Future-aware Safe Active Learning of Time Varying Systems using Gaussian Processes
Authors: Markus Lange-Hegermann
Christoph Zimmer
Venue: Accepted by TMLR ()

Abstract: Experimental exploration of high-cost systems with safety constraints, common in engineering applications, is a challenging endeavor. Data-driven models offer a promising solution, but acquiring the requisite data remains expensive and is potentially unsafe. Safe active learning techniques prove essential, enabling the learning of high-quality models with minimal expensive data points and high safety. This paper introduces a safe active learning framework tailored for time-varying systems, addressing drift, seasonal changes, and complexities due to dynamic behavior. The proposed Time-aware Integrated Mean Squared Prediction Error (T-IMSPE) method minimizes posterior variance over current and future states, optimizing information gathering also in the time domain. Empirical results highlight T-IMSPE's advantages in model quality through synthetic and real-world examples. State of the art Gaussian processes are compatible with T-IMSPE. Our theoretical contributions include a clear delineation which Gaussian process kernels, domains, and weighting measures are suitable for T-IMSPE and even beyond for its non-time aware predecessor IMSPE.

Published in Transactions on Machine Learning Research (04/2025)
Future-aware Safe Active Learning of Time Varying Systems using Gaussian Processes
Markus Sange-Hegernnn
Deportment of Electrical Engineering and Institute Industrial IT inI OWL University of Applied Sciences and Arts
Lemgo, Germany
Christoph Zimmer
bow.hhhhwwdbbbbboe.o.oooooo bosch. com
Bosch Center for Artificial Intelligence Renningen, Germany
Reviewed on OpenReview: http::://peeeevviee
normm? id= YBPbMKJ6bL
Abstract
Experimental exploration of high-cost systems with safety constraints, common in engineering applications, is challenging endeavor. Data-driven models offer a promising solution, but acquiring the requisite data remains expensive and is potentially unsafe. Safc active learning techniques prove essential, enabling the learning of high-quality models with minimal expensive data points and high safety. This paper introduces safe active learning framework tailored for time-varying systems, addressing drift, seasonal changes, and complexities due to dynamic behavior. The proposed Time-aware Integrated Mean Squared Prediction Error (T-IMSPF)) method minimizes posterior variance over current and future states, optimizing information gathering also in the time domain. Empirical results highlight T-IMSPE's advantages in model quality through synthetic and real-world examples. State of the art Gaussian processes are compatible with T-IMSPE. Our theoretical contributions include a clear delincation which Gaussian process kernels, domains, and weighting mcasures are suitable for T-IMSPE and even beyond for its non-time aware predecessor IMSPE.
1 Introduction
Many systems pose challenges due to their high costs for experimentation, whether in actual implementation in mumerical simulation of the underlying physics. This concern is particularly pronounced in enginecring applications. Data-driven models emerged as valuable solution for swiftly simulating such systems. However, acquiring the necessary data for these models can still be prohibitively expensive, e.g. in machine calibration [Badra et al.] (2200), biology [Sverchkov & Craven 2017), numerical sinulation (Al-Obaidi 2022), prototype building (Torres et al. 2016), or process optimization (Hernández Rodríguez et al. 2022). To address this situation where measurements can be taken almost arbitrary positions in continuous domain, active learning techniques bocome crucial, allowing the learning high-quality model using a minimal amount of expensive data points. This involves setting inputs or calibration parameters in manner that maximizes the information gathered from experiments.
Morcover, many real world systems are subject to safety constraints, such as critical temperatures, pressures, or restricted sensor working conditions, which must be adhered to during measurements. Often, these safety constraints are unknown and need to be modeled, leading to the application of safe active learning and related methodologics (Berkenkamp al. 2023 Cowen-Rivers al. 2022 Sui al. 2018 Schreiter et al.) 2015 While active learning usually superior to static Design of Experiments (DoE) approaches (Smith 1918
Pukelsheim 2006), static DoF. is not applicable in the face of learnable safety constraints outputs.

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high future variance roduction
prior variauce
posterior variance
variance
low future
future:
high
high
future:
future prior variance future posterior variance new data point variance reduction
variance reduction
varia
varia nee present:
present:
low
low
new i
varlance
HOW JE
variance
good current variance reduction
domain D
domain D
(a) Entropy, measure high current variance
(b) T-IMSPE (ours), reduce future variance
This illustrative sketch showcases the efficacy of our acquisition function T-IMSPE in variance reduction on domain D in present (dark color) and future (light color). The darker color at the bottom exhibits the transition from the prior blue line to the posterior green dashed line when conditioning on the red data point x at present time. The lighter colored top delineates the induced variance reduction in a future time step. The left subplot (a) highlights the suboptimal variance reduction when employing the entropy acquisition function: it leads to a measurement at the boundary and to a very low reduction the future variance. The right subplot (b) reveals the capability of T-IMSPE outperform entropy in reducing the average variance current and future time steps.
Figure 1:
Real-world scenarios frequently involve time- time-varying systems, presenting additional complexities. Examples include systems affected by drift (Talluru et al. (014),, possibly caused by continuously polluting sensors (Thewes et al., 2015); systems influenced by periodic trends like seasonal variations (Bacastow et al. 1985:
Scherliess & Fejer 1999); or dynamic systems, where current decisions influence future behavior, such as
is reflected in NX [Non-linear eXXgenous) model structures (Chen & Huang 2015 Zimmer et al. 2018);
Actively learning such systems proves challenging, as parts of the system may change or certain operation points might be currently unreachable. Despite these challenges, there is substantial demand for learning such systems. For example production systems with periodic changes may require accurate model-based anomaly detection, drift in sensors needs to be counteracted, or engines needs to be modeled for cost-efficient calibration. Dynamic systems necessitate an understanding of their dynamic behavior.
This paper introduces a safe active learning framework tailored for time-varying systems. The approach involves measuring at safe positions where the maximum information can be collected. Notably, we focus on also gathering information about future scenarios, rather than solely the current system state.
Technically, start from the acquisttion function IMSPE (Integrated Mcan Squared Prediction Error), also known under various other names such ALC (Active Learning Cohn) or IMSE (Integrated Mean Squared Error) [Sacks et al. 1989b). This acquisition function chooses measurement points that minimize the posterior prediction variances, integrated over the entire relevant domain. We extend IMSPE to the acquisition function Time-aware Integrated Mean Squared Prediction Error (T-IMSPE), which minimizes the posterior variance over curvent. and fulure states of the systems. Surprisingly, the necessary integration both IMSPE and T-IMSPE can be computed in closed form for many Gaussian process models, in particular the ones that are state of the art in active learning scenarios.
Our main contributions are:
We introduce the novel T-IMSPE acquisition function for safe active learning of time-varying systems, Figure We compute the defining integral in closed form, improving computational complexity and accuracy.
We demonstrate the advantages of T-IMSPE terms of model quality in a real world example about engine calibration for dynamic driving and two synthetic examples about drift and scasonal changes, Section 5
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(T-)IMSPE involves the computation of an integral. We consider computability of (T-)IMSPE in closed form via antiderivatives and provide cookbook of usage for various Gaussian processes kernels and domains to be used in (T-)IMSPE, see Section
We distinctly delineate the theoretical scope of closed-form integrals and show the boundaries of practical supremacy of T-IMSPE in various delincating scenarios in Appendix
2
Background
Active learning applications typically use Gaussian process (GP) models due to their Bayesian capabilities in dealing with few data points and their reliable uncertainty estimates through variances.
