Samuel Horváth

Sara Pieri, Jose Renato Restom, Samuel Horvath et al.

Federated Learning (FL) is a promising research paradigm that enables the collaborative training of machine learning models among various parties without the need for sensitive information exchange. Nonetheless, retaining data in individual clients introduces fundamental challenges to achieving performance on par with centrally trained models. Our study provides an extensive review of federated learning applied to visual recognition. It underscores the critical role of thoughtful architectural design choices in achieving optimal performance, a factor often neglected in the FL literature. Many existing FL solutions are tested on shallow or simple networks, which may not accurately reflect real-world applications. This practice restricts the transferability of research findings to large-scale visual recognition models. Through an in-depth analysis of diverse cutting-edge architectures such as convolutional neural networks, transformers, and MLP-mixers, we experimentally demonstrate that architectural choices can substantially enhance FL systems' performance, particularly when handling heterogeneous data. We study 19 visual recognition models from five different architectural families on four challenging FL datasets. We also re-investigate the inferior performance of convolution-based architectures in the FL setting and analyze the influence of normalization layers on the FL performance. Our findings emphasize the importance of architectural design for computer vision tasks in practical scenarios, effectively narrowing the performance gap between federated and centralized learning. Our source code is available at https://github.com/sarapieri/fed_het.git.

Abdurakhmon Sadiev, Marina Danilova, Eduard Gorbunov et al.

During recent years the interest of optimization and machine learning communities in high-probability convergence of stochastic optimization methods has been growing. One of the main reasons for this is that high-probability complexity bounds are more accurate and less studied than in-expectation ones. However, SOTA high-probability non-asymptotic convergence results are derived under strong assumptions such as the boundedness of the gradient noise variance or of the objective's gradient itself. In this paper, we propose several algorithms with high-probability convergence results under less restrictive assumptions. In particular, we derive new high-probability convergence results under the assumption that the gradient/operator noise has bounded central $α$-th moment for $α\in (1,2]$ in the following setups: (i) smooth non-convex / Polyak-Lojasiewicz / convex / strongly convex / quasi-strongly convex minimization problems, (ii) Lipschitz / star-cocoercive and monotone / quasi-strongly monotone variational inequalities. These results justify the usage of the considered methods for solving problems that do not fit standard functional classes studied in stochastic optimization.

Samuel Horváth, Konstantin Mishchenko, Peter Richtárik

In this work, we propose new adaptive step size strategies that improve several stochastic gradient methods. Our first method (StoPS) is based on the classical Polyak step size (Polyak, 1987) and is an extension of the recent development of this method for the stochastic optimization-SPS (Loizou et al., 2021), and our second method, denoted GraDS, rescales step size by "diversity of stochastic gradients". We provide a theoretical analysis of these methods for strongly convex smooth functions and show they enjoy deterministic-like rates despite stochastic gradients. Furthermore, we demonstrate the theoretical superiority of our adaptive methods on quadratic objectives. Unfortunately, both StoPS and GraDS depend on unknown quantities, which are only practical for the overparametrized models. To remedy this, we drop this undesired dependence and redefine StoPS and GraDS to StoP and GraD, respectively. We show that these new methods converge linearly to the neighbourhood of the optimal solution under the same assumptions. Finally, we corroborate our theoretical claims by experimental validation, which reveals that GraD is particularly useful for deep learning optimization.

Eduard Gorbunov, Abdurakhmon Sadiev, Marina Danilova et al.

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 naïvely, 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.

Samuel Horváth, Maziar Sanjabi, Lin Xiao et al.

The practice of applying several local updates before aggregation across clients has been empirically shown to be a successful approach to overcoming the communication bottleneck in Federated Learning (FL). Such methods are usually implemented by having clients perform one or more epochs of local training per round while randomly reshuffling their finite dataset in each epoch. Data imbalance, where clients have different numbers of local training samples, is ubiquitous in FL applications, resulting in different clients performing different numbers of local updates in each round. In this work, we propose a general recipe, FedShuffle, that better utilizes the local updates in FL, especially in this regime encompassing random reshuffling and heterogeneity. FedShuffle is the first local update method with theoretical convergence guarantees that incorporates random reshuffling, data imbalance, and client sampling - features that are essential in large-scale cross-device FL. We present a comprehensive theoretical analysis of FedShuffle and show, both theoretically and empirically, that it does not suffer from the objective function mismatch that is present in FL methods that assume homogeneous updates in heterogeneous FL setups, such as FedAvg (McMahan et al., 2017). In addition, by combining the ingredients above, FedShuffle improves upon FedNova (Wang et al., 2020), which was previously proposed to solve this mismatch. Similar to Mime (Karimireddy et al., 2020), we show that FedShuffle with momentum variance reduction (Cutkosky & Orabona, 2019) improves upon non-local methods under a Hessian similarity assumption.

Malik Shahid Sultan, Samuel Horvath, Hernando Ombao

Dependence between nodes in a network is an important concept that pervades many areas including finance, politics, sociology, genomics and the brain sciences. One way to characterize dependence between components of a multivariate time series data is via Granger Causality (GC). Standard traditional approaches to GC estimation / inference commonly assume linear dynamics, however such simplification does not hold in many real-world applications where signals are inherently non-linear. In such cases, imposing linear models such as vector autoregressive (VAR) models can lead to mis-characterization of true Granger Causal interactions. To overcome this limitation, Tank et al (IEEE Transactions on Pattern Analysis and Machine Learning, 2022) proposed a solution that uses neural networks with sparse regularization penalties. The regularization encourages learnable weights to be sparse, which enables inference on GC. This paper overcomes the limitations of current methods by leveraging advances in machine learning and deep learning which have been demonstrated to learn hidden patterns in the data. We propose novel classes of models that can handle underlying non-linearity in a computationally efficient manner, simultaneously providing GC and lag order selection. Firstly, we present the Learned Kernel VAR (LeKVAR) model that learns kernel parameterized by a shared neural net followed by penalization on learnable weights to discover GC structure. Secondly, we show one can directly decouple lags and individual time series importance via decoupled penalties. This is important as we want to select the lag order during the process of GC estimation. This decoupling acts as a filtering and can be extended to any DL model including Multi-Layer Perceptrons (MLP), Recurrent Neural Networks (RNN), Long Short Term Memory Networks (LSTM), Transformers etc, for simultaneous GC estimation and lag selection.

