Samuel Horváth

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.

Arman Bolatov, Artem Riabinin, Nikita Kornilov et al.

In large-scale optimization, the cheapness and effectiveness of update steps are the most crucial factors for a successful optimizer. Sign-based optimizers like Lion or Signum produce cheap per-step updates, whereas Muon's spectral matrix-sign update gives a much stronger direction at a substantially higher per-step cost. In this work, we propose LionMuon, which retains the effectiveness of Muon steps while considerably cutting the averaged iteration cost, similar to sign-based methods. It alternates between Lion's and Muon's updates on a fixed period P, sharing a single dual-EMA momentum buffer between them. The optimizer state memory therefore matches Lion and is exactly half of AdamW's. A simpler single-EMA variant, SignMuon, by itself already outperforms pure Muon. At P = 2, LionMuon Pareto-dominates Muon, Lion, Signum, and AdamW on every dataset and architecture we tested at 124M model size, reaching lower validation loss at lower compute, and the same advantage persists at 355M and 720M scale. On the theory side, we prove sharp complexity bounds under heavy-tailed noise which are governed by period-averaged smoothness and noise that interpolate between Muon's and Lion's constants. These bounds predict the compute-optimal period and the conditions under which LionMuon outruns Muon and Lion. Code: https://github.com/brain-lab-research/lion-muon

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.

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).

Cuong Anh Pham, Praneeth Vepakomma, Samuel Horváth

Alleviating catastrophic forgetting while enabling further learning is a primary challenge in continual learning (CL). Orthogonal-based training methods have gained attention for their efficiency and strong theoretical properties, and many existing approaches enforce orthogonality through gradient projection. In this paper, we revisit orthogonality and exploit the fact that small singular values correspond to directions that are nearly orthogonal to the input space of previous tasks. Building on this principle, we introduce NESS (Null-space Estimated from Small Singular values), a CL method that applies orthogonality directly in the weight space rather than through gradient manipulation. Specifically, NESS constructs an approximate null space using the smallest singular values of each layer's input representation and parameterizes task-specific updates via a compact low-rank adaptation (LoRA-style) formulation constrained to this subspace. The subspace basis is fixed to preserve the null-space constraint, and only a single trainable matrix is learned for each task. This design ensures that the resulting updates remain approximately in the null space of previous inputs while enabling adaptation to new tasks. Our theoretical analysis and experiments on three benchmark datasets demonstrate competitive performance, low forgetting, and stable accuracy across tasks, highlighting the role of small singular values in continual learning. The code is available at https://github.com/pacman-ctm/NESS.

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.

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.

Abdulla Jasem Almansoori, Maria Ivanova, Andrey Veprikov et al.

Low-Rank Adaptation (LoRA) fine-tunes large models by learning low-rank updates on top of frozen weights, dramatically reducing trainable parameters and memory. In this work, we address the gap between training with full steps with low-rank projections (SVDLoRA) and LoRA fine-tuning. We propose LoRSum, a memory-efficient subroutine that closes this gap for gradient descent by casting LoRA optimization as a proximal sub-problem and solving it efficiently with alternating least squares updates, which we prove to be an implicit block power method. We recover several recently proposed preconditioning methods for LoRA as special cases, and show that LoRSum can also be used for updating a low-rank momentum. In order to address full steps with preconditioned gradient descent, we propose a scaled variant of LoRSum that uses structured metrics such as K-FAC and Shampoo, and we show that storing the diagonal of these metrics still allows them to perform well while remaining memory-efficient. Experiments on a synthetic task, CIFAR-100, and language-model fine-tuning on GLUE, SQuAD v2, and WikiText-103, show that our method can match or improve LoRA baselines given modest compute overhead, while avoiding full-matrix SVD projections and retaining LoRA-style parameter efficiency.

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.

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.

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.

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.

Praneeth Vepakomma, Amirhossein Reisizadeh, Samuel Horváth et al.

This work focuses on the question of learning from a large number of devices with each device holding only a single sample of data. Several real-world applications exist to this one sample per client setup up including learning from fitness trackers, data/app usage aggregators, body-worn sensing devices, and daily event monitors to name a few. When a client has only one sample, the standard federated learning paradigm breaks down as a local update based on that single point is far from being useful, especially in the earlier rounds for estimation of the model coefficients. This utility is further weakened by the privacy-inducing noise applied at every round. This work caters to this problem to enable such clients to collaboratively contribute to effectively learn a global model without leaking the privacy of their data. The proposed approach injects a single, carefully calibrated noisy perturbation to transform the sample at each client, followed by a post-processed representation which is shared with the server. These representations aggregated at the server are processed to obtain an unbiased gradient update that in expectation matches the non-private centralized gradient while preserving data privacy. This approach is different than traditional private federated learning, where the communication payloads involve model coefficients as opposed to privately transformed data samples. This method enables devices with extremely limited data to collaborate and learn accurate, privacy-preserving models without requiring large local datasets or sacrificing individual privacy.

