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Alexander Gasnikov

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23

AAAI Conference 2026 Conference Paper

Stochastic Decentralized Optimization of Non-Smooth Convex and Convex-Concave Problems over Time-Varying Networks

  • Maxim Divilkovskiy
  • Alexander Gasnikov

We study non-smooth stochastic decentralized optimization problems over time-varying networks, where objective functions are distributed across nodes and network connections may intermittently appear or break. Specifically, we consider two settings: (i) stochastic non-smooth (strongly) convex optimization, and (ii) stochastic non-smooth (strongly) convex–(strongly) concave saddle point optimization. Convex problems of this type commonly arise in deep neural network training, while saddle point problems are central to machine learning tasks such as the training of generative adversarial networks (GANs). Prior works have primarily focused on the smooth setting, or time-invariant network scenarios. We extend the existing theory to the more general non-smooth and stochastic setting over time-varying networks and saddle point problems. Our analysis establishes upper bounds on both the number of stochastic oracle calls and communication rounds, matching lower bounds for both convex and saddle point optimization problems.

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NeurIPS Conference 2024 Conference Paper

Acceleration Exists! Optimization Problems When Oracle Can Only Compare Objective Function Values

  • Aleksandr Lobanov
  • Alexander Gasnikov
  • Andrei Krasnov

Frequently, the burgeoning field of black-box optimization encounters challenges due to a limited understanding of the mechanisms of the objective function. To address such problems, in this work we focus on the deterministic concept of Order Oracle, which only utilizes order access between function values (possibly with some bounded noise), but without assuming access to their values. As theoretical results, we propose a new approach to create non-accelerated optimization algorithms (obtained by integrating Order Oracle into existing optimization “tools”) in non-convex, convex, and strongly convex settings that are as good as both SOTA coordinate algorithms with first-order oracle and SOTA algorithms with Order Oracle up to logarithm factor. Moreover, using the proposed approach, we provide the first accelerated optimization algorithm using the Order Oracle. And also, using an already different approach we provide the asymptotic convergence of the first algorithm with the stochastic Order Oracle concept. Finally, our theoretical results demonstrate effectiveness of proposed algorithms through numerical experiments.

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NeurIPS Conference 2024 Conference Paper

Achieving Linear Convergence with Parameter-Free Algorithms in Decentralized Optimization

  • Ilya Kuruzov
  • Gesualdo Scutari
  • Alexander Gasnikov

This paper addresses the minimization of the sum of strongly convex, smoothfunctions over a network of agents without a centralized server. Existing decentralized algorithms require knowledge of functions and network parameters, such as the Lipschitz constant of the global gradient and/or network connectivity, forhyperparameter tuning. Agents usually cannot access this information, leadingto conservative selections and slow convergence or divergence. This paper introduces a decentralized algorithm that eliminates the need for specific parametertuning. Our approach employs an operator splitting technique with a novel variablemetric, enabling a local backtracking line-search to adaptively select the stepsizewithout global information or extensive communications. This results in favorableconvergence guarantees and dependence on optimization and network parameterscompared to existing nonadaptive methods. Notably, our method is the first adaptive decentralized algorithm that achieves linear convergence for strongly convex, smooth objectives. Preliminary numerical experiments support our theoreticalfindings, demonstrating superior performance in convergence speed and scalability.

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NeurIPS Conference 2024 Conference Paper

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

  • Artem Agafonov
  • Petr Ostroukhov
  • Roman Mozhaev
  • Konstantin Yakovlev
  • Eduard Gorbunov
  • Martin Takáč
  • Alexander Gasnikov
  • Dmitry Kamzolov

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.

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NeurIPS Conference 2024 Conference Paper

Lower Bounds and Optimal Algorithms for Non-Smooth Convex Decentralized Optimization over Time-Varying Networks

  • Dmitry Kovalev
  • Ekaterina Borodich
  • Alexander Gasnikov
  • Dmitrii Feoktistov

We consider the task of minimizing the sum of convex functions stored in a decentralized manner across the nodes of a communication network. This problem is relatively well-studied in the scenario when the objective functions are smooth, or the links of the network are fixed in time, or both. In particular, lower bounds on the number of decentralized communications and (sub)gradient computations required to solve the problem have been established, along with matching optimal algorithms. However, the remaining and most challenging setting of non-smooth decentralized optimization over time-varying networks is largely underexplored, as neither lower bounds nor optimal algorithms are known in the literature. We resolve this fundamental gap with the following contributions: (i) we establish the first lower bounds on the communication and subgradient computation complexities of solving non-smooth convex decentralized optimization problems over time-varying networks; (ii) we develop the first optimal algorithm that matches these lower bounds and offers substantially improved theoretical performance compared to the existing state of the art.