2.1 Gaussian processes (GPs)
A Gaussian process (GP) Gm k) defines a probability distribution on the evaluations of functions
D + R where the domain D c R = Rixd such that function values g(x1),...,g at points fi,-..,fT, E D are jointly (multivariate) Gaussian. Such a GP = GP(m, k) is specified by a mean function m D - IR:
x + E(g())) and a positive semidefinite covariance function
k: D2 - R (r, w) + E (Go()) - m(z)) m(w)))
Our experiments use the kind-of-default squared exponential covariance function KSE (x,x') + exp (---- [1==x118 ) Any finite list of evaluations of g at x = [pp follows the multivariate Gaus- sian distribution g(x) ~ ((m(x),k(x, for mxx) E IR" and k(x,,) E Re*a with g(x); = g(xx), m(x); = m(x;), and k(x,))i, = (Z,,,y)) Likelihoods and posteriors can now be computed via linear algebra from this multivariate Gaussian (Rasmussen & Williams 2006). We denote the posterior covariance function of GP conditioned on a dataset (z. y) (D R)" by k(-.-lz) and we define k(x) k(z, .) and k(r, w).. = k<2r,,9)) for w [101,-... Wm] The posterior covariance is independent of the output data y, which is used in active learning to plan the input
GPs are used to model dynamical systems in various ways. This paper uses NX structures in GP models (Chen & Huang, 2015 Crespi 2019) Additionally, GPs have been used to model state space systems FFigola ct al. 2014 Eleftheriadis et al. 2017), lincar differential equations (Besginow Lange-Hegermane Harkonen et al.] 2023), or periodic dynamics (Klenske el al. 2015).
2022
2.2
Time varying systems
We are motivated by learning certain time varying systems. For example, we consider systems with behavior depending on inputs x E D in addition to drift or other time dependent changes c.g. in daily or yearly cycles. Such behavior f is often modelled by including the time input t amongst the model inputs.
= f(x
(1)
In these systems, the input can be chosen arbitrarily in an active learning approach, whereas the time t is predetermined.
Another imporant kind of time varying systems is behavior y where the current behavior depends on current and previous inputs. A typical example is the modeling in the calibration of engines, whose temperature not only depends on current engine speed and engine load, but on previous values of speed and load as well, e.g. in case of braking speeding up. Other applications lie in calibration of engineering machinery like power turbines or yield prediction depending on fertilization in agriculture. Such behavior is often modelled by Nonlinear eXogenous (NX) structures, where in total l inputs are are used, often the current input x, and the l previous inputs Tt-1, 21-1111
= (1-1, T1-641)
(2)
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In such systems, the imput Fy can be chosen arbitrarily, whereas the inputs 24-1,21-21-.. 2...... .. 24-6++1 predetermined from previous choices.
In contrast, in dynamical systems, the combinations of state action cannot be chosen freely, as the system has to evolve to reach a certain state Σt. In our time varying systems, we have complete freedom to choose parts respectively 2t in (2), and the influence of the time comes from uncontrollable conditions respectively memory of the system that influences the system output.
of the inputs, ie. x in []]
2 with Bayesian optimization is common, e.g. (Brunzema et al. 2022 The goal of this paper is the construction of novel acquisition function for safe active learning in such dynamic systems, including the computation of closed form integrals and a cookbook for which GP covariance functions this is possible.
Optimizing such systems [
2.3 Safe active learning
Safe Active Learning (SAL) (Schreiter et al. 2015 Zimmer et al. 2018) is a sequential experimental design which strives for safety under previously unknown constraints. SAL iteratively determines a new point æ, for labolling that maximizes information [(x,), where I the so-called acquisition function, and is safe with high probability. Entropy is commonly used as measure for information and in case of Gaussian distribution proportional to the predictive variance of the regression GP. For comparison and elaboration of entropy, e-g. its other names in the literature, see Subsection 3.1
We consider the case of a safety critical quantity, e.g. a temperature should stay below critical temperature. Then, one can calculate a safety indicator z that indicates safety for non-negative values = 2 O. We assume that the safety critical quantity cannot be computed, but (potentially noisy) measurements on this safety critical quantity can be obtained. Then, a safety GP Isade = GP (maaee kaaee) modeling the values of gsate can be trained. Often but not necessary, the safety GP is the (or amongst the) model(s) that strive to learn. This yield the probability of safely obtaining obtained
point x. when already some data (xr, y) has been
{(xx) (ocx)=/a ///=//00 M(2:meatel meate (Z. (x, (xx, Ix (x)) dz.
820
Now, new point 2, be determined by maximizing (cf. Appendix F information constrained safety:
x.
I(x.)
where 0 a < indicates the desired safety level. The goal of this paper is the introduction of new acquisition function for measuring information / for time varying systems.
Remark 2.1. Our experiments use a 0.977 such that two standard deviations from the mean of a Gaussian is predicted as being safe. This safety criterion corresponds to the probability of a random and statistically independently drawn point being safe. In active learning, points are not drawn independently, since the data points are not drawn independently but chosen by an acquisition function. Due to this lack of statistical independence, points might have a higher probability of being unsafe. To describe our experiments, we chose the wording "satisfying" for safety percentages around 97.7 % or higher and 'accoptable' for slightly lower percentages that are still above 95 %. All experiments, independent of the acquisition functions, show similar and least acceptable safety.
After a new label y/ has been obtained, both the regression and the safety GP can be retrained and the next constrained optimization problem solved. More formally, SAL can be summarized the following Algorithm
We use GP hyperparameter training, which is fast and well-cstablished. If in certain real-time applications further speed up is required, techniques of amortized inference are at hand (Bizzer el al. 2023 Liu et al., 2020). Changing hyperparameters has a strong effect on the overall performance of all algorithms, including on safety (Fiodler et al. (024)) due to model misspocification, in particular case of few data points. Still, changing hyperparameters state of the art in active learning and Bayesian optimization (Garnett) 2023) and, in our experience, superior to fixing hyperparameters.
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Input: Budget n, initial data (Ir, v) (Iri, We 1=) Mi Output: Trained Gaussian Processes Train initial GPs; for i i= 1 to n do
Determine I. by solving (3) and get label ye for 24; Update (x, + ((x, y), (Z+: H+]) and retrain GPs;
end
Algorithm 1: Safe Active Learning (SAL) with Gaussian Processes
2.4 Integrated Mean Squared Prediction Error (IMSPE)
The Integrated mean squared prediction error (IMSPE) acquisition function strives for a model posterior with minimal average variance. Start with a (prior) Gaussian process GP(m, k) and have already measured at n positions x E Rnxdd IMSPE selects Ia E IR lxd with minimal
J k(z,/o,, ())
(4)
where (x,r/ar,, .)) is the posterior variance of the Gaussian processes g when conditioned on observations at and æ, and is suitable finite mcasure used to average over reference points Zr.
Consider IMSPE for the special case of the squared exponential covariance function with lengthscale of one and signal variance one and a Gaussian j with mean zero and variance 1. Assume furthermore one existing measurement point x Now let us consider which point z. E to choose next, according IMSPE. the appendix in Example A.1 we explicitely compute the ISMPE acquisition function Sz k(x, (x,, 1) dp((,)) plotted in Figure 2 Values for I, are most suitable to reduce the variance of the posterior GP in the area around zero with variance (as specified by the mcasure μ1) if they are slightly negative. The positive area is less suitable, since there already exists a measurement point at = 1.
Sye (1x. du(x,)
Figure 2: The IMSPE critcrion in the motivating example. Without the (groen) data at x = 1, the IMSPE criterion would be symmetric around zero. With the (green) data at x = 1, the optimal choice according to IMSPE for the (red) next measurement point xs the red position x, 0.5479200453..
