Error Feedback under $(L_0,L_1)$-Smoothness: Normalization and Momentum

We provide the first proof of convergence for normalized error feedback algorithms across a wide range of machine learning problems. Despite their popularity and efficiency in training deep neural networks, traditional analyses of error feedback algorithms rely on the smoothness assumption that does not capture the properties of objective functions in these problems. Rather, these problems have recently been shown to satisfy generalized smoothness assumptions, and the theoretical understanding of error feedback algorithms under these assumptions remains largely unexplored. Moreover, to the best of our knowledge, all existing analyses under generalized smoothness either i) focus on single-node settings or ii) make unrealistically strong assumptions for distributed settings, such as requiring data heterogeneity, and almost surely bounded stochastic gradient noise variance. In this paper, we propose distributed error feedback algorithms that utilize normalization to achieve the $O(1/\sqrt{K})$ convergence rate for nonconvex problems under generalized smoothness. Our analyses apply for distributed settings without data heterogeneity conditions, and enable stepsize tuning that is independent of problem parameters. Additionally, we provide strong convergence guarantees of normalized error feedback algorithms for stochastic settings. Finally, we show that due to their larger allowable stepsizes, our new normalized error feedback algorithms outperform their non-normalized counterparts on various tasks, including the minimization of polynomial functions, logistic regression, and ResNet-20 training.

Low-Resource Machine Translation through the Lens of Personalized Federated Learning

We present a new approach based on the Personalized Federated Learning algorithm MeritFed that can be applied to Natural Language Tasks with heterogeneous data. We evaluate it on the Low-Resource Machine Translation task, using the dataset from the Large-Scale Multilingual Machine Translation Shared Task (Small Track #2) and the subset of Sami languages from the multilingual benchmark for Finno-Ugric languages. In addition to its effectiveness, MeritFed is also highly interpretable, as it can be applied to track the impact of each language used for training. Our analysis reveals that target dataset size affects weight distribution across auxiliary languages, that unrelated languages do not interfere with the training, and auxiliary optimizer parameters have minimal impact. Our approach is easy to apply with a few lines of code, and we provide scripts for reproducing the experiments at https://github.com/VityaVitalich/MeritFed

Gradient Clipping Improves AdaGrad when the Noise Is Heavy-Tailed

Methods with adaptive stepsizes, such as AdaGrad and Adam, are essential for training modern Deep Learning models, especially Large Language Models. Typically, the noise in the stochastic gradients is heavy-tailed for the later ones. Gradient clipping provably helps to achieve good high-probability convergence for such noises. However, despite the similarity between AdaGrad/Adam and Clip-SGD, the high-probability convergence of AdaGrad/Adam has not been studied in this case. In this work, we prove that AdaGrad (and its delayed version) can have provably bad high-probability convergence if the noise is heavy-tailed. To fix this issue, we propose a new version of AdaGrad called Clip-RAdaGradD (Clipped Reweighted AdaGrad with Delay) and prove its high-probability convergence bounds with polylogarithmic dependence on the confidence level for smooth convex/non-convex stochastic optimization with heavy-tailed noise. Our empirical evaluations, including NLP model fine-tuning, highlight the superiority of clipped versions of AdaGrad/Adam in handling the heavy-tailed noise.

Remove that Square Root: A New Efficient Scale-Invariant Version of AdaGrad

Adaptive methods are extremely popular in machine learning as they make learning rate tuning less expensive. This paper introduces a novel optimization algorithm named KATE, which presents a scale-invariant adaptation of the well-known AdaGrad algorithm. We prove the scale-invariance of KATE for the case of Generalized Linear Models. Moreover, for general smooth non-convex problems, we establish a convergence rate of $O \left(\frac{\log T}{\sqrt{T}} \right)$ for KATE, matching the best-known ones for AdaGrad and Adam. We also compare KATE to other state-of-the-art adaptive algorithms Adam and AdaGrad in numerical experiments with different problems, including complex machine learning tasks like image classification and text classification on real data. The results indicate that KATE consistently outperforms AdaGrad and matches/surpasses the performance of Adam in all considered scenarios.

Federated Learning Can Find Friends That Are Beneficial

In Federated Learning (FL), the distributed nature and heterogeneity of client data present both opportunities and challenges. While collaboration among clients can significantly enhance the learning process, not all collaborations are beneficial; some may even be detrimental. In this study, we introduce a novel algorithm that assigns adaptive aggregation weights to clients participating in FL training, identifying those with data distributions most conducive to a specific learning objective. We demonstrate that our aggregation method converges no worse than the method that aggregates only the updates received from clients with the same data distribution. Furthermore, empirical evaluations consistently reveal that collaborations guided by our algorithm outperform traditional FL approaches. This underscores the critical role of judicious client selection and lays the foundation for more streamlined and effective FL implementations in the coming years.

