Efficiently Computable Safety Bounds for Gaussian Processes in Active Learning

TL;DR

This work tackles the challenge of enforcing safety in active learning with Gaussian processes along trajectories by deriving provable, efficiently computable bounds on the probability that a trajectory is unsafe. It introduces a centering transformation and Borell-TIS-based tail bounds, plus adaptive Monte Carlo (AMC), a semi-analytical bound (AB), and a hybrid ABM scheme to rapidly and safely bound trajectory safety probabilities. The methods are supported by rigorous theory (AMC correctness, Borell-based bounds, and ABM safety guarantees) and validated on toy, Himmelblau, and engine-control experiments, showing substantial reductions in computation time while enabling more exploration under tight safety constraints. The practical impact is faster, reliability-guaranteed safe exploration in dynamic systems, with code and algorithms provided for practitioners.

Abstract

Active learning of physical systems must commonly respect practical safety constraints, which restricts the exploration of the design space. Gaussian Processes (GPs) and their calibrated uncertainty estimations are widely used for this purpose. In many technical applications the design space is explored via continuous trajectories, along which the safety needs to be assessed. This is particularly challenging for strict safety requirements in GP methods, as it employs computationally expensive Monte-Carlo sampling of high quantiles. We address these challenges by providing provable safety bounds based on the adaptively sampled median of the supremum of the posterior GP. Our method significantly reduces the number of samples required for estimating high safety probabilities, resulting in faster evaluation without sacrificing accuracy and exploration speed. The effectiveness of our safe active learning approach is demonstrated through extensive simulations and validated using a real-world engine example.

Figures (10)

• Figure 1: By providing better estimates, we obtain accurate error bounds with much fewer MC samples. This reduction in computation time for the safety evaluation allows more time to obtain measurements for Safe Active Learning. The three left columns present visual representations of a Safe Active Learning task. Each column corresponds to a different algorithm runtime $t$, along with the respective number of training points $n$, resulting in $n-1$ explored trajectories. In these plots, the green dashed region marks the area classified as safe by the GP, while the space outside the red boundary indicates the ground truth unsafe region. The rightmost column provides comparisons of two crucial metrics, contingent on the number of iterations. These comparisons underscore the superiority of our approach in enhancing the effectiveness of the Safe Active Learning process by allowing more iteration in the same time.
• Figure 2: The left diagram illustrates the GP $Z_t$ from Subsection \ref{['subsection_toy_example']} via its mean function and pointwise two sigma bands; the right diagram shows the corresponding centered GP $X_t$ resulting from Remark \ref{['remark_transform_gp']}. The safety-relevant information of the mean of the original GP $Z$ moves to the covariance of the centered GP $X$. In particular, the variance of the centered GP $X$ rises where the mean of $Z$ approaches the safety bound zero.
• Figure 3: The diagram considers the toy GP from Section \ref{['subsection_toy_example']}, see also Figure \ref{['figure_centered_GP']} for a visualization of this GP. It shows the tail distribution (complementary cumulative distribution function) $P(X > x)$ for different values of $x$ of different estimations for the supremum of the centered GP. Note, that the safety condition here is $P(X < 1)$ The MC bound can be seen as optimal estimation of $P(X>x)$. Amongst the upper bounds to $P(X>x)$, we see, that the strong Borell inequality \ref{['eqn:Borell-1']} is the sharpest one.
• Figure 4: Consider the Himmelblau's function exploration from Section \ref{['subsection_himmelblau']}. (a) All our three novel adaptive methods (AMC, AB, ABM) improve upon the current state of the art MC with a sufficient sample size (see supplement for exact numbers), when considering the RMSE for high safety requirements $\alpha=0.001$. (b) Our method AMC improves over MC consistently in RMSE for three different safety requirements. Our favoured method ABM improves further over AMC both for (c) RMSE and (d) detecting the safe region correctly via the health coverage $c_h$. Results are averaged over 10 independent seeds.
• Figure 5: High-pressure fluid injection system with controllable inputs $v_k, n_k$ and measured output $\psi_k$ (picture taken from zimmer2018safetietze2014model)

Theorems & Definitions (6)

• Theorem 1
• Theorem 2: Borell-TIS inequality
• Remark 3
• Theorem 4
• Theorem 5
• Corollary 6