# A Variational Inequality Approach to Bayesian Regression Games

arXiv: https://arxiv.org/abs/2103.13509 · HTML: https://mlsanity.com/paper/2103.13509/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 72 references that are in the map.
- First posted on arXiv 24 Mar 2021.

## Builds on (most cited first, 12 of 12)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Robustness May Be at Odds with Accuracy](https://mlsanity.com/paper/1805.12152/) (2018)
- [Fast is better than free: Revisiting adversarial training](https://mlsanity.com/paper/2001.03994/) (2020)
- [Wild Patterns: Ten Years After the Rise of Adversarial Machine Learning](https://mlsanity.com/paper/1712.03141/) (2017)
- [FreeLB: Enhanced Adversarial Training for Natural Language Understanding](https://mlsanity.com/paper/1909.11764/) (2019)
- [Coordinate Descent Algorithms](https://mlsanity.com/paper/1502.04759/) (2015)
- [Projected Reflected Gradient Methods for Monotone Variational Inequalities](https://mlsanity.com/paper/1502.04968/) (2015)
- [Accelerated, Parallel and Proximal Coordinate Descent](https://mlsanity.com/paper/1312.5799/) (2013)
- [Inapproximability of Nash Equilibrium](https://mlsanity.com/paper/1405.3322/) (2014)

## Abstract (arXiv)

Bayesian regression games are a special class of two-player general-sum Bayesian games in which the learner is partially informed about the adversary's objective through a Bayesian prior. This formulation captures the uncertainty in regard to the adversary, and is useful in problems where the learner and adversary may have conflicting, but not necessarily perfectly antagonistic objectives. Although the Bayesian approach is a more general alternative to the standard minimax formulation, the applications of Bayesian regression games have been limited due to computational difficulties, and the existence and uniqueness of a Bayesian equilibrium are only known for quadratic cost functions. First, we prove the existence and uniqueness of a Bayesian equilibrium for a class of convex and smooth Bayesian games by regarding it as a solution of an infinite-dimensional variational inequality (VI) in Hilbert space. We consider two special cases in which the infinite-dimensional VI reduces to a high-dimensional VI or a nonconvex stochastic optimization, and provide two simple algorithms of solving them with strong convergence guarantees. Numerical results on real datasets demonstrate the promise of this approach.