ConservGP: Certified Directional Compression of Gaussian Process Uncertainty

Compressing a Gaussian process (GP) changes its predictive uncertainty, and the direction of the change matters: a planner needs the surrogate at least as uncertain as the teacher, a monitor no more uncertain. We present ConservGP, which fits a fixed-size sparse surrogate under a directional variance contract and certifies the contract over the continuous input domain, with every arithmetic step rounded outward on both models. Plain interval arithmetic loses the lower bound of the GP variance on most subintervals. The certifier instead encloses the mean coefficients through a verified linear solve and evaluates the variance in a form that keeps the cancellation between its two correlated terms. The certificate proves a signed inequality with a chosen margin, in either direction, between the teacher and the surrogate. Previous guarantees instead bound the range of one model or the size of an approximation error, or give a one-directional conservative posterior at a margin the construction fixes. In experiments, certified surrogates meet their contracts on every one-dimensional channel from synthetic data and two drone platforms, on the full domain volume of five problems with two to four inputs under certification-aware compression, and for six-output drone teachers under a matrix ordering on the joint covariance. On drone state channels the surrogate stores 54 times fewer parameters than the teacher and predicts about 34 times faster. In the selection step of safe Bayesian optimization the certified surrogate proposes only candidates inside the teacher's safe set, which unconstrained compression leaves in a fifth of the seeds. As a certified anomaly monitor it matches the teacher's discrimination at a matched false-alarm rate. For any input dimension, contracting the variance (the monitor contract) costs strictly more KL divergence than inflating it by the same margin (the envelope contract), with a gap that grows cubically at small margins and faster beyond them.