Data-Space Dirichlet Process Bayesian Neural Networks: Resolving Epistemic Uncertainty Collapse in the Data Void

Reliable epistemic uncertainty is critical for black-box optimization. However, standard parameter-space Bayesian Neural Networks (BNNs) suffer from epistemic uncertainty collapse in the data void: predictive variance vanishes in unobserved regions due to linear feature extrapolation, permanently trapping optimizers in suboptimal basins. We propose Data-Space Dirichlet Process Bayesian Neural Networks (DP-BNNs), placing a conjugate Dirichlet Process prior directly over the unknown data-generating distribution on the joint input-output space: P   DP(α, P₀). Crucially, optimizing a single deterministic neural network on a sampled posterior distribution directly instantiates an exact functional sample from the posterior function distribution: f_θ   Π(· | D_t). This enables hyperparameter-free, pure Non-Parametric Thompson Sampling (DP-TS) without multi-head ensembles, stochastic dropout passes, or heuristic exploration tuning. Theoretically, we prove non-vanishing epistemic void dispersion (predictive uncertainty strictly reverts to prior dispersion: lim_||x - x_i|| → ∞ σ(x) ≥ σ₀ > 0) and vanishing simple regret (S_T → 0 as T → ∞). Empirically, across 4 multimodal benchmarks (Ackley 2D, Levy 2D, Rosenbrock 4D, Rastrigin 4D), DP-TS escapes deceptive local traps where conventional BNNs fail (achieving up to 75× error reduction), outperforming Gaussian Processes with linear O(t) complexity, a rapid 21.6 ms step latency, and scalability to 32D and 60D control.