arXiv subjects

Aleksandar Armacki

Publications and source records attributed to Aleksandar Armacki.

13 recordsLinked to original sources

arXiv2026

A Unified Zeroth-Order Approach for Decentralized Minimax Optimization

We propose ZOMA, a unified Zeroth-Order decentralized accelerated MinimAx framework for multi-agent nonconvex Polyak--Łojasiewicz minimax optimization. The proposed framework only requires evaluating the function value and, as such, is tailored to gradient-free environments, where exact gradient information is either unavailable or computationally prohibitive to obtain. A central contribution of our \textbf{ZOMA} framework is a multi-level unification, along the following directions: (i) \emph{estimator} - our framework adopts a hybrid zeroth-order estimator, which accommodates, among others, both coordinate-wise and randomized uniform smoothing estimators; (ii) \emph{bias correction} - our framework subsumes a wide range of bias-correction strategies, including gradient tracking (GT), exact diffusion (ED), and EXTRA and (iii) \emph{acceleration} - our framework facilitates a broad class of acceleration techniques, including zeroth-order versions of STORM, PAGE, and L2S. The general nature of \textbf{ZOMA} leads to many novel decentralized zeroth-order minimax methods and allows us to establish unified convergence guarantees, matching the performance of state-of-the-art centralized zeroth-order minimax methods, while providing benefits, such as linear speed-up in the number of users. The unified framework also provides a systematic way to assess algorithmic suitability by specializing the convergence rates to specific problem structures and method designs. We validate the performance of the proposed algorithms via numerical simulations.

math.OC↗

arXiv2026

Tight Long-Term Tail Decay of (Clipped) SGD in Non-Convex Optimization

The study of tail behaviour of SGD-induced processes has been attracting a lot of interest, due to offering strong guarantees with respect to individual runs of an algorithm. While many works provide high-probability guarantees, quantifying the error rate for a fixed probability threshold, there is a lack of work directly studying the probability of failure, i.e., quantifying the tail decay rate for a fixed error threshold. Moreover, existing results are of finite-time nature, limiting their ability to capture the true long-term tail decay which is more informative for modern learning models, typically trained for millions of iterations. Our work closes these gaps, by studying the long-term tail decay of SGD-based methods through the lens of large deviations theory, establishing several strong results in the process. First, we provide an upper bound on the tails of the gradient norm-squared of the best iterate produced by (vanilla) SGD, for non-convex costs and bounded noise, with long-term decay at rate $e^{-t/\log(t)}$. Next, we relax the noise assumption by considering clipped SGD (c-SGD) under heavy-tailed noise with bounded moment of order $p \in (1,2]$, showing an upper bound with long-term decay at rate $e^{-t^{β_p}/\log(t)}$, where $β_p = \frac{4(p-1)}{3p-2}$ for $p \in (1,2)$ and $e^{-t/\log^2(t)}$ for $p = 2$. Finally, we provide lower bounds on the tail decay, at rate $e^{-t}$, showing that our rates for both SGD and c-SGD are tight, up to poly-logarithmic factors. Notably, our results demonstrate an order of magnitude faster long-term tail decay compared to existing work based on finite-time bounds, which show rates $e^{-\sqrt{t}}$ and $e^{-t^{β_p/2}}$, $p \in (1,2]$, for SGD and c-SGD, respectively. As such, we uncover regimes where the tails decay much faster than previously known, providing stronger long-term guarantees for individual runs.

