arXiv subjects

Artem Agafonov

Publications and source records attributed to Artem Agafonov.

At least 19 recordsLinked to original sources

arXiv2026

Smooth or Separable? Sparse Block Acceleration for Entropy-Regularized Linear Programs

We study entropy-regularized linear programs with sparse affine constraints and compare two exact dual representations induced by whether a redundant normalization constraint is retained or eliminated. Eliminating it yields a partially separable sum-exp dual with sparse affine factors but unbounded curvature. Our main result is an accelerated randomized block method whose acceleration preserves sparse data-access cost. The key ingredient is a global entropy-specific Hessian--gap bound that remains valid for signed constraint matrices, admits computable block-overlap refinements, and provides deterministic curvature certificates before sampling. This geometry supports square-root importance sampling and an exact lazy implementation in which sparse arithmetic cost is weighted by local curvature rather than by a worst-block factor. We further introduce a gap-dependent curvature dimension that connects the generalized-smooth regime to the classical effective-rank intuition for coordinate methods. Retaining the normalization constraint yields a log-sum-exp dual with globally bounded higher derivatives, enabling gradient-regularized and cubic-regularized Newton methods. Experiments compare the resulting methods in terms of coordinate epochs, arithmetic operations, and wall-clock time, separating the effects of acceleration, touched-data sparsity, and problem-specific structure.

math.OC↗

arXiv2026

Clipped or Unclipped? Finite-Sample Trade-offs for Averaged SGD under Heavy-Tailed Noise

Gradient clipping is widely used to stabilize training, but it need not improve the statistical accuracy of averaged SGD, even under heavy-tailed noise. We derive a finite-sample comparison of clipped and unclipped Polyak-Ruppert averaged SGD under finite conditional $p$-th moments, $p\ge2$. Our main result gives explicit accuracy and confidence conditions under which, for $p>2$, the Gaussian term dominates the unclipped deviation bound, so clipping need not improve its leading order. By balancing clipping bias and concentration, we obtain a bound in which the heavy-tail correction depends logarithmically rather than polynomially on the inverse failure probability. At $p=2$, this improves the confidence dependence of the leading bound. We establish sharpness of the unclipped heavy-tail term through an exact one-dimensional quadratic recursion and extend the comparison to projected convex SGD. We also prove concrete costs of clipping: every fixed finite threshold increases asymptotic variance on a scalar Gaussian quadratic, while whole-gradient clipping can shift the limiting point under asymmetric noise.

cs.LG↗

arXiv2026

Optimal Parameter-Free Second-order Acceleration with Inexact Hessians under Generalized Smoothness

We study second-order convex optimization under generalized Hessian smoothness, where the local Hessian Lipschitz constant may grow linearly with the gradient norm. For inexact Hessians, we develop adaptive algorithms, including an accelerated Monteiro-Svaiter-type method whose complexity matches the lower bound for second-order methods with $δ$-inexact Hessians in the classical Lipschitz-Hessian setting. Our methods are parameter-free in the sense that they require no problem-specific parameters as input, including the Hessian inexactness level $δ$ and smoothness constants. Instead, their backtracking adapts a single regularization parameter that jointly captures Hessian inexactness and local Hessian smoothness. The resulting rates split into two regimes. Far from the solution, the gradient-dependent smoothness term yields geometric convergence. Near the solution, the rate is governed by the ordinary Hessian smoothness and the inexactness level $δ$, recovering the classical Lipschitz-Hessian guarantees.

math.OC↗

arXiv2026

Decentralized Inexact Cubic Newton Method with Consensus Procedure

Distributed optimization is widely used in large-scale and privacy-preserving machine learning, where each agent stores a local objective and communicates only with its neighbors in a connected network. We study decentralized second-order optimization and focus on consensus procedures that approximately average local iterates, gradients, and Hessians through neighbor-to-neighbor communications. We propose a general Decentralized Cubic Newton method for convex optimization under $L_1$-smoothness of gradients and $L_2$-Lipschitz continuity of Hessians, and develop a theory that accurately tracks the inaccuracies caused by consensus and by disagreement between local iterates. Under these assumptions, the method matches the iteration complexity of the exact Cubic Newton method and requires only additional polylogarithmic communication-round overhead to reach the necessary consensus accuracy. We further propose an Accelerated Decentralized Cubic Newton method for strongly convex objectives and show that it matches the iteration complexity of the exact Accelerated Cubic Newton method, again with only additional polylogarithmic communication-round overhead. Finally, although the general method requires exchanging full $d \times d$ Hessian matrices, we show how it can be implemented for generalized linear models by transmitting only vectors, making the approach substantially more practical in high dimensions.

