Tropical Attention: Neural Algorithmic Reasoning for Combinatorial Algorithms

Dynamic programming (DP) algorithms for combinatorial optimization problems work with taking maximization, minimization, and classical addition in their recursion algorithms. The associated value functions correspond to convex polyhedra in the max plus semiring. Existing Neural Algorithmic Reasoning models, however, rely on softmax-normalized dot-product attention where the smooth exponential weighting blurs these sharp polyhedral structures and collapses when evaluated on out-of-distribution (OOD) settings. We introduce Tropical attention, a novel attention function that operates natively in the max-plus semiring of tropical geometry. We prove that Tropical attention can approximate tropical circuits of DP-type combinatorial algorithms. We then propose that using Tropical transformers enhances empirical OOD performance in both length generalization and value generalization, on algorithmic reasoning tasks, surpassing softmax baselines while remaining stable under adversarial attacks. We also present adversarial-attack generalization as a third axis for Neural Algorithmic Reasoning benchmarking. Our results demonstrate that Tropical attention restores the sharp, scale-invariant reasoning absent from softmax.

Can Transformers Do Enumerative Geometry?

How can Transformers model and learn enumerative geometry? What is a robust procedure for using Transformers in abductive knowledge discovery within a mathematician-machine collaboration? In this work, we introduce a new paradigm in computational enumerative geometry in analyzing the $\psi$-class intersection numbers on the moduli space of curves. By formulating the enumerative problem as a continuous optimization task, we develop a Transformer-based model for computing $\psi$-class intersection numbers based on the underlying quantum Airy structure. For a finite range of genera, our model is capable of regressing intersection numbers that span an extremely wide range of values, from $10^{-45}$ to $10^{45}$. To provide a proper inductive bias for capturing the recursive behavior of intersection numbers, we propose a new activation function, Dynamic Range Activator (DRA). Moreover, given the severe heteroscedasticity of $\psi$-class intersections and the required precision, we quantify the uncertainty of the predictions using Conformal Prediction with a dynamic sliding window that is aware of the number of marked points. Next, we go beyond merely computing intersection numbers and explore the enumerative "world-model" of the Transformers. Through a series of causal inference and correlational interpretability analyses, we demonstrate that Transformers are actually modeling Virasoro constraints in a purely data-driven manner. Additionally, we provide evidence for the comprehension of several values appearing in the large genus asymptotic of $\psi$-class intersection numbers through abductive hypothesis testing.