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NeurIPS Conference 2024 Conference Paper

Optimal Flow Matching: Learning Straight Trajectories in Just One Step

  • Nikita Kornilov
  • Petr Mokrov
  • Alexander Gasnikov
  • Alexander Korotin

Over the several recent years, there has been a boom in development of Flow Matching (FM) methods for generative modeling. One intriguing property pursued by the community is the ability to learn flows with straight trajectories which realize the Optimal Transport (OT) displacements. Straightness is crucial for the fast integration (inference) of the learned flow's paths. Unfortunately, most existing flow straightening methods are based on non-trivial iterative FM procedures which accumulate the error during training or exploit heuristics based on minibatch OT. To address these issues, we develop and theoretically justify the novel Optimal Flow Matching approach which allows recovering the straight OT displacement for the quadratic transport in just one FM step. The main idea of our approach is the employment of vector field for FM which are parameterized by convex functions. The code of our OFM implementation and the conducted experiments is available at https: //github. com/Jhomanik/Optimal-Flow-Matching

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NeurIPS Conference 2023 Conference Paper

Accelerated Zeroth-order Method for Non-Smooth Stochastic Convex Optimization Problem with Infinite Variance

  • Nikita Kornilov
  • Ohad Shamir
  • Aleksandr Lobanov
  • Darina Dvinskikh
  • Alexander Gasnikov
  • Innokentiy Shibaev
  • Eduard Gorbunov
  • Samuel Horváth

In this paper, we consider non-smooth stochastic convex optimization with two function evaluations per round under infinite noise variance. In the classical setting when noise has finite variance, an optimal algorithm, built upon the batched accelerated gradient method, was proposed in (Gasnikov et. al. , 2022). This optimality is defined in terms of iteration and oracle complexity, as well as the maximal admissible level of adversarial noise. However, the assumption of finite variance is burdensome and it might not hold in many practical scenarios. To address this, we demonstrate how to adapt a refined clipped version of the accelerated gradient (Stochastic Similar Triangles) method from (Sadiev et al. , 2023) for a two-point zero-order oracle. This adaptation entails extending the batching technique to accommodate infinite variance — a non-trivial task that stands as a distinct contribution of this paper.

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NeurIPS Conference 2023 Conference Paper

First Order Methods with Markovian Noise: from Acceleration to Variational Inequalities

  • Aleksandr Beznosikov
  • Sergey Samsonov
  • Marina Sheshukova
  • Alexander Gasnikov
  • Alexey Naumov
  • Eric Moulines

This paper delves into stochastic optimization problems that involve Markovian noise. We present a unified approach for the theoretical analysis of first-order gradient methods for stochastic optimization and variational inequalities. Our approach covers scenarios for both non-convex and strongly convex minimization problems. To achieve an optimal (linear) dependence on the mixing time of the underlying noise sequence, we use the randomized batching scheme, which is based on the multilevel Monte Carlo method. Moreover, our technique allows us to eliminate the limiting assumptions of previous research on Markov noise, such as the need for a bounded domain and uniformly bounded stochastic gradients. Our extension to variational inequalities under Markovian noise is original. Additionally, we provide lower bounds that match the oracle complexity of our method in the case of strongly convex optimization problems.

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NeurIPS Conference 2023 Conference Paper

Similarity, Compression and Local Steps: Three Pillars of Efficient Communications for Distributed Variational Inequalities

  • Aleksandr Beznosikov
  • Martin Takac
  • Alexander Gasnikov

Variational inequalities are a broad and flexible class of problems that includes minimization, saddle point, and fixed point problems as special cases. Therefore, variational inequalities are used in various applications ranging from equilibrium search to adversarial learning. With the increasing size of data and models, today's instances demand parallel and distributed computing for real-world machine learning problems, most of which can be represented as variational inequalities. Meanwhile, most distributed approaches have a significant bottleneck -- the cost of communications. The three main techniques to reduce the total number of communication rounds and the cost of one such round are the similarity of local functions, compression of transmitted information, and local updates. In this paper, we combine all these approaches. Such a triple synergy did not exist before for variational inequalities and saddle problems, nor even for minimization problems. The methods presented in this paper have the best theoretical guarantees of communication complexity and are significantly ahead of other methods for distributed variational inequalities. The theoretical results are confirmed by adversarial learning experiments on synthetic and real datasets.

