The Critical Horizon: Inspection Design Principles for Multi-Stage Operations and Deep Reasoning
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866915795715489792 |
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| author | Emadi, Seyed Morteza |
| author_facet | Emadi, Seyed Morteza |
| contents | Manufacturing lines, service journeys, supply chains, and AI reasoning chains share a common challenge: attributing a terminal outcome to the intermediate stage that caused it. We establish an information-theoretic barrier to this credit assignment problem: the signal connecting early steps to final outcomes decays exponentially with depth, creating a critical horizon beyond which reliable learning from endpoint data alone requires exponentially many samples. We prove four results. First, a Signal Decay Bound: sample complexity for attributing outcomes to early stages grows exponentially in the number of intervening steps. Second, Width Limits: parallel rollouts provide only logarithmic relief, with correlation capping the effective number of independent samples. Third, an Objective Mismatch: additive reward aggregation optimizes the wrong quantity when sequential validity requires all steps to be correct. Fourth, Optimal Inspection Design: uniform checkpoint spacing is minimax-optimal under homogeneous signal attenuation, while a greedy algorithm yields optimal non-uniform schedules under heterogeneous attenuation. Together, these results provide a common analytical foundation for inspection design in operations and supervision design in AI. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_09394 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Critical Horizon: Inspection Design Principles for Multi-Stage Operations and Deep Reasoning Emadi, Seyed Morteza Machine Learning Artificial Intelligence Information Theory Manufacturing lines, service journeys, supply chains, and AI reasoning chains share a common challenge: attributing a terminal outcome to the intermediate stage that caused it. We establish an information-theoretic barrier to this credit assignment problem: the signal connecting early steps to final outcomes decays exponentially with depth, creating a critical horizon beyond which reliable learning from endpoint data alone requires exponentially many samples. We prove four results. First, a Signal Decay Bound: sample complexity for attributing outcomes to early stages grows exponentially in the number of intervening steps. Second, Width Limits: parallel rollouts provide only logarithmic relief, with correlation capping the effective number of independent samples. Third, an Objective Mismatch: additive reward aggregation optimizes the wrong quantity when sequential validity requires all steps to be correct. Fourth, Optimal Inspection Design: uniform checkpoint spacing is minimax-optimal under homogeneous signal attenuation, while a greedy algorithm yields optimal non-uniform schedules under heterogeneous attenuation. Together, these results provide a common analytical foundation for inspection design in operations and supervision design in AI. |
| title | The Critical Horizon: Inspection Design Principles for Multi-Stage Operations and Deep Reasoning |
| topic | Machine Learning Artificial Intelligence Information Theory |
| url | https://arxiv.org/abs/2602.09394 |