Understanding Hidden Spatial Preferences in Bayesian Optimization

Maria Bånkestad; Jens Sjölund

cs.LGOct 10, 2026

Understanding Hidden Spatial Preferences in Bayesian Optimization

Authors: Maria Bånkestad, Jens Sjölund

Abstract

Bayesian optimization uses posterior uncertainty to decide where to explore, but this uncertainty is not spatially neutral. On bounded domains, Gaussian-process posterior variance is typically larger near faces, edges, and corners, giving variance-based acquisition functions a systematic boundary bias. This spatial preference is usually inherited from the surrogate rather than chosen as part of the exploration strategy. We separate these roles by keeping the predictive model fixed and modifying the spatial preference of the exploration term. We control the preference with spatial weight fields and amortize their construction with neural surrogates. Controlled experiments show that the Gaussian process' default boundary bias changes which parts of the domain are explored and which optima are reached. We compare three alternative exploration principles and find, across standard benchmark functions and stochastic robot-pushing tasks, that they can often lower regret, particularly early in the optimization. The spatial structure of posterior uncertainty is therefore not a neutral default: it shapes the exploration strategy and can materially affect optimization performance.

Figures & tables

Figure 1: A Gaussian process fitted on a bounded domain (top) and its posterior standard deviation (bottom). In this example, uncertainty is highest near the domain boundaries, so pure variance exploration selects a boundary point despite the larger gap between observations in the interior.

Figure 2: Near edges and corners, fewer observations lie within the effective correlation neighborhood, leading to higher posterior uncertainty.

Figure 3: The three exploration principles. Top: the quantity targeted by each principle in one dimension. Geometric equalization flattens the expected weighted exploration term, uniform allocation targets the distribution of the next pure-exploration selection, and efficiency-oriented exploration favors locations whose uncertainty produces greater reduction across the domain. Bottom: the corresponding weight fields in $D=2$ for $N=10$ and $\ell_{e}=0.27$ , shown on a common scale with $g=1$ denoting no reweighting.

Figure 4: The controlled landscape in

D=2

, with local minima of identical shape near a corner, a face, and the interior. One minimum is made deeper and serves as the global minimum. Circles mark the neighborhoods used to determine whether each minimum is reached, and dotted lines separate the spatial classes used to categorize evaluation locations.

Figure 5: Effect of the spatial exploration preference on sampling and optimization. Each row shows one dimension, with

\lambda=4

and

25

paired seeds. Left: fraction of the first

D+2

evaluations after the initial design that fall in the interior, on a face, on an edge, or near a corner, together with the corresponding fraction of domain volume in grey. Whiskers are

90\%

bootstrap intervals over seeds and the vertical axis is logarithmic. Right: simple regret per BO round for each location of the optimum, shown as the mean with

90\%

confidence bands. Vertical dotted lines mark

t=D+2

and

t=2(D+2)

Figure 6: Regret difference to the unweighted rule on the benchmark tasks, after

2(D+2)

3(D+2)

, and

4(D+2)

evaluations (left to right), as a percentage of the regret shared by all rules after the initial design; negative values indicate lower regret than

g=1

. Tasks are ordered by the distance of the optimum to the boundary, with the dimension beside the name; the two robot-pushing simulators are shown to the right of the divider. Points are means over

100

paired seeds, with

90\%

bootstrap intervals over the paired differences. All rules use

\lambda=2

and the same exploration lengthscale. Beale, which every rule solves to zero regret, is omitted.

Figure 7: Simple regret on six representative tasks. Curves show means over

100

paired seeds with

90\%

confidence bands. The dotted line marks

2(D+2)

evaluations, and uniform random sampling is shown in grey. Full trajectories are given in Appendix D .

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Table 8

Figure 8: Early sampling as a function of the exploration weight

\lambda

. The relative sampling rate is the share of the first

D+2

evaluations in a location class divided by the share of the domain that the class occupies; a value of

1

corresponds to uniform sampling. Solid lines show corners and dashed lines the interior.

benchmark suite
task	$D$	$d_{\partial}$	class	$g=1$	std. UCB	$g_{\mathrm{geo}}$	$g_{\mathrm{eff}}$	$g_{\mathrm{uniform}}$	random
Hartmann 3D	3	0.11	face	0.088 $\pm$ 0.025	0.044 $\pm$ 0.026	0.052 $\pm$ 0.016	0.035 $\pm$ 0.014	0.065 $\pm$ 0.017	0.495 $\pm$ 0.052
Hartmann 6D	6	0.15	edge	0.286 $\pm$ 0.085	0.153 $\pm$ 0.031	0.191 $\pm$ 0.045	0.166 $\pm$ 0.043	0.201 $\pm$ 0.047	3.359 $\pm$ 0.175
Beale 2D	2	0.17	interior	0.000 $\pm$ 0.000	0.000 $\pm$ 0.000	0.000 $\pm$ 0.000	0.000 $\pm$ 0.000	0.000 $\pm$ 0.000	0.000 $\pm$ 0.000
Hartmann 4D	4	0.19	interior	0.059 $\pm$ 0.018	0.053 $\pm$ 0.017	0.055 $\pm$ 0.017	0.044 $\pm$ 0.016	0.056 $\pm$ 0.017	0.635 $\pm$ 0.044
Styblinski-Tang 3D	3	0.21	interior	0.253 $\pm$ 0.028	0.179 $\pm$ 0.025	0.104 $\pm$ 0.020	0.115 $\pm$ 0.021	0.192 $\pm$ 0.033	0.465 $\pm$ 0.029

Appendix

Table 3: Simple regret at the end of the budget, as the mean and

90\%

bootstrap half-width over

100

paired seeds. The columns

d_{\partial}

and class give the distance of the global optimum to the boundary and its location class. The column std. UCB is standard GP-LCB, which evaluates the exploration term at the fitted lengthscale.

Figure 9: Simple regret on every benchmark function, shown as the mean over

100

paired seeds with

90\%

confidence bands.

Figure 10: As Figure 6 , with the normalizing average

\bar{g}_{N}

taken over the domain instead of over the evaluated points.

Figure 11: Simple regret on the two robot-pushing simulators, shown as the mean over

100

paired seeds with

90\%

confidence bands. Regret below

0.001

is drawn at

0.001

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