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NeurIPS Conference 2022 Conference Paper

A Damped Newton Method Achieves Global $\mathcal O \left(\frac{1}{k^2}\right)$ and Local Quadratic Convergence Rate

  • Slavomír Hanzely
  • Dmitry Kamzolov
  • Dmitry Pasechnyuk
  • Alexander Gasnikov
  • Peter Richtarik
  • Martin Takac

In this paper, we present the first stepsize schedule for Newton method resulting in fast global and local convergence guarantees. In particular, we a) prove an $\mathcal O \left( 1/{k^2} \right)$ global rate, which matches the state-of-the-art global rate of cubically regularized Newton method of Polyak and Nesterov (2006) and of regularized Newton method of Mishchenko (2021), and the later variant of Doikov and Nesterov (2021), b) prove a local quadratic rate, which matches the best-known local rate of second-order methods, and c) our stepsize formula is simple, explicit, and does not require solving any subproblem. Our convergence proofs hold under affine-invariant assumptions closely related to the notion of self-concordance. Finally, our method has competitive performance when compared to existing baselines which share the same fast global convergence guarantees.

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NeurIPS Conference 2022 Conference Paper

Accelerated Primal-Dual Gradient Method for Smooth and Convex-Concave Saddle-Point Problems with Bilinear Coupling

  • Dmitry Kovalev
  • Alexander Gasnikov
  • Peter Richtarik

In this paper we study the convex-concave saddle-point problem $\min_x \max_y f(x) + y^\top\mathbf{A}x - g(y)$, where $f(x)$ and $g(y)$ are smooth and convex functions. We propose an Accelerated Primal-Dual Gradient Method (APDG) for solving this problem, achieving (i) an optimal linear convergence rate in the strongly-convex-strongly-concave regime, matching the lower complexity bound (Zhang et al. , 2021), and (ii) an accelerated linear convergence rate in the case when only one of the functions $f(x)$ and $g(y)$ is strongly convex or even none of them are. Finally, we obtain a linearly convergent algorithm for the general smooth and convex-concave saddle point problem $\min_x \max_y F(x, y)$ without the requirement of strong convexity or strong concavity.

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NeurIPS Conference 2022 Conference Paper

Clipped Stochastic Methods for Variational Inequalities with Heavy-Tailed Noise

  • Eduard Gorbunov
  • Marina Danilova
  • David Dobre
  • Pavel Dvurechenskii
  • Alexander Gasnikov
  • Gauthier Gidel

Stochastic first-order methods such as Stochastic Extragradient (SEG) or Stochastic Gradient Descent-Ascent (SGDA) for solving smooth minimax problems and, more generally, variational inequality problems (VIP) have been gaining a lot of attention in recent years due to the growing popularity of adversarial formulations in machine learning. While high-probability convergence bounds are known to more accurately reflect the actual behavior of stochastic methods, most convergence results are provided in expectation. Moreover, the only known high-probability complexity results have been derived under restrictive sub-Gaussian (light-tailed) noise and bounded domain assumptions [Juditsky et al. , 2011]. In this work, we prove the first high-probability complexity results with logarithmic dependence on the confidence level for stochastic methods for solving monotone and structured non-monotone VIPs with non-sub-Gaussian (heavy-tailed) noise and unbounded domains. In the monotone case, our results match the best known ones in the light-tails case [Juditsky et al. , 2011], and are novel for structured non-monotone problems such as negative comonotone, quasi-strongly monotone, and/or star-cocoercive ones. We achieve these results by studying SEG and SGDA with clipping. In addition, we numerically validate that the gradient noise of many practical GAN formulations is heavy-tailed and show that clipping improves the performance of SEG/SGDA.