cs.LG↗

arXiv2026

High-Probability Convergence Guarantees of Decentralized SGD

Convergence in high-probability (HP) has attracted increasing interest, due to implying exponentially decaying tail bounds and strong guarantees for individual runs of an algorithm. While many works study HP guarantees in centralized settings, much less is understood in the decentralized setup, where existing works require strong assumptions, like uniformly bounded gradients, or asymptotically vanishing noise. This results in a significant gap between the assumptions used to establish convergence in the HP and the mean-squared error (MSE) sense, and is also contrary to centralized settings, where it is known that $\mathtt{SGD}$ converges in HP under the same conditions on the cost function as needed for MSE convergence. Motivated by these observations, we study the HP convergence of Decentralized $\mathtt{SGD}$ ($\mathtt{DSGD}$) in the presence of light-tailed noise, providing several strong results. First, we show that $\mathtt{DSGD}$ converges in HP under the same conditions on the cost as in the MSE sense, removing the restrictive assumptions used in prior works. Second, our sharp analysis yields order-optimal rates for both non-convex and strongly convex costs. Third, we establish a linear speed-up in the number of users, leading to matching or strictly better transient times than those obtained from MSE results, further underlining the tightness of our analysis. To the best of our knowledge, this is the first work that shows $\mathtt{DSGD}$ achieves a linear speed-up in the HP sense. Our relaxed assumptions and sharp rates stem from several technical results of independent interest, including a result on the variance-reduction effect of decentralized methods in the HP sense, as well as a novel bound on the moment-generating function of strongly convex costs, of interest even in centralized settings. Numerical experiments validate our theory.

cs.LG↗

arXiv2026

High-Probability Convergence in Decentralized Stochastic Optimization with Gradient Tracking

We study high-probability (HP) convergence guarantees in decentralized stochastic optimization, where multiple agents collaborate to jointly train a model over a network. Existing HP results in decentralized settings almost exclusively focus on the Decentralized Stochastic Gradient Descent ($\mathtt{DSGD}$) algorithm, which requires strong assumptions, such as bounded data heterogeneity, or strong convexity of each agent's cost. This is contrary to the mean-squared error (MSE) results, where methods incorporating bias-correction techniques are known to converge under relaxed assumptions and achieve better practical performance. In this paper we provide the first step toward bridging the gap, by studying HP convergence of $\mathtt{DSGD}$ incorporating the gradient tracking technique, in the presence of noise satisfying a relaxed sub-Gaussian condition. We show that the resulting method, dubbed $\mathtt{GT-DSGD}$, achieves order-optimal HP convergence rates for both non-convex and Polyak-Łojasiewicz costs, of order $\mathcal{O}\Big(\frac{\log(1/δ)}{\sqrt{nT}}\Big)$ and $\mathcal{O}\Big(\frac{\log(1/δ)}{nT}\Big)$, respectively, where $n$ is the number of agents, $T$ is the time horizon and $δ\in (0,1)$ is the confidence parameter. Our results establish that $\mathtt{GT-DSGD}$ converges in the HP sense under the same conditions on the cost as in the MSE sense, while achieving comparable transient times. To the best of our knowledge, these are the first HP guarantees for decentralized optimization methods incorporating bias-correction. Numerical experiments on real and synthetic data verify our theoretical findings, underlining the superior performance of $\mathtt{GT-DSGD}$ and highlighting that the benefits of incorporating bias-correction are also maintained in the HP sense.

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arXiv2026

Distributed Gradient Clustering: Convergence and the Effect of Initialization

We study the effects of center initialization on the performance of a family of distributed gradient-based clustering algorithms introduced in [1], that work over connected networks of users. In the considered scenario, each user contains a local dataset and communicates only with its immediate neighbours, with the aim of finding a global clustering of the joint data. We perform extensive numerical experiments, evaluating the effects of center initialization on the performance of our family of methods, demonstrating that our methods are more resilient to the effects of initialization, compared to centralized gradient clustering [2]. Next, inspired by the $K$-means++ initialization [3], we propose a novel distributed center initialization scheme, which is shown to improve the performance of our methods, compared to the baseline random initialization.

