arXiv subjects

Alexander Tyurin

Publications and source records attributed to Alexander Tyurin.

At least 19 recordsLinked to original sources

arXiv2026

Global Convergence of Gradient Descent for Score Matching in Gaussian Mixtures via Reverse Fisher Divergence

The score matching problem is a central training objective in modern generative modeling, diffusion models, fitting unnormalized statistical models, and inverse problems. A standard approach is to minimize the forward Fisher divergence, where the expectation is taken with respect to the teacher distribution. However, recent results show that even in simple Gaussian mixture model settings, this objective can lead to undesirable and initialization-dependent convergence behavior. In this paper, we study an alternative objective: the reverse Fisher divergence, where the expectation is taken with respect to the student distribution. We analyze gradient descent (GD) for fitting Gaussian mixture models and show that this change in the objective leads to significantly better optimization properties. First, when the teacher distribution is a single Gaussian and the student is a Gaussian mixture model with fixed weights and identity covariances, we prove the global convergence of GD from arbitrary initializations. Second, we extend the analysis to the case where the teacher is also a Gaussian mixture model and prove global convergence guarantees under a global random initialization scheme and a $\widetilde{\Omega}(1)$-separation assumption on the target means. In particular, with high probability, each student component converges near its closest teacher component, and we provide conditions under which the student distribution converges in total variation distance. Our proofs rely on a new Lyapunov-based analysis of the gradient descent dynamics, showing that the reverse Fisher divergence has a much more favorable optimization landscape than the forward Fisher divergence.

cs.LG↗

arXiv2026

Scalable Distributed Stochastic Optimization via Bidirectional Compression: Beyond Pessimistic Limits

In centralized, distributed, and federated learning with stochastic gradients and $n$ workers, it was recently shown that it is infeasible to find an $\varepsilon$-stationary point faster than $\tilde{\Omega}\left(\min\left\{\frac{d \kappa L \Delta}{\varepsilon} + \frac{h L \Delta}{\varepsilon} + \frac{h \sigma^2 L \Delta}{n \varepsilon^2},\; \frac{h \sigma^2 L \Delta}{\varepsilon^2} + \frac{h L \Delta}{\varepsilon}\right\}\right)$ seconds in both homogeneous and heterogeneous settings under standard assumptions: $L$-smoothness, $\sigma^2$-bounded unbiased stochastic gradients, and lower boundedness of the function, i.e., $f(x) \geq f^*$ for all $x \in \mathbb{R}^d$, where $\Delta = f(x^0) - f^*$, $h$ is the computation time, $\kappa$ is the communication speed between the workers and the server, and $d$ is the dimension of the iterates and gradients. This result is pessimistic since it does not allow a complexity in which both $\frac{d \kappa L \Delta}{\varepsilon}$ and $\frac{h \sigma^2 L \Delta}{\varepsilon^2}$ improve with $n$, even when using random sparsification techniques; moreover, this lower bound can be matched by either non-distributed SGD or vanilla Synchronous SGD, which reduces the impact of recent progress in the design of compression-based methods. In this work, we challenge this limitation and propose new compressed methods, Inkheart SGD and M4, and show that under an additional structural assumption, which is necessary due to the lower bound and which does not restrict the class of considered problems, we achieve new state-of-the-art time complexities that break this pessimistic barrier and allow scaling with the number of workers $n$.

math.OC↗

arXiv2026

Optimality in Decentralized Optimization under Bandwidth Constraints

We consider a realistic decentralized setup with bandwidth-constrained communication and derive optimal time complexities for non-convex stochastic parallel and asynchronous optimization (up to logarithmic factors). We develop the corresponding methods, Grace SGD and Leon SGD, for both homogeneous and heterogeneous settings. Unlike previous work, our optimal bounds are characterized in terms of min-cut/max-flow quantities and rely on tools from Gomory-Hu trees and Steiner Tree Packing problems, providing tighter and more practical complexities.