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NeurIPS Conference 2022 Conference Paper

Decentralized Local Stochastic Extra-Gradient for Variational Inequalities

  • Aleksandr Beznosikov
  • Pavel Dvurechenskii
  • Anastasiia Koloskova
  • Valentin Samokhin
  • Sebastian U. Stich
  • Alexander Gasnikov

We consider distributed stochastic variational inequalities (VIs) on unbounded domains with the problem data that is heterogeneous (non-IID) and distributed across many devices. We make a very general assumption on the computational network that, in particular, covers the settings of fully decentralized calculations with time-varying networks and centralized topologies commonly used in Federated Learning. Moreover, multiple local updates on the workers can be made for reducing the communication frequency between the workers. We extend the stochastic extragradient method to this very general setting and theoretically analyze its convergence rate in the strongly-monotone, monotone, and non-monotone (when a Minty solution exists) settings. The provided rates explicitly exhibit the dependence on network characteristics (e. g. , mixing time), iteration counter, data heterogeneity, variance, number of devices, and other standard parameters. As a special case, our method and analysis apply to distributed stochastic saddle-point problems (SPP), e. g. , to the training of Deep Generative Adversarial Networks (GANs) for which decentralized training has been reported to be extremely challenging. In experiments for the decentralized training of GANs we demonstrate the effectiveness of our proposed approach.

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NeurIPS Conference 2022 Conference Paper

Distributed Methods with Compressed Communication for Solving Variational Inequalities, with Theoretical Guarantees

  • Aleksandr Beznosikov
  • Peter Richtarik
  • Michael Diskin
  • Max Ryabinin
  • Alexander Gasnikov

Variational inequalities in general and saddle point problems in particular are increasingly relevant in machine learning applications, including adversarial learning, GANs, transport and robust optimization. With increasing data and problem sizes necessary to train high performing models across various applications, we need to rely on parallel and distributed computing. However, in distributed training, communication among the compute nodes is a key bottleneck during training, and this problem is exacerbated for high dimensional and over-parameterized models. Due to these considerations, it is important to equip existing methods with strategies that would allow to reduce the volume of transmitted information during training while obtaining a model of comparable quality. In this paper, we present the first theoretically grounded distributed methods for solving variational inequalities and saddle point problems using compressed communication: MASHA1 and MASHA2. Our theory and methods allow for the use of both unbiased (such as Rand$k$; MASHA1) and contractive (such as Top$k$; MASHA2) compressors. New algorithms support bidirectional compressions, and also can be modified for stochastic setting with batches and for federated learning with partial participation of clients. We empirically validated our conclusions using two experimental setups: a standard bilinear min-max problem, and large-scale distributed adversarial training of transformers.

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NeurIPS Conference 2022 Conference Paper

Optimal Algorithms for Decentralized Stochastic Variational Inequalities

  • Dmitry Kovalev
  • Aleksandr Beznosikov
  • Abdurakhmon Sadiev
  • Michael Persiianov
  • Peter Richtarik
  • Alexander Gasnikov

Variational inequalities are a formalism that includes games, minimization, saddle point, and equilibrium problems as special cases. Methods for variational inequalities are therefore universal approaches for many applied tasks, including machine learning problems. This work concentrates on the decentralized setting, which is increasingly important but not well understood. In particular, we consider decentralized stochastic (sum-type) variational inequalities over fixed and time-varying networks. We present lower complexity bounds for both communication and local iterations and construct optimal algorithms that match these lower bounds. Our algorithms are the best among the available literature not only in the decentralized stochastic case, but also in the decentralized deterministic and non-distributed stochastic cases. Experimental results confirm the effectiveness of the presented algorithms.

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NeurIPS Conference 2022 Conference Paper

Optimal Gradient Sliding and its Application to Optimal Distributed Optimization Under Similarity

  • Dmitry Kovalev
  • Aleksandr Beznosikov
  • Ekaterina Borodich
  • Alexander Gasnikov
  • Gesualdo Scutari

We study structured convex optimization problems, with additive objective $r: =p + q$, where $r$ is ($\mu$-strongly) convex, $q$ is $L_q$-smooth and convex, and $p$ is $L_p$-smooth, possibly nonconvex. For such a class of problems, we proposed an inexact accelerated gradient sliding method that can skip the gradient computation for one of these components while still achieving optimal complexity of gradient calls of $p$ and $q$, that is, $\mathcal{O}(\sqrt{L_p/\mu})$ and $\mathcal{O}(\sqrt{L_q/\mu})$, respectively. This result is much sharper than the classic black-box complexity $\mathcal{O}(\sqrt{(L_p+L_q)/\mu})$, especially when the difference between $L_p$ and $L_q$ is large. We then apply the proposed method to solve distributed optimization problems over master-worker architectures, under agents' function similarity, due to statistical data similarity or otherwise. The distributed algorithm achieves for the first time lower complexity bounds on both communication and local gradient calls, with the former having being a long-standing open problem. Finally the method is extended to distributed saddle-problems (under function similarity) by means of solving a class of variational inequalities, achieving lower communication and computation complexity bounds.

