The Critical Horizon: Inspection Design Principles for Multi-Stage Operations and Deep Reasoning
This provides a foundational analysis for inspection design in operations and supervision in AI, addressing a common challenge in manufacturing, supply chains, and deep reasoning.
The paper tackles the credit assignment problem in multi-stage operations and AI reasoning by establishing an information-theoretic barrier where signal decays exponentially with depth, proving that sample complexity grows exponentially and uniform checkpoint spacing is optimal under homogeneous conditions.
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 no algorithm can learn from endpoint data alone. 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.