math.OC↗

arXiv2025

Simple Stepsize for Quasi-Newton Methods with Global Convergence Guarantees

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.

math.OC↗

arXiv2025

Multi-Agent Path Finding For Large Agents Is Intractable

The multi-agent path finding (MAPF) problem asks to find a set of paths on a graph such that when synchronously following these paths the agents never encounter a conflict. In the most widespread MAPF formulation, the so-called Classical MAPF, the agents sizes are neglected and two types of conflicts are considered: occupying the same vertex or using the same edge at the same time step. Meanwhile in numerous practical applications, e.g. in robotics, taking into account the agents' sizes is vital to ensure that the MAPF solutions can be safely executed. Introducing large agents yields an additional type of conflict arising when one agent follows an edge and its body overlaps with the body of another agent that is actually not using this same edge (e.g. staying still at some distinct vertex of the graph). Until now it was not clear how harder the problem gets when such conflicts are to be considered while planning. Specifically, it was known that Classical MAPF problem on an undirected graph can be solved in polynomial time, however no complete polynomial-time algorithm was presented to solve MAPF with large agents. In this paper we, for the first time, establish that the latter problem is NP-hard and, thus, if P!=NP no polynomial algorithm for it can, unfortunately, be presented. Our proof is based on the prevalent in the field technique of reducing the seminal 3SAT problem (which is known to be an NP-complete problem) to the problem at hand. In particular, for an arbitrary 3SAT formula we procedurally construct a dedicated graph with specific start and goal vertices and show that the given 3SAT formula is satisfiable iff the corresponding path finding instance has a solution.

cs.MA↗

arXiv2025

Learning Confident Classifiers in the Presence of Label Noise

The success of Deep Neural Network (DNN) models significantly depends on the quality of provided annotations. In medical image segmentation, for example, having multiple expert annotations for each data point is common to minimize subjective annotation bias. Then, the goal of estimation is to filter out the label noise and recover the ground-truth masks, which are not explicitly given. This paper proposes a probabilistic model for noisy observations that allows us to build a confident classification and segmentation models. To accomplish it, we explicitly model label noise and introduce a new information-based regularization that pushes the network to recover the ground-truth labels. In addition, for segmentation task we adjust the loss function by prioritizing learning in high-confidence regions where all the annotators agree on labeling. We evaluate the proposed method on a series of classification tasks such as noisy versions of MNIST, CIFAR-10, Fashion-MNIST datasets as well as CIFAR-10N, which is real-world dataset with noisy human annotations. Additionally, for segmentation task, we consider several medical imaging datasets, such as, LIDC and RIGA that reflect real-world inter-variability among multiple annotators. Our experiments show that our algorithm outperforms state-of-the-art solutions for the considered classification and segmentation problems.

cs.CV↗

arXiv2024

OPTAMI: Global Superlinear Convergence of High-order Methods

Second-order methods for convex optimization outperform first-order methods in terms of theoretical iteration convergence, achieving rates up to $O(k^{-5})$ for highly-smooth functions. However, their practical performance and applications are limited due to their multi-level structure and implementation complexity. In this paper, we present new results on high-order optimization methods, supported by their practical performance. First, we show that the basic high-order methods, such as the Cubic Regularized Newton Method, exhibit global superlinear convergence for $μ$-strongly star-convex functions, a class that includes $μ$-strongly convex functions and some non-convex functions. Theoretical convergence results are both inspired and supported by the practical performance of these methods. Secondly, we propose a practical version of the Nesterov Accelerated Tensor method, called NATA. It significantly outperforms the classical variant and other high-order acceleration techniques in practice. The convergence of NATA is also supported by theoretical results. Finally, we introduce an open-source computational library for high-order methods, called OPTAMI. This library includes various methods, acceleration techniques, and subproblem solvers, all implemented as PyTorch optimizers, thereby facilitating the practical application of high-order methods to a wide range of optimization problems. We hope this library will simplify research and practical comparison of methods beyond first-order.

math.OC↗

arXiv2024

Exploring Jacobian Inexactness in Second-Order Methods for Variational Inequalities: Lower Bounds, Optimal Algorithms and Quasi-Newton Approximations

