Further reading

Kan invariance and categorical presentation are classical themes in Mac Lane [ 1971 ] and Riehl [ 2016 ] . The concrete equivalence between FTRL and mirror descent is developed by McMahan [ 2011 ] ; the wider online convex setting is surveyed by Hazan [ 2023 ] . Localization and homotopy-invariant semantics are treated by Riehl [ 2014 ] . The observer-preserving decision quotient is the UODL refinement proposed here.

Classical online boosting and its convex-analytic ingredients are presented in Hazan [ 2023 ] . The distribution monad and convex-algebra viewpoint used for free completion is part of the operadic theory developed by Haderi et al. [ 2024 ] . Compositional learners as symmetric monoidal objects motivate the open-interface view [ Fong et al. , 2019 ] . The amplification certificate and its persistence interpretation are proposed here.

The classical vector-payoff theorem is due to Blackwell [ 1956 ] . Online convex optimization and game-theoretic duality are reviewed by Hazan [ 2023 ] . Nonclassical information structures and the limits they place on coordinated decisions originate in Witsenhausen [ 1971 , 1975 ] . The interpretation of approachability as persistent, localized consistency is developed here.