Tropical Gradient Descent

TL;DR

This paper introduces a novel gradient descent method tailored for tropical geometry optimization problems, leveraging polyhedral structures to efficiently find local minima and outperform classical methods in tropical convexity scenarios.

Contribution

The paper presents a new tropical gradient descent algorithm that exploits tropical geometric structures, with proven global solvability and convergence properties, and demonstrates superior empirical performance.

Findings

Outperforms classical gradient descent on tropical convex problems

Ensures global solvability for 1-sample tropical problems

Integrates seamlessly with advanced optimizers like Adam

Abstract

We propose a gradient descent method for solving optimization problems arising in settings of tropical geometry - a variant of algebraic geometry that has attracted growing interest in applications such as computational biology, economics, and computer science. Our approach takes advantage of the polyhedral and combinatorial structures arising in tropical geometry to propose a versatile method for approximating local minima in tropical statistical optimization problems - a rapidly growing body of work in recent years. Theoretical results establish global solvability for 1-sample problems and a convergence rate matching classical gradient descent. Numerical experiments demonstrate the method's superior performance compared to classical gradient descent for tropical optimization problems which exhibit tropical convexity but not classical convexity. We also demonstrate the seamless integration of tropical descent into advanced optimization methods, such as Adam, offering improved overall accuracy.