Tropical gradient descent

Roan Talbut () and Anthea Monod ()
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Roan Talbut: Imperial College London
Anthea Monod: Imperial College London

Journal of Global Optimization, 2025, vol. 93, issue 2, No 3, 413-449

Abstract: 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.

Keywords: Tropical quasi-convexity; Tropical projective torus; Gradient descent; Location problems; Statistical optimization; Phylogenetics; 49J53; 49K99; 14T90; 62R01; 62R30 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10898-025-01533-1

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