math.OC↗

arXiv2026

Do We Need Asynchronous SGD? On the Near-Optimality of Synchronous Methods

Modern distributed optimization methods mostly rely on traditional synchronous approaches, despite substantial recent progress in asynchronous optimization. We revisit Synchronous SGD and its robust variant, called $m$-Synchronous SGD, and theoretically show that they are nearly optimal in many heterogeneous computation scenarios, which is somewhat unexpected. We analyze the synchronous methods under random computation times and adversarial partial participation of workers, and prove that their time complexities are optimal in many practical regimes, up to logarithmic factors. While synchronous methods are not universal solutions and there exist tasks where asynchronous methods may be necessary, we show that they are sufficient for many modern heterogeneous computation scenarios.

cs.DC↗

arXiv2025

Gradient Descent as a Perceptron Algorithm: Understanding Dynamics and Implicit Acceleration

Even for the gradient descent (GD) method applied to neural network training, understanding its optimization dynamics, including convergence rate, iterate trajectories, function value oscillations, and especially its implicit acceleration, remains a challenging problem. We analyze nonlinear models with the logistic loss and show that the steps of GD reduce to those of generalized perceptron algorithms (Rosenblatt, 1958), providing a new perspective on the dynamics. This reduction yields significantly simpler algorithmic steps, which we analyze using classical linear algebra tools. Using these tools, we demonstrate on a minimalistic example that the nonlinearity in a two-layer model can provably yield a faster iteration complexity $\tilde{O}(\sqrt{d})$ compared to $\Omega(d)$ achieved by linear models, where $d$ is the number of features. This helps explain the optimization dynamics and the implicit acceleration phenomenon observed in neural networks. The theoretical results are supported by extensive numerical experiments. We believe that this alternative view will further advance research on the optimization of neural networks.

cs.LG↗

arXiv2025

Asynchronous Policy Gradient Aggregation for Efficient Distributed Reinforcement Learning

We study distributed reinforcement learning (RL) with policy gradient methods under asynchronous and parallel computations and communications. While non-distributed methods are well understood theoretically and have achieved remarkable empirical success, their distributed counterparts remain less explored, particularly in the presence of heterogeneous asynchronous computations and communication bottlenecks. We introduce two new algorithms, Rennala NIGT and Malenia NIGT, which implement asynchronous policy gradient aggregation and achieve state-of-the-art efficiency. In the homogeneous setting, Rennala NIGT provably improves the total computational and communication complexity while supporting the AllReduce operation. In the heterogeneous setting, Malenia NIGT simultaneously handles asynchronous computations and heterogeneous environments with strictly better theoretical guarantees. Our results are further corroborated by experiments, showing that our methods significantly outperform prior approaches.

cs.LG↗

arXiv2025

Local SGD and Federated Averaging Through the Lens of Time Complexity

We revisit the classical Local SGD and Federated Averaging (FedAvg) methods for distributed optimization and federated learning. While prior work has primarily focused on iteration complexity, we analyze these methods through the lens of time complexity, taking into account both computation and communication costs. Our analysis reveals that, despite its favorable iteration complexity, the time complexity of canonical Local SGD is provably worse than that of Minibatch SGD and Hero SGD (locally executed SGD). We introduce a corrected variant, Dual Local SGD, and further improve it by increasing the local step sizes, leading to a new method called Decaying Local SGD. Our analysis shows that these modifications, together with Hero SGD, are optimal in the nonconvex setting (up to logarithmic factors), closing the time complexity gap. Finally, we use these insights to improve the theory of a number of other asynchronous and local methods.

math.OC↗

arXiv2025

Near-Optimal Convergence of Accelerated Gradient Methods under Generalized and $(L_0, L_1)$-Smoothness