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.

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.

Riccardo Zaccone, Stefanos Laskaridis, Marco Ciccone et al.

The growing scale of deep neural networks, encompassing large language models (LLMs) and vision transformers (ViTs), has made training from scratch prohibitively expensive and deployment increasingly costly. These models are often used as computational monoliths with fixed cost, a rigidity that does not leverage overparametrized architectures and largely hinders adaptive deployment across different cost budgets. We argue that importance-ordered nested components can be extracted from pretrained models, and selectively activated on the available computational budget. To this end, our proposed FlexRank method leverages low-rank weight decomposition with nested, importance-based consolidation to extract submodels of increasing capabilities. Our approach enables a "train-once, deploy-everywhere" paradigm that offers a graceful trade-off between cost and performance without training from scratch for each budget - advancing practical deployment of large models.

Artem Agafonov, Vladislav Ryspayev, Samuel Horváth et al.

Quasi-Newton methods are widely used for solving convex optimization problems due to their ease of implementation, practical efficiency, and strong local convergence guarantees. However, their global convergence is typically established only under specific line search strategies and the assumption of strong convexity. In this work, we extend the theoretical understanding of Quasi-Newton methods by introducing a simple stepsize schedule that guarantees a global convergence rate of ${O}(1/k)$ for the convex functions. Furthermore, we show that when the inexactness of the Hessian approximation is controlled within a prescribed relative accuracy, the method attains an accelerated convergence rate of ${O}(1/k^2)$ -- matching the best-known rates of both Nesterov's accelerated gradient method and cubically regularized Newton methods. We validate our theoretical findings through empirical comparisons, demonstrating clear improvements over standard Quasi-Newton baselines. To further enhance robustness, we develop an adaptive variant that adjusts to the function's curvature while retaining the global convergence guarantees of the non-adaptive algorithm.

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.

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.

Abdulla Jasem Almansoori, Maria Ivanova, Andrey Veprikov et al.

Low-Rank Adaptation (LoRA) fine-tunes large models by learning low-rank updates on top of frozen weights, dramatically reducing trainable parameters and memory. However, there is still a gap between full training with low-rank projections (SVDLoRA) and LoRA fine-tuning, indicating that LoRA steps can be further improved. In this study, we propose OPLoRA, a memory-efficient optimizer that closes this gap by casting LoRA optimization as an interpretable sub-problem and solving it efficiently with alternating least squares updates, where 1-2 alternating steps are empirically found to be sufficient to closely match truncated SVD without ever forming the full matrix. We also retrieve the recently proposed preconditioning methods for LoRA as a special case. OPLoRA supports momentum by maintaining a low-rank estimate using the same subroutine (LoRSum) for computing the step, with a memory budget of 3 times the number of LoRA parameters (i.e., same as Adam). We also propose an experimental scaled variant that uses the K-FAC metric, which could be of interest. Across a linear task, MNIST, CIFAR-100, and RoBERTa-base (MNLI), OPLoRA consistently approaches SVDLoRA's performance using significantly less memory.

Viktor Kovalchuk, Nikita Kotelevskii, Maxim Panov et al.

Federated learning (FL) usually shares model weights or gradients, which is costly for large models. Logit-based FL reduces this cost by sharing only logits computed on a public proxy dataset. However, aggregating information from heterogeneous clients is still challenging. This paper studies this problem, introduces and compares three logit aggregation methods: simple averaging, uncertainty-weighted averaging, and a learned meta-aggregator. Evaluated on MNIST and CIFAR-10, these methods reduce communication overhead, improve robustness under non-IID data, and achieve accuracy competitive with centralized training.

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.

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.

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.

Samuel Horváth, Aaron Klein, Peter Richtárik et al.

