Colosseum of Mechanisms: A Unified Framework for Unsupervised Multi-Mechanism Learning via Quasi-Gradient Descent

An unsupervised multi-mechanism model has $|J|$ mechanisms and $|I|$ observations. Fitting it optimally means an intractable search over $|J{|}^{|I|}$ assignments. The \`divide-and-conquer\' (DaC) approach assumes a segmentation form and searches only the assignments that form expresses: tractable, but broken when the assumption is incorrect. We propose the Colosseum of Mechanisms (CoM), a \`conquer-then-divide\' framework: it fits the parameters first, then reads off the segmentation. At every observation, CoM stages a competition among all $|J|$ mechanisms, then aggregates only the winners. CoM assumes no segmentation form and therefore accommodates any. Optimizing the CoM objective $\ell$ solves the combinatorial search without traversing those assignments. An algebraic identity makes this possible. This identity assumes nothing about the noise, nor the correctness of the supplied mechanism forms. But the combinatorial difficulty does not disappear; it reappears as nonconvexity and nonsmoothness in $\ell$. Our Quasi-Gradient Descent algorithm guarantees $\epsilon$-Clarke stationarity in $\mathcal{O}({\epsilon }^{-2})$ iterations, exactly what standard gradient descent achieves on a single smooth function. Even the simplest CoM case is NP-hard to minimize. We therefore give a necessary condition for global optimality. In DaC's best case, CoM matches its accuracy about 1000$\times$ faster. In a non-ideal DaC case, CoM's accuracy and speed are unchanged, while DaC loses eleven accuracy points and runs slower still. On U.S. GNP data, CoM halves the out-of-sample forecast error of a published benchmark. CoM also reconstructs entangled high-frequency dynamics in EEG.