We study first-order methods for convex optimization problems with functions $f$ satisfying the recently proposed $\ell$-smoothness condition $||\nabla^{2}f(x)|| \le \ell\left(||\nabla f(x)||\right),$ which generalizes the $L$-smoothness and $(L_{0},L_{1})$-smoothness. While accelerated gradient descent AGD is known to reach the optimal complexity $O(\sqrt{L} R / \sqrt{\varepsilon})$ under $L$-smoothness, where $\varepsilon$ is an error tolerance and $R$ is the distance between a starting and an optimal point, existing extensions to $\ell$-smoothness either incur extra dependence on the initial gradient, suffer exponential factors in $L_{1} R$, or require costly auxiliary sub-routines, leaving open whether an AGD-type $O(\sqrt{\ell(0)} R / \sqrt{\varepsilon})$ rate is possible for small-$\varepsilon$, even in the $(L_{0},L_{1})$-smoothness case. We resolve this open question. Leveraging a new Lyapunov function and designing new algorithms, we achieve $O(\sqrt{\ell(0)} R / \sqrt{\varepsilon})$ oracle complexity for small-$\varepsilon$ and virtually any $\ell$. For instance, for $(L_{0},L_{1})$-smoothness, our bound $O(\sqrt{L_0} R / \sqrt{\varepsilon})$ is provably optimal in the small-$\varepsilon$ regime and removes all non-constant multiplicative factors present in prior accelerated algorithms.

math.OC↗

arXiv2025

Proving the Limited Scalability of Centralized Distributed Optimization via a New Lower Bound Construction

We consider centralized distributed optimization in the classical federated learning setup, where $n$ workers jointly find an $\varepsilon$-stationary point of an $L$-smooth, $d$-dimensional nonconvex function $f$, having access only to unbiased stochastic gradients with variance $\sigma^2$. Each worker requires at most $h$ seconds to compute a stochastic gradient, and the communication times from the server to the workers and from the workers to the server are $\tau_{s}$ and $\tau_{w}$ seconds per coordinate, respectively. One of the main motivations for distributed optimization is to achieve scalability with respect to $n$. For instance, it is well known that the distributed version of SGD has a variance-dependent runtime term $\frac{h \sigma^2 L \Delta}{n \varepsilon^2},$ which improves with the number of workers $n,$ where $\Delta = f(x^0) - f^*,$ and $x^0 \in R^d$ is the starting point. Similarly, using unbiased sparsification compressors, it is possible to reduce both the variance-dependent runtime term and the communication runtime term. However, once we account for the communication from the server to the workers $\tau_{s}$, we prove that it becomes infeasible to design a method using unbiased random sparsification compressors that scales both the server-side communication runtime term $\tau_{s} d \frac{L \Delta}{\varepsilon}$ and the variance-dependent runtime term $\frac{h \sigma^2 L \Delta}{\varepsilon^2},$ better than poly-logarithmically in $n$, even in the homogeneous (i.i.d.) case, where all workers access the same distribution. To establish this result, we construct a new "worst-case" function and develop a new lower bound framework that reduces the analysis to the concentration of a random sum, for which we prove a concentration bound. These results reveal fundamental limitations in scaling distributed optimization, even under the homogeneous assumption.

math.OC↗

arXiv2025

Learning of Population Dynamics: Inverse Optimization Meets JKO Scheme

Learning population dynamics involves recovering the underlying process that governs particle evolution, given evolutionary snapshots of samples at discrete time points. Recent methods frame this as an energy minimization problem in probability space and leverage the celebrated JKO scheme for efficient time discretization. In this work, we introduce $\texttt{iJKOnet}$, an approach that combines the JKO framework with inverse optimization techniques to learn population dynamics. Our method relies on a conventional $\textit{end-to-end}$ adversarial training procedure and does not require restrictive architectural choices, e.g., input-convex neural networks. We establish theoretical guarantees for our methodology and demonstrate improved performance over prior JKO-based methods. The code of $\texttt{iJKOnet}$ is available at https://github.com/MuXauJl11110/iJKOnet.

