Many applications, e.g. in content recommendation, sports, or recruitment, leverage the comparisons of alternatives to score those alternatives. The classical Bradley-Terry model and its variants have been widely used to do so. The historical model considers binary comparisons (victory/defeat) between alternatives, while more recent developments allow finer comparisons to be taken into account. In this article, we introduce a probabilistic model encompassing a broad variety of paired comparisons that can take discrete or continuous values. We do so by considering a well-behaved subset of the exponential family, which we call the family of generalized Bradley-Terry (GBT) models, as it includes the classical Bradley-Terry model and many of its variants. Remarkably, we prove that all GBT models are guaranteed to yield a strictly convex negative log-likelihood. Moreover, assuming a Gaussian prior on alternatives' scores, we prove that the maximum a posteriori (MAP) of GBT models, whose existence, uniqueness and fast computation are thus guaranteed, varies monotonically with respect to comparisons (the more A beats B, the better the score of A) and is Lipschitz-resilient with respect to each new comparison (a single new comparison can only have a bounded effect on all the estimated scores). These desirable properties make GBT models appealing for practical use. We illustrate some features of GBT models on simulations.

The possibility of adversarial (a.k.a., Byzantine) clients makes federated learning (FL) prone to arbitrary manipulation. The natural approach to robustify FL against adversarial clients is to replace the simple averaging operation at the server in the standard $\mathsf{FedAvg}$ algorithm by a robust averaging rule. While a significant amount of work has been devoted to studying the convergence of federated robust averaging (which we denote by $\mathsf{FedRo}$), prior work has largely ignored the impact of client subsampling and local steps, two fundamental FL characteristics. While client subsampling increases the effective fraction of Byzantine clients, local steps increase the drift between the local updates computed by honest (i.e., non-Byzantine) clients. Consequently, a careless deployment of $\mathsf{FedRo}$ could yield poor performance. We validate this observation by presenting an in-depth analysis of $\mathsf{FedRo}$ tightly analyzing the impact of client subsampling and local steps. Specifically, we present a sufficient condition on client subsampling for nearly-optimal convergence of $\mathsf{FedRo}$ (for smooth non-convex loss). Also, we show that the rate of improvement in learning accuracy diminishes with respect to the number of clients subsampled, as soon as the sample size exceeds a threshold value. Interestingly, we also observe that under a careful choice of step-sizes, the learning error due to Byzantine clients decreases with the number of local steps. We validate our theory by experiments on the FEMNIST and CIFAR-$10$ image classification tasks.

The success of machine learning (ML) has been intimately linked with the availability of large amounts of data, typically collected from heterogeneous sources and processed on vast networks of computing devices (also called workers). Beyond accuracy, the use of ML in critical domains such as healthcare and autonomous driving calls for robustness against data poisoning and faulty workers. The problem of Byzantine ML formalizes these robustness issues by considering a distributed ML environment in which workers (storing a portion of the global dataset) can deviate arbitrarily from the prescribed algorithm. Although the problem has attracted a lot of attention from a theoretical point of view, its practical importance for addressing realistic faults (where the behavior of any worker is locally constrained) remains unclear. It has been argued that the seemingly weaker threat model where only workers' local datasets get poisoned is more reasonable. We highlight here some important results on the efficacy of Byzantine robustness for tackling data poisoning. In particular, we discuss cases where, while tolerating a wider range of faulty behaviors, Byzantine ML yields solutions that are optimal even under the weaker threat model of data poisoning.

Byzantine machine learning (ML) aims to ensure the resilience of distributed learning algorithms to misbehaving (or Byzantine) machines. Although this problem received significant attention, prior works often assume the data held by the machines to be homogeneous, which is seldom true in practical settings. Data heterogeneity makes Byzantine ML considerably more challenging, since a Byzantine machine can hardly be distinguished from a non-Byzantine outlier. A few solutions have been proposed to tackle this issue, but these provide suboptimal probabilistic guarantees and fare poorly in practice. This paper closes the theoretical gap, achieving optimality and inducing good empirical results. In fact, we show how to automatically adapt existing solutions for (homogeneous) Byzantine ML to the heterogeneous setting through a powerful mechanism, we call nearest neighbor mixing (NNM), which boosts any standard robust distributed gradient descent variant to yield optimal Byzantine resilience under heterogeneity. We obtain similar guarantees (in expectation) by plugging NNM in the distributed stochastic heavy ball method, a practical substitute to distributed gradient descent. We obtain empirical results that significantly outperform state-of-the-art Byzantine ML solutions.