Variational inequalities represent a broad class of problems, including minimization and min-max problems, commonly found in machine learning. Existing second-order and high-order methods for variational inequalities require precise computation of derivatives, often resulting in prohibitively high iteration costs. In this work, we study the impact of Jacobian inaccuracy on second-order methods. For the smooth and monotone case, we establish a lower bound with explicit dependence on the level of Jacobian inaccuracy and propose an optimal algorithm for this key setting. When derivatives are exact, our method converges at the same rate as exact optimal second-order methods. To reduce the cost of solving the auxiliary problem, which arises in all high-order methods with global convergence, we introduce several Quasi-Newton approximations. Our method with Quasi-Newton updates achieves a global sublinear convergence rate. We extend our approach with a tensor generalization for inexact high-order derivatives and support the theory with experiments.

math.OC↗

arXiv2024

Advancing the lower bounds: An accelerated, stochastic, second-order method with optimal adaptation to inexactness

We present a new accelerated stochastic second-order method that is robust to both gradient and Hessian inexactness, which occurs typically in machine learning. We establish theoretical lower bounds and prove that our algorithm achieves optimal convergence in both gradient and Hessian inexactness in this key setting. We further introduce a tensor generalization for stochastic higher-order derivatives. When the oracles are non-stochastic, the proposed tensor algorithm matches the global convergence of Nesterov Accelerated Tensor method. Both algorithms allow for approximate solutions of their auxiliary subproblems with verifiable conditions on the accuracy of the solution.

math.OC↗

arXiv2024

Exploiting higher-order derivatives in convex optimization methods

Exploiting higher-order derivatives in convex optimization is known at least since 1970's. In each iteration higher-order (also called tensor) methods minimize a regularized Taylor expansion of the objective function, which leads to faster convergence rates if the corresponding higher-order derivative is Lipschitz-continuous. Recently a series of lower iteration complexity bounds for such methods were proved, and a gap between upper an lower complexity bounds was revealed. Moreover, it was shown that such methods can be implementable since the appropriately regularized Taylor expansion of a convex function is also convex and, thus, can be minimized in polynomial time. Only very recently an algorithm with optimal convergence rate $1/k^{(3p+1)/2}$ was proposed for minimizing convex functions with Lipschitz $p$-th derivative. For convex functions with Lipschitz third derivative, these developments allowed to propose a second-order method with convergence rate $1/k^5$, which is faster than the rate $1/k^{3.5}$ of existing second-order methods.

math.OC↗

arXiv2023

Cubic Regularization is the Key! The First Accelerated Quasi-Newton Method with a Global Convergence Rate of $O(k^{-2})$ for Convex Functions

In this paper, we propose the first Quasi-Newton method with a global convergence rate of $O(k^{-1})$ for general convex functions. Quasi-Newton methods, such as BFGS, SR-1, are well-known for their impressive practical performance. However, they may be slower than gradient descent for general convex functions, with the best theoretical rate of $O(k^{-1/3})$. This gap between impressive practical performance and poor theoretical guarantees was an open question for a long period of time. In this paper, we make a significant step to close this gap. We improve upon the existing rate and propose the Cubic Regularized Quasi-Newton Method with a convergence rate of $O(k^{-1})$. The key to achieving this improvement is to use the Cubic Regularized Newton Method over the Damped Newton Method as an outer method, where the Quasi-Newton update is an inexact Hessian approximation. Using this approach, we propose the first Accelerated Quasi-Newton method with a global convergence rate of $O(k^{-2})$ for general convex functions. In special cases where we can improve the precision of the approximation, we achieve a global convergence rate of $O(k^{-3})$, which is faster than any first-order method. To make these methods practical, we introduce the Adaptive Inexact Cubic Regularized Newton Method and its accelerated version, which provide real-time control of the approximation error. We show that the proposed methods have impressive practical performance and outperform both first and second-order methods.

math.OC↗

arXiv2022

Inexact Tensor Methods and Their Application to Stochastic Convex Optimization

We propose general non-accelerated and accelerated tensor methods under inexact information on the derivatives of the objective, analyze their convergence rate. Further, we provide conditions for the inexactness in each derivative that is sufficient for each algorithm to achieve the desired accuracy. As a corollary, we propose stochastic tensor methods for convex optimization and obtain sufficient mini-batch sizes for each derivative.

math.OC↗

arXiv2022

FLECS-CGD: A Federated Learning Second-Order Framework via Compression and Sketching with Compressed Gradient Differences

In the recent paper FLECS (Agafonov et al, FLECS: A Federated Learning Second-Order Framework via Compression and Sketching), the second-order framework FLECS was proposed for the Federated Learning problem. This method utilize compression of sketched Hessians to make communication costs low. However, the main bottleneck of FLECS is gradient communication without compression. In this paper, we propose the modification of FLECS with compressed gradient differences, which we call FLECS-CGD (FLECS with Compressed Gradient Differences) and make it applicable for stochastic optimization. Convergence guarantees are provided in strongly convex and nonconvex cases. Experiments show the practical benefit of proposed approach.

