Non-euclidean proximal methods for convex-concave saddle-point problems

Eyal Cohen, Shoham Sabach, Marc Teboulle

Research output: Contribution to journalArticlepeer-review

Abstract

Motivated by the flexibility of the Proximal Alternating Predictor Corrector (PAPC) algorithm which can tackle a broad class of structured constrained convex optimization problems via their convex-concave saddle-point reformulation, in this paper, we extend the scope of the PAPC algorithm to include non-Euclidean proximal steps. This allows for adapting to the geometry of the problem at hand to produce simpler computational steps. We prove a sublinear convergence rate of the produced ergodic sequence, and under additional natural assumptions on the non-Euclidean distances, we also prove that the algorithm globally converges to a saddle-point. We demonstrate the performance and simplicity of the proposed algorithm through its application to the multinomial logistic regression problem.

Original languageEnglish
Pages (from-to)43-60
Number of pages18
JournalJournal of Applied and Numerical Optimization
Volume3
Issue number1
DOIs
StatePublished - Apr 2021

Keywords

  • Bregman and ϕ-divergences
  • Convergence rate
  • Iteration complexity
  • Non-Euclidean proximal distances and algorithms
  • Nonsmooth convex minimization
  • Saddle-point problems

All Science Journal Classification (ASJC) codes

  • Numerical Analysis
  • Modelling and Simulation
  • Control and Optimization
  • Computational Mathematics

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