一种用于非平衡最优传输的非精确Bregman近端点方法及其加速版本

Bregman Proximal Point Algorithm Revisited: A New Inexact Version and its Inertial Variant

We study a general convex optimization problem, which covers various classic problems in different areas and particularly includes many optimal transport related problems arising in recent years. To solve this problem, we revisit the classic Bregman proximal point algorithm (BPPA) and introduce a new inexact stopping condition for solving the subproblems, which can circumvent the underlying feasibility difficulty often appearing in existing inexact conditions when the problem has a complex feasible set. Our inexact condition also covers several existing inexact conditions as special cases and …

具有两类相对停止准则的非精确Bregman近邻凸差算法

An Inexact Bregman Proximal Difference-of-Convex Algorithm with Two Types of Relative Stopping Criteria

本文针对凸差优化问题提出非精确Bregman近邻凸差算法（iBPDCA），通过引入两类相对停止准则放宽子问题精确求解要求，在Kurdyka-Łojasiewicz性质下建立全局收敛性，数值实验验证了其优越性能。

Online Alternating Direction Method (longer version)

Online optimization has emerged as powerful tool in large scale optimization. In this pa- per, we introduce efficient online optimization algorithms based on the alternating direction method (ADM), which can solve online convex optimization under linear constraints where the objective could be non-smooth. We introduce new proof techniques for ADM in the batch setting, which yields a O(1/T) convergence rate for ADM and forms the basis for regret anal- ysis in the online setting. We consider two scenarios in the online setting, based on whether an additional Bregman divergence is needed or not. …

相对光滑与相对Lipschitz连续极小化问题的加速Bregman梯度方法

Accelerated Bregman gradient methods for relatively smooth and relatively Lipschitz continuous minimization problems

本文针对相对光滑与相对Lipschitz连续优化问题，提出多种带不精确oracle的加速Bregman梯度方法及自适应中间方法，并通过Poisson逆问题数值实验验证其优势。

不精确Bregman近端梯度方法及其带绝对和部分相对停止准则的惯性变体

Inexact Bregman Proximal Gradient Method and its Inertial Variant with Absolute and Partial Relative Stopping Criteria

针对BPGM子问题精确求解不现实的问题，本文开发带绝对/部分相对停止准则的不精确版本iBPGM及惯性变体v-iBPGM，建立收敛速率并通过数值实验验证其在不同不精确性设置下的收敛行为。

牛顿法的原始加速

Primal Acceleration of Newton's Method

提出一种仅用原始变量、每次迭代仅解一个线性系统的直接加速牛顿法，其函数残差收敛速率达 $O(1/k^3)$，可扩展至任意几何与复合优化问题。

Revisiting Global Pooling through the Lens of Optimal Transport

Global pooling is one of the most significant operations in many machine learning models and tasks, whose implementation, however, is often empirical in practice. In this study, we develop a novel and solid global pooling framework through the lens of optimal transport. We demonstrate that most existing global pooling methods are equivalent to solving some specializations of an unbalanced optimal transport (UOT) problem. Making the parameters of the UOT problem learnable, we unify various global pooling methods in the same framework, and accordingly, propose a generalized global pooling layer …

Gradient Methods for Problems with Inexact Model of the Objective

We consider optimization methods for convex minimization problems under inexact information on the objective function. We introduce inexact model of the objective, which as a particular cases includes $(δ,L)$ inexact oracle and relative smoothness condition. We analyze gradient method which uses this inexact model and obtain convergence rates for convex and strongly convex problems. To show potential applications of our general framework we consider three particular problems. The first one is clustering by electorial model introduced in [Nesterov, 2018]. The second one is approximating optimal…

带动量的退火Sinkhorn：线性内存下带认证的无正则化最优传输

Annealed Sinkhorn with Momentum: Certified Unregularized Optimal Transport in Linear Memory

本文提出带动量的退火Sinkhorn（BDRS）方法，用于无正则化离散最优传输，在线性内存中实现原始-对偶认证，并在颜色转移任务中显著加速和降低对偶间隙。

On Inexact Solution of Auxiliary Problems in Tensor Methods for Convex Optimization

In this paper we study the auxiliary problems that appear in $p$-order tensor methods for unconstrained minimization of convex functions with $ν$-Hölder continuous $p$th derivatives. This type of auxiliary problems corresponds to the minimization of a $(p+ν)$-order regularization of the $p$th order Taylor approximation of the objective. For the case $p=3$, we consider the use of Gradient Methods with Bregman distance. When the regularization parameter is sufficiently large, we prove that the referred methods take at most $\mathcal{O}(\log(ε^{-1}))$ iterations to find either a suitable approxim…

基于可调 Oracle 方法的最优非精确性调度

Optimal inexactness schedules for Tunable Oracle based Methods

本文研究可调非精确 oracle 的一阶优化方法，提出最优非精确性调度策略，在收敛性与计算成本间取得最佳折中，并通过 Fast Gradient 算法的数值实验验证了其实际有效性。

A Bregman Splitting Algorithm for Distributed Optimization over Networks

We consider distributed optimization problems in which a group of agents are to collaboratively seek the global optimum through peer-to-peer communication networks. The problem arises in various application areas, such as resource allocation, sensor fusion and distributed learning. We propose a general efficient distributed algorithm--termed Distributed Forward-Backward Bregman Splitting (D-FBBS)--to simultaneously solve the above primal problem as well as its dual based on Bregman method and operator splitting. The proposed algorithm allows agents to communicate asynchronously and thus lends …

逆最优传输与Bregman正则化问题的适定性及高效算法

Well-Posedness and Efficient Algorithms for Inverse Optimal Transport with Bregman Regularization

本文研究了在Bregman正则化下逆最优传输问题的适定性，分析了成本矩阵的结构假设下的存在性、唯一性及稳定性，并提出了一种近似块坐标下降法以高效求解具有强凸惩罚项的问题。

时变复合优化的非精确在线近端镜像下降

Inexact Online Proximal Mirror Descent for time-varying composite optimization

本文研究含梯度与近端算子误差的在线近端镜像下降算法，在凸光滑条件下获得了动态遗憾精确估计，消除了先前结果的$O(T)$项并将适用域扩展至全空间$\mathbb{R}^n$。