Eduard Gorbunov, Nazarii Tupitsa, Sayantan Choudhury et al.

Due to the non-smoothness of optimization problems in Machine Learning, generalized smoothness assumptions have been gaining a lot of attention in recent years. One of the most popular assumptions of this type is $(L_0,L_1)$-smoothness (Zhang et al., 2020). In this paper, we focus on the class of (strongly) convex $(L_0,L_1)$-smooth functions and derive new convergence guarantees for several existing methods. In particular, we derive improved convergence rates for Gradient Descent with (Smoothed) Gradient Clipping and for Gradient Descent with Polyak Stepsizes. In contrast to the existing results, our rates do not rely on the standard smoothness assumption and do not suffer from the exponential dependency from the initial distance to the solution. We also extend these results to the stochastic case under the over-parameterization assumption, propose a new accelerated method for convex $(L_0,L_1)$-smooth optimization, and derive new convergence rates for Adaptive Gradient Descent (Malitsky and Mishchenko, 2020).

Grigory Malinovsky, Samuel Horváth, Konstantin Burlachenko et al.

Federated Learning (FL) is a distributed machine learning approach where multiple clients work together to solve a machine learning task. One of the key challenges in FL is the issue of partial participation, which occurs when a large number of clients are involved in the training process. The traditional method to address this problem is randomly selecting a subset of clients at each communication round. In our research, we propose a new technique and design a novel regularized client participation scheme. Under this scheme, each client joins the learning process every $R$ communication rounds, which we refer to as a meta epoch. We have found that this participation scheme leads to a reduction in the variance caused by client sampling. Combined with the popular FedAvg algorithm (McMahan et al., 2017), it results in superior rates under standard assumptions. For instance, the optimization term in our main convergence bound decreases linearly with the product of the number of communication rounds and the size of the local dataset of each client, and the statistical term scales with step size quadratically instead of linearly (the case for client sampling with replacement), leading to better convergence rate $\mathcal{O}\left(\frac{1}{T^2}\right)$ compared to $\mathcal{O}\left(\frac{1}{T}\right)$, where $T$ is the total number of communication rounds. Furthermore, our results permit arbitrary client availability as long as each client is available for training once per each meta epoch.

Nazarii Tupitsa, Abdulla Jasem Almansoori, Yanlin Wu et al.

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.

Samuel Horvath, Stefanos Laskaridis, Shashank Rajput et al.

Deep Neural Networks (DNNs) have been a large driver for AI breakthroughs in recent years. However, these models have been getting increasingly large as they become more accurate and safe. This means that their training becomes increasingly costly and time-consuming and typically yields a single model to fit all targets. Various techniques have been proposed in the literature to mitigate this, including pruning, sparsification, or quantization of model weights and updates. While achieving high compression rates, they often incur significant computational overheads at training or lead to non-negligible accuracy penalty. Alternatively, factorization methods have been leveraged for low-rank compression of DNNs. Similarly, such techniques (e.g., SVD) frequently rely on heavy iterative decompositions of layers and are potentially sub-optimal for non-linear models, such as DNNs. We take a further step in designing efficient low-rank models and propose Maestro, a framework for trainable low-rank layers. Instead of iteratively applying a priori decompositions, the low-rank structure is baked into the training process through LoD, a low-rank ordered decomposition. Not only is this the first time importance ordering via sampling is applied on the decomposed DNN structure, but it also allows selecting ranks at a layer granularity. Our theoretical analysis demonstrates that in special cases LoD recovers the SVD decomposition and PCA. Applied to DNNs, Maestro enables the extraction of lower footprint models that preserve performance. Simultaneously, it enables the graceful trade-off between accuracy-latency for deployment to even more constrained devices without retraining.

Xiangjian Hou, Sarit Khirirat, Mohammad Yaqub et al.

Federated learning (FL) is a distributed machine learning (ML) framework where multiple clients collaborate to train a model without exposing their private data. FL involves cycles of local computations and bi-directional communications between the clients and server. To bolster data security during this process, FL algorithms frequently employ a differential privacy (DP) mechanism that introduces noise into each client's model updates before sharing. However, while enhancing privacy, the DP mechanism often hampers convergence performance. In this paper, we posit that an optimal balance exists between the number of local steps and communication rounds, one that maximizes the convergence performance within a given privacy budget. Specifically, we present a proof for the optimal number of local steps and communication rounds that enhance the convergence bounds of the DP version of the ScaffNew algorithm. Our findings reveal a direct correlation between the optimal number of local steps, communication rounds, and a set of variables, e.g the DP privacy budget and other problem parameters, specifically in the context of strongly convex optimization. We furthermore provide empirical evidence to validate our theoretical findings.

Grigory Malinovsky, Peter Richtárik, Samuel Horváth et al.

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 general assumptions, we prove convergence rates for the proposed method that match the existing state-of-the-art (SOTA) theoretical results. We also propose a heuristic on adjusting any Byzantine-robust method to a partial participation scenario via clipping.

Eduard Gorbunov, Samuel Horváth, Peter Richtárik et al.

Byzantine-robustness has been gaining a lot of attention due to the growth of the interest in collaborative and federated learning. However, many fruitful directions, such as the usage of variance reduction for achieving robustness and communication compression for reducing communication costs, remain weakly explored in the field. This work addresses this gap and proposes Byz-VR-MARINA - a new Byzantine-tolerant method with variance reduction and compression. A key message of our paper is that variance reduction is key to fighting Byzantine workers more effectively. At the same time, communication compression is a bonus that makes the process more communication efficient. We derive theoretical convergence guarantees for Byz-VR-MARINA outperforming previous state-of-the-art for general non-convex and Polyak-Lojasiewicz loss functions. Unlike the concurrent Byzantine-robust methods with variance reduction and/or compression, our complexity results are tight and do not rely on restrictive assumptions such as boundedness of the gradients or limited compression. Moreover, we provide the first analysis of a Byzantine-tolerant method supporting non-uniform sampling of stochastic gradients. Numerical experiments corroborate our theoretical findings.

