We study the deterministic global optimization of trained Gaussian process posterior mean functions over hyperrectangular domains. Although the posterior mean function has a compact closed-form representation, its global optimization is challenging because it remains nonlinear and nonconvex. Existing exact deterministic approaches become increasingly difficult to scale as the number of training data points grows, leading to approximation-based methods that improve tractability by optimizing a modified (inexact) objective. In this work, we propose PALM-Mean, a piecewise-analytic lower-bounding framework embedded in reduced-space spatial branch-and-bound. At each node, kernel terms that are locally important are replaced by a sign-aware piecewise-linear relaxation in an appropriate scalar distance variable, while the remaining terms are bounded analytically in closed form. We show this hybrid approach yields a valid lower bound for the posterior mean, while limiting the size of the branch-and-bound subproblems. We establish validity of the node lower bounds and $\varepsilon$-global convergence of the resulting algorithm. Computational results on synthetic benchmarks and real-world application problems show that PALM-Mean improves scalability relative to representative general-purpose deterministic global solvers, particularly as the number of training data points increases.

The proof is an induction on k. Consider the general case p2k+1. It is easy to see that g (x) = ex p2k(x) and g (x) = ex p2k 1(x). By the induction hypothesis, g 0 and therefore g is convex. Thus, the minimum of g is given by its stationary points. It is easy to observe that x = 0 is indeed a stationary point. Thus, minx R g(x) = g(0) = 0, which finishes the proof.

Let k (,) be the ν = 2 .5 Matérn kernel.

The "folk wisdom" in the literature is that the focus on local optimization sidesteps the curse of dimensionality;

Providing guarantees on the decision-making processes of autonomous systems, often based on complex black-box machine learning models, is paramount for their safe deployment. This need motivates efforts towards responsible artificial intelligence, which broadly entails questions of reliability, robustness, fairness, and interpretability.

Neural Information Processing Systems http://nips.cc/

This appendix contains the proofs of the results found in Section 4. We start by introducing a useful The claim follows then directly from (4) and the definition of mutual information.Lemma 2. We then can compute the derivative and ask under which conditions it is non negative. The function b defined in (18) is monotonically increasing for positive arguments. Finally, let us fix ε > 0. Combining Lemmas 7 and 8, we obtain: b( σ The following result makes this statement precise. The following lemma makes this statement precise. In this Appendix, we collect details about the experiment presented in Section 6. Code for the used acquisition functions can be found at ISE selects the next parameter to evaluate according to (6), which is a non convex optimization problem constrained in one of the variables.