cs.LG↗

arXiv2025

Birch SGD: A Tree Graph Framework for Local and Asynchronous SGD Methods

We propose a new unifying framework, Birch SGD, for analyzing and designing distributed SGD methods. The central idea is to represent each method as a weighted directed tree, referred to as a computation tree. Leveraging this representation, we introduce a general theoretical result that reduces convergence analysis to studying the geometry of these trees. This perspective yields a purely graph-based interpretation of optimization dynamics, offering a new and intuitive foundation for method development. Using Birch SGD, we design eight new methods and analyze them alongside previously known ones, with at least six of the new methods shown to have optimal computational time complexity. Our research leads to two key insights: (i) all methods share the same "iteration rate" of $O\left(\frac{(R + 1) L \Delta}{\varepsilon} + \frac{\sigma^2 L \Delta}{\varepsilon^2}\right)$, where $R$ the maximum "tree distance" along the main branch of a tree; and (ii) different methods exhibit different trade-offs-for example, some update iterates more frequently, improving practical performance, while others are more communication-efficient or focus on other aspects. Birch SGD serves as a unifying framework for navigating these trade-offs. We believe these results provide a unified foundation for understanding, analyzing, and designing efficient asynchronous and parallel optimization methods.

cs.LG↗

arXiv2025

Ringmaster ASGD: The First Asynchronous SGD with Optimal Time Complexity

Asynchronous Stochastic Gradient Descent (Asynchronous SGD) is a cornerstone method for parallelizing learning in distributed machine learning. However, its performance suffers under arbitrarily heterogeneous computation times across workers, leading to suboptimal time complexity and inefficiency as the number of workers scales. While several Asynchronous SGD variants have been proposed, recent findings by Tyurin & Richt\'arik (NeurIPS 2023) reveal that none achieve optimal time complexity, leaving a significant gap in the literature. In this paper, we propose Ringmaster ASGD, a novel Asynchronous SGD method designed to address these limitations and tame the inherent challenges of Asynchronous SGD. We establish, through rigorous theoretical analysis, that Ringmaster ASGD achieves optimal time complexity under arbitrarily heterogeneous and dynamically fluctuating worker computation times. This makes it the first Asynchronous SGD method to meet the theoretical lower bounds for time complexity in such scenarios.

cs.LG↗

arXiv2024

Toward a Unified Theory of Gradient Descent under Generalized Smoothness

We study the classical optimization problem $\min_{x \in \mathbb{R}^d} f(x)$ and analyze the gradient descent (GD) method in both nonconvex and convex settings. It is well-known that, under the $L$-smoothness assumption ($\|\nabla^2 f(x)\| \leq L$), the optimal point minimizing the quadratic upper bound $f(x_k) + \langle\nabla f(x_k), x_{k+1} - x_k\rangle + \frac{L}{2} \|x_{k+1} - x_k\|^2$ is $x_{k+1} = x_k - \gamma_k \nabla f(x_k)$ with step size $\gamma_k = \frac{1}{L}$. Surprisingly, a similar result can be derived under the $\ell$-generalized smoothness assumption ($\|\nabla^2 f(x)\| \leq \ell(\|\nabla f(x)\|)$). In this case, we derive the step size $$\gamma_k = \int_{0}^{1} \frac{d v}{\ell(\|\nabla f(x_k)\| + \|\nabla f(x_k)\| v)}.$$ Using this step size rule, we improve upon existing theoretical convergence rates and obtain new results in several previously unexplored setups.

math.OC↗

arXiv2024

From Logistic Regression to the Perceptron Algorithm: Exploring Gradient Descent with Large Step Sizes

We focus on the classification problem with a separable dataset, one of the most important and classical problems from machine learning. The standard approach to this task is logistic regression with gradient descent (LR+GD). Recent studies have observed that LR+GD can find a solution with arbitrarily large step sizes, defying conventional optimization theory. Our work investigates this phenomenon and makes three interconnected key observations about LR+GD with large step sizes. First, we find a remarkably simple explanation of why LR+GD with large step sizes solves the classification problem: LR+GD reduces to a batch version of the celebrated perceptron algorithm when the step size $\gamma \to \infty.$ Second, we observe that larger step sizes lead LR+GD to higher logistic losses when it tends to the perceptron algorithm, but larger step sizes also lead to faster convergence to a solution for the classification problem, meaning that logistic loss is an unreliable metric of the proximity to a solution. Surprisingly, high loss values can actually indicate faster convergence. Third, since the convergence rate in terms of loss function values of LR+GD is unreliable, we examine the iteration complexity required by LR+GD with large step sizes to solve the classification problem and prove that this complexity is suboptimal. To address this, we propose a new method, Normalized LR+GD - based on the connection between LR+GD and the perceptron algorithm - with much better theoretical guarantees.

