# Gradient Methods for Problems with Inexact Model of the Objective

**Authors:** Fedor Stonyakin, Darina Dvinskikh, Pavel Dvurechensky, Alexey Kroshnin, Olesya Kuznetsova, Artem Agafonov, Alexander Gasnikov, Alexander Tyurin, César A. Uribe, Dmitry Pasechnyuk, Sergei Artamonov  
**arXiv:** [1902.09001](https://arxiv.org/abs/1902.09001)  
**Date:** 2019-02-24  
**Summary page:** https://www.summarizepaper.com/en/paper/1902.09001v2/

## Key points

- 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.

## Summary

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.