# 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…

## 12 Citations

### Zero-Sum Stochastic Stackelberg Games

- EconomicsArXiv
- 2022

This paper proves the existence of recursive (i.e., Markov perfect) Stackelberg equilibria (recSE) in zero-sum stochastic games, provides necessary and sufﬁcient conditions for a policy to be a recSE, and shows that recSE can be computed in (weakly) polynomial time via value iteration.

### Robust No-Regret Learning in Min-Max Stackelberg Games

- Computer ScienceAAMAS
- 2022

This paper investigates the behavior of no-regret learning in min-max games with dependent strategy sets, and shows that if both players minimize their Lagrangian regrets, then play converges to a Stackelberg equilibrium.

### Gradient Descent Ascent in Min-Max Stackelberg Games

- Computer Science
- 2022

It is shown that solving Fisher markets, a canonical example of a min-max Stackelberg game, using a novel algorithm, corresponds to buyers and sellers using myopic best-response dynamics in a repeated market, allowing the convergence of these dynamics in 𝑂 ( 1 / 𝜀 2 ) iterations in Fisher markets.

### Exploitability Minimization in Games and Beyond

- Computer Science, EconomicsArXiv
- 2022

The exploitability-minimization problem can be recast as a min-max optimization problem, and polynomial-time first-order methods are obtained to compute a refinement of GNE, namely the variational equilibria (VE), in convex-concave cumulative regret pseudo-games with jointly convex constraints.

### Synthesizing Reactive Test Environments for Autonomous Systems: Testing Reach-Avoid Specifications with Multi-Commodity Flows

- Computer ScienceArXiv
- 2022

An optimization problem, framed as a multi- commodity network ﬂow problem, that solves for constraints on the virtual product graph which can then be projected to the test environment and the result of the optimization problem is reactive test synthesis that ensures that the system meets the test speciﬁcations along with satisfying the system speci-cations.

### Learning Stackelberg Equilibria and Applications to Economic Design Games

- Economics
- 2022

We study the use of reinforcement learning to learn the optimal leader’s strategy in Stackelberg games. Learning a leader’s strategy has an innate stationarity problem—when optimizing the leader’s…

### Learning Autonomous Vehicle Safety Concepts from Demonstrations

- Computer Science
- 2022

A data-driven AV safety design methodology is proposed that first learns “reasonable” behavioral assumptions from data, and then synthesizes an AV safety concept using these learned behavioral assumptions.

### Examining Responsibility and Deliberation in AI Impact Statements and Ethics Reviews

- PsychologyAIES
- 2022

The artificial intelligence research community is continuing to grapple with the ethics of its work by encouraging researchers to discuss potential positive and negative consequences. Neural…

### A Survey of Decision Making in Adversarial Games

- Computer ScienceArXiv
- 2022

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.

### Minimally Constrained Testing for Autonomy with Temporal Logic Specifications

- Computer Science
- 2022

This paper considers a subset of Linear Temporal Logic to represent formal requirements on the system and the test environment, and presents a framework to construct a minimally constrained test.

## References

SHOWING 1-10 OF 90 REFERENCES

### Near-Optimal Algorithms for Minimax Optimization

- Computer ScienceCOLT 2020
- 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…