We give a short discussion on IMSPE and entropy as aquisition functions for safe active learning without time awareness. We introduce related work on IMSPE in Section 3 and for further theoretical comparisons of IMSPE against other acquisition functions, we refer to Appendix D In the realm of statisties, IMSPE stands out the state of the art, where the superior theoretical properties of IMSPE are highly regarded. In contrast, in the field of machine learning, entropy takes precedence state of the art. Our preliminary experiments have indicated that both communitics make valid choices, and the divergence in statc-of-tho-art criteria arises from varied applications ions of GPs. In statistical communities, there is lendency to mainlain fixed GP hyperparameters in active learning methods, while in machine learning, hyperparameters are regularly optimized. Our experiments on timo-varying systems, where we do optimize hyperparameters, contradict these preliminary experiments, IMSPE is superior over entropy. In contrast to this, the ablation experiment in Appendix E.1 shows that without seasonal change (a 0), entropy outperforms T-IMSPF,
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which is identical to IMSPE for such time-constant systems. We note that these findings based on initial experiments, lack scientific rigor, need thorough follow-up, and are out of scope of this paper.
3 Related work
3.1 Acquisition functions for active learning
In machine learning, the standard acquisition function is entropy. This approach is often called ALM (Active
Learning Mackay) (MacKay 1992). The acquisition function BALD (Houlsby et al. 2011 was originally presented for classification tasks. For regression tasks, it is equivalent to the entropy criterion Riis et al.
2022a, A.3) (Riis et al. 2022b G.2).
The acquisition function mulual information has gained prominence in safe active learning scenarios. Originally applied in the context of sensor placement for fixed sensors (Krause et al. 2008), mutual information-based acquisition functions have found applications in active learning scenarios with discrete domain D (Vasisht
et al. 2014 Kirsch et al. 2019 Li et al. 2022b). For detailed comparison of IMSPE with averaging mutual
information (Krause et al. 2008 and B-diversity (Leinster) 2021 refer to Appendix
D.2
While classical criteria from the Design of Experiments (DoE) theory (Smith 1918 Pukelsheim 2006), such as D-optimal designs, could be potential acquisition functions, they have not gained traction in machine learning due to their emphasis on learning parameters rather than making predictions. IMSPE closely connected to the classical V-optimality criterion DoE thcory, where measurement points are chosen minimize the model's variance at finite number of points (Fedorov & Hackl, 1997, (2.1.18).. IMSPE can be considered variant of A-optional designs, where it minimizes the average variance of the response, while A-optimality minimizes the average variance of the model paramcters. Other geometric criteria have also been proposed in practice (Thewes et al. 2016). None of these DoE criteria allow to include safety constraints.
IMSPE has a long history. The seminal work by (Sacks et al. 1989b), see also (Sacks et al. 1989ah and the authors' book (Santner et al. 2003), introduced the acquisition function IMSE (Integrated Mcan Squared Error), estimating mean squared error via posterior variance for polynomial covariance functions. YYing 1991
renamed TMSE into IMSPE and conducts extensive theoretical investigations into its asymptotic properties.
The second time the name IMSPE appears in the literature is in (Fang 2000). (Ankenman et al.
2010
proposes the use of IMSE, for which they use a closed form formula, for active learning. (Leatherman et al. 2014 gives a thorough overview about techniques for static designs with IMSPE. (Leatherman et al. 2018 computes the IMSPE for certain finite dimensional GPs. For modern references to IMSPE we refer to
|Gramacy 2020: Binois et al. 2019), with exlensive implementation hetGP (Binois Gramacy 2021 in the
system R.
IMSPE has been reintroduced under different names. Cohn et al. 1996) introduced active learning for mixtures of Gaussian with the goal of minimizing IV (Integrated Variance), for which they found a simple closed form; this is another special case of IMSPE later named A (Active Learning Cohn). (Seo et al., 2000) introduced active learning to GPs, where they used VI from ALC while approximating the integral using sampled points. (Burnaev & Panov 2015) independently introduced IMSPE as IntegratedMSEGGain providing an unproven closed form. (Gorodetsky & MMrzzouk] (0166) independently introduce IMSPE under the name IVAR (Integrated VARiance), where they compute the integral numerically. Recent literature has extensively utilized IMSPE its variants. (Vernon et al. 2019) focuses on safety boundaries, (Meka et al. 2020) applies ALC in closed-set active learning, (Lee et al.
2022 investigates IMSE, ALC, and ALM in safe active learning, and (Lee et al. 2023) employs a discrete approximation to ALC/MMSE for local GPs.
None of these approaches use IMSPE or its variants under different names to specifically collect information for future time steps.
Some authors distinguish between IMSPE and ALC (auuer al. 2023 noting that IMSPE computes the intogral in equation in closed form, while ALC approximates it numerically. However, this distinction is not widely recognized in the literature. For an example, see GGramacy 2020 $6) by the second author (Suuer et 2023

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3.2 Active learning in dynamic systems
Active learning has also been explored in dynamic systems. (Schneider 1995 Umlauft et al. 2020 consider dropping old data points when their information content becomes irrelevant. Lughofer 2017] provides an overview of active learning in data streams, while (Jain et al., 2018 applics active learning to GP models for non-linear control during closed-loop control. Safety considerations in dynamic systems are addressed by (Zimmer et al. (2018), and (Buisson-Fenet et al. 2020) proposes an iterative active learning scheme dynamical systems based on mutual information. (Yu al. 2021) conducts active learning specifically for
GP state space models, and (Heim et al.) 2020) learns safety constraints in dynamical systems. Similar strategies can be found in reinforcement learning for non-stationary environments (Padakandla, el al. 2020
Padakandla 2021 Ritto et al.) 2019: Capone et al.
2022 Cowen-Rivers et al. 2022). control (Fisac ct al. 2018 Agarwal et al.
2020), concept drift (Han et al. 2022). and theoretical work on regret in non-stationary
systems (Zhag 2021
Zhao et al. 2022). The papers (Fiducioso et al. 2019 Krause & Ong
2011 consider
regret in the current time/context and show that their choice of acquisition function still sublinear changing contexts.
Notably, none of these approaches considers acquisition functions that account for information prediction accuracy at future time steps.
3.3
Safe learning
Safe exploration has been employed in robotics (Sui ct al.
2018 Berkenkamp et al. 2016
Baumann et al.
2021),
energy management (Galichet et al. 2013). terrain exploration, (Moldovan & Abbeel 2012 Turchetta
let al. 2019h and engine modeling (Schreiter et al. 2015 Zimmer et al. 2018 Schillinger et al. 2017 Li
2022a). While most of this Safe Learning work is on Bayesian optimization, some is on active learning iet al.
but does not consider future information or future prediction accuracy. (Li et al. 2024 considers Bayesian optimization (but not active learning) a dynamic setting, with focus on safety guarantees.
Active learning aims learning model over the whole input space perform some post-hoc or worst-case analysis. In contrast, Bayesian optimization is only interested in optimal point of operation. Once this point or the area is found, the rest of the input space is not of any interest anymore. In this regard, active learning and Bayesian optimization are conceptually different. Specifically, IMSPE reduces the variance globally and seems unreasonable to be integrated meaningfully into a Bayesian optimization framework.
4 T-IMSPE - Time-aware Integrated Mean Squared Prediction Error
Our criterion T-IMSPE uses the concepts and formulas from IMSPE time-dependent models, to ensure variance reduction not only for the current time step, but also for future time steps. We first consider GP models with time as an additional input for both continuous and discrete time domains, and afterwards we consider NX-GP models for dynamie systems in discrete time domain. In both cases, T-IMSPE work with wide range of GP covariance structures.
