Direct Inference on Admissible Sets (Diona)

Deep Gaussian processes (DGPs) are flexible surrogate models for computer experiments with nonstationary response surfaces, but inference over their latent representations remains computationally challenging. Existing methods typically rely on variational inference or Markov chain Monte Carlo (MCMC), which impose variational-family assumptions or require costly posterior sampling. We propose the latent maximum-a-posteriori deep Gaussian process (LaMP-DGP), which uses gradient-based optimization to obtain MAP estimates of the latent representations and enables efficient probabilistic prediction through linked-GP calculations. To scale LaMP-DGP to large datasets, we develop a mini-batch optimization algorithm and, under a polynomial mini-batch bias-decay condition, establish a stationarity bound for latent-state optimization that motivates a growing batch-size schedule. Across numerical experiments and real-data analyses, LaMP-DGP provides consistently competitive predictive performance relative to doubly stochastic variational inference, stochastic gradient Hamiltonian Monte Carlo, and fully Bayesian Vecchia DGP inference. Computationally, LaMP-DGP is substantially faster than the Vecchia MCMC approach while remaining broadly competitive with the stochastic-gradient alternatives. These results position LaMP-DGP as a competitive and computationally efficient alternative for DGP inference. An R package, "LaMPDGP", is available for implementing the proposed method.

Gradient observations can substantially improve Gaussian process (GP) surrogates, particularly in high-dimensional settings where function evaluations are expensive. However, exact inference with $n$ function values and $n$ full gradients in $d$ dimensions scales cubically in the joint state size, imposing an intractable $\mathcal{O}(n^3 d^3)$ computational bottleneck. We introduce TERA, a highly scalable derivative GP method based on target-specific exact gradient reduction. We prove that for stationary kernels, the gradient components orthogonal to the directions connecting the target and conditioning points are conditionally independent of the target function value; consequently, the exact conditional density is fully characterized by at most $m^2$ directional derivatives once a conditioning set of size $m$ is specified. By using these reduced, dimension-free conditionals as local factors in a Vecchia approximation, TERA effectively decouples $n$ and $d$ from the dense matrix inversion. This reduces the per-target evaluation cost to $\mathcal{O}(dm^2 + m^6)$ time and $\mathcal{O}(dm^2 + m^4)$ memory, leaving the underlying derivative GP model mathematically unchanged. Empirical evaluations demonstrate that TERA achieves state-of-the-art predictive accuracy while operating orders of magnitude faster than standard derivative GPs. Crucially, both computation time and peak GPU memory remain essentially flat with respect to $d$, enabling highly scalable inference in high-dimensional spaces.

Deep learning has proven highly effective for nonlinear system identification, but heavily parameterized neural networks are prone to overfitting in low-data regimes and lack reliable uncertainty quantification. The recently developed manifold meta-learning framework addresses the data efficiency problem by restricting the model parameters to a meta-learned low-dimensional manifold. However, that method is purely deterministic. We propose a fully probabilistic extension of the manifold meta-learning framework, based on amortized Variational Inference, where a generative prior over the low-dimensional parameter manifold is learned. During task-specific adaptation, we combine Maximum A Posteriori estimation with the Laplace approximation to yield a mathematically grounded posterior approximation. Evaluated on a static regression task and the Bouc--Wen dynamical system benchmark, the proposed approach achieves predictive accuracy comparable to its deterministic counterpart while successfully providing calibrated uncertainty bounds in severely low-data regimes.