No-regret dynamics in the Fenchel game: a unified framework for algorithmic convex optimization

Jun Kun Wang, Jacob Abernethy, Kfir Y. Levy

Research output: Contribution to journalArticlepeer-review


We develop an algorithmic framework for solving convex optimization problems using no-regret game dynamics. By converting the problem of minimizing a convex function into an auxiliary problem of solving a min–max game in a sequential fashion, we can consider a range of strategies for each of the two-players who must select their actions one after the other. A common choice for these strategies are so-called no-regret learning algorithms, and we describe a number of such and prove bounds on their regret. We then show that many classical first-order methods for convex optimization—including average-iterate gradient descent, the Frank–Wolfe algorithm, Nesterov’s acceleration methods, the accelerated proximal method—can be interpreted as special cases of our framework as long as each player makes the correct choice of no-regret strategy. Proving convergence rates in this framework becomes very straightforward, as they follow from plugging in the appropriate known regret bounds. Our framework also gives rise to a number of new first-order methods for special cases of convex optimization that were not previously known.

Original languageEnglish
JournalMathematical Programming
StateAccepted/In press - 2023


  • Convex optimization
  • Frank–Wolfe method
  • Momentum methods
  • Nesterov’s accelerated gradient methods
  • No-regret learning
  • Online learning
  • Zero-sum game

All Science Journal Classification (ASJC) codes

  • Software
  • General Mathematics


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