4.1 T-IMSPE for GPs with time amongst its inputs
Here, we consider time varying systems as in where the time is one of the model inputs. Consider GP g gP(m,k) k) defined the domain R x D, where R represents the time domain and D Rd represents the remaining inputs, e.g. "spatial" inputs. For example, when not encoding any specific time dependent behavior in the GP prior, k might be squared exponential covariance function on [B++1, Let be a finite measure on R. x D. We define T-IMPSE for choosing a new data point x, E D at time t,, E R after already obtained data (r,z)= E (R x D)" for such GPs as
T-IMSPE(s.) =
((())>))))__))))))))())))) ((((,,,
(T, dp((,;;;)
5
By Theorem 4.4 below, this integral is computable in closed form for most common covariance functions. The time-aware aspect be included in the measure μ. Assume have suitable measure, ju on the
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domain D. Then, we encode desire to accumulate information in the relevant time interval [t,, + At) by a uniform distribution 1t 1 [t.,e.+r1]- Then, defining as the product measure / == 1r & ju strives to choose a new data point as follow: the posterior variance over the domain D is reduced being weighed by μD, while this reduction not only achieved at t, but aware of future times in all of [t.. t, + Ar]. The relevant time domain might also be considered discrete via μ = 1 (t,,tt +h..)+ then μ Mr @ DD Eist. 5, & UD for the Kronecker delta 62.
4.2 T-IMSPE for GPs with NX-structure
Here, we consider time varying systems in 2 where the time dependency is modeled via an NX structure. Consider y/2 = f (2t, .,,
for some positive & E N by placing a GP prior g GP k) on the function ʃ: D' + RR. Such models work in discrete time domain. Let be a finite measure defined on D Assume we already obtained n data points at 2 E (Df)", where the condition Zi,t = 2441.++1 E D for the NX structure for i E f1,2,..... ........11 1} and t E (1,2...., - 1}. We define T-IMPSE for choosing new data point Is D for such NX GPs as
T-IMSPE(2.) = / - 2(I dp(x,),
(6)
1.. E De is the new measurement, position with previous measurement positions to predict the model dynamics. Since , are the measurement positions from previous steps, they are well known values, to which we add the vector I, of new inputs. We now use the closed formula for T-IMSPE in k(a, x,, from Theorem 4.4 where instead of x, we substitute V,. This formula allows optimization for F..
where v.
In our experiments we chose the ê-times product measure μ := 0 PD for a fixed finite measure Ap on the domain D. Note that if pD is Gaussian or a uniform measure on cimttidiie interval, then so is μ. This acquisition function is timc-aware, points are chosen that reduce the variance over all possible dynamical behaviour.
This version of T-IMSPE is a special case of the previous one: start with the version from Subsection 4.1 ignore the time dependency in the GP and intepret the GP to have the NX-time inputs. Now, the additional average over time in T-IMPSE is trivial.
4.3 P-elementary functions for closed form (T-)IMSPE
A key step to applicability of our new T-IMSPE criterion is the evaluation of the integral in Equation (4) and hence in the Equations (5) and 6 This section shows that the integral can be computed in closed form for a variety of covariance functions and measures. This decreases computational load improves numerical stability of values and gradient. Even though this work uses the squared exponential kernel, work on efficient kernel selection (Bitzer al. 2022 allows the modeler to quickly pick suitable composed kernel. Our theoretical contributinn ion actually extends beyond our T-IMSPE criterion to the general situation of IMSPE. We summarize and extend existing results (Sacks et al. 1989b Leatherman et al. 2018 that this integral can be computed in closed form, putting them into a more abstract and widely applicable framework.
Recall the class of elementary functions, which was defined in the 19th century [Liouville] 1833a e Bronstein 2005) to c.g. describe the well known result that while erf(x: == f exx(===² dx analytical and even entire, it is not expressible in closed form using classical functions. The class of elementary functions not suitable to describe functions being computable in PyTorc,, e.g. it is too big in the sense that it is closed under inverse functions and it is too small as functions like (r)((:) are nowadays casily computable. Furthermore, the class of elementary functions only considers univariate scalar function f U + IR for U C R, whereas multivariate multi-ouput functions are common in machine learning. Now, we adapt this definit ion of elementary functions to formally define what it means to compute functions in closed form in a system like PyTorch.
Fix programming framework P. We think of as PyTorch, but one might as well think of P as JAX, TF any other current or future framework. Then, view the following operation computable in closed
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form in P: [a] they have access to functions in P. [b] one compose functions, and perform arithmetic operations. More formally, we define the following.
Definition 4.1. We define the set of P-elementary functions as the intersection of all sets of partia functions U - d for U _ Rdd d', d" E Ezy, that
(a) include all constants (considered as constant functions) and functions available in P.
(b) are closed under function composition o, and
(c) closed under (multivariate and multi-output) rational maps; in particular additions, subtractions, multiplications, divisions, and linear maps.
Example 4.2. The following functions are PyTorch-olementary:
i. 22) + 201 - 42x² by Definition
ii. an - ex1( by Definition 4.1
ii (2) + erf( (21) by using all Definition 4.1 as (I1 H erf(x11) o ((x1, + x1).
iv. (I1, + moffmmmma + 22, (( by using all of Delinition 4.1
We now show that IMSPE and T-IMSPE are PyTorch-elementary for a wide range of covariance functions, making safe active learning with them is computationally efficient and numerically stable.
4.4 Computability in closed form
IMSPE and T-IMSPE can be computed in closed form with mild assumption on the GP covariance. Definition 4.3. Let μ be a finite measure on Rd. We say a covariance function & P-elementary covariance marginalizable w.r.t. i if the integral f k(x1,, kr,, x22) du(x,) exists as a P-elementary function in 31 and 22-
This is a non-trivial definition. In the appendices, we show or recall from the existing literature that constant (Example A.3 ARD squared exponential (Examples A.4), polynomial (Example A.5 half-integer Matérn (Example A.5 Wiener process (Example A.5 and random Fourier feature (Example A.11 covariance functions are Pytorch-elementary covariance, while rational quadratic and periodic covariance functions are probably not (Example A. all when considering as non-degenerate continuous uniform distribution on a multidimensional interval or non-degenerate Gaussian distribution.
The class of P-elementary covariance marginalizable covariance functions closed under various operations: scaling (Proposition A.8 multiplication on independent inputs (Proposition A.9), and also under sums when the covariance functions satisfy another condition (Proposition A.10 Furthermore, if covariance function is P-elementary covariance marginalizable w.r.t. two measures, then so it is w.r.t. the sum of these two measures (Proposition A.12 This cookbook allows to construct new GPs from previous ones, such that T-IMSPE stays applicable. For more details on this cookbook see Appendix A.3 From this discussion we conclude that the following theorem is widely applicable.
Theorem 4.4. Assume the prior covariance function k of prior GP to be P-elementary covariance marginalizable w.r.t. a measure μ. Then, IMSPE and T-IMSPE are P-elementary, i.e. they can be computed in closed form in the programming framework P.
These results have the major advantage that it is sufficient for the prior covariance function to be P-elementary covariance marginalizable, and we do not need properties for the posterior covariance functions. For a proof
PIntersecting way of constructing mathematical objects when direct constructions inconvenient. E.g, the span of of vectors be defined intersection of all lincar subspaces containing this set of vectors. 3partial functions are functions that defined in all their domain. This is necessary, e.g. divide through functions having zeros logarithms.
