Geometry of Regularized Dirichlet Likelihood: Structural Theorem, SOSC Characterization, and Certified Face-Enumeration Algorithm

Abstract Evidential Deep Learning (EDL) is a modern approach to uncertainty estimation where a neural network predicts Dirichlet distribution parameters. Recent works have shown the statistical inconsistency of the standard EDL objective for out-of-distribution (OOD) detection. This work investigates the optimization landscape geometry of the regularized likelihood on a box domain for a fixed evidence vector. \textbf{Main contributions:} (1) rigorous proof of the algebraic equivalence between second-order sufficient conditions (SOSC) and the strict positivity of the self-consistency function's derivative; (2) derivation of an exact local comparative statics formula extended to simplex perturbations; (3) proof of nonconvexity and justification of the global optimality guarantee via face enumeration; (4) a deterministic algorithm with an unconditional guarantee of finite-convergence; (5) the Asymptotic Active-Set Saturation Theorem and a rigorous proof of the unreachability of the saturation threshold over the entire simplex (the Evidence Ceiling effect), explaining the sparsification of the active set and the degradation of uncertainty metrics at high classifier confidence. We also show that the revealed topology and the Evidence Ceiling effect are inherent in the original digamma EDL objective, for which the saturation threshold is found analytically. Experimental verification is performed using synthetic benchmarks and logit data.