# Convex-Concave Min-Max Stackelberg Games

@inproceedings{Goktas2021ConvexConcaveMS, title={Convex-Concave Min-Max Stackelberg Games}, author={Denizalp Goktas and Amy Greenwald}, booktitle={Neural Information Processing Systems}, year={2021} }

Min-max optimization problems (i.e., min-max games) have been attracting a great deal of attention because of their applicability to a wide range of machine learning problems. Although signiﬁcant progress has been made recently, the literature to date has focused on games with independent strategy sets; little is known about solving games with dependent strategy sets, which can be interpreted as min-max Stackelberg games. We introduce two ﬁrst-order methods that solve a large class of convex…

## 13 Citations

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This paper provides a systematic survey on three main game models widely employed in adversarial games, i.e., zero-sum normal-form and extensive-form games, Stackelberg (security) games, zero- sum differential games, from an array of perspectives, including basic knowledge of game models, (approximate) equilibrium concepts, problem classiﬁcations, research frontiers, (assumed) optimal strategy seeking techniques, prevailing algorithms, and practical applications.

## References

SHOWING 1-10 OF 90 REFERENCES

### Near-Optimal Algorithms for Minimax Optimization

- Computer ScienceCOLT
- 2020

The first algorithm with $\tilde{O}(\sqrt{\kappa_{\mathbf x}\kappa- y}})$ gradient complexity is presented, matching the lower bound up to logarithmic factors.

### On Gradient Descent Ascent for Nonconvex-Concave Minimax Problems

- Computer ScienceICML
- 2020

This is the first nonasymptotic analysis for two-time-scale GDA in this setting, shedding light on its superior practical performance in training generative adversarial networks (GANs) and other real applications.

### Block Alternating Optimization for Non-convex Min-max Problems: Algorithms and Applications in Signal Processing and Communications

- Computer ScienceICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)
- 2019

This work proposes two simple algorithms, which alternatingly perform one gradient descent-type step for each minimization block and one gradient ascent- type step for the maximization problem, and shows that such simple alternating min-max algorithms converge to first-order stationary solutions.

### Tatonnement beyond gross substitutes?: gradient descent to the rescue

- EconomicsSTOC '13
- 2013

A class of markets for which tatonnement is equivalent to gradient descent is defined and all processes in this family converge to an equilibrium for any Convex Potential Function (CPF) market.

### Minimax Problems with Coupled Linear Constraints: Computational Complexity, Duality and Solution Methods

- Computer Science
- 2021

This work studies a special minimax problem where there are linear constraints that couple both the minimization and maximization decision variables, and shows that the considered problem is challenging, in the sense that it violates the classical max-min inequality.

### What is Local Optimality in Nonconvex-Nonconcave Minimax Optimization?

- Computer ScienceICML
- 2020

A proper mathematical definition of local optimality for this sequential setting---local minimax is proposed, as well as its properties and existence results are presented.

### Generalized Nash Equilibrium Problems

- EconomicsAnn. Oper. Res.
- 2010

The Generalized Nash Equilibrium Problem is an important model that has its roots in the economic sciences but is being fruitfully used in many different fields and its main properties and solution algorithms are discussed.

### EXISTENCE OF AN EQUILIBRIUM FOR A COMPETITIVE ECONOMY

- Economics
- 1954

A. Wald has presented a model of production and a model of exchange and proofs of the existence of an equilibrium for each of them. Here proofs of the existence of an equilibrium are given for an…

### Envelope Theorems for Arbitrary Choice Sets

- Mathematics, Economics
- 2002

The standard envelope theorems apply to choice sets with convex and topological structure, providing sufficient conditions for the value function to be differentiable in a parameter and…