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P2
1:25
32
0
-G.
0.0
T1
0.35
o
6.0
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-0.5
-0.0
-0.220
-0.5
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2
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4
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-03
-25
Figure 3: Plots for the scasonal change experiment (Subscction 5.1 for = 0 (left) and t = 6 (right).
of Theorem 1.4 we refer to Appendix [ which also provides a very concrete formula for the computation of these criteria. We do not claim novelty these proofs, which are known for the special case of polynomial covariance functions since (Sacks et al.) 1989b) and additional special cases spread over the subsequent
lileralure, see Subsection 3.1
We follow the state of the art approach in active learning, where GPs are trained after each addition of a new data point, including hyperparameter training. After this training, IMSPE and T-IMSPE have the same computational complexity as computing the entropy and can be evaluted on On for n training data points.
Corollary 4.5. IMSPE and T-IMSPE as computed in the proof of Theorem 4.4 need a one-time Cholesky decomposition (which is computed anyway during the GP training) of the data covariance matriz k(æ, x) in O(n²), independent of the new data point Ta, where n the number of data points. Afterwards, these criteria can be evaluated in O(n²) a function in T..
The proof of this corollary in Appendix is very explicit and yields direct formula for computations. For us, the most important special case of the above general theory is the following.
Corollary 4.6. IMSPE and T-IMSPE can be computed in closed form in Pyfoorc for prior GP with squared exponential covariance, provided the measure μ is Gaussian continuous uniform.
5 Experiments
We conduct three experiments for safe active learning (Algorithm 1 in time varying systems. Thereby, demonstrate the superiority of the modeling quality achieved by T-IMSPE over both entropy, which is currently the state of the art (sce Appendix D in active lcarning, and IMSPE, while keeping the same safcty standard. For further results of our experiments resp. further details on the setup of our experiments see Appendix E resp. Appendix and the attached code. We use the paired Wilcox signed rank test to show statistical significance.
5.1 Experiment: seasonal change
We consider learning system with strong periodic seasonal changes. The system is given as by rotating the tro-dimensional domain in the function from Equation 7 in Appendix F.1 and plotted in Figure 3 with additional plots in the Appendix in Figure 11 It particular challenging due to the quick period in time of 4T 12.6 and high variation of around 50 % in its range. These changes also strongly affect the position of the safe area.
We start with 8 initial measurements at times O..... positioned at the inital points of Sobol sequence in the safe area. Afterwards, 100 further measurements at times 8,..., 107 are conducted according to the respective safe active learning criteria T-IMSPE (from Equation 5) entropy, and IMSPE.
For the upcoming experiment and the subsoquent experiment, we utilize a grid as our test datasot, restricting to grid points where the behavior is currently safe. The rationale behind restricting our test data to this safe area. is rooted assessing the model's quality there. The test data varies across different time steps, reflecting
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T-IMSPE (ours)
RMSE gain in safe arca
4
RMSE in safe area
Entropy IMSPE
3
5
4
3
2
average
10%
20%
30%
%00
%60
%0y
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90%
100%
average
10%
20%
30%
%00
%60
%02
%08
%606
0%
16011
time in experiment
time in experiment
Figure 4: Box plots of RMSE value in the safe area of the experiments from active learning with seasonal changes (left, Subsection 5.1 and under drift (right, Subsection 5.2h. The x-axis shows these value averaged over all time steps and at certain ascending time steps from 10% to 100%. The left diagram compares the results of 450 runs between T-IMSPE (bluc), entropy (red), and IMSPE (green), where T-IMSPE is highly significantly (p < 2.2e-16) superior to entropy in all eleven comparison, whereas T-IMSPE is superior over IMSPE on average (p < 2.1e-3). The median error reached by T-IMSPE at 30 % of the time is reached by entropy at 50 % of the time, which is a % reduction of mcasuremt time and cost. The right diagram shows the gain of RMSE values in 500 runs when choosing T-IMSPE over entropy (red) and over IMSPE (green). The results are significantly (p < 1e-10) better for T-IMSPE over entropy in all 11 comparisons and and significantly better in 10 out of 11 comparisons for T-IMSPE over IMSPE.
the changing state of the system. Consequently, analysis is confined to the current time, deliberately excluding future time steps. This intentional exclusion prevents any unwarranted advantage of T-IMSPE over entropy in the evaluation criteria, as T-IMSPE optimizes over future time steps. This precaution ensures a fair and unbiased evaluation of model performance within the immediate temporal context. Comparisons are done between runs of equal random seeds, in particular the same noise is added to the initial measurement points.
Due to changing domains, we strive for good RMSE not only the final time step, but during all of the measurement time. Figure summarizes the results in terms RMSE in 50 runs with different seeds by the above protocol after 10, 20, .... 100 time steps, and the average over all time steps. In this experiment, T-IMSPE is vastly superior in reducing RMSE in comparison to the state of the art entropy. The superiority of T-IMSPF is highly significant (p < 2.2e-16) for all time steps and the average model quality. This number 2.2e-16 is the smallest representable p-value in and indicates a result that is statistically highly significant, essentially suggesting that the probability of the observed result occurring under the null hypothesis is vanishingly small. Comparing T-IMSPE IMSPE, we have T-IMSPE being superior IMSPE on average (p < 2.1e-3) and over the first three time steps at 10 % to 30 % (p < 2.2c-4, p < 1.1e- p 1.5c-7). 7). Afterwards, when IMSPE and T-IMSPE have both mostly converged, both methods perform comparably, with advantages for IMSPE at time steps 40 % to 90 %.
With all acquisiiion functions, over 99.5 % of all points were safe; this is satisfying by Remark 2.1
5.2 Experiment: drift
We consider lcarning a system with strong temporal drift. The system is given as by the formula in Equation (8) in Appendix F.1 and plotted in Figure 5 with additional plots in the Appendix in Figure 12 and Figure This system particularly challenging due to the strong drift effect: the safe area shrinks in size by approximately factor of 2 and the range of the function increases by factor of more than 150. With the exception of the changed formula, our setup is the same as for the experiment with seasonal drift in the previous Subsection 5.1
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T2
0.0
T1
31
2
4
2
2
4
4
-2
2
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-0.1
Figure 5: Plots of the drift experiment (Subscction 5.2 for t t= 0 (left) and t = 100 (right).
Due to changing domains, we strive for a good RMSE not only at the final time step, but during all of the measurement time. Figure summarizes the results in terms RMSE in 500 runs with different seeds by the above protocol after 10, 20, .... 100 time steps, and the average over all time steps. We see that at almost all time steps, and in particular on average (left), the RMSE of T-IMSPE is smaller than that of the entropy. In
comparisons. The superiority of T-IMSPE over IMSPE is highly significant (p < 2c-16) 16) on average, highly significant (p < le-5) for time steps (10 %, 40 %, 50 %, 60 %, 80 %), significant (p < 5e-2) for time steps (20 %, 70 %, 90 %, 100 %), and inconclusive at one time step (30 %).
We have 96.2 %-9..5 % safe points for all acquisition functions; this is acceptable by Remark 2.1
5.3 Dynamic real world system: rail pressure
This experiment considers the rail pressure example from (Tictze et al., 2014). The system consists of an actuation Us and an engine (rotational) speed ngy which yield the rail pressure 4/a This system is partieularly challenging, measurement positions at one time step strongly affect the measurements at subsequent time steps. Furthermore, such measurements are quite expensive in real world scenarios. There exists a formula, given in the code, such that ubx- = (n Mg-1. R::2; n Uhs Ub-1, Ux-3) with U2-2 missing, and we will try to find a GP model such that the z g(n,, n fgg 71-3, Uk, Uh-1, 2 Uk-3)- Here, the final RMSE values in the safe domain are most important practice. To obtain test data in the safe area, we constructed random safe trajectory of length 2024, where the next point is always a random point that turned out to be safe.
