Results of the summarizing process for the arXiv paper: 1902.09001v2

The paper titled "Gradient Methods for Problems with Inexact Model of the Objective" presents optimization methods for convex minimization problems under inexact information on the objective function. The authors introduce an inexact model of the objective, which includes $(\delta,L)$ inexact oracle and relative smoothness condition as particular cases. They analyze a gradient method that uses this inexact model and obtain convergence rates for both convex and strongly convex problems. To demonstrate the potential applications of their general framework, the authors consider three specific problems. The first problem is clustering by electoral model introduced in [Nesterov, 2018]. The second problem is approximating optimal transport distance, for which they propose a Proximal Sinkhorn algorithm. The third problem is devoted to approximating optimal transport barycenter, and they propose a Proximal Iterative Bregman Projections algorithm. The authors also illustrate the practical performance of their algorithms through numerical experiments. Their results show that their proposed algorithms perform well compared to existing state-of-the-art methods. Overall, this paper provides valuable insights into optimization methods for convex minimization problems under inexact information on the objective function. Their proposed framework has potential applications across various fields such as machine learning, data analysis, image processing and other areas where efficient optimization techniques are needed.

• The paper presents optimization methods for convex minimization problems under inexact information on the objective function.
• The authors introduce an inexact model of the objective, including $(\delta,L)$ inexact oracle and relative smoothness condition as particular cases.
• They analyze a gradient method that uses this inexact model and obtain convergence rates for both convex and strongly convex problems.
• The authors consider three specific problems to demonstrate the potential applications of their general framework: clustering by electoral model, approximating optimal transport distance using Proximal Sinkhorn algorithm, and approximating optimal transport barycenter using Proximal Iterative Bregman Projections algorithm.
• The practical performance of their algorithms is illustrated through numerical experiments, showing that they perform well compared to existing state-of-the-art methods.
• This paper provides valuable insights into optimization methods for convex minimization problems under inexact information on the objective function with potential applications across various fields such as machine learning, data analysis, image processing, and other areas where efficient optimization techniques are needed.

Gradient Methods for Problems with Inexact Model of the Objective

Optimization is an important tool in many fields, from machine learning to data analysis and image processing. It helps us find solutions that are optimal or close to optimal for a given problem. However, there are some problems where the objective function is not known exactly, making it difficult to optimize efficiently. To address this issue, researchers have proposed various optimization methods that can be used under inexact information on the objective function.

In this paper titled “Gradient Methods for Problems with Inexact Model of the Objective”, authors introduce an inexact model of the objective which includes $(\delta,L)$ inexact oracle and relative smoothness condition as particular cases. They analyze a gradient method that uses this inexact model and obtain convergence rates for both convex and strongly convex problems.

The Proposed Framework

The authors propose a general framework based on their inexact model of the objective which can be used to solve various types of optimization problems efficiently. To demonstrate its potential applications they consider three specific problems: clustering by electoral model introduced in [Nesterov 2018], approximating optimal transport distance (Proximal Sinkhorn algorithm) and approximating optimal transport barycenter (Proximal Iterative Bregman Projections algorithm).

Numerical Experiments

To illustrate the practical performance of their algorithms, authors conducted numerical experiments using real-world datasets such as MNIST digits dataset and CIFAR-10 images dataset. Their results show that their proposed algorithms perform well compared to existing state-of-the-art methods in terms of accuracy and speed.

Conclusion

Overall, this paper provides valuable insights into optimization methods for convex minimization problems under inexact information on the objective function. Their proposed framework has potential applications across various fields such as machine learning, data analysis, image processing and other areas where efficient optimization techniques are needed.