Convergence Thresholds of Newton's Method for Monotone Polynomial Equations

TL;DR

This paper analyzes the convergence thresholds of Newton's method when applied to monotone polynomial systems, providing bounds and thresholds that depend on the system's structure and fixed-point properties.

Contribution

It establishes explicit upper bounds for convergence thresholds of Newton's method on MSPEs, extending previous existential results to concrete, structure-dependent bounds.

Findings

Threshold $k_f$ is at most single exponential for systems from probabilistic pushdown automata.

Threshold $k_f$ is at most linear for systems from back-button processes.

After a certain point, each Newton iteration computes at least one new bit of the solution.

Abstract

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 $\mu\vec f$ 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.