arXiv subjects

Eduard Gorbunov

Publications and source records attributed to Eduard Gorbunov.

At least 19 recordsLinked to original sources

arXiv2026

High-Probability Convergence of Clipped SGD under Heavy-Tailed Noise and $(L_0,L_1)$-Smoothness

Gradient clipping is widely used in language-model training to control heavy-tailed gradient noise and can improve convergence guarantees over stochastic gradient descent (SGD) under $(L_0,L_1)$-smoothness. Under these joint conditions, a central challenge is to obtain high-probability guarantees without exponential dependence on $L_1R_0$, where $R_0$ bounds the initial distance to a minimizer. We resolve this challenge for convex objectives, establishing, to the best of our knowledge, the first such guarantees for standard Clip-SGD. We assume unbiased stochastic gradients with bounded central $α$-th moment, $α\in(1,2]$. Our bounds have only polylogarithmic dependence on the inverse failure probability and recover known deterministic generalized-smoothness rates when the noise vanishes, as well as the standard high-probability rate under heavy-tailed noise in the classical $L$-smooth setting. The convex rate is attained by a computable output that averages iterates whose sampled stochastic gradients are not clipped, requiring neither function values nor extra oracle calls. We establish a lower bound for Clip-SGD with any fixed stepsize and clipping level, showing that, under the stated iteration conditions, our convex stochastic rate is optimal up to logarithmic factors in the iteration budget. Our upper-bound analysis uses a directional clipping-bias bound to absorb part of the bias into the progress generated by the clipped population gradient, avoiding an exponentially large local smoothness constant. We also obtain nonconvex high-probability guarantees that recover the noiseless rate, match the classical-smooth stochastic iteration rate when $L_1=0$, and have no explicit $L_1$ dependence in the asymptotically dominant stochastic term.

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arXiv2026

Accelerated and Stable Convergence with Anchored Generalized Optimistic Method

We study first-order methods for solving monotone variational inequalities arising in min-max optimization. Classical approaches such as the extragradient method rely on two gradient queries per iteration, which limits their analysis and applicability in the online and stochastic settings. We propose a family of Generalized Optimistic Methods with Anchoring (GOMA), which combine two-time-scale optimistic updates with an anchoring term inspired by Halpern iteration. In the deterministic setting, GOMA achieves the optimal accelerated last-iterate rate $O(1/k^2)$ on the squared gradient norm for monotone Lipschitz operators. In the stochastic setting with unbounded variance, a simplified single-call variant of GOMA achieves a last-iterate convergence rate of $O(1/\sqrt{k})$ on the squared gradient norm. To the best of our knowledge, this is the first such guarantee for stochastic monotone Lipschitz variational inequalities in the unconstrained setting without variance reduction or growing batches.

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arXiv2026

What's in a Smoothness Constant? Tighter Rates for Local SGD with Bounded Second-order Heterogeneity

Local SGD, also known as Federated Averaging, is a widely used distributed optimization algorithm. Although Local SGD often outperforms alternatives such as Mini-batch SGD in practice, theory still only partially explains when and why local updates help under realistic data heterogeneity. Recent work by [Patel et al., 2025] shows that a bounded second-order heterogeneity assumption captures the efficiency of Local SGD for strongly convex objectives, and conjectures that the same principle extends to the general convex setting. In this paper, we prove this conjecture by establishing an improved convergence guarantee for Local SGD on general convex objectives under bounded second-order heterogeneity. We also improve the best-known lower bounds for Local SGD in this setting, showing that our upper bounds are nearly tight. Together, these results provide a sharper, more fine-grained convergence theory for Local SGD. As a further application of our techniques, we provide a lower bound for serial SGD with replacement, showing how second-order heterogeneity captures the impact of rare high-curvature clients.

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arXiv2026

On the Interaction of Batch Noise, Adaptivity, and Compression, under $(L_0,L_1)$-Smoothness: An SDE Approach

Distributed stochastic optimization intertwines (i) stochastic gradient noise, (ii) communication compression, and (iii) adaptive/normalized updates. While each factor has been studied in isolation, their joint effect under realistic assumptions remains poorly understood. In this work, we develop a unified theoretical framework for Distributed Compressed SGD (DCSGD) and its sign variant Distributed SignSGD (DSignSGD) under the recently introduced $(L_0, L_1)$-smoothness condition. From a conceptual perspective, we show that the first- and second-order modified equations from the literature do not accurately model the discrete-time step-size/stability restrictions, especially under $(L_0,L_1)$-smoothness. From a technical perspective, we propose new first-order SDEs by carefully incorporating curvature-dependent terms into their drift: This helps capture the fine-grained relationship between learning rate restrictions, gradient noise, compression, and the geometry of the loss landscape. Importantly, we do so under general gradient noise assumptions, including heavy-tailed and affine-variance regimes, which extend beyond the classical bounded-variance setting. Our results suggest that normalizing the updates of DCSGD emerges as a natural condition for stability, with the degree of normalization precisely determined by the gradient noise structure, the landscape's regularity, and the compression rate. In contrast, DSignSGD converges even under heavy-tailed noise with standard learning rate schedules. Together, these findings offer both new theoretical insights and perspectives, and practical guidance.

