Generalised Bellman recurrence and three dualities in sequential decision-making

Abstract:What gives the Bellman equation its form? We show that the recursive properties of optimal value functions follow from three conditions: that the dynamics decomposes through sufficient statistics, that the return decomposes recursively, and that the aggregation of uncertainty is compatible with both. When all three conditions hold on a common state, the Bellman equation arises from their mutual consistency; when one fails, tractability can often be recovered by augmenting the state or by deforming return or dynamics. The same conditions are shown to give rise to three dualities: one between probability and return, one between return and aggregation, and one between aggregation and probability. Our framework reveals these dualities as arising from a single construction, unifying methods developed separately across reinforcement learning, control, and decision theory.

Machine Learning (cs.LG); Artificial Intelligence (cs.AI)

arXiv:2607.18077 [cs.LG]

(or arXiv:2607.18077v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2607.18077 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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