Byzantine Robustness and Partial Participation Can Be Achieved Simultaneously: Just Clip Gradient Differences

Distributed learning has emerged as a leading paradigm for training large machine learning models. However, in real-world scenarios, participants may be unreliable or malicious, posing a significant challenge to the integrity and accuracy of the trained models. Byzantine fault tolerance mechanisms have been proposed to address these issues, but they often assume full participation from all clients, which is not always practical due to the unavailability of some clients or communication constraints. In our work, we propose the first distributed method with client sampling and provable tolerance to Byzantine workers. The key idea behind the developed method is the use of gradient clipping to control stochastic gradient differences in recursive variance reduction. This allows us to bound the potential harm caused by Byzantine workers, even during iterations when all sampled clients are Byzantine. Furthermore, we incorporate communication compression into the method to enhance communication efficiency. Under quite general assumptions, we prove convergence rates for the proposed method that match the existing state-of-the-art (SOTA) theoretical results.

Byzantine-Tolerant Methods for Distributed Variational Inequalities

Robustness to Byzantine attacks is a necessity for various distributed training scenarios. When the training reduces to the process of solving a minimization problem, Byzantine robustness is relatively well-understood. However, other problem formulations, such as min-max problems or, more generally, variational inequalities, arise in many modern machine learning and, in particular, distributed learning tasks. These problems significantly differ from the standard minimization ones and, therefore, require separate consideration. Nevertheless, only one work (Adibi et al., 2022) addresses this important question in the context of Byzantine robustness. Our work makes a further step in this direction by providing several (provably) Byzantine-robust methods for distributed variational inequality, thoroughly studying their theoretical convergence, removing the limitations of the previous work, and providing numerical comparisons supporting the theoretical findings.

Breaking the Heavy-Tailed Noise Barrier in Stochastic Optimization Problems

We consider stochastic optimization problems with heavy-tailed noise with structured density. For such problems, we show that it is possible to get faster rates of convergence than $\mathcal{O}(K^{-2(\alpha - 1)/\alpha})$, when the stochastic gradients have finite moments of order $\alpha \in (1, 2]$. In particular, our analysis allows the noise norm to have an unbounded expectation. To achieve these results, we stabilize stochastic gradients, using smoothed medians of means. We prove that the resulting estimates have negligible bias and controllable variance. This allows us to carefully incorporate them into clipped-SGD and clipped-SSTM and derive new high-probability complexity bounds in the considered setup.

Communication Compression for Byzantine Robust Learning: New Efficient Algorithms and Improved Rates

Byzantine robustness is an essential feature of algorithms for certain distributed optimization problems, typically encountered in collaborative/federated learning. These problems are usually huge-scale, implying that communication compression is also imperative for their resolution. These factors have spurred recent algorithmic and theoretical developments in the literature of Byzantine-robust learning with compression. In this paper, we contribute to this research area in two main directions. First, we propose a new Byzantine-robust method with compression -- Byz-DASHA-PAGE -- and prove that the new method has better convergence rate (for non-convex and Polyak-Lojasiewicz smooth optimization problems), smaller neighborhood size in the heterogeneous case, and tolerates more Byzantine workers under over-parametrization than the previous method with SOTA theoretical convergence guarantees (Byz-VR-MARINA). Secondly, we develop the first Byzantine-robust method with communication compression and error feedback -- Byz-EF21 -- along with its bidirectional compression version -- Byz-EF21-BC -- and derive the convergence rates for these methods for non-convex and Polyak-Lojasiewicz smooth case. We test the proposed methods and illustrate our theoretical findings in the numerical experiments.

High-Probability Convergence for Composite and Distributed Stochastic Minimization and Variational Inequalities with Heavy-Tailed Noise

High-probability analysis of stochastic first-order optimization methods under mild assumptions on the noise has been gaining a lot of attention in recent years. Typically, gradient clipping is one of the key algorithmic ingredients to derive good high-probability guarantees when the noise is heavy-tailed. However, if implemented na\"ively, clipping can spoil the convergence of the popular methods for composite and distributed optimization (Prox-SGD/Parallel SGD) even in the absence of any noise. Due to this reason, many works on high-probability analysis consider only unconstrained non-distributed problems, and the existing results for composite/distributed problems do not include some important special cases (like strongly convex problems) and are not optimal. To address this issue, we propose new stochastic methods for composite and distributed optimization based on the clipping of stochastic gradient differences and prove tight high-probability convergence results (including nearly optimal ones) for the new methods. Using similar ideas, we also develop new methods for composite and distributed variational inequalities and analyze the high-probability convergence of these methods.