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arXiv2026

Last Iterate Convergence of AdaGrad-Norm for Convex Non-Smooth Optimization

We study the convergence of the last iterate (i.e., the $(N+1)$-th iterate) of the AdaGrad method. Although AdaGrad -- an adaptive subgradient method -- underpins a wide class of algorithms, most existing convergence analyses focus on averaged (or best) iterates. We derive worst-case upper bounds on the suboptimality of the final point and show that, with an optimally tuned stepsize parameter, the last iterate converges at the rate $O(1/N^{1/4})$. We complement this guarantee with matching lower-bound constructions, proving that this rate is tight and that AdaGrad's last-iterate rate is strictly worse than the classical $O(1/N^{1/2})$ rate for its averaged iterate. Technically, our analysis introduces an exponent parameter that captures the growth of the cumulative squared subgradients; combined with the last-iterate inequality of Zamani and Glineur (2025), this reduces the problem to bounding a particular series.

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arXiv2026

Byzantine-Robust and Differentially Private Federated Optimization under Weaker Assumptions

Federated Learning (FL) enables heterogeneous clients to collaboratively train a shared model without centralizing their raw data, offering an inherent level of privacy. However, gradients and model updates can still leak sensitive information, while malicious servers may mount adversarial attacks such as Byzantine manipulation. These vulnerabilities highlight the need to address differential privacy (DP) and Byzantine robustness within a unified framework. Existing approaches, however, often rely on unrealistic assumptions such as bounded gradients, require auxiliary server-side datasets, or fail to provide convergence guarantees. We address these limitations by proposing Byz-Clip21-SGD2M, a new algorithm that integrates robust aggregation with double momentum and carefully designed clipping. We prove high-probability convergence guarantees under standard $L$-smoothness and $σ$-sub-Gaussian gradient noise assumptions, thereby relaxing conditions that dominate prior work. Our analysis recovers state-of-the-art convergence rates in the absence of adversaries and improves utility guarantees under Byzantine and DP settings. Empirical evaluations on CNN and MLP models trained on MNIST further validate the effectiveness of our approach.

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arXiv2026

Who to Trust? Aggregating Client Predictions in Federated Distillation

Under data heterogeneity (e.g., $\textit{class mismatch}$), clients may produce unreliable predictions for instances belonging to unfamiliar classes. An equally weighted combination of such predictions can corrupt the teacher signal used for distillation. In this paper, we provide a theoretical analysis of Federated Distillation and show that aggregating client predictions on a shared public dataset converges to a neighborhood of the optimum, where the neighborhood size is governed by the aggregation quality. We further propose two uncertainty-aware aggregation methods, $\mathbf{UWA}$ and $\mathbf{sUWA}$, which leverage density-based uncertainty estimates to down-weight unreliable client predictions. Experiments on image and text classification benchmarks demonstrate that our methods are particularly effective under high data heterogeneity, while matching standard averaging when heterogeneity is low.

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arXiv2026

On the Role of Batch Size in Stochastic Conditional Gradient Methods

We study the role of batch size in stochastic conditional gradient methods under a $μ$-Kurdyka-Łojasiewicz ($μ$-KL) condition. Focusing on momentum-based stochastic conditional gradient algorithms (e.g., Scion), we derive a new analysis that explicitly captures the interaction between stepsize, batch size, and stochastic noise. Our study reveals a regime-dependent behavior: increasing the batch size initially improves optimization accuracy but, beyond a critical threshold, the benefits saturate and can eventually degrade performance under a fixed token budget. Notably, the theory predicts the magnitude of the optimal stepsize and aligns well with empirical practices observed in large-scale training. Leveraging these insights, we derive principled guidelines for selecting the batch size and stepsize, and propose an adaptive strategy that increases batch size and sequence length during training while preserving convergence guarantees. Experiments on NanoGPT are consistent with the theoretical predictions and illustrate the emergence of the predicted scaling regimes. Overall, our results provide a theoretical framework for understanding batch size scaling in stochastic conditional gradient methods and offer guidance for designing efficient training schedules in large-scale optimization.

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arXiv2026

Byzantine-Robust Optimization under $(L_0, L_1)$-Smoothness

We consider distributed optimization under Byzantine attacks in the presence of $(L_0,L_1)$-smoothness, a generalization of standard $L$-smoothness that captures functions with state-dependent gradient Lipschitz constants. We propose Byz-NSGDM, a normalized stochastic gradient descent method with momentum that achieves robustness against Byzantine workers while maintaining convergence guarantees. Our algorithm combines momentum normalization with Byzantine-robust aggregation enhanced by Nearest Neighbor Mixing (NNM) to handle both the challenges posed by $(L_0,L_1)$-smoothness and Byzantine adversaries. We prove that Byz-NSGDM achieves a convergence rate of $O(K^{-1/4})$ up to a Byzantine bias floor proportional to the robustness coefficient and gradient heterogeneity. Experimental validation on heterogeneous MNIST classification, synthetic $(L_0,L_1)$-smooth optimization, and character-level language modeling with a small GPT model demonstrates the effectiveness of our approach against various Byzantine attack strategies. An ablation study further shows that Byz-NSGDM is robust across a wide range of momentum and learning rate choices.