Previous papers considered this dynamic example using piecewise linear trajectories as in Zimmer et al. 2018
where given history of measurement position (na, vx), (nR-1, vk-1), new point (R Ck-+5) was chosen 5-1 and the measurement position between were linearly interpolated as (Mo+d, Uk+4) = 3 (Nx, uk) g (1++55 Uk+5). We consider fully dynamic lcarning, where given the above history, (Mx+1, Uk+1) is chosen. Hence, we are in the setting of NX models as in Subsection 4.2 where IMSPE is not feasible as a comparison.
The results of the RMSE of comparing 100 runs of entropy to 100 runs of T-IMSPE (from Equation 6 over 1000 itcrations are shown in Figure 6 The superiority in RMSE values shows the superiority of the T-IMSPE over entropy for dynamical systems. The Wilcox signed rank test conducted 10 times (every 100 steps) shows that at all time steps TIMSPE is significantly better than entropy with < 1.1e-6 after 100 time steps and < 3.2e-15 (sic)) at all remaining tested time steps.
Entropy had 97.5 % safe points and T-IMSPE had 99.2 % safe points in these experiments, see also Table 2 in Appendix which is satisfying for both acquisition functions by Remark 2.1
6 Conclusion
This paper introduced an algorithm for safe active learning that specifically suited for time varying systems. Our T-IMSPE acquisition function is able to capture the time dependence and does not only collect information for the current. time step, but also for future time steps.
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RRMS
8
Rail Pressure the
Actuation Die
6
4
2
entropy
Fluid
T-IMSPE (ours)
steps
Engine Speed na
250
500
750
1000
Figure 6: Left: Common rail pressure system (system for high-pressure fuel injection) with controllable inputs Uk, B& and measured output th, taken from Zimmer et al. (2018) Tietze et al. (2014) Right: This diagram show the decline in RMSE in the rail pressure model from Subsection 5.3 Entropy (red) shows a slow decline over the 1000 steps, whereas our approach T-IMSPE declines more consistensly, faster, and ends in much smaller RMSE values. The dashed line is the mean of 100 runs, the area shows the 2a area and the solid lines show 5 examplatory runs. The mean error reached by T-IMSE in the rail pressure example after 250 steps is the same as the one reached by entropy after 1000 steps. This is an 75 % reduction of measurement time and cost
Our T-IMSPE acquisition function can be evaluated in closed-form at comparable complexity as the widely used entropy acquisition function. This holds for commonly used kernels and we theoretically show the boundary of computability for the IMSPE and T-IMSPE criteria and construct cookbook of GP constructions for IMSPE and T-IMSPE. The superiority of TIMSPE is highly significant.
Additionally, the theoretical contributions this paper allow further applications which need more control over information collection; for example one can specify where and when information important and one can see for which covariance functions such information gathering possible.
The experimental results show a clear and significant advantage for the model quality obtained by T-IMSPE over the state of the art entropy and over IMSPE in various settings of time changing systems, while keeping the safety. In Appendix E.1 we discuss what happens if systems are only slightly time changing.
References
Naman Agarwal, Brian Bullins, Elad Hazan, Sham Kakade, and Karan Singh. Online control with adversarial disturbances. International Conference on Machine Learning (ICML), 2019.
Ahmed Ramadhan Al-Obaidi. Characterization of internal thermohydraulic flow and heat transfer improvement in a threc-dimensional circular corrugated tube surfaces based on numerical simulation and design of experiment. Heat Transfer, 5165)84688 4713, 2022.
Bruce Ankenman, Barry L. Nelson, and Jeremy Staum. Stochastic kriging for simulation metamodeling. Operations Research, 58(2):371-382 2010.
RB Bacastow, ChD Keeling, and TP Whorf. Seasonal amplitude increase in atmospheric CO2 concentration at Mauna Loa, Hawaii, 1985.
Journal of Geophysical Research: Atmospheres, 90(D6):10529 -05440,
Jihad Badra, Jaeheon Sim, Yuanjiang Pei, Yoann Viollet, Pinaki Pal, Carsten Futterer, Mattia Brenner, Sibendu Som, Aamir Farooq, Junseok Chang, et al. Combustion system optimization of light-duty GCT engine using CFD and machine learning. SAE World Congress Experience, 2020.
Dominik Baumann, Alonso Marco, Matteo Turchetta, and Sebastian Trimpe. Gosafe: Globally optimal safe robot. learning. International Conference Robolics and Automation. (ICRA), 2021.
13

---

Published in Transactions on Machine Learning Research (04/2025)
Felix Berkenkamp, Angela P. Schoellig, and Andreas Krause. Safe controller optimization for quadrotors with Gaussian processes. International Conference on Robolics and Automation (TCRA), 2016.
Felix Berkenkamp, Andreas Krause, and Angela P Schoellig. Bayesian oplimization with salety constraints: safe and automatic parameter tuning in robotics. Machine Learning, 1171137)3713-3777 2023.
Andreas Besginow and Markus Tange-Hegermamn. Constraining Gaussian processes to systems of linear ordinary differential equations. Advances in Newral Information Processing Systems (NeurIPS), 2022.
Mickaël Binois, Jiangeng Huang, Robert Gramacy, and Mike Ludkovski. Replication or exploration? Sequential design for stochastic simulation experiments. Technomelrics, 61(1):7 23, 2019.
Mickaël Binois and Robert B. Gramacy. hetGP: Heteroskedastic Gaussian process modeling and sequential design in R. Journal of Statislical Software, 98(13):1 44, 2021.
Matthias Bitzer, Mona Meister, and Christoph Zimmer. Structural kernel search via Bayesian optimization and symbolical optimal transport. Advances in Neural Information Processing Systems (NeurIPS), 2022.
Matthias Bitzer, Mona Meister, and Christoph Zimmer. Amortized inference for Gaussian process hyperpa- rameters of structured kernels. Uncertainty in Artificial Intelligence (UAI), 2023.
Manuel Bronstein. Symbolic integration I: transcendental functions. Springer, 2005.
Paul Brunzema, Alexander Von Rohr, and Sebastian Trimpe. On controller tuning with time-varying Bayesian optimization. In 2022 IEEE 61st Conference on Decision and Control (CDC), 2022. doi: 2022199999.022299924999999999999
Mona Fenet, Friedrich Solowjow, and Sebastian Trimpe. Actively learning Gaussian process dynamics. Learning for Dynamics and Control, 2020.
Evgeny Burnaev and Maxim Panov. Adaptive design of experiments based on Gaussian processes. Statistionl Learning and Data Sciences (SLDS), 2015.
Alexandre Capone, Armin Lederer, Jonas Umlauft, and Sandra Hirche. Data selection for multi-task lcarning under dynamic constraints. Control Systems Lellers, 5(3):959 964, 2020.
Zhiyong Chen and Jie Huang. Stabilization and regulation of nonlinear systems. Springer, 2015.
Chun-Huo Chiu, Lou Jost, and Anne Chao. Phylogenetic beta diversity, similarity, and differentiation measures based on hill numbers. Ecological Monographs, 84(1):21-44, 2014.
David A Cohn, Zoubin Ghahramani, and Michael I Jordan. Active learning with statistical models. Journal of artificial intelligence research, 4:129 145, 1996.
