17.3.1 Program A: non-anticipating information structures

17.3.1 Program A: non-anticipating information structures

Classical OCO and its bandit variants retain a global clock even when feedback is partial. Chapters 12–13 replace the chain \(\mathbb N\) by an event category whose arrows encode causal availability. The open problem is to prove online Kan invariance when this index is an asynchronous partial order, a distributed event category, or an endogenous information shape altered by the learner’s actions.

The first obligation is to characterize the Beck–Chevalley defect for nonlinear information shapes: which squares express legitimate causal restriction, and when does extension commute with that restriction? The target is a simplicial online UDL theorem in which homotopy-coherent decision diagrams stabilize without being linearized into a fictitious global history.