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arXiv2026

Double Momentum and Error Feedback for Clipping with Fast Rates and Differential Privacy

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.

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arXiv2025

Median Clipping for Zeroth-order Non-Smooth Convex Optimization and Multi-Armed Bandit Problem with Heavy-tailed Symmetric Noise

In this paper, we consider non-smooth convex optimization with a zeroth-order oracle corrupted by symmetric stochastic noise. Unlike the existing high-probability results requiring the noise to have bounded $κ$-th moment with $κ\in (1,2]$, our results allow even heavier noise with any $κ> 0$, e.g., the noise distribution can have unbounded expectation. Our convergence rates match the best-known ones for the case of the bounded variance, namely, to achieve function accuracy $\varepsilon$ our methods with Lipschitz oracle require $\tilde{O}(d^2\varepsilon^{-2})$ iterations for any $κ> 0$. We build the median gradient estimate with bounded second moment as the mini-batched median of the sampled gradient differences. We apply this technique to the stochastic multi-armed bandit problem with heavy-tailed distribution of rewards and achieve $\tilde{O}(\sqrt{dT})$ regret. We demonstrate the performance of our zeroth-order and MAB algorithms for various $κ\in (0,2]$ on synthetic and real-world data. Our methods do not lose to SOTA approaches and dramatically outperform them for $κ\leq 1$.

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arXiv2025

Initialization using Update Approximation is a Silver Bullet for Extremely Efficient Low-Rank Fine-Tuning

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.

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arXiv2025

Differentially Private Clipped-SGD: High-Probability Convergence with Arbitrary Clipping Level

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.

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arXiv2025

Clipping Improves Adam-Norm and AdaGrad-Norm when the Noise Is Heavy-Tailed

Methods with adaptive stepsizes, such as AdaGrad and Adam, are essential for training modern Deep Learning models, especially Large Language Models. Typically, the noise in the stochastic gradients is heavy-tailed for the later ones. Gradient clipping provably helps to achieve good high-probability convergence for such noises. However, despite the similarity between AdaGrad/Adam and Clip-SGD, the 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.

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arXiv2025

EF21 with Bells & Whistles: Six Algorithmic Extensions of Modern Error Feedback

First proposed by Seide (2014) as a heuristic, error feedback (EF) is a very popular mechanism for enforcing convergence of distributed gradient-based optimization methods enhanced with communication compression strategies based on the application of contractive compression operators. However, existing theory of EF relies on very strong assumptions (e.g., bounded gradients), and provides pessimistic convergence rates (e.g., while the best known rate for EF in the smooth nonconvex regime, and when full gradients are compressed, is $O(1/T^{2/3})$, the rate of gradient descent in the same regime is $O(1/T)$). Recently, Richtárik et al. (2021) proposed a new error feedback mechanism, EF21, based on the construction of a Markov compressor induced by a contractive compressor. EF21 removes the aforementioned theoretical deficiencies of EF and at the same time works better in practice. In this work we propose six practical extensions of EF21, all supported by strong convergence theory: partial participation, stochastic approximation, variance reduction, proximal setting, momentum, and bidirectional compression. To the best of our knowledge, several of these techniques have not been previously analyzed in combination with EF, and in cases where prior analysis exists -- such as for bidirectional compression -- our theoretical convergence guarantees significantly improve upon existing results.

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arXiv2025

Methods with Local Steps and Random Reshuffling for Generally Smooth Non-Convex Federated Optimization

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.

math.OC↗

arXiv2025

Linear Convergence Rate in Convex Setup is Possible! Gradient Descent Method Variants under $(L_0,L_1)$-Smoothness

The gradient descent (GD) method -- is a fundamental and likely the most popular optimization algorithm in machine learning (ML), with a history traced back to a paper in 1847 (Cauchy, 1847). It was studied under various assumptions, including so-called $(L_0,L_1)$-smoothness, which received noticeable attention in the ML community recently. In this paper, we provide a refined convergence analysis of gradient descent and its variants, assuming generalized smoothness. In particular, we show that $(L_0,L_1)$-GD has the following behavior in the convex setup: as long as $\|\nabla f(x^k)\| \geq \frac{L_0}{L_1}$ the algorithm has linear convergence in function suboptimality, and when $\|\nabla f(x^k)\| < \frac{L_0}{L_1}$ is satisfied, $(L_0,L_1)$-GD has standard sublinear rate. Moreover, we also show that this behavior is common for its variants with different types of oracle: Normalized Gradient Descent as well as Clipped Gradient Descent (the case when the full gradient $\nabla f(x)$ is available); Random Coordinate Descent (when the gradient component $\nabla_{i} f(x)$ is available); Random Coordinate Descent with Order Oracle (when only $\text{sign} [f(y) - f(x)]$ is available). In addition, we also extend our analysis of $(L_0,L_1)$-GD to the strongly convex case.

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arXiv2025

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

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

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