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NeurIPS Conference 2022 Conference Paper

The First Optimal Acceleration of High-Order Methods in Smooth Convex Optimization

  • Dmitry Kovalev
  • Alexander Gasnikov

In this paper, we study the fundamental open question of finding the optimal high-order algorithm for solving smooth convex minimization problems. Arjevani et al. (2019) established the lower bound $\Omega\left(\epsilon^{-2/(3p+1)}\right)$ on the number of the $p$-th order oracle calls required by an algorithm to find an $\epsilon$-accurate solution to the problem, where the $p$-th order oracle stands for the computation of the objective function value and the derivatives up to the order $p$. However, the existing state-of-the-art high-order methods of Gasnikov et al. (2019b); Bubeck et al. (2019); Jiang et al. (2019) achieve the oracle complexity $\mathcal{O}\left(\epsilon^{-2/(3p+1)} \log (1/\epsilon)\right)$, which does not match the lower bound. The reason for this is that these algorithms require performing a complex binary search procedure, which makes them neither optimal nor practical. We fix this fundamental issue by providing the first algorithm with $\mathcal{O}\left(\epsilon^{-2/(3p+1)}\right)$ $p$-th order oracle complexity.

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NeurIPS Conference 2022 Conference Paper

The First Optimal Algorithm for Smooth and Strongly-Convex-Strongly-Concave Minimax Optimization

  • Dmitry Kovalev
  • Alexander Gasnikov

In this paper, we revisit the smooth and strongly-convex-strongly-concave minimax optimization problem. Zhang et al. (2021) and Ibrahim et al. (2020) established the lower bound $\Omega\left(\sqrt{\kappa_x\kappa_y} \log \frac{1}{\epsilon}\right)$ on the number of gradient evaluations required to find an ϵ-accurate solution, where κx and κy are condition numbers for the strong convexity and strong concavity assumptions. However, the existing state-of-the-art methods do not match this lower bound: algorithms of Lin et al. (2020) and Wang and Li (2020) have gradient evaluation complexity $\mathcal{O}\left(\sqrt{\kappa_x\kappa_y} \log^3 \frac{1}{\epsilon}\right)$ and $\mathcal{O}\left( \sqrt{\kappa_x\kappa_y}\log^3 (\kappa_x\kappa_y)\log\frac{1}{\epsilon}\right)$, respectively. We fix this fundamental issue by providing the first algorithm with $\mathcal{O}\left(\sqrt{\kappa_x\kappa_y} \log \frac{1}{\epsilon}\right)$ gradient evaluation complexity. We design our algorithm in three steps: (i) we reformulate the original problem as a minimization problem via the pointwise conjugate function; (ii) we apply a specific variant of the proximal point algorithm to the reformulated problem; (iii) we compute the proximal operator inexactly using the optimal algorithm for operator norm reduction in monotone inclusions.

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NeurIPS Conference 2021 Conference Paper

Distributed Saddle-Point Problems Under Data Similarity

  • Aleksandr Beznosikov
  • Gesualdo Scutari
  • Alexander Rogozin
  • Alexander Gasnikov

We study solution methods for (strongly-)convex-(strongly)-concave Saddle-Point Problems (SPPs) over networks of two type--master/workers (thus centralized) architectures and mesh (thus decentralized) networks. The local functions at each node are assumed to be \textit{similar}, due to statistical data similarity or otherwise. We establish lower complexity bounds for a fairly general class of algorithms solving the SPP. We show that a given suboptimality $\epsilon>0$ is achieved over master/workers networks in $\Omega\big(\Delta\cdot \delta/\mu\cdot \log (1/\varepsilon)\big)$ rounds of communications, where $\delta>0$ measures the degree of similarity of the local functions, $\mu$ is their strong convexity constant, and $\Delta$ is the diameter of the network. The lower communication complexity bound over mesh networks reads $\Omega\big(1/{\sqrt{\rho}} \cdot {\delta}/{\mu}\cdot\log (1/\varepsilon)\big)$, where $\rho$ is the (normalized) eigengap of the gossip matrix used for the communication between neighbouring nodes. We then propose algorithms matching the lower bounds over either types of networks (up to log-factors). We assess the effectiveness of the proposed algorithms on a robust regression problem.

