Graph-Based Ascent Algorithms for Function Maximization

TL;DR

This paper introduces two novel graph-based ascent algorithms using Metropolis-Hastings dynamics for maximizing functions on graph nodes, with proven convergence rates and comparative simulations.

Contribution

It proposes new descent algorithms for graph function maximization based on Metropolis-Hastings, with theoretical convergence analysis and empirical performance evaluation.

Findings

Convergence rates derived for both algorithms.

Algorithms outperform unbiased random walk in simulations.

Performance depends on function smoothness.

Abstract

We study the problem of finding the maximum of a function defined on the nodes of a connected graph. The goal is to identify a node where the function obtains its maximum. We focus on local iterative algorithms, which traverse the nodes of the graph along a path, and the next iterate is chosen from the neighbors of the current iterate with probability distribution determined by the function values at the current iterate and its neighbors. We study two algorithms corresponding to a Metropolis-Hastings random walk with different transition kernels: (i) The first algorithm is an exponentially weighted random walk governed by a parameter $\gamma$. (ii) The second algorithm is defined with respect to the graph Laplacian and a smoothness parameter $k$. We derive convergence rates for the two algorithms in terms of total variation distance and hitting times. We also provide simulations showing the relative convergence rates of our algorithms in comparison to an unbiased random walk, as a function of the smoothness of the graph function. Our algorithms may be categorized as a new class of "descent-based" methods for function maximization on the nodes of a graph.