Samuel Horváth

Federated learning (FL) is an emerging machine learning paradigm involving multiple clients, e.g., mobile phone devices, with an incentive to collaborate in solving a machine learning problem coordinated by a central server. FL was proposed in 2016 by Konečný et al. and McMahan et al. as a viable privacy-preserving alternative to traditional centralized machine learning since, by construction, the training data points are decentralized and never transferred by the clients to a central server. Therefore, to a certain degree, FL mitigates the privacy risks associated with centralized data collection. Unfortunately, optimization for FL faces several specific issues that centralized optimization usually does not need to handle. In this thesis, we identify several of these challenges and propose new methods and algorithms to address them, with the ultimate goal of enabling practical FL solutions supported with mathematically rigorous guarantees.

Kaustubh Ponkshe, Raghav Singhal, Eduard Gorbunov et al.

Low-rank adapters have become standard for efficiently fine-tuning large language models, but they often fall short of achieving the performance of full fine-tuning. We propose a method, LoRA Silver Bullet or LoRA-SB, that approximates full fine-tuning within low-rank subspaces using a carefully designed initialization strategy. We theoretically demonstrate that the architecture of LoRA-XS, which inserts a learnable r x r matrix between B and A while keeping other matrices fixed, provides the precise conditions needed for this approximation. We leverage its constrained update space to achieve optimal scaling for high-rank gradient updates while removing the need for scaling factor tuning. We prove that our initialization offers an optimal low-rank approximation of the initial gradient and preserves update directions throughout training. Extensive experiments across mathematical reasoning, commonsense reasoning, and language understanding tasks demonstrate that our approach exceeds the performance of LoRA (and baselines) while using 27-90 times fewer learnable parameters, and comprehensively outperforms LoRA-XS. Our findings establish that it is possible to simulate full fine-tuning in low-rank subspaces, and achieve significant parameter efficiency gains without sacrificing performance. Our code is publicly available at: https://github.com/CERT-Lab/lora-sb.

Abdulla Jasem Almansoori, Samuel Horváth, Martin Takáč

Federated learning has become a popular machine learning paradigm with many potential real-life applications, including recommendation systems, the Internet of Things (IoT), healthcare, and self-driving cars. Though most current applications focus on classification-based tasks, learning personalized generative models remains largely unexplored, and their benefits in the heterogeneous setting still need to be better understood. This work proposes a novel architecture combining global client-agnostic and local client-specific generative models. We show that using standard techniques for training federated models, our proposed model achieves privacy and personalization by implicitly disentangling the globally consistent representation (i.e. content) from the client-dependent variations (i.e. style). Using such decomposition, personalized models can generate locally unseen labels while preserving the given style of the client and can predict the labels for all clients with high accuracy by training a simple linear classifier on the global content features. Furthermore, disentanglement enables other essential applications, such as data anonymization, by sharing only the content. Extensive experimental evaluation corroborates our findings, and we also discuss a theoretical motivation for the proposed approach.

Xingyu Qu, Samuel Horvath

Model merging offers an efficient way to combine pre-trained neural networks but often suffers from inconsistent performance, especially when merging models with different initializations. We identify the ``vanishing feature'' phenomenon, where input-induced features diminish during propagation through the merged model, degrading performance. Through theoretical and empirical analysis, we reveal that this phenomenon underpins challenges like variance collapse and explains techniques like permutation-based merging, post-merging normalization, etc. We show that existing normalization strategies can be enhanced by precisely targeting the vanishing feature issue. Leveraging these insights, we propose the ``Preserve-First Merging'' (PFM) strategy, which focuses on preserving early-layer features, enabling the merged models, for the first time, to outperform the original models in advanced settings without post-training. Furthermore, we demonstrate that the vanishing feature phenomenon extends to other contexts, such as model pruning. Applying post-pruning normalization to mitigate the issue significantly improves one-shot pruning performance at high sparsity, offering a simple and effective post-pruning solution. The code is available at https://github.com/XingyuQu/VF.

Andrey Veprikov, Arman Bolatov, Samuel Horváth et al.

Optimization lies at the core of modern deep learning, yet existing methods often face a fundamental trade-off between adapting to problem geometry and leveraging curvature utilization. Steepest descent algorithms adapt to different geometries through norm choices but remain strictly first-order, whereas quasi-Newton and adaptive optimizers incorporate curvature information but are restricted to Frobenius geometry, limiting their applicability across diverse architectures. In this work, we propose a unified framework generalizing steepest descent, quasi-Newton methods, and adaptive methods through the novel notion of preconditioned matrix norms. This abstraction reveals that widely used optimizers such as SGD and Adam, as well as more advanced approaches like Muon and KL-Shampoo, and recent hybrids including SOAP and SPlus, all emerge as special cases of the same principle. Within this framework, we provide the first systematic treatment of affine and scale invariance in the matrix-parameterized setting, establishing necessary and sufficient conditions under generalized norms. Building on this foundation, we introduce two new methods, $\texttt{MuAdam}$ and $\texttt{MuAdam-SANIA}$, which combine the spectral geometry of Muon with Adam-style preconditioning. Our experiments demonstrate that these optimizers are competitive with, and in some cases outperform, existing state-of-the-art methods. Our code is available at https://github.com/brain-lab-research/LIB/tree/quasi_descent

Viktor Moskvoretskii, Nazarii Tupitsa, Chris Biemann et al.

We present a new approach called MeritOpt 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 datasets of South East Asian and Finno-Ugric languages. In addition to its effectiveness, MeritOpt 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/MeritOpt.

