Abstract Machine learning (ML) solutions are nowadays distributed, according to the so-called server/worker architecture. One server holds the model parameters while several workers train the model. Clearly, such architecture is prone to various types of component failures, which can be all encompassed within the spectrum of a Byzantine behavior. Several approaches have been proposed recently to tolerate Byzantine workers. Yet all require trusting a central parameter server. We initiate in this paper the study of the “general” Byzantine-resilient distributed machine learning problem where no individual component is trusted. In particular, we distribute the parameter server computation on several nodes. We show that this problem can be solved in an asynchronous system, despite the presence of $$\frac{1}{3}$$ 1 3 Byzantine parameter servers (i.e., $$n_{ps} > 3f_{ps}+1$$ n ps > 3 f ps + 1 ) and $$\frac{1}{3}$$ 1 3 Byzantine workers (i.e., $$n_w > 3f_w$$ n w > 3 f w ), which is asymptotically optimal. We present a new algorithm, ByzSGD, which solves the general Byzantine-resilient distributed machine learning problem by relying on three major schemes. The first, scatter/gather, is a communication scheme whose goal is to bound the maximum drift among models on correct servers. The second, distributed median contraction (DMC), leverages the geometric properties of the median in high dimensional spaces to bring parameters within the correct servers back close to each other, ensuring safe and lively learning. The third, Minimum-diameter averaging (MDA), is a statistically-robust gradient aggregation rule whose goal is to tolerate Byzantine workers. MDA requires a loose bound on the variance of non-Byzantine gradient estimates, compared to existing alternatives [e.g., Krum (Blanchard et al., in: Neural information processing systems, pp 118-128, 2017)]. Interestingly, ByzSGD ensures Byzantine resilience without adding communication rounds (on a normal path), compared to vanilla non-Byzantine alternatives. ByzSGD requires, however, a larger number of messages which, we show, can be reduced if we assume synchrony. We implemented ByzSGD on top of both TensorFlow and PyTorch, and we report on our evaluation results. In particular, we show that ByzSGD guarantees convergence with around 32% overhead compared to vanilla SGD. Furthermore, we show that ByzSGD’s throughput overhead is 24–176% in the synchronous case and 28–220% in the asynchronous case.

We present GARFIELD, a library to transparently make machine learning (ML) applications, initially built with popular (but fragile) frameworks, e.g., TensorFlow and PyTorch, Byzantine-resilient. GARFIELD relies on a novel object-oriented design, reducing the coding effort, and addressing the vulnerability of the shared-graph architecture followed by classical ML frameworks. GARFIELD encompasses various communication patterns and supports computations on CPUs and GPUs, allowing addressing the general question of the practical cost of Byzantine resilience in ML applications. We report on the usage of GARFIELD on three main ML architectures: (a) a single server with multiple workers, (b) several servers and workers, and (c) peer-to-peer settings. Using GARFIELD, we highlight interesting facts about the cost of Byzantine resilience. In particular, (a) Byzantine resilience, unlike crash resilience, induces an accuracy loss, (b) the throughput overhead comes more from communication than from robust aggregation, and (c) tolerating Byzantine servers costs more than tolerating Byzantine workers.

We study \emph{Byzantine collaborative learning}, where $n$ nodes seek to collectively learn from each others' local data. The data distribution may vary from one node to another. No node is trusted, and $f < n$ nodes can behave arbitrarily. We prove that collaborative learning is equivalent to a new form of agreement, which we call \emph{averaging agreement}. In this problem, nodes start each with an initial vector and seek to approximately agree on a common vector, which is close to the average of honest nodes' initial vectors. We present two asynchronous solutions to averaging agreement, each we prove optimal according to some dimension. The first, based on the minimum-diameter averaging, requires $n \geq 6f+1$, but achieves asymptotically the best-possible averaging constant up to a multiplicative constant. The second, based on reliable broadcast and coordinate-wise trimmed mean, achieves optimal Byzantine resilience, i.e., $n \geq 3f+1$. Each of these algorithms induces an optimal Byzantine collaborative learning protocol. In particular, our equivalence yields new impossibility theorems on what any collaborative learning algorithm can achieve in adversarial and heterogeneous environments.

This paper addresses the problem of combining Byzantine resilience with privacy in machine learning (ML). Specifically, we study if a distributed implementation of the renowned Stochastic Gradient Descent (SGD) learning algorithm is feasible withboth differential privacy (DP) and (α,f)-Byzantine resilience. To the best of our knowledge, this is the first work to tackle this problem from a theoretical point of view. A key finding of our analyses is that the classical approaches to these two (seemingly) orthogonal issues are incompatible. More precisely, we show that a direct composition of these techniques makes the guarantees of the resulting SGD algorithm depend unfavourably upon the number of parameters of the ML model, making the training of large models practically infeasible. We validate our theoretical results through numerical experiments on publicly-available datasets; showing that it is impractical to ensure DP and Byzantine resilience simultaneously.

