Evidential deep learning hits an evidence ceiling: as confidence nears one, the active set collapses to a single class

Stationary points of the regularized Dirichlet likelihood on a box domain have a threshold structure: classes whose evidence exceeds a threshold are active, the rest sit at the boundary value 1, and the active set forms an upper slice ordered by evidence. Earlier critiques of evidential deep learning targets focused on statistical inconsistency, the behavior of the estimator relative to the true data distribution, leaving the geometry of the solution set for a fixed evidence vector uncharacterized. The structure follows from two monotonicity facts, a threshold function strictly decreasing in total evidence and a per-class equation strictly decreasing in its argument, with degenerate cases treated separately.

The second-order sufficient conditions collapse into a scalar inequality, positivity of the self-consistency function's derivative, turning a high-dimensional Hessian test into a one-dimensional one. Convergence results for standard projected Newton methods assume the Hessian is positive definite throughout, and the nonlinear self-consistent threshold structure blocks a direct transfer of those results. Via the Sherman-Morrison formula, positive definiteness of the reduced Hessian on an active face is equivalent to a positive self-consistency derivative, in both directions.

The objective is not globally convex on the box domain, yet enumerating faces by active-set size guarantees the global minimum for almost all evidence vectors. No global optimality guarantee existed for this objective, and generic quasi-Newton methods misidentify the active set under the ill-conditioned geometry near the box constraints. On 1000 synthetic configurations with K=10 and λ=0.1, the Hybrid Compact-Core Newton method reached 100.0% active-set accuracy, versus 63.8% for L-BFGS-B and 67.4% for projected gradient descent.

Total evidence has an impassable saturation threshold that every stationary point stays below, explaining active-set sparsification and degraded uncertainty metrics at high confidence. Evidence contraction had been treated as a training anomaly and patched with heuristic regularization rather than attributed to the Dirichlet-KL landscape itself. On logits from a ResNet-18 trained on CIFAR-10 and tested on out-of-distribution data including SVHN, the highest mean total evidence across confidence bins was 19.14, never crossing the threshold near 19.75, with a 0% violation rate.

The analysis targets the per-sample optimization landscape for a fixed evidence vector, with experiments on synthetic configurations at K=10 and λ=0.1 and on logits from a ResNet-18 trained on CIFAR-10 and tested on SVHN, CIFAR-100, Textures and adversarial examples. The authors propose extending the solver to evidential frameworks such as subjective logic and Dempster-Shafer theory and integrating it into real-time safety-critical inference pipelines.

The analysis does not cover training dynamics, that is, how stochastic gradient descent traverses this nonconvex landscape over multiple epochs. The global optimality guarantee excludes a measure-zero set of degenerate cases where the self-consistency derivative vanishes, and a complete algebraic classification of those strata remains open.