Computing Fixpoints of Learned Functions: Chaotic Iteration and Simple Stochastic Games

2026-01-22

AI Summary

This work addresses the challenge of efficiently computing (least) fixed points over the non-negative reals when the underlying function is only approximately accessible. We propose an improved damped Mann iteration scheme that accommodates chaotic update strategies—updating only a subset of variables per iteration—and significantly relaxes constraints on the learning rate sequence, allowing it to be non-convergent or not tending to zero. This flexibility enables effective handling of high-dimensional systems and heterogeneous approximation rates. Notably, we extend damped Mann iteration for the first time to chaotic settings and probabilistic models such as simple stochastic games, achieving stable computation of fixed points for approximate functions in high dimensions. The method successfully applies to expected payoff computation, overcoming limitations of traditional approaches in both convergence speed and update mechanisms.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Social networks and social learning
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

The problem of determining the (least) fixpoint of (higher-dimensional) functions over the non-negative reals frequently occurs when dealing with systems endowed with a quantitative semantics. We focus on the situation in which the functions of interest are not known precisely but can only be approximated. As a first contribution we generalize an iteration scheme called dampened Mann iteration, recently introduced in the literature. The improved scheme relaxes previous constraints on parameter sequences, allowing learning rates to converge to zero or not converge at all. While seemingly minor, this flexibility is essential to enable the implementation of chaotic iterations, where only a subset of components is updated in each step, allowing to tackle higher-dimensional problems. Additionally, by allowing learning rates to converge to zero, we can relax conditions on the convergence speed of function approximations, making the method more adaptable to various scenarios. We also show that dampened Mann iteration applies immediately to compute the expected payoff in various probabilistic models, including simple stochastic games, not covered by previous work.

Problem

• fixpoint computation
• function approximation
• quantitative semantics
• higher-dimensional systems
• stochastic games

Innovation

• dampened Mann iteration
• chaotic iteration
• fixpoint computation
• simple stochastic games
• function approximation