge to a rescaled Brownian motion. We next provide two iterative inference methods: the {it plug-in} and the {it random scaling}. Random scaling constructs an asymptotically pivotal statistic for inference by using the information along the whole Local SGD path. Both the methods are communication efficient and applicable to online data. Our theoretical and empirical results show that Local SGD simultaneously achieves both statistical efficiency and communication efficiency.
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In this paper, we propose texttt{FedGLOMO}, the first (first-order) FL algorithm that achieves the optimal iteration complexity (i.e matching the known lower bound) on smooth non-convex objectives -- without using clients full gradient in each round. Our key algorithmic idea that enables attaining this optimal complexity is applying judicious momentum terms that promote variance reduction in both the local updates at the clients, and the global update at the server. Our algorithm is also provably optimal even with compressed communication between the clients and the server, which is an important consideration in the practical deployment of FL algorithms. Our experiments illustrate the intrinsic variance reduction effect of texttt{FedGLOMO} which implicitly suppresses client-drift in heterogeneous data distribution settings and promotes communication-efficiency. As a prequel to texttt{FedGLOMO}, we propose texttt{FedLOMO} which applies momentum only in the local client updates. We establish that texttt{FedLOMO} enjoys improved convergence rates under common non-convex settings compared to prior work, and with fewer assumptions.
Stochastic gradient descent (SGD) and projected stochastic gradient descent (PSGD) are scalable algorithms to compute model parameters in unconstrained and constrained optimization problems. In comparison with stochastic gradient descent (SGD), PSGD forces its iterative values into the constrained parameter space via projection. The convergence rate of PSGD-type estimates has been exhaustedly studied, while statistical properties such as asymptotic distribution remain less explored. From a purely statistical point of view, this paper studies the limiting distribution of PSGD-based estimate when the true parameters satisfying some linear-equality constraints. Our theoretical findings reveal the role of projection played in the uncertainty of the PSGD estimate. As a byproduct, we propose an online hypothesis testing procedure to test the linear-equality constraints. Simulation studies on synthetic data and an application to a real-world dataset confirm our theory.
This work presents a new distributed Byzantine tolerant federated learning algorithm, HoldOut SGD, for Stochastic Gradient Descent (SGD) optimization. HoldOut SGD uses the well known machine learning technique of holdout estimation, in a distributed fashion, in order to select parameter updates that are likely to lead to models with low loss values. This makes it more effective at discarding Byzantine workers inputs than existing methods that eliminate outliers in the parameter-space of the learned model. HoldOut SGD first randomly selects a set of workers that use their private data in order to propose gradient updates. Next, a voting committee of workers is randomly selected, and each voter uses its private data as holdout data, in order to select the best proposals via a voting scheme. We propose two possible mechanisms for the coordination of workers in the distributed computation of HoldOut SGD. The first uses a truthful central server and corresponds to the typical setting of current federated learning. The second is fully distributed and requires no central server, paving the way to fully decentralized federated learning. The fully distributed version implements HoldOut SGD via ideas from the blockchain domain, and specifically the Algorand committee selection and consensus processes. We provide formal guarantees for the HoldOut SGD process in terms of its convergence to the optimal model, and its level of resilience to the fraction of Byzantine workers. Empirical evaluation shows that HoldOut SGD is Byzantine-resilient and efficiently converges to an effectual model for deep-learning tasks, as long as the total number of participating workers is large and the fraction of Byzantine workers is less than half (<1/3 for the fully distributed variant).
In this paper, we consider the estimation of a low Tucker rank tensor from a number of noisy linear measurements. The general problem covers many specific examples arising from applications, including tensor regression, tensor completion, and tensor PCA/SVD. We propose a Riemannian Gauss-Newton (RGN) method with fast implementations for low Tucker rank tensor estimation. Different from the generic (super)linear convergence guarantee of RGN in the literature, we prove the first quadratic convergence guarantee of RGN for low-rank tensor estimation under some mild conditions. A deterministic estimation error lower bound, which matches the upper bound, is provided that demonstrates the statistical optimality of RGN. The merit of RGN is illustrated through two machine learning applications: tensor regression and tensor SVD. Finally, we provide the simulation results to corroborate our theoretical findings.
When the data are stored in a distributed manner, direct application of traditional statistical inference procedures is often prohibitive due to communication cost and privacy concerns. This paper develops and investigates two Communication-Efficient Accurate Statistical Estimators (CEASE), implemented through iterative algorithms for distributed optimization. In each iteration, node machines carry out computation in parallel and communicate with the central processor, which then broadcasts aggregated information to node machines for new updates. The algorithms adapt to the similarity among loss functions on node machines, and converge rapidly when each node machine has large enough sample size. Moreover, they do not require good initialization and enjoy linear converge guarantees under general conditions. The contraction rate of optimization errors is presented explicitly, with dependence on the local sample size unveiled. In addition, the improved statistical accuracy per iteration is derived. By regarding the proposed method as a multi-step statistical estimator, we show that statistical efficiency can be achieved in finite steps in typical statistical applications. In addition, we give the conditions under which the one-step CEASE estimator is statistically efficient. Extensive numerical experiments on both synthetic and real data validate the theoretical results and demonstrate the superior performance of our algorithms.
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