单调多项式方程组的牛顿法收敛阈值
Convergence Thresholds of Newton's Method for Monotone Polynomial Equations
AI总结：
针对单调多项式方程组，研究牛顿迭代法收敛阈值的存在性，给出基于最小不动点分量的上界，并证明对概率下推自动机和回退过程导出的强连通系统，阈值分别为单指数和线性。
摘要：
单调多项式方程组（MSPEs）是形如 $X_1 = f_1(X_1, ..., X_n),$ $..., X_n = f_n(X_1, ..., X_n)$ 的不动点方程组，其中每个 $f_i$ 是系数为正实数的多项式。计算给定 MSPE $\vec X = \vec f(\vec X)$ 的最小非负解的问题自然出现在随机模型的分析中，例如随机上下文无关文法、概率下推自动机和回退过程。Etessami 和 Yannakakis 最近将牛顿迭代法应用于 MSPEs。在之前的一篇论文中，我们证明了对于强连通 MSPEs 存在阈值 $k_{\vec f}$，使得在牛顿法的 $k_{\vec f}$ 次迭代后，每次新迭代至少计算解的新 1 位。然而，该证明纯粹是存在性的。在本文中，我们给出了 $k_{\vec f}$ 的上界，该上界是 $\vec f(\vec X)$ 的最小不动点 $\mu\vec f$ 的最小分量的函数。利用这一结果，我们证明了对于从概率下推自动机和回退过程导出的强连通 MSPEs，$k_{\vec f}$ 分别至多为单指数和线性。此外，我们证明了对于任意 MSPEs 存在一个阈值，之后每次新迭代至少计算解的 $1/w2^h$ 新位，其中 $w$ 和 $h$ 是强连通分量 DAG 的宽度和高度。
英文摘要：
Monotone systems of polynomial equations (MSPEs) are systems of fixed-point equations $X_1 = f_1(X_1, ..., X_n),$ $..., X_n = f_n(X_1, ..., X_n)$ where each $f_i$ is a polynomial with positive real coefficients. The question of computing the least non-negative solution of a given MSPE $\vec X = \vec f(\vec X)$ arises naturally in the analysis of stochastic models such as stochastic context-free grammars, probabilistic pushdown automata, and back-button processes. Etessami and Yannakakis have recently adapted Newton's iterative method to MSPEs. In a previous paper we have proved the existence of a threshold $k_{\vec f}$ for strongly connected MSPEs, such that after $k_{\vec f}$ iterations of Newton's method each new iteration computes at least 1 new bit of the solution. However, the proof was purely existential. In this paper we give an upper bound for $k_{\vec f}$ as a function of the minimal component of the least fixed-point $μ\vec f$ of $\vec f(\vec X)$. Using this result we show that $k_{\vec f}$ is at most single exponential resp. linear for strongly connected MSPEs derived from probabilistic pushdown automata resp. from back-button processes. Further, we prove the existence of a threshold for arbitrary MSPEs after which each new iteration computes at least $1/w2^h$ new bits of the solution, where $w$ and $h$ are the width and height of the DAG of strongly connected components.
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