# Operator Splitting for Learning to Predict Equilibria in Convex Games

arXiv: https://arxiv.org/abs/2106.00906 · HTML: https://mlsanity.com/paper/2106.00906/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 30 of its 80 references that are in the map.
- First posted on arXiv 2 Jun 2021.

## Builds on (most cited first, 30 of 30)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Differentiable Convex Optimization Layers](https://mlsanity.com/paper/1910.12430/) (2019)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [The Little Engine that Could: Regularization by Denoising (RED)](https://mlsanity.com/paper/1611.02862/) (2016)
- [Melding the Data-Decisions Pipeline: Decision-Focused Learning for Combinatorial Optimization](https://mlsanity.com/paper/1809.05504/) (2018)
- [Learning to Optimize: A Primer and A Benchmark](https://mlsanity.com/paper/2103.12828/) (2021)
- [Universal Approximation with Deep Narrow Networks](https://mlsanity.com/paper/1905.08539/) (2019)
- [End-to-End Constrained Optimization Learning: A Survey](https://mlsanity.com/paper/2103.16378/) (2021)
- [Multiscale Deep Equilibrium Models](https://mlsanity.com/paper/2006.08656/) (2020)
- [Deep Equilibrium Architectures for Inverse Problems in Imaging](https://mlsanity.com/paper/2102.07944/) (2021)
- [Implicit Deep Learning](https://mlsanity.com/paper/1908.06315/) (2019)
- [Monotone operator equilibrium networks](https://mlsanity.com/paper/2006.08591/) (2020)
- [JFB: Jacobian-Free Backpropagation for Implicit Networks](https://mlsanity.com/paper/2103.12803/) (2021)
- [Learning Maximally Monotone Operators for Image Recovery](https://mlsanity.com/paper/2012.13247/) (2020)
- [A Three-Operator Splitting Scheme and its Optimization Applications](https://mlsanity.com/paper/1504.01032/) (2015)
- [Data-Driven Estimation in Equilibrium Using Inverse Optimization](https://mlsanity.com/paper/1308.3397/) (2013)
- [SHINE: SHaring the INverse Estimate from the forward pass for bi-level optimization and implicit models](https://mlsanity.com/paper/2106.00553/) (2021)
- [Computational Rationalization: The Inverse Equilibrium Problem](https://mlsanity.com/paper/1308.3506/) (2013)
- [What game are we playing? End-to-end learning in normal and extensive form games](https://mlsanity.com/paper/1805.02777/) (2018)
- [End-to-End Learning and Intervention in Games](https://mlsanity.com/paper/2010.13834/) (2020)
- [Feasibility-based Fixed Point Networks](https://mlsanity.com/paper/2104.14090/) (2021)
- [Adaptive Three Operator Splitting](https://mlsanity.com/paper/1804.02339/) (2018)
- [Large Scale Learning of Agent Rationality in Two-Player Zero-Sum Games](https://mlsanity.com/paper/1903.04101/) (2019)
- [Data-Driven Estimation of Travel Latency Cost Functions via Inverse Optimization in Multi-Class Transportation Networks](https://mlsanity.com/paper/1703.04010/) (2017)
- [Using Inverse Optimization to Learn Cost Functions in Generalized Nash Games](https://mlsanity.com/paper/2102.12415/) (2021)
- [Projection Methods: An Annotated Bibliography of Books and Reviews](https://mlsanity.com/paper/1406.6143/) (2014)
- [The Price of Anarchy in Transportation Networks: Data-Driven Evaluation and Reduction Strategies](https://mlsanity.com/paper/1606.02194/) (2016)

## Abstract (arXiv)

Systems of competing agents can often be modeled as games. Assuming rationality, the most likely outcomes are given by an equilibrium (e.g. a Nash equilibrium). In many practical settings, games are influenced by context, i.e. additional data beyond the control of any agent (e.g. weather for traffic and fiscal policy for market economies). Often the exact game mechanics are unknown, yet vast amounts of historical data consisting of (context, equilibrium) pairs are available, raising the possibility of learning a solver which predicts the equilibria given only the context. We introduce Nash Fixed Point Networks (N-FPNs), a class of neural networks that naturally output equilibria. Crucially, N- FPNs employ a constraint decoupling scheme to handle complicated agent action sets while avoiding expensive projections. Empirically, we find N-FPNs are compatible with the recently developed Jacobian-Free Backpropagation technique for training implicit networks, making them significantly faster and easier to train than prior models. Our experiments show N-FPNs are capable of scaling to problems orders of magnitude larger than existing learned game solvers.