Bayesian optimization (BO) is a sample efficient approach to automatically tune the hyperparameters of machine learning models. In practice, one frequently has to solve similar hyperparameter tuning problems sequentially. For example, one might have to tune a type of neural network learned across a series of different classification problems. Recent work on multi-task BO exploits knowledge gained from previous tuning tasks to speed up a new tuning task. However, previous approaches do not account for the fact that BO is a sequential decision making procedure. Hence, there is in general a mismatch between the number of evaluations collected in the current tuning task compared to the number of evaluations accumulated in all previously completed tasks. In this work, we enable multi-task BO to compensate for this mismatch, such that the transfer learning procedure is able to handle different data regimes in a principled way. We propose a new multi-task BO method that learns a set of ordered, non-linear basis functions of increasing complexity via nested drop-out and automatic relevance determination. Experiments on a variety of hyperparameter tuning problems show that our method improves the sample ef

Filip Hanzely, Slavomír Hanzely, Samuel Horváth et al.

In this work, we consider the optimization formulation of personalized federated learning recently introduced by Hanzely and Richtárik (2020) which was shown to give an alternative explanation to the workings of local {\tt SGD} methods. Our first contribution is establishing the first lower bounds for this formulation, for both the communication complexity and the local oracle complexity. Our second contribution is the design of several optimal methods matching these lower bounds in almost all regimes. These are the first provably optimal methods for personalized federated learning. Our optimal methods include an accelerated variant of {\tt FedProx}, and an accelerated variance-reduced version of {\tt FedAvg}/Local {\tt SGD}. We demonstrate the practical superiority of our methods through extensive numerical experiments.

Samuel Horváth, Peter Richtárik

Modern large-scale machine learning applications require stochastic optimization algorithms to be implemented on distributed compute systems. A key bottleneck of such systems is the communication overhead for exchanging information across the workers, such as stochastic gradients. Among the many techniques proposed to remedy this issue, one of the most successful is the framework of compressed communication with error feedback (EF). EF remains the only known technique that can deal with the error induced by contractive compressors which are not unbiased, such as Top-$K$. In this paper, we propose a new and theoretically and practically better alternative to EF for dealing with contractive compressors. In particular, we propose a construction which can transform any contractive compressor into an induced unbiased compressor. Following this transformation, existing methods able to work with unbiased compressors can be applied. We show that our approach leads to vast improvements over EF, including reduced memory requirements, better communication complexity guarantees and fewer assumptions. We further extend our results to federated learning with partial participation following an arbitrary distribution over the nodes, and demonstrate the benefits thereof. We perform several numerical experiments which validate our theoretical findings.

Aleksandr Beznosikov, Samuel Horváth, Peter Richtárik et al.

In the last few years, various communication compression techniques have emerged as an indispensable tool helping to alleviate the communication bottleneck in distributed learning. However, despite the fact biased compressors often show superior performance in practice when compared to the much more studied and understood unbiased compressors, very little is known about them. In this work we study three classes of biased compression operators, two of which are new, and their performance when applied to (stochastic) gradient descent and distributed (stochastic) gradient descent. We show for the first time that biased compressors can lead to linear convergence rates both in the single node and distributed settings. We prove that distributed compressed SGD method, employed with error feedback mechanism, enjoys the ergodic rate $O\left( δL \exp \left[-\frac{μK}{δL}\right] + \frac{(C + δD)}{Kμ}\right)$, where $δ\ge 1$ is a compression parameter which grows when more compression is applied, $L$ and $μ$ are the smoothness and strong convexity constants, $C$ captures stochastic gradient noise ($C=0$ if full gradients are computed on each node) and $D$ captures the variance of the gradients at the optimum ($D=0$ for over-parameterized models). Further, via a theoretical study of several synthetic and empirical distributions of communicated gradients, we shed light on why and by how much biased compressors outperform their unbiased variants. Finally, we propose several new biased compressors with promising theoretical guarantees and practical performance.

Samuel Horváth, Lihua Lei, Peter Richtárik et al.

Adaptivity is an important yet under-studied property in modern optimization theory. The gap between the state-of-the-art theory and the current practice is striking in that algorithms with desirable theoretical guarantees typically involve drastically different settings of hyperparameters, such as step-size schemes and batch sizes, in different regimes. Despite the appealing theoretical results, such divisive strategies provide little, if any, insight to practitioners to select algorithms that work broadly without tweaking the hyperparameters. In this work, blending the "geometrization" technique introduced by Lei & Jordan 2016 and the \texttt{SARAH} algorithm of Nguyen et al., 2017, we propose the Geometrized \texttt{SARAH} algorithm for non-convex finite-sum and stochastic optimization. Our algorithm is proved to achieve adaptivity to both the magnitude of the target accuracy and the Polyak-Łojasiewicz (PL) constant if present. In addition, it achieves the best-available convergence rate for non-PL objectives simultaneously while outperforming existing algorithms for PL objectives.