Alexander Coven-Rivers, Daniel Palenicek, Vincent Moens, Mohammed Amin Abdullah, Aivar Sootla, Jun Wang, and Haitham Bou-Ammar. Samba: Safe model-basod & active reinforcement Icarning. Machine Learning, 111(1):173-200) 2022.
Veronica Crespi. Dynamic safe active learning with NARX Gaussian processes. Master's thesis, Universität Stultgart, 2019.
Frank E Curtis, Tim Mitchell, and Michael L Overton. bfgs-sqp method for nonsmooth, nonconvex, constrained optimization and its evaluation using relative minimization profiles. Optimization Methods and Software, 32(1):148 181, 2017.
Stefanos Eleftheriadis, Tom Nicholson, Mare Deiscnroth, and James Hensman. Identification of Gaussian process state space models. Advances in neural information processing systems (NewrIPS), 2017.
Zhide Fang. Robust extrapolation designs for biased polynomial models. Journal of Statistical Planning and Inference, 13() 147, 2000.
14

---

Published in Transactions on Machine Learning Research (04/2025)
Valerii V. Fedorov and Peter Hackl. Model-Oriented Design of Experiments. Lecture Notes in Statistics 125. Springer, 1997.
Marcello Fiducioso, Sebastian Curi, Benedikt Schumacher, Markus Gwerder, and Andreas Krause. Safe contextual Bayesian optimization for sustainable room temperature pid control tuning. International Joint Conference on Artificial Intelligence (IJCAI), 2019.
Christian Fiedler, Johanna Menn, Lukas Kreisköther, and Sebastian Trimpe. On safety in sale Bayesian optimization. TMLR, 2024.
Jaime F Fisac, Anayo K Akametalu, Melanie N Zeilinger, Shahab Kaynama, Jeremy Gillula, and Claire J Tomlin. A general safety framework for learning- learning-based based control in uncertain robotic systems. Transactions on Aulomalic Control, 64(7):2737-2733 2018.
Roger Frigola, Yutian Chen, and Carl Edward Rasmussen. Variational Gaussian process state-space models. Advances in neural information processing systems (NeurIPS), 2014.
N. Galichet, M. Scbag, and O. Teytaud. Exploration vs exploitation vs safety: Risk-aware multi-armed bandits. Asian Conference on Machine Learning, 2013.
Jacob Gardner, Geoff Pleiss, Kilian Q Weinberger, David Bindel, and Andrew G Wilson. GPyTorch: Blackbox matrix-matrix Gaussian process inference with gpu acceleration. Advances in neural information processing systems (NeurIPS), 2018.
Roman Garnett. Bayesian oplimizalion. Cambridge University Press, 2023.
Alex Gorodetsky and Youssef Marzouk. Mercer kernels and integrated variance experimental design: con- nections between Gaussian process regression and polynomial approximation. Journal on Uncerlainty Quantification, 4(1):796-828, 2016.
Robert Gramacy. Surrogates; Gaussian process modeling, design, and optimization for the applied sciences. CRC press, 2020.
Meng Han, Zhiqiang Chen, Muhang Li, Hongxin Wu, and Xilong Zhang. A survey of active and passive concept drift handling methods. Computational Intelligence, 38(4):1492-1535, 2022.
Mare Harkonen, Markus Lange-Hegermaan and Bogdan Raita. Gaussian process priors for systems of linear partial differential equations with constant coefficients. International Conference on Machine Learning (ICML), 2023.
Steve Heim, Alexander von Rohr, Sebastian Trimpe, and Alexander Badri-Spröwitz. A learnable safety measure. Conference on Robot Learning, 2020.
James Hensman, Nicolas Durrande, and Arno Solin. Variational fourier fcatures for Gaussian processes. Journal of Machine Learning Research, 18(15)):1-52, 2018
Tanja Hernández Rodríguez, Anton Sekulic, Markus Lange- Hegermann, and Björn Frahm. Designing robust biotechnological processes regarding variabilities using multi-objective optimization applied to a biopharmacoutical seed train design. Processes, 10(5):883, 2022.
Neil Houlsby, Ferenc Huszár, Zoubin Ghahramani, and Máté Lengyel. Bayesian active learning for classification and preference learning. 1 5745, 2011.
Achin Jain, Truong Nghiem, Manfred Morari, and Rahul Mangharam. Learning and control using Gaussian processes. International conference on cyber-physical systems (ICCPS), 2018.
Andreas Kirsch, Joost Van Amersfoort, and Yarin Gal. Batchbald: Efficient and diverse batch acquisition for deep Bayesian active learning. Advances in neural information processing systems (NeurIPS), 32, 2019.
15

---

Published in Transactions on Machine Learning Research (04/2025)
Edgar D Klenske, Melanie N Zeilinger, Bernhard Schölkopf, and Philipp Hennig. Gaussian process-based predictive control for periodic error correction. Transactions on Control Systems Technology. 24(1):110-211, 2015.
Andreas Krause and Cheng Ong, Contextual Gaussian process bandit optimization. Advances in neural information processing systems (NeurIPS), 24, 2011.
Andreas Krause, Ajit Singh, and Carlos Guestrin. Near-optimal sensor placements in Gaussian processes:
Theory, efficient algorithms and empirical studies. Journal of Machine Learning Research, 9(2). 2008.
Erin R Leatherman, Angela M Dean, and Thomas J Santner. Designs for computer experiments that minimize the weighted integrated mean square prediction error. Technical report, Tech. Rep. 875, The Ohio State University, Columbus, Ohio, 2014.
Erin R Leatherman, Thomas J Santner, and Angela M Dean. Computer experiment designs for accurate prodiction. Statistics and Computing, 28:739-751, 2018.
Cheolhei Lee, Xing Wang, Jianguo Wu, and Xiaowei Yue. Faihure-averse active learning for physics-- systems. Transactions on Aulomation Science and Engineering, 2022.
Cheolhei Lee, Kaiwen Wang, Jianguo Wu, Wenjun Cai, and Xiaowei Yue. Partitioned active learning for heterogeneous systems. Journal of Computing and Information Science in Engineering, 23(4):041009 2023.
Tom Leinster. Entropy and diversity: the ariomatic approach. Cambridge university press, 2021.
Con-You Li, Barbara Rakitsch, and Christoph Zimmer. Safe active learning for multi-output Gaussian processes. International Conference Artificial Intelligence and Statistics (ATSTATS), 2022a.
Jialin Li, Marta Zagorowska, Gíulia De Pasquale, Alisa Rupenyan, and John Lygeros. Safe time-varying optimization based Gaussian processes with spatio-temporal kernel. 4rXiv:2409.18000, 2024.
Shibo Li, Jeff M Phillips, Xin Yu, Robert Kirby, and Shandian Zhe. Batch multi-fidelity active learning with budget constraints. Advances in Neural Information Processing Systems (NeurIPS), 2022b.
Buyun Liang Tim Mitchell, and Ju Sun. NCVX: A general-purpose optimization solver for constrained machine and deop learning. arXim:2210.0099 NewrIPS workshow OPT 2022: Optimization for Machine Learning, 2022.
Joseph Liouville. Premier mémoire sur la délermination des intégrales donl la voleur est algébrique. Impr. Royale, 1833a.
Joseph Liouville. Rapport sur deux mémoires de mr. j. liouville, ayant pour titre: Mémoires sur la détermination des intégrales dont la valeur est algêbrique., 1833b.
Joseph Liouville. Second mémoire sur la détermination des intégrales dont la valeur est algébrique. Impr. Royale, 1833c.
