[Paper Review] Operator Splitting for Learning to Predict Equilibria in Convex Games

Daniel McKenzie, Howard Heaton+4 more|arXiv (Cornell University)|Jun 2, 2021

Model Reduction and Neural Networks50 references4 citations

TL;DR

This paper introduces Nash Fixed Point Networks (N-FPNs), a deep learning framework that predicts equilibria in convex games using context-dependent data. By leveraging operator splitting with constraint decoupling and Jacobian-free backpropagation, N-FPNs efficiently compute equilibria without costly projections, enabling scalable training and inference on large-scale problems such as traffic routing, outperforming prior methods in accuracy and speed.

ABSTRACT

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.

Motivation & Objective

• To develop a scalable, data-driven framework for predicting equilibria in contextual convex games where cost functions are unknown but (context, equilibrium) pairs are available.
• To address the computational bottleneck of projection-based methods in prior learned game solvers by decoupling constraints using Three-Operator splitting.
• To enable end-to-end training of neural networks that output valid equilibria while respecting complex action set constraints.
• To demonstrate the universality of N-FPNs in approximating Nash equilibria for games with diagonal strict convexity.
• To extend applicability to related equilibria such as Wardrop and quantal response equilibria using the same framework.

Proposed method

• N-FPNs are implicit neural networks that define an operator whose fixed points correspond to Nash equilibria, computed via iterative application until convergence.
• The method employs a constraint decoupling scheme derived from Three-Operator splitting to avoid expensive projections onto agent action sets.
• The operator is parameterized by a neural network that maps context $d$ to the game's gradient, enabling end-to-end learning of equilibrium prediction.
• Forward propagation uses explicit formulae for projections via decoupling, enabling efficient inference without iterative projections.
• Backward propagation leverages Jacobian-free backpropagation (JFB) to train the network efficiently, avoiding explicit Hessian computation.
• The framework is generalized to handle variational inequalities, making it applicable to convex optimization and physical simulation problems.

Experimental results

Research questions

• RQ1 Can a deep learning model be trained to predict equilibria in contextual convex games without knowing the underlying cost functions?
• RQ2 Can constraint handling in learned game solvers be made computationally efficient without sacrificing accuracy or constraint satisfaction?
• RQ3 Is the N-FPN architecture capable of universal approximation for a class of games with diagonal strict convexity?
• RQ4 Can N-FPNs scale to large-scale problems such as traffic routing with high-dimensional action sets?
• RQ5 Can the framework be extended to predict other types of equilibria, such as Wardrop or quantal response equilibria?

Key findings

• N-FPNs achieve near-perfect prediction of Wardrop equilibria on the TRAFIX benchmark, with relative MSE and TRAFIX scores converging during training.
• The N-FPN framework scales to problems orders of magnitude larger than existing learned game solvers, including traffic networks with hundreds of high-dimensional polytopes.
• Constraint decoupling enables explicit projection formulae, eliminating the need for iterative projections and significantly accelerating forward and backward passes.
• Jacobian-free backpropagation allows efficient training of the implicit N-FPN architecture, making it feasible to train on large-scale data.
• Empirical results show N-FPNs outperform traditional feedforward networks in both accuracy and scalability on large-scale traffic routing problems.
• The N-FPN model is a universal approximator for contextual games with diagonal strict convexity, proving theoretical expressiveness.