cs.LG↗

arXiv2024

Tighter Performance Theory of FedExProx

We revisit FedExProx - a recently proposed distributed optimization method designed to enhance convergence properties of parallel proximal algorithms via extrapolation. In the process, we uncover a surprising flaw: its known theoretical guarantees on quadratic optimization tasks are no better than those offered by the vanilla Gradient Descent (GD) method. Motivated by this observation, we develop a novel analysis framework, establishing a tighter linear convergence rate for non-strongly convex quadratic problems. By incorporating both computation and communication costs, we demonstrate that FedExProx can indeed provably outperform GD, in stark contrast to the original analysis. Furthermore, we consider partial participation scenarios and analyze two adaptive extrapolation strategies - based on gradient diversity and Polyak stepsizes - again significantly outperforming previous results. Moving beyond quadratics, we extend the applicability of our analysis to general functions satisfying the Polyak-Lojasiewicz condition, outperforming the previous strongly convex analysis while operating under weaker assumptions. Backed by empirical results, our findings point to a new and stronger potential of FedExProx, paving the way for further exploration of the benefits of extrapolation in federated learning.

math.OC↗

arXiv2024

Tight Time Complexities in Parallel Stochastic Optimization with Arbitrary Computation Dynamics

In distributed stochastic optimization, where parallel and asynchronous methods are employed, we establish optimal time complexities under virtually any computation behavior of workers/devices/CPUs/GPUs, capturing potential disconnections due to hardware and network delays, time-varying computation powers, and any possible fluctuations and trends of computation speeds. These real-world scenarios are formalized by our new universal computation model. Leveraging this model and new proof techniques, we discover tight lower bounds that apply to virtually all synchronous and asynchronous methods, including Minibatch SGD, Asynchronous SGD (Recht et al., 2011), and Picky SGD (Cohen et al., 2021). We show that these lower bounds, up to constant factors, are matched by the optimal Rennala SGD and Malenia SGD methods (Tyurin & Richt\'arik, 2023).

math.OC↗

arXiv2024

On the Optimal Time Complexities in Decentralized Stochastic Asynchronous Optimization

We consider the decentralized stochastic asynchronous optimization setup, where many workers asynchronously calculate stochastic gradients and asynchronously communicate with each other using edges in a multigraph. For both homogeneous and heterogeneous setups, we prove new time complexity lower bounds under the assumption that computation and communication speeds are bounded. We develop a new nearly optimal method, Fragile SGD, and a new optimal method, Amelie SGD, that converge under arbitrary heterogeneous computation and communication speeds and match our lower bounds (up to a logarithmic factor in the homogeneous setting). Our time complexities are new, nearly optimal, and provably improve all previous asynchronous/synchronous stochastic methods in the decentralized setup.

math.OC↗

arXiv2024

Freya PAGE: First Optimal Time Complexity for Large-Scale Nonconvex Finite-Sum Optimization with Heterogeneous Asynchronous Computations

In practical distributed systems, workers are typically not homogeneous, and due to differences in hardware configurations and network conditions, can have highly varying processing times. We consider smooth nonconvex finite-sum (empirical risk minimization) problems in this setup and introduce a new parallel method, Freya PAGE, designed to handle arbitrarily heterogeneous and asynchronous computations. By being robust to "stragglers" and adaptively ignoring slow computations, Freya PAGE offers significantly improved time complexity guarantees compared to all previous methods, including Asynchronous SGD, Rennala SGD, SPIDER, and PAGE, while requiring weaker assumptions. The algorithm relies on novel generic stochastic gradient collection strategies with theoretical guarantees that can be of interest on their own, and may be used in the design of future optimization methods. Furthermore, we establish a lower bound for smooth nonconvex finite-sum problems in the asynchronous setup, providing a fundamental time complexity limit. This lower bound is tight and demonstrates the optimality of Freya PAGE in the large-scale regime, i.e., when $\sqrt{m} \geq n$, where $n$ is # of workers, and $m$ is # of data samples.

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