On the Representational Geometry of Dynamic Programs
Standard neural architectures struggle to generalize to longer inputs for dynamic programming (DP) targets, and this paper investigates the geometric root causes Every finite min-plus DP can be equivalently described as a shortest path on a DAG, a tropical polynomial, or an extended Newton polyhedron — and these three descriptions form isomorphic semirings at both formal polynomial and computed function levels The paper proves that the two native dimension-reduction operations in the semiring (s
Analysis
TL;DR
- Standard neural architectures struggle to generalize to longer inputs for dynamic programming (DP) targets, and this paper investigates the geometric root causes
- Every finite min-plus DP can be equivalently described as a shortest path on a DAG, a tropical polynomial, or an extended Newton polyhedron — and these three descriptions form isomorphic semirings at both formal polynomial and computed function levels
- The paper proves that the two native dimension-reduction operations in the semiring (setting variables to identities) are neither injective nor always closed within DP, explaining a fundamental generalization barrier
- Series and parallel composition of DAGs fail to construct all possible DAG topologies from smaller sub-DAGs, and even terminal-only operations cannot capture all DP compositions
- These structural negatives provide a rigorous geometric explanation for why length generalization in DP tasks remains an open challenge for neural networks
Why It Matters
This work bridges discrete mathematics, algebraic geometry, and machine learning by providing a formal geometric framework to understand a persistent practical problem: why neural networks trained on DP tasks fail to generalize beyond their training sequence lengths. For AI practitioners building models on algorithmic reasoning tasks, this paper offers theoretical grounding for observed failures and points toward structural limitations that architectural changes alone may not overcome.
Technical Details
- Three isomorphic descriptions: The paper establishes that finite min-plus DP problems admit three equivalent formulations — graph-theoretic (shortest path on a DAG), algebraic (tropical polynomial), and geometric (extended Newton polyhedron encoding decision boundaries) — with isomorphic semiring structures at both the formal polynomial level and the computed function level
- Structural redundancy characterization: Operations connecting the two semiring levels are shown to characterize all structural redundancies in DP representations, providing a complete algebraic taxonomy of DP equivalences
- Negative generalization results: Two key structural negatives are proven — (1) dimension reduction via identity substitution is neither injective nor closed within the DP semiring, and (2) series/parallel composition and terminal-only operations are insufficient to construct all DAG topologies or capture all DP compositions
- Decision boundary analysis: The extended Newton polyhedron encodes the decision boundary of which path wins in a DP, and the paper frames the length-generalization question as whether the decision boundary at length T determines the boundary at length T+1
Industry Insight
- Researchers developing neural solvers for algorithmic reasoning should treat length generalization in DP as a fundamental structural limitation rather than a mere training or architectural deficiency; model design should account for the proven incompleteness of series/parallel composition in capturing DP structure
- The isomorphic semiring framework could inform the design of inductive biases or structured neural architectures that respect the algebraic structure of DP problems, potentially improving generalization by aligning model capacity with the true complexity class of the target
- This theoretical groundwork suggests that hybrid approaches combining neural networks with explicit symbolic or algorithmic components may be necessary for robust DP generalization, as pure end-to-end learning faces provable geometric barriers
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