Nurbek Tastan, Samar Fares, Toluwani Aremu et al.

Federated learning (FL) has emerged as a pivotal approach in machine learning, enabling multiple participants to collaboratively train a global model without sharing raw data. While FL finds applications in various domains such as healthcare and finance, it is challenging to ensure global model convergence when participants do not contribute equally and/or honestly. To overcome this challenge, principled mechanisms are required to evaluate the contributions made by individual participants in the FL setting. Existing solutions for contribution assessment rely on general accuracy evaluation, often failing to capture nuanced dynamics and class-specific influences. This paper proposes a novel contribution assessment method called ShapFed for fine-grained evaluation of participant contributions in FL. Our approach uses Shapley values from cooperative game theory to provide a granular understanding of class-specific influences. Based on ShapFed, we introduce a weighted aggregation method called ShapFed-WA, which outperforms conventional federated averaging, especially in class-imbalanced scenarios. Personalizing participant updates based on their contributions further enhances collaborative fairness by delivering differentiated models commensurate with the participant contributions. Experiments on CIFAR-10, Chest X-Ray, and Fed-ISIC2019 datasets demonstrate the effectiveness of our approach in improving utility, efficiency, and fairness in FL systems. The code can be found at https://github.com/tnurbek/shapfed.

Ivo Gollini Navarrete, Nicolas Mauricio Cuadrado, Jose Renato Restom et al.

Pruning offers a promising solution to mitigate the associated costs and environmental impact of deploying large deep neural networks (DNNs). Traditional approaches rely on computationally expensive trained models or time-consuming iterative prune-retrain cycles, undermining their utility in resource-constrained settings. To address this issue, we build upon the established principles of saliency (LeCun et al., 1989) and connection sensitivity (Lee et al., 2018) to tackle the challenging problem of one-shot pruning neural networks (NNs) before training (PBT) at initialization. We introduce Fisher-Taylor Sensitivity (FTS), a computationally cheap and efficient pruning criterion based on the empirical Fisher Information Matrix (FIM) diagonal, offering a viable alternative for integrating first- and second-order information to identify a model's structurally important parameters. Although the FIM-Hessian equivalency only holds for convergent models that maximize the likelihood, recent studies (Karakida et al., 2019) suggest that, even at initialization, the FIM captures essential geometric information of parameters in overparameterized NNs, providing the basis for our method. Finally, we demonstrate empirically that layer collapse, a critical limitation of data-dependent pruning methodologies, is easily overcome by pruning within a single training epoch after initialization. We perform experiments on ResNet18 and VGG19 with CIFAR-10 and CIFAR-100, widely used benchmarks in pruning research. Our method achieves competitive performance against state-of-the-art techniques for one-shot PBT, even under extreme sparsity conditions. Our code is made available to the public.

Nicolas M Cuadrado Avila, Samuel Horváth, Martin Takáč

This work studies the challenge of optimal energy management in building-based microgrids through a collaborative and privacy-preserving framework. We evaluated two common RL algorithms (PPO and TRPO) in different collaborative setups to manage distributed energy resources (DERs) efficiently. Using a customized version of the CityLearn environment and synthetically generated data, we simulate and design net-zero energy scenarios for microgrids composed of multiple buildings. Our approach emphasizes reducing energy costs and carbon emissions while ensuring privacy. Experimental results demonstrate that Federated TRPO is comparable with state-of-the-art federated RL methodologies without hyperparameter tuning. The proposed framework highlights the feasibility of collaborative learning for achieving optimal control policies in energy systems, advancing the goals of sustainable and efficient smart grids. Our code is accessible \href{https://github.com/Optimization-and-Machine-Learning-Lab/energy_fed_trpo.git}{\textit{this repo}}.

Konstantin Burlachenko, Samuel Horváth, Peter Richtárik

Federated Learning (FL) has emerged as a promising technique for edge devices to collaboratively learn a shared machine learning model while keeping training data locally on the device, thereby removing the need to store and access the full data in the cloud. However, FL is difficult to implement, test and deploy in practice considering heterogeneity in common edge device settings, making it fundamentally hard for researchers to efficiently prototype and test their optimization algorithms. In this work, our aim is to alleviate this problem by introducing FL_PyTorch : a suite of open-source software written in python that builds on top of one the most popular research Deep Learning (DL) framework PyTorch. We built FL_PyTorch as a research simulator for FL to enable fast development, prototyping and experimenting with new and existing FL optimization algorithms. Our system supports abstractions that provide researchers with a sufficient level of flexibility to experiment with existing and novel approaches to advance the state-of-the-art. Furthermore, FL_PyTorch is a simple to use console system, allows to run several clients simultaneously using local CPUs or GPU(s), and even remote compute devices without the need for any distributed implementation provided by the user. FL_PyTorch also offers a Graphical User Interface. For new methods, researchers only provide the centralized implementation of their algorithm. To showcase the possibilities and usefulness of our system, we experiment with several well-known state-of-the-art FL algorithms and a few of the most common FL datasets.

Nikita Kotelevskii, Samuel Horváth, Karthik Nandakumar et al.

In modern federated learning, one of the main challenges is to account for inherent heterogeneity and the diverse nature of data distributions for different clients. This problem is often addressed by introducing personalization of the models towards the data distribution of the particular client. However, a personalized model might be unreliable when applied to the data that is not typical for this client. Eventually, it may perform worse for these data than the non-personalized global model trained in a federated way on the data from all the clients. This paper presents a new approach to federated learning that allows selecting a model from global and personalized ones that would perform better for a particular input point. It is achieved through a careful modeling of predictive uncertainties that helps to detect local and global in- and out-of-distribution data and use this information to select the model that is confident in a prediction. The comprehensive experimental evaluation on the popular real-world image datasets shows the superior performance of the model in the presence of out-of-distribution data while performing on par with state-of-the-art personalized federated learning algorithms in the standard scenarios.

