Learn to Predict Equilibria via Fixed Point Networks

Abstract

Systems of interacting agents can often be modeled as contextual games, where the context encodes additional information, beyond the control of any agent (e.g. weather for traffic and fiscal policy for market economies). In such systems, the most likely outcome is given by a Nash equilibrium. In many practical settings, only game equilibria are observed, while the optimal parameters for a game model are unknown. This work introduces Nash Fixed Point Networks (N-FPNs), a class of implicit-depth neural networks that output Nash equilibria of contextual games. The N-FPN architecture fuses data-driven modeling with provided constraints. Given equilibrium observations of a contextual game, N-FPN parameters are learnt to predict equilibria outcomes given only the context. We present an end-to-end training scheme for N-FPNs that is simple and memory efficient to implement with existing autodifferentiation tools. N-FPNs also exploit a novel constraint decoupling scheme to avoid costly projections. Provided numerical examples show the efficacy of N-FPNs on atomic and non-atomic games (e.g. traffic routing). 2 * Equal contribution 2 Codes are available on Github: github.com/howardheaton/nash_fixed_point_networks Preprint. Under review.

Data-Driven Behaviour Estimation in Parametric Games

“…More formally, we denote such factors as the context ξ ∈ Ξ of the game, where Ξ ⊆ R c is the set of possible contexts (Sessa et al, 2020;Heaton et al, 2021). We assume that each agent's parametric utility depends also on a known context ξ, as formalized next.…”

“…swarms of birds (Molloy et al, 2018), automatic control, e.g. mobility systems (Censi et al, 2019), internet of things (Chi et al, 2021), and power systems (Belgioioso et al, 2020), and machine learning (Heaton et al, 2021).…”

“…More specifically, (Li et al, 2019) infers a nonlinear interaction cost function from observed noisy matching data via optimal transport while (Trivedi et al, 2020) learns a reward function which serves as an explanation for an observed snapshot of a graph via inverse reinforcement learning. Alternatively, other works including (Ling et al, 2018;Li et al, 2020;Heaton et al, 2021) present a learning-theoretic approach, where an operator approximating the game gradient is estimated within a deep learning framework, to infer the unknown utility parameters from equilibria observations. (Ling et al, 2018) and (Ling et al, 2019) propose a parametric end-to-end framework to learn the utilities in a two-player zero-sum game from an observed quantal response equilibrium and nested logit quan-tal response equilibrium, respectively, using a differentiable game solver layer.…”

“…(Ling et al, 2018) and (Ling et al, 2019) propose a parametric end-to-end framework to learn the utilities in a two-player zero-sum game from an observed quantal response equilibrium and nested logit quan-tal response equilibrium, respectively, using a differentiable game solver layer. (Heaton et al, 2021) and (Li et al, 2020) generalize the above two approaches to any games by casting the observed Nash equilibrium as a solution to a VI and solving the parametric utility learning problem via gradient descent methods.…”

Explainable AI via Learning to Optimize

“…This enables inferences to be computed by iterating until convergence, thereby enabling theoretical guarantees. Memory-efficient training techniques were also developed for this class of models, which have been applied successfully in games, 34 music source separation, 41 language modeling, 11 , segmentation, 12 and inverse problems. 30;32 The recent work 30 most closely aligns with our L2O methodology.…”