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arXiv2026

Sharp High-Probability Rates for Nonlinear SGD under Heavy-Tailed Noise via Symmetrization

We study convergence in high-probability of SGD-type methods in non-convex optimization and the presence of heavy-tailed noise. To combat the heavy-tailed noise, a general black-box nonlinear framework is considered, subsuming nonlinearities like sign, clipping, normalization and their smooth counterparts. Our first result shows that nonlinear SGD (N-SGD) achieves the rate $\widetilde{\mathcal{O}}(t^{-1/2})$, for any noise with unbounded moments and a symmetric probability density function (PDF). Crucially, N-SGD has exponentially decaying tails, matching the performance of linear SGD under light-tailed noise. To handle non-symmetric noise, we propose two novel estimators, based on the idea of noise symmetrization. The first, dubbed Symmetrized Gradient Estimator (SGE), assumes a noiseless gradient at any reference point is available at the start of training, while the second, dubbed Mini-batch SGE (MSGE), uses mini-batches to estimate the noiseless gradient. Combined with the nonlinear framework, we get N-SGE and N-MSGE methods, respectively, both achieving the same convergence rate and exponentially decaying tails as N-SGD, while allowing for non-symmetric noise with unbounded moments and PDF satisfying a mild technical condition, with N-MSGE additionally requiring bounded noise moment of order $p \in (1,2]$. Compared to works assuming noise with bounded $p$-th moment, our results: 1) are based on a novel symmetrization approach; 2) provide a unified framework and relaxed moment conditions; 3) imply optimal oracle complexity of N-SGD and N-SGE, strictly better than existing works when $p < 2$, while the complexity of N-MSGE is close to existing works. Compared to works assuming symmetric noise with unbounded moments, we: 1) provide a sharper analysis and improved rates; 2) facilitate state-dependent symmetric noise; 3) extend the strong guarantees to non-symmetric noise.

stat.ML↗

arXiv2025

Large Deviation Upper Bounds and Improved MSE Rates of Nonlinear SGD: Heavy-tailed Noise and Power of Symmetry

We study large deviation upper bounds and mean-squared error (MSE) guarantees of a general framework of nonlinear stochastic gradient methods in the online setting, in the presence of heavy-tailed noise. Unlike existing works that rely on the closed form of a nonlinearity (typically clipping), our framework treats the nonlinearity in a black-box manner, allowing us to provide unified guarantees for a broad class of bounded nonlinearities, including many popular ones, like sign, quantization, normalization, as well as component-wise and joint clipping. We provide several strong results for a broad range of step-sizes in the presence of heavy-tailed noise with symmetric probability density function, positive in a neighbourhood of zero and potentially unbounded moments. In particular, for non-convex costs we provide a large deviation upper bound for the minimum norm-squared of gradients, showing an asymptotic tail decay on an exponential scale, at a rate $\sqrt{t} / \log(t)$. We establish the accompanying rate function, showing an explicit dependence on the choice of step-size, nonlinearity, noise and problem parameters. Next, for non-convex costs and the minimum norm-squared of gradients, we derive the optimal MSE rate $\widetilde{\mathcal{O}}(t^{-1/2})$. Moreover, for strongly convex costs and the last iterate, we provide an MSE rate that can be made arbitrarily close to the optimal rate $\mathcal{O}(t^{-1})$, improving on the state-of-the-art results in the presence of heavy-tailed noise. Finally, we establish almost sure convergence of the minimum norm-squared of gradients, providing an explicit rate, which can be made arbitrarily close to $o(t^{-1/4})$.

cs.LG↗

arXiv2025

Nonlinear Stochastic Gradient Descent and Heavy-tailed Noise: A Unified Framework and High-probability Guarantees