Mohammed Aljahdali, Ahmed M. Abdelmoniem, Marco Canini et al.

In Federated Learning (FL), forgetting, or the loss of knowledge across rounds, hampers algorithm convergence, particularly in the presence of severe data heterogeneity among clients. This study explores the nuances of this issue, emphasizing the critical role of forgetting in FL's inefficient learning within heterogeneous data contexts. Knowledge loss occurs in both client-local updates and server-side aggregation steps; addressing one without the other fails to mitigate forgetting. We introduce a metric to measure forgetting granularly, ensuring distinct recognition amid new knowledge acquisition. Leveraging these insights, we propose Flashback, an FL algorithm with a dynamic distillation approach that is used to regularize the local models, and effectively aggregate their knowledge. Across different benchmarks, Flashback outperforms other methods, mitigates forgetting, and achieves faster round-to-target-accuracy, by converging in 6 to 16 rounds.

Yury Demidovich, Petr Ostroukhov, Grigory Malinovsky et al.

Non-convex Machine Learning problems typically do not adhere to the standard smoothness assumption. Based on empirical findings, Zhang et al. (2020b) proposed a more realistic generalized $(L_0, L_1)$-smoothness assumption, though it remains largely unexplored. Many existing algorithms designed for standard smooth problems need to be revised. However, in the context of Federated Learning, only a few works address this problem but rely on additional limiting assumptions. In this paper, we address this gap in the literature: we propose and analyze new methods with local steps, partial participation of clients, and Random Reshuffling without extra restrictive assumptions beyond generalized smoothness. The proposed methods are based on the proper interplay between clients' and server's stepsizes and gradient clipping. Furthermore, we perform the first analysis of these methods under the Polyak-Ł ojasiewicz condition. Our theory is consistent with the known results for standard smooth problems, and our experimental results support the theoretical insights.

Sayantan Choudhury, Nazarii Tupitsa, Nicolas Loizou et al.

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.

Vincent Plassier, Nikita Kotelevskii, Aleksandr Rubashevskii et al.

Conformal Prediction (CP) stands out as a robust framework for uncertainty quantification, which is crucial for ensuring the reliability of predictions. However, common CP methods heavily rely on data exchangeability, a condition often violated in practice. Existing approaches for tackling non-exchangeability lead to methods that are not computable beyond the simplest examples. This work introduces a new efficient approach to CP that produces provably valid confidence sets for fairly general non-exchangeable data distributions. We illustrate the general theory with applications to the challenging setting of federated learning under data heterogeneity between agents. Our method allows constructing provably valid personalized prediction sets for agents in a fully federated way. The effectiveness of the proposed method is demonstrated in a series of experiments on real-world datasets.

Nurbek Tastan, Stefanos Laskaridis, Martin Takac et al.

Large pre-trained models are commonly adapted to downstream tasks using parameter-efficient fine-tuning methods such as Low-Rank Adaptation (LoRA), which injects small trainable low-rank matrices instead of updating all weights. While LoRA dramatically reduces trainable parameters with little overhead, it can still underperform full fine-tuning in accuracy and often converges more slowly. We introduce LoFT, a novel low-rank adaptation method that behaves like full fine-tuning by aligning the optimizer's internal dynamics with those of updating all model weights. LoFT not only learns weight updates in a low-rank subspace (like LoRA) but also properly projects the optimizer's first and second moments (Adam's momentum and variance) into the same subspace, mirroring full-model updates. By aligning the low-rank update itself with the full update, LoFT eliminates the need for tuning extra hyperparameters, e.g., LoRA scaling factor $α$. Empirically, this approach substantially narrows the performance gap between adapter-based tuning and full fine-tuning and consistently outperforms standard LoRA-style methods, all without increasing inference cost.

Savelii Chezhegov, Aleksandr Beznosikov, Samuel Horváth et al.

Gradient clipping is a widely used technique in Machine Learning and Deep Learning (DL), known for its effectiveness in mitigating the impact of heavy-tailed noise, which frequently arises in the training of large language models. Additionally, first-order methods with clipping, such as Clip-SGD, exhibit stronger convergence guarantees than SGD under the $(L_0,L_1)$-smoothness assumption, a property observed in many DL tasks. However, the high-probability convergence of Clip-SGD under both assumptions -- heavy-tailed noise and $(L_0,L_1)$-smoothness -- has not been fully addressed in the literature. In this paper, we bridge this critical gap by establishing the first high-probability convergence bounds for Clip-SGD applied to convex $(L_0,L_1)$-smooth optimization with heavy-tailed noise. Our analysis extends prior results by recovering known bounds for the deterministic case and the stochastic setting with $L_1 = 0$ as special cases. Notably, our rates avoid exponentially large factors and do not rely on restrictive sub-Gaussian noise assumptions, significantly broadening the applicability of gradient clipping.

Philip Zmushko, Aleksandr Beznosikov, Martin Takáč et al.

With the increase in the number of parameters in large language models, the process of pre-training and fine-tuning increasingly demands larger volumes of GPU memory. A significant portion of this memory is typically consumed by the optimizer state. To overcome this challenge, recent approaches such as low-rank adaptation (LoRA (Hu et al., 2021)), low-rank gradient projection (GaLore (Zhao et al., 2024)), and blockwise optimization (BAdam (Luo et al., 2024)) have been proposed. However, in all these algorithms, the $\textit{effective rank of the weight updates remains low-rank}$, which can lead to a substantial loss of information from the gradient. This loss can be critically important, especially during the pre-training stage. In this paper, we introduce $\texttt{FRUGAL}$ ($\textbf{F}$ull-$\textbf{R}$ank $\textbf{U}$pdates with $\textbf{G}$r$\textbf{A}$dient sp$\textbf{L}$itting), a new memory-efficient optimization framework. $\texttt{FRUGAL}$ leverages gradient splitting to perform low-dimensional updates using advanced algorithms (such as Adam), while updates along the remaining directions are executed via state-free methods like SGD or signSGD (Bernstein et al., 2018). Our framework can be integrated with various low-rank update selection techniques, including GaLore and BAdam. We provide theoretical convergence guarantees for our framework when using SGDM for low-dimensional updates and SGD for state-free updates. Additionally, our method consistently outperforms concurrent approaches across various fixed memory budgets, achieving state-of-the-art results in pre-training and fine-tuning tasks while balancing memory efficiency and performance metrics.

