Gradient Descent Ascent in MinMax Stackelberg Games
Abstract
Minmax optimization problems (i.e., minmax games) have attracted a great deal of attention recently as their applicability to a wide range of machine learning problems has become evident. In this paper, we study minmax games with dependent strategy sets, where the strategy of the first player constrains the behavior of the second. Such games are best understood as sequential, i.e., Stackelberg, games, for which the relevant solution concept is Stackelberg equilibrium, a generalization of Nash. One of the most popular algorithms for solving minmax games is gradient descent ascent (GDA). We present a straightforward generalization of GDA to minmax Stackelberg games with dependent strategy sets, but show that it may not converge to a Stackelberg equilibrium. We then introduce two variants of GDA, which assume access to a solution oracle for the optimal Karush Kuhn Tucker (KKT) multipliers of the games' constraints. We show that such an oracle exists for a large class of convexconcave minmax Stackelberg games, and provide proof that our GDA variants with such an oracle converge in $O(\frac{1}{\varepsilon^2})$ iterations to an $\varepsilon$Stackelberg equilibrium, improving on the most efficient algorithms currently known which converge in $O(\frac{1}{\varepsilon^3})$ iterations. We then show that solving Fisher markets, a canonical example of a minmax Stackelberg game, using our novel algorithm, corresponds to buyers and sellers using myopic bestresponse dynamics in a repeated market, allowing us to prove the convergence of these dynamics in $O(\frac{1}{\varepsilon^2})$ iterations in Fisher markets. We close by describing experiments on Fisher markets which suggest potential ways to extend our theoretical results, by demonstrating how different properties of the objective function can affect the convergence and convergence rate of our algorithms.
 Publication:

arXiv eprints
 Pub Date:
 August 2022
 arXiv:
 arXiv:2208.09690
 Bibcode:
 2022arXiv220809690G
 Keywords:

 Computer Science  Computer Science and Game Theory;
 Economics  Theoretical Economics
 EPrint:
 13 pages, 1 figure, Games, Agents, and Incentives Workshop (AAMAS'22). arXiv admin note: text overlap with arXiv:2110.05192, arXiv:2203.14126