Modern machine learning architectures distinguish servers and workers. Typically, a d-dimensional model is hosted by a server and trained by n workers, using a distributed stochastic gradient descent (SGD) optimization scheme. At each SGD step, the goal is to estimate the gradient of a cost function. The simplest way to do this is to average the gradients estimated by the workers. However, averaging is not resilient to even one single Byzantine failure of a worker. Many alternative gradient aggregation rules (GARs) have recently been proposed to tolerate a maximum number f of Byzantine workers. These GARs differ according to (1) the complexity of their computation time, (2) the maximal number of Byzantine workers despite which convergence can still be ensured (breakdown point), and (3) their accuracy, which can be captured by (3.1) their angular error, namely the angle with the true gradient, as well as (3.2) their ability to aggregate full gradients. In particular, many are not full gradients for they operate on each dimension separately, which results in a coordinate-wise blended gradient, leading to low accuracy in practical situations where the number (s) of workers that are actually Byzantine in an execution is small (s < < f). We propose Aksel, a new scalable median-based GAR with optimal time complexity (𝒪(nd)), optimal breakdown point (n > 2f) and the lowest upper bound on the expected angular error (𝒪(√d)) among full gradient approaches. We also study the actual angular error of Aksel when the gradient distribution is normal and show that it only grows in 𝒪(√dlog{n}), which is the first logarithmic upper bound ever proven on the number of workers n assuming an optimal breakdown point. We also report on an empirical evaluation of Aksel on various classification tasks, which we compare to alternative GARs against state-of-the-art attacks. Aksel is the only GAR reaching top accuracy when there is actually none or few Byzantine workers while maintaining a good defense even under the extreme case (s = f). For simplicity of presentation, we consider a scheme with a single server. However, as we explain in the paper, Aksel can also easily be adapted to multi-server architectures that tolerate the Byzantine behavior of a fraction of the servers.

Machine Learning (ML) solutions are nowadays distributed, according to the so-called server/worker architecture. One server holds the model parameters while several workers train the model. Clearly, such architecture is prone to various types of component failures, which can be all encompassed within the spectrum of a Byzantine behavior. Several approaches have been proposed recently to tolerate Byzantine workers. Yet all require trusting a central parameter server. We initiate in this paper the study of the "general" Byzantine-resilient distributed machine learning problem where no individual component is trusted. In particular, we distribute the parameter server computation on several nodes.

Could a gradient aggregation rule (GAR) for distributed machine learning be both robust and fast? This paper answers by the affirmative through Multi-Bulyan. Given n workers, f of which are arbitrary malicious (Byzantine) and m = n - f are not, we prove that Multi-Bulyan can ensure a strong form of Byzantine resilience, as well as an m/n slowdown, compared to averaging, the fastest (but non Byzantine resilient) rule for distributed machine learning. When m ≈ n (almost all workers are correct), Multi-Bulyan reaches the speed of averaging. We also prove that Multi-Bulyan's cost in local computation is O(d) (like averaging), an important feature for ML where d commonly reaches 109, while robust alternatives have at least quadratic cost in d. Our theoretical findings are complemented with an experimental evaluation which, in addition to supporting the linear O(d) complexity argument, conveys the fact that Multi-Bulyan's parallelisability further adds to its efficiency.

While machine learning is going through an era of celebrated success, concerns have been raised about the vulnerability of its backbone: stochastic gradient descent (SGD). Recent approaches have been proposed to ensure the robustness of distributed SGD against adversarial (Byzantine) workers sending poisoned gradients during the training phase. Some of these approaches have been proven Byzantine–resilient: they ensure the convergence of SGD despite the presence of a minority of adversarial workers. We show in this paper that convergence is not enough. In high dimension $d \gg 1$, an adver\-sary can build on the loss function’s non–convexity to make SGD converge to ineffective models. More precisely, we bring to light that existing Byzantine–resilient schemes leave a margin of poisoning of $\bigOmega\left(f(d)\right)$, where $f(d)$ increases at least like $\sqrt[p]{d }$. Based on this leeway, we build a simple attack, and experimentally show its strong to utmost effectivity on CIFAR–10 and MNIST. We introduce Bulyan, and prove it significantly reduces the attackers leeway to a narrow $\bigO\,( \sfrac{1}{\sqrt{d }})$ bound. We empirically show that Bulyan does not suffer the fragility of existing aggregation rules and, at a reasonable cost in terms of required batch size, achieves convergence as if only non–Byzantine gradients had been used to update the model.

With the development of neural networks based machine learning and their usage in mission critical applications, voices are rising against the black box aspect of neural networks as it becomes crucial to understand their limits and capabilities. With the rise of neuromorphic hardware, it is even more critical to understand how a neural network, as a distributed system, tolerates the failures of its computing nodes, neurons, and its communication channels, synapses. Experimentally assessing the robustness of neural networks involves the quixotic venture of testing all the possible failures, on all the possible inputs, which ultimately hits a combinatorial explosion for the first, and the impossibility to gather all the possible inputs for the second.In this paper, we prove an upper bound on the expected error of the output when a subset of neurons crashes. This bound involves dependencies on the network parameters that can be seen as being too pessimistic in the average case. It involves a polynomial dependency on the Lipschitz coefficient of the neurons' activation function, and an exponential dependency on the depth of the layer where a failure occurs. We back up our theoretical results with experiments illustrating the extent to which our prediction matches the dependencies between the network parameters and robustness. Our results show that the robustness of neural networks to the average crash can be estimated without the need to neither test the network on all failure configurations, nor access the training set used to train the network, both of which are practically impossible requirements.