Nazarii Tupitsa, Samuel Horváth, Martin Takáč et al.

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.

Nurbek Tastan, Samuel Horvath, Karthik Nandakumar

Collaborative learning (CL) enables multiple participants to jointly train machine learning (ML) models on decentralized data sources without raw data sharing. While the primary goal of CL is to maximize the expected accuracy gain for each participant, it is also important to ensure that the gains are fairly distributed: no client should be negatively impacted, and gains should reflect contributions. Most existing CL methods require central coordination and focus only on gain maximization, overlooking fairness. In this work, we first show that the existing measure of collaborative fairness based on the correlation between accuracy values without and with collaboration has drawbacks because it does not account for negative collaboration gain. We argue that maximizing mean collaboration gain (MCG) while simultaneously minimizing the collaboration gain spread (CGS) is a fairer alternative. Next, we propose the CYCle protocol that enables individual participants in a private decentralized learning (PDL) framework to achieve this objective through a novel reputation scoring method based on gradient alignment between the local cross-entropy and distillation losses. We further extend the CYCle protocol to operate on top of gossip-based decentralized algorithms such as Gossip-SGD. We also theoretically show that CYCle performs better than standard FedAvg in a two-client mean estimation setting under high heterogeneity. Empirical experiments demonstrate the effectiveness of the CYCle protocol to ensure positive and fair collaboration gain for all participants, even in cases where the data distributions of participants are highly skewed.

Alex Iacob, Lorenzo Sani, Mher Safaryan et al.

Scaling foundation model training with Distributed Data Parallel (DDP) methods is bandwidth-limited. Existing infrequent communication methods like Local SGD were designed to synchronize only model parameters and cannot be trivially applied to adaptive optimizers due to additional optimizer states. Current approaches extending Local SGD either lack convergence guarantees or require synchronizing all optimizer states, tripling communication costs. We propose Desynced Low Communication Adaptive Optimizers (DES-LOC), a family of optimizers assigning independent synchronization periods to parameters and momenta, enabling lower communication costs while preserving convergence. Through extensive experiments on language models of up to 1.7B, we show that DES-LOC can communicate 170x less than DDP and 2x less than the previous state-of-the-art Local ADAM. Furthermore, unlike previous heuristic approaches, DES-LOC is suited for practical training scenarios prone to system failures. DES-LOC offers a scalable, bandwidth-efficient, and fault-tolerant solution for foundation model training.

Nurbek Tastan, Samuel Horvath, Karthik Nandakumar

Collaborative learning enables multiple participants to learn a single global model by exchanging focused updates instead of sharing data. One of the core challenges in collaborative learning is ensuring that participants are rewarded fairly for their contributions, which entails two key sub-problems: contribution assessment and reward allocation. This work focuses on fair reward allocation, where the participants are incentivized through model rewards - differentiated final models whose performance is commensurate with the contribution. In this work, we leverage the concept of slimmable neural networks to collaboratively learn a shared global model whose performance degrades gracefully with a reduction in model width. We also propose a post-training fair allocation algorithm that determines the model width for each participant based on their contributions. We theoretically study the convergence of our proposed approach and empirically validate it using extensive experiments on different datasets and architectures. We also extend our approach to enable training-time model reward allocation.

Alex Iacob, Andrej Jovanovic, Mher Safaryan et al.

Training large models with distributed data parallelism (DDP) requires frequent communication of gradients across workers, which can saturate bandwidth. Infrequent communication strategies (e.g., Local SGD) reduce this overhead but, when applied to adaptive optimizers, often suffer a performance gap relative to fully synchronous DDP. We trace this gap to a time-scale mismatch: the optimizer's fast-moving momentum, tuned for frequent updates, decays too quickly to smooth gradients over long intervals, leading to noise-dominated optimization. To address this, we propose MT-DAO, a family of optimizers that employs multiple slow- and fast-moving first momenta or the gradient to track update dynamics across different time scales, for which we provide the first convergence guarantees. Empirically, for language-model pre-training, this eliminates the performance gap with DDP, outperforming infrequent-communication baselines in perplexity and reducing iso-token wall-clock time by 6-27% on Ethernet interconnects. At the 720M scale, MT-DAO reaches a target perplexity in 24% fewer steps and 35% less time than the single-momentum DDP baseline. MT-DAO enables effective cross-datacenter training and training over wide geographic areas.

Nurbek Tastan, Samuel Horvath, Karthik Nandakumar

Multi-agent systems (MAS) based on Large Language Models (LLMs) have the potential to solve tasks that are beyond the reach of any single LLM. However, this potential can only be realized when the collaboration mechanism between agents is optimized. Specifically, optimizing the communication structure between agents is critical for fruitful collaboration. Most existing approaches rely on fixed topologies, pretrained graph generators, optimization over edges, or employ external LLM judges, thereby adding to the complexity. In this work, we introduce a response-conditioned framework that adapts communication on-the-fly. Agents independently generate responses to the user query and assess peer contributions using an approximation of the Shapley value. A directed acyclic graph (DAG) is then constructed to regulate the propagation of the responses among agents, which ensures stable and efficient message transmission from high-contributing agents to others. This graph is dynamically updated based on the agent responses from the previous collaboration round. Since the proposed framework enables the self-organization of agents without additional supervision or training, we refer to it as SelfOrg. The SelfOrg framework goes beyond task- and query-level optimization and takes into account the stochastic nature of agent responses. Experiments with both strong and weak LLM backends demonstrate robust performance, with significant gains in the weak regime where prior methods collapse. We also theoretically show that multiple agents increase the chance of correctness and that the correct responses naturally dominate the information flow.