We study high-probability convergence in online learning, in the presence of heavy-tailed noise. To combat the heavy tails, a general framework of nonlinear SGD methods is considered, subsuming several popular nonlinearities like sign, quantization, component-wise and joint clipping. In our work the nonlinearity is treated in a black-box manner, allowing us to establish unified guarantees for a broad range of nonlinear methods. For symmetric noise and non-convex costs we establish convergence of gradient norm-squared, at a rate $\widetilde{\mathcal{O}}(t^{-1/4})$, while for the last iterate of strongly convex costs we establish convergence to the population optima, at a rate $\mathcal{O}(t^{-ζ})$, where $ζ\in (0,1)$ depends on noise and problem parameters. Further, if the noise is a (biased) mixture of symmetric and non-symmetric components, we show convergence to a neighbourhood of stationarity, whose size depends on the mixture coefficient, nonlinearity and noise. Compared to state-of-the-art, who only consider clipping and require unbiased noise with bounded $p$-th moments, $p \in (1,2]$, we provide guarantees for a broad class of nonlinearities, without any assumptions on noise moments. While the rate exponents in state-of-the-art depend on noise moments and vanish as $p \rightarrow 1$, our exponents are constant and strictly better whenever $p < 6/5$ for non-convex and $p < 8/7$ for strongly convex costs. Experiments validate our theory, showing that clipping is not always the optimal nonlinearity, further underlining the value of a general framework.

cs.LG↗

arXiv2024

A Unified Framework for Center-based Clustering of Distributed Data

We develop a family of distributed center-based clustering algorithms that work over networks of users. In the proposed scenario, users contain a local dataset and communicate only with their immediate neighbours, with the aim of finding a clustering of the full, joint data. The proposed family, termed Distributed Gradient Clustering (DGC-$\mathcal{F}_ρ$), is parametrized by $ρ\geq 1$, controling the proximity of users' center estimates, with $\mathcal{F}$ determining the clustering loss. Our framework allows for a broad class of smooth convex loss functions, including popular clustering losses like $K$-means and Huber loss. Specialized to popular clustering losses like $K$-means and Huber loss, DGC-$\mathcal{F}_ρ$ gives rise to novel distributed clustering algorithms DGC-KM$_ρ$ and DGC-HL$_ρ$, while novel clustering losses based on Logistic and Fair functions lead to DGC-LL$_ρ$ and DGC-FL$_ρ$. We provide a unified analysis and establish several strong results, under mild assumptions. First, we show that the sequence of centers generated by the methods converges to a well-defined notion of fixed point, under any center initialization and value of $ρ$. Second, we prove that, as $ρ$ increases, the family of fixed points produced by DGC-$\mathcal{F}_ρ$ converges to a notion of consensus fixed points. We show that consensus fixed points of DGC-$\mathcal{F}_ρ$ are equivalent to fixed points of gradient clustering over the full data, guaranteeing a clustering of the full data is produced. For the special case of Bregman losses, we show that our fixed points converge to the set of Lloyd points. Extensive numerical experiments on synthetic and real data confirm our theoretical findings, show strong performance of our methods and demonstrate the usefulness and wide range of potential applications of our general framework, such as outlier detection.

cs.LG↗

arXiv2024

High-probability Convergence Bounds for Nonlinear Stochastic Gradient Descent Under Heavy-tailed Noise

We study high-probability convergence guarantees of learning on streaming data in the presence of heavy-tailed noise. In the proposed scenario, the model is updated in an online fashion, as new information is observed, without storing any additional data. To combat the heavy-tailed noise, we consider a general framework of nonlinear stochastic gradient descent (SGD), providing several strong results. First, for non-convex costs and component-wise nonlinearities, we establish a convergence rate arbitrarily close to $\mathcal{O}\left(t^{-\frac{1}{4}}\right)$, whose exponent is independent of noise and problem parameters. Second, for strongly convex costs and component-wise nonlinearities, we establish a rate arbitrarily close to $\mathcal{O}\left(t^{-\frac{1}{2}}\right)$ for the weighted average of iterates, with exponent again independent of noise and problem parameters. Finally, for strongly convex costs and a broader class of nonlinearities, we establish convergence of the last iterate, with a rate $\mathcal{O}\left(t^{-ζ} \right)$, where $ζ\in (0,1)$ depends on problem parameters, noise and nonlinearity. As we show analytically and numerically, $ζ$ can be used to inform the preferred choice of nonlinearity for given problem settings. Compared to state-of-the-art, who only consider clipping, require bounded noise moments of order $η\in (1,2]$, and establish convergence rates whose exponents go to zero as $η\rightarrow 1$, we provide high-probability guarantees for a much broader class of nonlinearities and symmetric density noise, with convergence rates whose exponents are bounded away from zero, even when the noise has finite first moment only. Moreover, in the case of strongly convex functions, we demonstrate analytically and numerically that clipping is not always the optimal nonlinearity, further underlining the value of our general framework.

