Take Goodhart Seriously: Principled Limit on General-Purpose AI Optimization

Abstract: A common but rarely examined assumption in machine learning is that training yields models that actually satisfy their specified objective function. We call this the Objective Satisfaction Assumption (OSA). Although deviations from OSA are acknowledged, their implications are overlooked. We argue, in a learning-paradigm-agnostic framework, that OSA fails in realistic conditions: approximation, estimation, and optimization errors guarantee systematic deviations from the intended objective, regardless of the quality of its specification. Beyond these technical limitations, perfectly capturing and translating the developer's intent, such as alignment with human preferences, into a formal objective is practically impossible, making misspecification inevitable. Building on recent mathematical results, absent a mathematical characterization of these gaps, they are indistinguishable from those that collapse into Goodhart's law failure modes under strong optimization pressure. Because the Goodhart breaking point cannot be located ex ante, a principled limit on the optimization of General-Purpose AI systems is necessary. Absent such a limit, continued optimization is liable to push systems into predictable and irreversible loss of control.