Saleh Vatan Khah, Savelii Chezhegov, Shahrokh Farahmand et al.

Gradient clipping is a fundamental tool in Deep Learning, improving the high-probability convergence of stochastic first-order methods like SGD, AdaGrad, and Adam under heavy-tailed noise, which is common in training large language models. It is also a crucial component of Differential Privacy (DP) mechanisms. However, existing high-probability convergence analyses typically require the clipping threshold to increase with the number of optimization steps, which is incompatible with standard DP mechanisms like the Gaussian mechanism. In this work, we close this gap by providing the first high-probability convergence analysis for DP-Clipped-SGD with a fixed clipping level, applicable to both convex and non-convex smooth optimization under heavy-tailed noise, characterized by a bounded central $α$-th moment assumption, $α\in (1,2]$. Our results show that, with a fixed clipping level, the method converges to a neighborhood of the optimal solution with a faster rate than the existing ones. The neighborhood can be balanced against the noise introduced by DP, providing a refined trade-off between convergence speed and privacy guarantees.

Rustem Islamov, Samuel Horvath, Aurelien Lucchi et al.

Strong Differential Privacy (DP) and Optimization guarantees are two desirable properties for a method in Federated Learning (FL). However, existing algorithms do not achieve both properties at once: they either have optimal DP guarantees but rely on restrictive assumptions such as bounded gradients/bounded data heterogeneity, or they ensure strong optimization performance but lack DP guarantees. To address this gap in the literature, we propose and analyze a new method called Clip21-SGD2M based on a novel combination of clipping, heavy-ball momentum, and Error Feedback. In particular, for non-convex smooth distributed problems with clients having arbitrarily heterogeneous data, we prove that Clip21-SGD2M has optimal convergence rate and also near optimal (local-)DP neighborhood. Our numerical experiments on non-convex logistic regression and training of neural networks highlight the superiority of Clip21-SGD2M over baselines in terms of the optimization performance for a given DP-budget.

Ruichen Luo, Sebastian U Stich, Samuel Horváth et al.

LocalSGD and SCAFFOLD are widely used methods in distributed stochastic optimization, with numerous applications in machine learning, large-scale data processing, and federated learning. However, rigorously establishing their theoretical advantages over simpler methods, such as minibatch SGD (MbSGD), has proven challenging, as existing analyses often rely on strong assumptions, unrealistic premises, or overly restrictive scenarios. In this work, we revisit the convergence properties of LocalSGD and SCAFFOLD under a variety of existing or weaker conditions, including gradient similarity, Hessian similarity, weak convexity, and Lipschitz continuity of the Hessian. Our analysis shows that (i) LocalSGD achieves faster convergence compared to MbSGD for weakly convex functions without requiring stronger gradient similarity assumptions; (ii) LocalSGD benefits significantly from higher-order similarity and smoothness; and (iii) SCAFFOLD demonstrates faster convergence than MbSGD for a broader class of non-quadratic functions. These theoretical insights provide a clearer understanding of the conditions under which LocalSGD and SCAFFOLD outperform MbSGD.

Salma Kharrat, Marco Canini, Samuel Horvath

This work tackles the challenges of data heterogeneity and communication limitations in decentralized federated learning. We focus on creating a collaboration graph that guides each client in selecting suitable collaborators for training personalized models that leverage their local data effectively. Our approach addresses these issues through a novel, communication-efficient strategy that enhances resource efficiency. Unlike traditional methods, our formulation identifies collaborators at a granular level by considering combinatorial relations of clients, enhancing personalization while minimizing communication overhead. We achieve this through a bi-level optimization framework that employs a constrained greedy algorithm, resulting in a resource-efficient collaboration graph for personalized learning. Extensive evaluation against various baselines across diverse datasets demonstrates the superiority of our method, named DPFL. DPFL consistently outperforms other approaches, showcasing its effectiveness in handling real-world data heterogeneity, minimizing communication overhead, enhancing resource efficiency, and building personalized models in decentralized federated learning scenarios.

Savelii Chezhegov, Yaroslav Klyukin, Andrei Semenov et al.

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 current understanding of the high-probability convergence of AdaGrad/Adam-type methods is limited in this case. In this work, we prove that AdaGrad/Adam (and their delayed version) can have provably bad high-probability convergence if the noise is heavy-tailed. We also show that gradient clipping fixes this issue, i.e., we derive new high-probability convergence bounds with polylogarithmic dependence on the confidence level for AdaGrad-Norm and Adam-Norm with clipping and with/without delay for smooth convex/non-convex stochastic optimization with heavy-tailed noise. We extend our results to the case of AdaGrad/Adam with delayed stepsizes. Our empirical evaluations highlight the superiority of clipped versions of AdaGrad/Adam in handling the heavy-tailed noise.

Yunxiang Li, Rui Yuan, Chen Fan et al.

Policy gradient is a widely utilized and foundational algorithm in the field of reinforcement learning (RL). Renowned for its convergence guarantees and stability compared to other RL algorithms, its practical application is often hindered by sensitivity to hyper-parameters, particularly the step-size. In this paper, we introduce the integration of the Polyak step-size in RL, which automatically adjusts the step-size without prior knowledge. To adapt this method to RL settings, we address several issues, including unknown f* in the Polyak step-size. Additionally, we showcase the performance of the Polyak step-size in RL through experiments, demonstrating faster convergence and the attainment of more stable policies.

Yunxiang Li, Nicolas Mauricio Cuadrado, Samuel Horváth et al.

The smart grid domain requires bolstering the capabilities of existing energy management systems; Federated Learning (FL) aligns with this goal as it demonstrates a remarkable ability to train models on heterogeneous datasets while maintaining data privacy, making it suitable for smart grid applications, which often involve disparate data distributions and interdependencies among features that hinder the suitability of linear models. This paper introduces a framework that combines FL with a Trust Region Policy Optimization (FL TRPO) aiming to reduce energy-associated emissions and costs. Our approach reveals latent interconnections and employs personalized encoding methods to capture unique insights, understanding the relationships between features and optimal strategies, allowing our model to generalize to previously unseen data. Experimental results validate the robustness of our approach, affirming its proficiency in effectively learning policy models for smart grid challenges.

