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A Measure-Theoretic Analysis of Reasoning: Structural Generalization and Approximation Limits

NeurIPS 2026 · Learning theory and algorithms › Kernels and regression theory · approach: Empirical generalization measures · 0 citations (OpenAlex)

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Proposes a measure-theoretic/optimal-transport framework bounding OOD reasoning risk via Wasserstein-1 shift, Lipschitz constants of APE vs RoPE, and TC0/Barron depth-width limits, validated on 54 Transformer configs trained on Countdown search traces.

Problem

Theoretical understanding of which architectural conditions let autoregressive Transformers generalize search-based reasoning across structural (OOD) domain shifts, bridging circuit complexity with distribution shift.

Approach

Projects discrete search trajectories into a 12-dimensional structural feature space via a pushforward map, bounds OOD risk by Kantorovich-Rubinstein duality as K_f times W1 plus an approximation error, and argues APE yields Omega(1) Lipschitz constant under shifts while RoPE is translation-equivariant; also maps backtracking to Dyck-k for a TC0 depth bound.

Key results

• Large wide mixed APE reaches 0.4215 accuracy on the Mixed test set versus 0.5425 for large wide mixed RoPE.
• Large deep mixed APE reaches 0.533 versus 0.4215 for the large wide mixed APE configuration.
• Estimated regularized W1 is about 0.42 between BFS and Mixed and about 0.81 between BFS and DFS.

Surprising findings

• Under DFS-RoPE, the large wide configuration (0.518) slightly outperforms the large deep configuration (0.5145), contrary to the 'deep is better' law.

Limitations

• Relies on an epsilon-relaxed structural sufficiency assumption that may fail for open-ended natural language reasoning.
• Key steps (Lipschitz gap, Omega(1) APE risk) rest on assumed margin and routing-deviation conditions; proofs are sketches.
• Empirical Wasserstein estimates are regularized proxies in R^12 and the TC0 bottleneck covers only single-pass routing, not inference-time search.

Experimental setup

Datasets: Countdown trajectories synthesized with Stream of Search (BFS, DFS, Mixed)

Benchmarks: Countdown (BFS/DFS/Mixed test sets)

Metrics: exact-match accuracy regularized Wasserstein-1 estimate

Baselines: APE RoPE

Builds on: Stream of Search RoPE Sinkhorn-Knopp Kantorovich-Rubinstein duality

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