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NeurIPS Conference 2021 Conference Paper

Lower Bounds and Optimal Algorithms for Smooth and Strongly Convex Decentralized Optimization Over Time-Varying Networks

  • Dmitry Kovalev
  • Elnur Gasanov
  • Alexander Gasnikov
  • Peter Richtarik

We consider the task of minimizing the sum of smooth and strongly convex functions stored in a decentralized manner across the nodes of a communication network whose links are allowed to change in time. We solve two fundamental problems for this task. First, we establish {\em the first lower bounds} on the number of decentralized communication rounds and the number of local computations required to find an $\epsilon$-accurate solution. Second, we design two {\em optimal algorithms} that attain these lower bounds: (i) a variant of the recently proposed algorithm ADOM (Kovalev et al, 2021) enhanced via a multi-consensus subroutine, which is optimal in the case when access to the dual gradients is assumed, and (ii) a novel algorithm, called ADOM+, which is optimal in the case when access to the primal gradients is assumed. We corroborate the theoretical efficiency of these algorithms by performing an experimental comparison with existing state-of-the-art methods.

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NeurIPS Conference 2020 Conference Paper

Stochastic Optimization with Heavy-Tailed Noise via Accelerated Gradient Clipping

  • Eduard Gorbunov
  • Marina Danilova
  • Alexander Gasnikov

In this paper, we propose a new accelerated stochastic first-order method called clipped-SSTM for smooth convex stochastic optimization with heavy-tailed distributed noise in stochastic gradients and derive the first high-probability complexity bounds for this method closing the gap in the theory of stochastic optimization with heavy-tailed noise. Our method is based on a special variant of accelerated Stochastic Gradient Descent (SGD) and clipping of stochastic gradients. We extend our method to the strongly convex case and prove new complexity bounds that outperform state-of-the-art results in this case. Finally, we extend our proof technique and derive the first non-trivial high-probability complexity bounds for SGD with clipping without light-tails assumption on the noise.

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NeurIPS Conference 2018 Conference Paper

Decentralize and Randomize: Faster Algorithm for Wasserstein Barycenters

  • Pavel Dvurechenskii
  • Darina Dvinskikh
  • Alexander Gasnikov
  • Cesar Uribe
  • Angelia Nedich

We study the decentralized distributed computation of discrete approximations for the regularized Wasserstein barycenter of a finite set of continuous probability measures distributedly stored over a network. We assume there is a network of agents/machines/computers, and each agent holds a private continuous probability measure and seeks to compute the barycenter of all the measures in the network by getting samples from its local measure and exchanging information with its neighbors. Motivated by this problem, we develop, and analyze, a novel accelerated primal-dual stochastic gradient method for general stochastic convex optimization problems with linear equality constraints. Then, we apply this method to the decen- tralized distributed optimization setting to obtain a new algorithm for the distributed semi-discrete regularized Wasserstein barycenter problem. Moreover, we show explicit non-asymptotic complexity for the proposed algorithm. Finally, we show the effectiveness of our method on the distributed computation of the regularized Wasserstein barycenter of univariate Gaussian and von Mises distributions, as well as some applications to image aggregation.

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NeurIPS Conference 2016 Conference Paper

Learning Supervised PageRank with Gradient-Based and Gradient-Free Optimization Methods

  • Lev Bogolubsky
  • Pavel Dvurechenskii
  • Alexander Gasnikov
  • Gleb Gusev
  • Yurii Nesterov
  • Andrei Raigorodskii
  • Aleksey Tikhonov
  • Maksim Zhukovskii

In this paper, we consider a non-convex loss-minimization problem of learning Supervised PageRank models, which can account for features of nodes and edges. We propose gradient-based and random gradient-free methods to solve this problem. Our algorithms are based on the concept of an inexact oracle and unlike the state-of-the-art gradient-based method we manage to provide theoretically the convergence rate guarantees for both of them. Finally, we compare the performance of the proposed optimization methods with the state of the art applied to a ranking task.

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