The geometric median, an instrumental component of the secure machine learning toolbox, is known to be effective when robustly aggregating models (or gradients), gathered from potentially malicious (or strategic) users. What is less known is the extent to which the geometric median incentivizes dishonest behaviors. This paper addresses this fundamental question by quantifying its strategyproofness. While we observe that the geometric median is not even approximately strategyproof, we prove that it is asymptotically $\alpha$-strategyproof: when the number of users is large enough, a user that misbehaves can gain at most a multiplicative factor $\alpha$, which we compute as a function of the distribution followed by the users. We then generalize our results to the case where users actually care more about specific dimensions, determining how this impacts $\alpha$. We also show how the skewed geometric medians can be used to improve strategyproofness.

Collaborative learning algorithms, such as distributed SGD (or D-SGD), are prone to faulty machines that may deviate from their prescribed algorithm because of software or hardware bugs, poisoned data or malicious behaviors. While many solutions have been proposed to enhance the robustness of D-SGD to such machines, previous works either resort to strong assumptions (trusted server, homogeneous data, specific noise model) or impose a gradient computational cost that is several orders of magnitude higher than that of D-SGD. We present MoNNA, a new algorithm that (a) is provably robust under standard assumptions and (b) has a gradient computation overhead that is linear in the fraction of faulty machines, which is conjectured to be tight. Essentially, MoNNA uses Polyak’s momentum of local gradients for local updates and nearest-neighbor averaging (NNA) for global mixing, respectively. While MoNNA is rather simple to implement, its analysis has been more challenging and relies on two key elements that may be of independent interest. Specifically, we introduce the mixing criterion of $(\alpha, \lambda)$-reduction to analyze the non-linear mixing of non-faulty machines, and present a way to control the tension between the momentum and the model drifts. We validate our theory by experiments on image classification and make our code available at https://github.com/LPD-EPFL/robust-collaborative-learning.

We present Epidemic Learning (EL), a simple yet powerful decentralized learning (DL) algorithm that leverages changing communication topologies to achieve faster model convergence compared to conventional DL approaches. At each round of EL, each node sends its model updates to a random sample of $s$ other nodes (in a system of $n$ nodes). We provide an extensive theoretical analysis of EL, demonstrating that its changing topology culminates in superior convergence properties compared to the state-of-the-art (static and dynamic) topologies. Considering smooth non-convex loss functions, the number of transient iterations for EL, i.e., the rounds required to achieve asymptotic linear speedup, is in $O(n^3/s^2)$ which outperforms the best-known bound $O(n^3)$ by a factor of $s^2$, indicating the benefit of randomized communication for DL. We empirically evaluate EL in a 96-node network and compare its performance with state-of-the-art DL approaches. Our results illustrate that EL converges up to $ 1.7\times$ quicker than baseline DL algorithms and attains $2.2 $\% higher accuracy for the same communication volume.

Byzantine resilience emerged as a prominent topic within the distributed machine learning community. Essentially, the goal is to enhance distributed optimization algorithms, such as distributed SGD, in a way that guarantees convergence despite the presence of some misbehaving (a.k.a., Byzantine) workers. Although a myriad of techniques addressing the problem have been proposed, the field arguably rests on fragile foundations. These techniques are hard to prove correct and rely on assumptions that are (a) quite unrealistic, i.e., often violated in practice, and (b) heterogeneous, i.e., making it difficult to compare approaches. We present RESAM (RESilient Averaging of Momentums), a unified framework that makes it simple to establish optimal Byzantine resilience, relying only on standard machine learning assumptions. Our framework is mainly composed of two operators: resilient averaging at the server and distributed momentum at the workers. We prove a general theorem stating the convergence of distributed SGD under RESAM. Interestingly, demonstrating and comparing the convergence of many existing techniques become direct corollaries of our theorem, without resorting to stringent assumptions. We also present an empirical evaluation of the practical relevance of RESAM.

To study the resilience of distributed learning, the “Byzantine" literature considers a strong threat model where workers can report arbitrary gradients to the parameter server. Whereas this model helped obtain several fundamental results, it has sometimes been considered unrealistic, when the workers are mostly trustworthy machines. In this paper, we show a surprising equivalence between this model and data poisoning, a threat considered much more realistic. More specifically, we prove that every gradient attack can be reduced to data poisoning, in any personalized federated learning system with PAC guarantees (which we show are both desirable and realistic). This equivalence makes it possible to obtain new impossibility results on the resilience of any “robust” learning algorithm to data poisoning in highly heterogeneous applications, as corollaries of existing impossibility theorems on Byzantine machine learning. Moreover, using our equivalence, we derive a practical attack that we show (theoretically and empirically) can be very effective against classical personalized federated learning models.

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.