cs.LG↗

arXiv2023

A One-shot Framework for Distributed Clustered Learning in Heterogeneous Environments

The paper proposes a family of communication efficient methods for distributed learning in heterogeneous environments in which users obtain data from one of $K$ different distributions. In the proposed setup, the grouping of users (based on the data distributions they sample), as well as the underlying statistical properties of the distributions, are apriori unknown. A family of One-shot Distributed Clustered Learning methods (ODCL-$\mathcal{C}$) is proposed, parametrized by the set of admissible clustering algorithms $\mathcal{C}$, with the objective of learning the true model at each user. The admissible clustering methods include $K$-means (KM) and convex clustering (CC), giving rise to various one-shot methods within the proposed family, such as ODCL-KM and ODCL-CC. The proposed one-shot approach, based on local computations at the users and a clustering based aggregation step at the server is shown to provide strong learning guarantees. In particular, for strongly convex problems it is shown that, as long as the number of data points per user is above a threshold, the proposed approach achieves order-optimal mean-squared error (MSE) rates in terms of the sample size. An explicit characterization of the threshold is provided in terms of problem parameters. The trade-offs with respect to selecting various clustering methods (ODCL-CC, ODCL-KM) are discussed and significant improvements over state-of-the-art are demonstrated. Numerical experiments illustrate the findings and corroborate the performance of the proposed methods.

cs.LG↗

arXiv2022

Gradient Based Clustering

We propose a general approach for distance based clustering, using the gradient of the cost function that measures clustering quality with respect to cluster assignments and cluster center positions. The approach is an iterative two step procedure (alternating between cluster assignment and cluster center updates) and is applicable to a wide range of functions, satisfying some mild assumptions. The main advantage of the proposed approach is a simple and computationally cheap update rule. Unlike previous methods that specialize to a specific formulation of the clustering problem, our approach is applicable to a wide range of costs, including non-Bregman clustering methods based on the Huber loss. We analyze the convergence of the proposed algorithm, and show that it converges to the set of appropriately defined fixed points, under arbitrary center initialization. In the special case of Bregman cost functions, the algorithm converges to the set of centroidal Voronoi partitions, which is consistent with prior works. Numerical experiments on real data demonstrate the effectiveness of the proposed method.

cs.LG↗

arXiv2022

Personalized Federated Learning via Convex Clustering

We propose a parametric family of algorithms for personalized federated learning with locally convex user costs. The proposed framework is based on a generalization of convex clustering in which the differences between different users' models are penalized via a sum-of-norms penalty, weighted by a penalty parameter $λ$. The proposed approach enables "automatic" model clustering, without prior knowledge of the hidden cluster structure, nor the number of clusters. Analytical bounds on the weight parameter, that lead to simultaneous personalization, generalization and automatic model clustering are provided. The solution to the formulated problem enables personalization, by providing different models across different clusters, and generalization, by providing models different than the per-user models computed in isolation. We then provide an efficient algorithm based on the Parallel Direction Method of Multipliers (PDMM) to solve the proposed formulation in a federated server-users setting. Numerical experiments corroborate our findings. As an interesting byproduct, our results provide several generalizations to convex clustering.

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