Tropical Gradient Descent

TL;DR

The paper introduces tropical gradient descent, a steepest-descent method formulated with the tropical norm to optimize problems in tropical geometry. It establishes global solvability for 1-sample tropical location problems and a convergence rate of $O\left(\dfrac{1}{\sqrt{m}}\right)$, with empirical evidence showing superiority over classical gradient methods on tropical yet non-classical-convex problems. The authors integrate tropical descent with Adam variants to improve accuracy and demonstrate effectiveness on centrality statistics (Fermat--Weber, Fréchet means), tropical Wasserstein projections, and tropical linear regression. They further showcase applications to phylogenetic analysis under the multi-species coalescent model and a hidden-factor auction model, underscoring the practical impact in computational biology and economics. Overall, the work provides a rigorous tropical optimization framework with strong theoretical guarantees and compelling empirical performance for tropical data science tasks."

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.

Figures (13)

• Figure 1: The $d_{\mathrm{tr}}$, $d_{\triangle_{\min}}$ and $d_{\triangle_{\max}}$ tropical unit balls in $\mathbb{R}^{3}/\mathbb{R} \mathbf{1} \cong \mathcal{H} = \{ \sum x_i = 0 \}$. The solid lines show coordinate directions.
• Figure 2: A tropical hyperplane, tropical line segment and tropical convex hull in $\mathbb{R}^{3}/\mathbb{R} \mathbf{1} \cong \mathcal{H} = \{\mathbf{x}: \sum x_i = 0 \}$. The dashed lines show coordinate directions.
• Figure 3: A heat map of a 3-sample linear regression objective on $\mathbb{R}^{3}/\mathbb{R} \mathbf{1} \cong \mathcal{H} = \{ \mathbf{x}: \sum x_i = 0 \}$.
• Figure 4: A heat map of a 3-sample $\infty$-Wasserstein projection objective on $\mathbb{R}^{3}/\mathbb{R} \mathbf{1} \cong \mathcal{H} = \{ \mathbf{x}: \sum x_i = 0 \}$.
• Figure 5: The mean log error bound ${\rm log} \, \epsilon_m'$ and the mean log tropical distance to the estimated minimum $\mathbf{t}^*$ when applying tropical descent (TD) with $\alpha = 1$ to solve the tropical linear regression problem. For each dataset, we take 50 random initialization points and compute ${\rm log} \, \epsilon_m'$ and ${\rm log} \, d_{\mathrm{tr}}(\mathbf{t}_m, \mathbf{t}^*)$ independently, then take an average. We also record the 10th and 90th percentile values to estimate an 80% confidence interval.

Theorems & Definitions (38)

• Definition 1: Tropical Algebra
• Definition 2: Tropical Projective Torus
• Remark 3
• Definition 4: Tropical Asymmetric Distance
• Remark 5
• Definition 6: Tropical Hyperplane
• Definition 7: Tropical Line Segment
• Definition 8: Tropical Convex Hull develin2004tropical
• Definition 9: Tropical Linear Regression akian2023tropical
• Definition 10: Oriented Geodesic Segment