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.

翻译：在具有定量语义的系统分析中，确定非负实数上（高维）函数的（最小）不动点是一个常见问题。本文关注目标函数无法精确获知、仅能通过近似得到的情形。作为第一项贡献，我们推广了近期文献中提出的阻尼Mann迭代方案。改进后的方案放宽了对参数序列的原有约束，允许学习率收敛至零或根本不收敛。尽管看似细微，这种灵活性对于实现混沌迭代至关重要——在混沌迭代中，每一步仅更新部分分量，从而能够处理更高维的问题。此外，通过允许学习率收敛至零，我们可以放宽对函数近似收敛速度的条件要求，使该方法能更灵活地适应不同场景。我们还证明，阻尼Mann迭代可直接用于计算各类概率模型中的期望收益，包括先前工作未涵盖的简单随机博弈。