We consider the problem of distributed statistical machine learning in adversarial settings, where some unknown and time-varying subset of working machines may be compromised and behave arbitrarily to prevent an accurate model from being learned. This setting captures the potential adversarial attacks faced by Federated Learning -- a modern machine learning paradigm that is proposed by Google researchers and has been intensively studied for ensuring user privacy. Formally, we focus on a distributed system consisting of a parameter server and @math working machines. Each working machine keeps @math data samples, where @math is the total number of samples. The goal is to collectively learn the underlying true model parameter of dimension @math . In classical batch gradient descent methods, the gradients reported to the server by the working machines are aggregated via simple averaging, which is vulnerable to a single Byzantine failure. In this paper, we propose a Byzantine gradient descent method based on the geometric median of means of the gradients. We show that our method can tolerate @math Byzantine failures, and the parameter estimate converges in @math rounds with an estimation error of @math , hence approaching the optimal error rate @math in the centralized and failure-free setting. The total computational complexity of our algorithm is of @math at each working machine and @math at the central server, and the total communication cost is of @math . We further provide an application of our general results to the linear regression problem. A key challenge arises in the above problem is that Byzantine failures create arbitrary and unspecified dependency among the iterations and the aggregated gradients. We prove that the aggregated gradient converges uniformly to the true gradient function.
On the technical front, a crucial step in our convergence proof is to show the geometric median of means of @math i.i.d. random gradients converges to the underlying gradient function @math uniformly over @math . Our proof builds on several ideas from the empirical process theory, which guarantees uniform convergence of the empirical risk function @math to the population risk @math . However, what we need is the uniform convergence of empirical function @math , as well as its version, to the population gradient function @math . To this end, we use concentration inequalities to first establish point-wise convergence and then boost it to uniform convergence via the celebrated @math -net argument. Similar ideas have been used recently in the work @cite_11 , which studies the stationary points of the empirical risk function.
@cite_11
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Most high-dimensional estimation and prediction methods propose to minimize a cost function (empirical risk) that is written as a sum of losses associated to each data point. In this paper we focus on the case of non-convex losses, which is practically important but still poorly understood. Classical empirical process theory implies uniform convergence of the empirical risk to the population risk. While uniform convergence implies consistency of the resulting M-estimator, it does not ensure that the latter can be computed efficiently. In order to capture the complexity of computing M-estimators, we propose to study the landscape of the empirical risk, namely its stationary points and their properties. We establish uniform convergence of the gradient and Hessian of the empirical risk to their population counterparts, as soon as the number of samples becomes larger than the number of unknown parameters (modulo logarithmic factors). Consequently, good properties of the population risk can be carried to the empirical risk, and we can establish one-to-one correspondence of their stationary points. We demonstrate that in several problems such as non-convex binary classification, robust regression, and Gaussian mixture model, this result implies a complete characterization of the landscape of the empirical risk, and of the convergence properties of descent algorithms. We extend our analysis to the very high-dimensional setting in which the number of parameters exceeds the number of samples, and provide a characterization of the empirical risk landscape under a nearly information-theoretically minimal condition. Namely, if the number of samples exceeds the sparsity of the unknown parameters vector (modulo logarithmic factors), then a suitable uniform convergence result takes place. We apply this result to non-convex binary classification and robust regression in very high-dimension.