On the Representational Geometry of Dynamic Programs

2026-08-25

AI Summary

研究解决了标准神经架构在处理动态规划目标时难以泛化到更长输入的问题，通过几何方法分析了决策边界的特性及其长度泛化问题。

Abstract

Standard neural architectures often fail to generalize to longer inputs for dynamic programming (DP) targets. We investigate what makes this hard geometrically. Every finite min-plus DP is a shortest path on a DAG, which is equivalently a tropical polynomial whose extended Newton polyhedron encodes the decision boundary of which path wins. We prove these three descriptions (graph, polynomial, polyhedron) form isomorphic semirings at two levels --- formal polynomials and their computed functions --- connected by operations that characterize all structural redundancies. We then address the length-generalization question geometrically: does the decision boundary at length $T$ decide the boundary at $T+1$? We present two structural negatives. The semiring's two native ways to reduce dimension (setting a variable to each identity) are neither injective nor always closed within the DP. Series and parallel composition fail to construct all DAG topologies from smaller sub-DAGs, and even all terminal-only operations do not capture all DP compositions.

Problem

• dynamic programming
• generalization
• neural architectures
• geometric analysis
• decision boundary

Innovation

• dynamic programming
• geometric representation
• tropical polynomial
• length generalization
• semiring isomorphism