Sarit Khirirat, Eduard Gorbunov, Samuel Horváth et al.

Motivated by the increasing popularity and importance of large-scale training under differential privacy (DP) constraints, we study distributed gradient methods with gradient clipping, i.e., clipping applied to the gradients computed from local information at the nodes. While gradient clipping is an essential tool for injecting formal DP guarantees into gradient-based methods [1], it also induces bias which causes serious convergence issues specific to the distributed setting. Inspired by recent progress in the error-feedback literature which is focused on taming the bias/error introduced by communication compression operators such as Top-$k$ [2], and mathematical similarities between the clipping operator and contractive compression operators, we design Clip21 -- the first provably effective and practically useful error feedback mechanism for distributed methods with gradient clipping. We prove that our method converges at the same $\mathcal{O}\left(\frac{1}{K}\right)$ rate as distributed gradient descent in the smooth nonconvex regime, which improves the previous best $\mathcal{O}\left(\frac{1}{\sqrt{K}}\right)$ rate which was obtained under significantly stronger assumptions. Our method converges significantly faster in practice than competing methods.

Jihao Xin, Marco Canini, Peter Richtárik et al.

Distributed training enables large-scale deep learning, but suffers from high communication overhead, especially as models and datasets grow. Gradient compression, particularly quantization, is a promising approach to mitigate this bottleneck. However, existing quantization schemes are often incompatible with Allreduce, the dominant communication primitive in distributed deep learning, and many prior solutions rely on heuristics without theoretical guarantees. We introduce Global-QSGD, an Allreduce-compatible gradient quantization method that leverages global norm scaling to reduce communication overhead while preserving accuracy. Global-QSGD is backed by rigorous theoretical analysis, extending standard unbiased compressor frameworks to establish formal convergence guarantees. Additionally, we develop a performance model to evaluate its impact across different hardware configurations. Extensive experiments on NVLink, PCIe, and large-scale cloud environments show that Global-QSGD accelerates distributed training by up to 3.51% over baseline quantization methods, making it a practical and efficient solution for large-scale deep learning workloads.

Konstantin Mishchenko, Rustem Islamov, Eduard Gorbunov et al.

We present a partially personalized formulation of Federated Learning (FL) that strikes a balance between the flexibility of personalization and cooperativeness of global training. In our framework, we split the variables into global parameters, which are shared across all clients, and individual local parameters, which are kept private. We prove that under the right split of parameters, it is possible to find global parameters that allow each client to fit their data perfectly, and refer to the obtained problem as overpersonalized. For instance, the shared global parameters can be used to learn good data representations, whereas the personalized layers are fine-tuned for a specific client. Moreover, we present a simple algorithm for the partially personalized formulation that offers significant benefits to all clients. In particular, it breaks the curse of data heterogeneity in several settings, such as training with local steps, asynchronous training, and Byzantine-robust training.

Elnur Gasanov, Ahmed Khaled, Samuel Horváth et al.

Federated Learning (FL) is an increasingly popular machine learning paradigm in which multiple nodes try to collaboratively learn under privacy, communication and multiple heterogeneity constraints. A persistent problem in federated learning is that it is not clear what the optimization objective should be: the standard average risk minimization of supervised learning is inadequate in handling several major constraints specific to federated learning, such as communication adaptivity and personalization control. We identify several key desiderata in frameworks for federated learning and introduce a new framework, FLIX, that takes into account the unique challenges brought by federated learning. FLIX has a standard finite-sum form, which enables practitioners to tap into the immense wealth of existing (potentially non-local) methods for distributed optimization. Through a smart initialization that does not require any communication, FLIX does not require the use of local steps but is still provably capable of performing dissimilarity regularization on par with local methods. We give several algorithms for solving the FLIX formulation efficiently under communication constraints. Finally, we corroborate our theoretical results with extensive experimentation.

Jianyu Wang, Zachary Charles, Zheng Xu et al.

Federated learning and analytics are a distributed approach for collaboratively learning models (or statistics) from decentralized data, motivated by and designed for privacy protection. The distributed learning process can be formulated as solving federated optimization problems, which emphasize communication efficiency, data heterogeneity, compatibility with privacy and system requirements, and other constraints that are not primary considerations in other problem settings. This paper provides recommendations and guidelines on formulating, designing, evaluating and analyzing federated optimization algorithms through concrete examples and practical implementation, with a focus on conducting effective simulations to infer real-world performance. The goal of this work is not to survey the current literature, but to inspire researchers and practitioners to design federated learning algorithms that can be used in various practical applications.

Samuel Horvath, Stefanos Laskaridis, Mario Almeida et al.

Federated Learning (FL) has been gaining significant traction across different ML tasks, ranging from vision to keyboard predictions. In large-scale deployments, client heterogeneity is a fact and constitutes a primary problem for fairness, training performance and accuracy. Although significant efforts have been made into tackling statistical data heterogeneity, the diversity in the processing capabilities and network bandwidth of clients, termed as system heterogeneity, has remained largely unexplored. Current solutions either disregard a large portion of available devices or set a uniform limit on the model's capacity, restricted by the least capable participants. In this work, we introduce Ordered Dropout, a mechanism that achieves an ordered, nested representation of knowledge in deep neural networks (DNNs) and enables the extraction of lower footprint submodels without the need of retraining. We further show that for linear maps our Ordered Dropout is equivalent to SVD. We employ this technique, along with a self-distillation methodology, in the realm of FL in a framework called FjORD. FjORD alleviates the problem of client system heterogeneity by tailoring the model width to the client's capabilities. Extensive evaluation on both CNNs and RNNs across diverse modalities shows that FjORD consistently leads to significant performance gains over state-of-the-art baselines, while maintaining its nested structure.