动态规划的表示几何
On the Representational Geometry of Dynamic Programs
- Bowdoin College（鲍登学院）
- Naval Postgraduate School（海军研究生院）
- Aalborg University（奥尔堡大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究从几何角度分析标准神经网络对动态规划长输入泛化困难的问题，证明动态规划的图、多项式、多面体描述构成同构半环，且长度泛化的决策边界不存在层级决定关系。
AI中文摘要：
标准神经网络架构通常无法对动态规划（DP）目标生成长输入的泛化能力，我们从几何角度探究其难点所在。每个有限的 min-plus DP 都是有向无环图（DAG）上的最短路径，等价于一个热带多项式，其扩展牛顿多面体编码了哪条路径获胜的决策边界。我们证明这三种描述（图、多项式、多面体）在两个层面构成同构半环——形式多项式及其计算函数——由表征所有结构冗余的操作连接。随后我们从几何角度解决长度泛化问题：长度为 T 的决策边界是否决定长度为 T+1 的边界？我们给出两个结构性否定结论：半环的两种原生降维方式（将变量设为各单位元）既非单射也不总是在 DP 内封闭；串行与并行组合无法由更小的子 DAG 构造所有 DAG 拓扑，甚至仅含终端的操作也无法覆盖所有 DP 组合。
英文摘要：
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.