cs.LG↗

arXiv2022

FLECS: A Federated Learning Second-Order Framework via Compression and Sketching

Inspired by the recent work FedNL (Safaryan et al, FedNL: Making Newton-Type Methods Applicable to Federated Learning), we propose a new communication efficient second-order framework for Federated learning, namely FLECS. The proposed method reduces the high-memory requirements of FedNL by the usage of an L-SR1 type update for the Hessian approximation which is stored on the central server. A low dimensional `sketch' of the Hessian is all that is needed by each device to generate an update, so that memory costs as well as number of Hessian-vector products for the agent are low. Biased and unbiased compressions are utilized to make communication costs also low. Convergence guarantees for FLECS are provided in both the strongly convex, and nonconvex cases, and local linear convergence is also established under strong convexity. Numerical experiments confirm the practical benefits of this new FLECS algorithm.

math.OC↗

arXiv2021

Inexact Relative Smoothness and Strong Convexity for Optimization and Variational Inequalities by Inexact Model

In this paper, we propose a general algorithmic framework for first-order methods in optimization in a broad sense, including minimization problems, saddle-point problems, and variational inequalities. This framework allows obtaining many known methods as a special case, the list including accelerated gradient method, composite optimization methods, level-set methods, Bregman proximal methods. The idea of the framework is based on constructing an inexact model of the main problem component, i.e. objective function in optimization or operator in variational inequalities. Besides reproducing known results, our framework allows constructing new methods, which we illustrate by constructing a universal conditional gradient method and a universal method for variational inequalities with a composite structure. This method works for smooth and non-smooth problems with optimal complexity without a priori knowledge of the problem's smoothness. As a particular case of our general framework, we introduce relative smoothness for operators and propose an algorithm for variational inequalities (VIs) with such operators. We also generalize our framework for relatively strongly convex objectives and strongly monotone variational inequalities. This paper is an extended and updated version of [arXiv:1902.00990]. In particular, we add an extension of relative strong convexity for optimization and variational inequalities.

math.OC↗

arXiv2021

An Accelerated Second-Order Method for Distributed Stochastic Optimization

We consider distributed stochastic optimization problems that are solved with master/workers computation architecture. Statistical arguments allow to exploit statistical similarity and approximate this problem by a finite-sum problem, for which we propose an inexact accelerated cubic-regularized Newton's method that achieves lower communication complexity bound for this setting and improves upon existing upper bound. We further exploit this algorithm to obtain convergence rate bounds for the original stochastic optimization problem and compare our bounds with the existing bounds in several regimes when the goal is to minimize the number of communication rounds and increase the parallelization by increasing the number of workers.

math.OC↗

arXiv2020

Lower bounds for conditional gradient type methods for minimizing smooth strongly convex functions

In this paper, we consider conditional gradient methods. These are methods that use a linear minimization oracle, which, for a given vector $p \in \mathbb{R}^n$, computes the solution of the subproblem $$\arg \min_{x\in X}{\langle p,x \rangle}.$$ There are a variety of conditional gradient methods that have a linear convergence rate in a strongly convex case. However, in all these methods, the dimension of the problem is included in the rate of convergence, which in modern applications can be very large. In this paper, we prove that in the strongly convex case, the convergence rate of the conditional gradient methods in the best case depends on the dimension of the problem $ n $ as $ \widetilde Ω (\sqrt {n}) $. Thus, the conditional gradient methods may turn out to be ineffective for solving strongly convex optimization problems of large dimensions. Also, the application of conditional gradient methods to minimization problems of a quadratic form is considered. The effectiveness of the Frank-Wolfe method for solving the quadratic optimization problem in the convex case on a simplex (PageRank) has already been proved. This work shows that the use of conditional gradient methods to solve the minimization problem of a quadratic form in a strongly convex case is ineffective due to the presence of dimension in the convergence rate of these methods. Therefore, the Shrinking Conditional Gradient method is considered. Its difference from the conditional gradient methods is that it uses a modified linear minimization oracle. The convergence rate of such an algorithm does not depend on dimension. Using the Shrinking Conditional Gradient method the complexity of solving the minimization problem of quadratic form on a $ \infty $-ball is obtained. The resulting evaluation of the method is comparable to the complexity of the gradient method.

math.OC↗