In this paper, we present a new Hyperfast Second-Order Method with convergence rate $O(N^{-5})$ up to a logarithmic factor for the convex function with Lipshitz the third derivative. This method based on two ideas. The first comes from the superfast second-order scheme of Yu. Nesterov (CORE Discussion Paper 2020/07, 2020). It allows implementing the third-order scheme by solving subproblem using only the second-order oracle. This method converges with rate $O(N^{-4})$. The second idea comes from the work of Kamzolov et al. (arXiv:2002.01004). It is the inexact near-optimal third-order method. In this work, we improve its convergence and merge it with the scheme of solving subproblem using only the second-order oracle. As a result, we get convergence rate $O(N^{-5})$ up to a logarithmic factor. This convergence rate is near-optimal and the best known up to this moment. Further, we investigate the situation when there is a sum of two functions and improve the sliding framework from Kamzolov et al. (arXiv:2002.01004) for the second-order methods.
Statistical preconditioning enables fast methods for distributed large-scale empirical risk minimization problems. In this approach, multiple worker nodes compute gradients in parallel, which are then used by the central node to update the parameter by solving an auxiliary (preconditioned) smaller-scale optimization problem. The recently proposed Statistically Preconditioned Accelerated Gradient (SPAG) method has complexity bounds superior to other such algorithms but requires an exact solution for computationally intensive auxiliary optimization problems at every iteration. In this paper, we …
In this paper, we present a first-order projection-free method, namely, the universal conditional gradient sliding (UCGS) method, for solving $\varepsilon$-approximate solutions to convex differentiable optimization problems. For objective functions with Hölder continuous gradients, we show that UCGS is able to terminate with $\varepsilon$-solutions with at most $O((M_νD_X^{1+ν}/{\varepsilon})^{2/(1+3ν)})$ gradient evaluations and $O((M_νD_X^{1+ν}/{\varepsilon})^{4/(1+3ν)})$ linear objective optimizations, where $ν\in (0,1]$ and $M_ν>0$ are the exponent and constant of the Hölder condition. Fu…
We study structured convex optimization problems, with additive objective $r:=p + q$, where $r$ is ($μ$-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, $\mathcal{O}(\sqrt{L_p/μ})$ and $\mathcal{O}(\sqrt{L_q/μ})$, respectively. This result is much sharper than the classic black-box complexity $\mathcal{O}(\sqrt{(…
The goal of this paper is to reduce the total complexity of gradient-based methods for two classes of problems: affine-constrained composite convex optimization and bilinear saddle-point structured non-smooth convex optimization. Our technique is based on a double-loop inexact accelerated proximal gradient (APG) method for minimizing the summation of a non-smooth but proximable convex function and two smooth convex functions with different smoothness constants and computational costs. Compared to the standard APG method, the inexact APG method can reduce the total computation cost if one smoot…
Nanjing University of Information Science & Technology（南京信息工程大学）；MBZUAI（穆罕默德·本·扎耶德人工智能大学）；JD Finance America Corporation（京东金融美国公司）；University of Pittsburgh（匹兹堡大学）
The conditional gradient algorithm (also known as the Frank-Wolfe algorithm) has recently regained popularity in the machine learning community due to its projection-free property to solve constrained problems. Although many variants of the conditional gradient algorithm have been proposed to improve performance, they depend on first-order information (gradient) to optimize. Naturally, these algorithms are unable to function properly in the field of increasingly popular zeroth-order optimization, where only zeroth-order information (function value) is available. To fill in this gap, we propose…
This paper studies the communication complexity of convex risk-averse optimization over a network. The problem generalizes the well-studied risk-neutral finite-sum distributed optimization problem and its importance stems from the need to handle risk in an uncertain environment. For algorithms in the literature, there exists a gap in communication complexities for solving risk-averse and risk-neutral problems. We propose two distributed algorithms, namely the distributed risk averse optimization (DRAO) method and the distributed risk averse optimization with sliding (DRAO-S) method, to close t…
We consider stochastic convex optimization problems with affine constraints and develop several methods using either primal or dual approach to solve it. In the primal case, we use a special penalization technique to make the initial problem more convenient for using optimization methods. We propose algorithms to solve it based on Similar Triangles Method with Inexact Proximal Step for the convex smooth and strongly convex smooth objective functions and methods based on Gradient Sliding algorithm to solve the same problems in the non-smooth case. We prove the convergence guarantees in the smoo…
In this paper, we propose a new method based on the Sliding Algorithm from Lan(2016, 2019) for the convex composite optimization problem that includes two terms: smooth one and non-smooth one. Our method uses the stochastic noised zeroth-order oracle for the non-smooth part and the first-order oracle for the smooth part. To the best of our knowledge, this is the first method in the literature that uses such a mixed oracle for the composite optimization. We prove the convergence rate for the new method that matches the corresponding rate for the first-order method up to a factor proportional to…
Optimal Rates for Federated Variational Inequalities under different relaxations of bounded variance assumption
Innopolis University（伊诺波利斯大学）；Lomonosov Moscow State University（莫斯科国立罗蒙诺索夫大学）；HSE University（高等经济大学）；Trusted AI Research Center, RAS（俄罗斯科学院可信人工智能研究中心）；Steklov Mathematical Institute, RAS（俄罗斯科学院斯捷克洛夫数学研究所）
Stochastic Distributed Optimization under Average Second-order Similarity: Algorithms and Analysis
Academy for Advanced Interdisciplinary Studies（北京大学前沿交叉学科研究院）；Peking University（北京大学数学科学学院）；School of Mathematical Sciences（国家电网有限公司SGIT人工智能实验室）；Peking University；SGIT AI Lab, State Grid Corporation of China