Sulin Liu, Xingyuan Sun, Peter J Ramadge, and Ryan P Adams. Task-agnostic amortized inference of Gaussian process hyperparameters. Advances in Newral Information Processing Systems (NewrIPS), 2020.
Edwin Lughofer. On-line active learning: new paradigm to improve practical useability of data stream modeling methods. Information Sciences, 415:356 376, 2017.
David JC MacKay. The evidence framework applied to classification networks. Neural computation, 4(5): 720-736, 1992.
Garth McCormick. Computability of global solutions to factorable nonconvex programs: Part i convex underestimating problems. Mathemalical programming, 10(1):147-175, 1976.
16

---

Published in Transactions on Machine Learning Research (04/2025)
Rajitha Meka, Adel Alaeddini, Sakiko Oyama, and Kristina Langer. An active learning methodology for efficient estimation of expensive noisy black-box functions using Gaussian process regression. TEFF Access, 8:111460-11 111474, 2020.
T. M. Moldovan and P. Abbeel. Safe Exploration in Markov Decision Processes. International Conference Machine Learning (ICML), 2012.
Sindhu Padakandla. A survey of reinforcement learning algorithms for dynamically varying environments. ACM Computing Surveys (CSUR), 54(6))1-25, 2021.
Sindhu Padakandla, Prabuchandran KJ, and Shalabh Bhatnagar. Reinforcement learning algorithm for non-stationary environments. Applied Intelligence, 50:3590 3606, 2020.
Adam Paszke, Sam Gross, Soumith Chintala, Gregory Chanan, Edward Yang, Zachary DeVito, Zeming Lin, Alban Desmaison, Luca Antiga, and Adam Lerer. Automatic differentiation in pytorch. Advances in neural information processing systems (NeurIPS), 2017.
Friedrich Pukelsheim. Optimal design of experiments. SIAM, 2006.
Carl Edward Rasmussen and Christopher K. I. Williams. Gaussian processes for machine learning. MIT Press, 2006.
Christoffer Riis, Francisco Antunes, Frederik Hüttel, Carlos Lima Azevodo, and Francisco Percira. Bayesian active learning with fully Bayesian Gaussian processes. Advances in Neural Information Processing Systems (NeurIPS), 2022a.
Christoffer Ris, Francisco Antunes, Frederik Hüttel, and Francisco Pereira. Mixture of Gaussian processes for Bayesian active learning. Advances in Neural Information Processing Systems (NewrPPS). 2022b.
TG Ritto, S Beregi, and DAW Barton. Reinforcement learning and approximate Bayesian computation for model selection and parameter calibration applied to nonlinear dynamical system. Mechanical Systems and Signal Processing, 181:109485, 2022.
HoHo Rosenbrock. An automatic method for finding the greatest or least value of function. The computer journal, 3(3):175-184, 1960.
Sacks, WJ Welch, TJ Mitchell, and HP Wynn. Design and analysis of computer experiments (with discussion). Statistical Science, 4(4):409-43) 435, 1989a.
Jerome Sacks, Susannah B Schiller, and William J Welch. Designs for computer experiments. Technometrics, 31(1):41-41 47, 1989b.
Thomas J Santner, Brian J Williams, William I Notz, and Brain J Williams. The design and analysis of computer experiments. Springer, 2003.
Annie Sauer, Robert B Gramacy, and David Higdon. Active learning for deep Gaussian process surrogates. Technometrics, 2023.
L Scherliess and Bela G Fejer. Radar and satellite global equatorial f region vertical drift model. Journal of Geophysical Research: Space Physics, 104(A4):6829-66) -844889996262,,-- 1999.
Mark Schillinger, Benjamin Hartmann, Patric Skalecki, Mona Meister, Duy Nguyen-Tuong, and Oliver Nelles. Safe active learning and safe Bayesian optimization for tuning Pl-controller. IFAC-PapersOn Online, 50(1): 5967 5972, 2017.
Je Schneider. Active learning on non-stationary functions, 1995. URL https://www.rd: cmu. www..duuuuu..cc. Filler/poo//scheeeeeee________
Jens Schreiter, Duy Nguycn-Tuong, Mona Eberts, Bastian Bischoff, Heiner Markert, and Marc Toussaint. Safe exploration for active learning with Gaussian processes. Machine Learning and Knowledge Discovery in Dalabases: European Conference (ECML PKDD), 2015.
17

---

Published in Transactions on Machine Learning Research (04/2025)
Sambu Seo, Marko Wallat, Thore Graepel, and Klaus Obermayer. Gaussian process regression: Active data selection and test point rejection. Mustererkenrungg 2000: 22. DAGM-Symposium, pp. 27-34, 2000.
Kirstine Smith. On the standard deviations of adjusted and interpolated values of an observed polynomial function and its constants and the guidance they give towards proper choice of the distribution of observations. Biometrika, 1211/2) 1918.
Yanan Sui, Vincent Zhuang, Joel Burdick, and Yisong Yue. Stagewise safe Bayesian optimization with Gaussian processes. International conference on machine learning (ICML), 2018.
Yuriy Sverchkov and Mark Craven. A review of active learning approaches to experimental design for uncovering biological networks. PLoS compulational biology, 13( 2017.
KM Talluru, V Kulandaivelu, N Hutchins, and I Marusic. A calibration technique to correct sensor drift issues in hot-wire anemometry. Measurement Science and Technology, 25(10)) 2014.
Jörn Tebbe, Christoph Zimmer, Ansgar Steland, Markus Henggmnn,,,,,n,, and Fabian Mies. Efficiently computable safety bounds for Gaussian processes in active learning. In International Conference on Artificial Intelligence and Statistics (AISTATS), 2024.
Silja Thewes, Markus Hange-Hegermam,, Christoph Reuber, and Ralf Bock. Advanced Gaussian process modeling techniques. Design of Experiments (DoE) in Powertrain Development, 2015.
Silja Thewes, Matthias Krause, Christoph Reuber, Markus Lange-Hegermam, Rafacl Dziadck, and Martin Rebbert. Efficient in-vehicle calibration by the usage of automation and enhanced online DoE approaches. Simulation and Testing for Vehicle Technology, 2016.
Nils Tietze, U Konigorski, C Fleck, and D Sguyen-Tuung Model-based calibration of engine controller using automated transient design of experiment. Internationales Stuttgarter Symposium: Automobil-und Motorentechnil; 2014.
Jonathan Torres, Matthew Cole, Allen Owji, Zachary DeMastry, and Ali P Gordon. An approach for mechanical property optimization of fused deposition modeling with polylactic acid via design of experiments. Rapid Prototyping Journal, 22(2):387-404, 2016.
Matteo Turchetta, Felix Berkenkamp, and Andreas Krause. Safe exploration for interactive machine learning. Advances in Neural Information Processing Systems (NeurIPS), 2019.
Jonas Umlauft, Thomas Beckers, Alexandre Capone, Armin Lederer, and Sandra Hirche. Smart forgetting for safe online learning with Gaussian processes. Learning for Dynamics and Control, pp. 160-169, 2020.
Alje van Dam. Diversity and its decomposition into variety, balance and disparity. Royal Society open science, 6(7):190452, 2019.
Decpak Vasisht, Andreas Damianou, Manik Varma, and Ashish Kapoor. Active learning for sparse Bayesian multilabel classification. International conference on Knowledge discovery and dala méning (KDD), 2014.
Ian Vernon Samuel E Jackson, and Jonathan A Cumming. Known boundary emulation of complex computer models. Journal on Uncertainty Quantification, 7(3):838-88) 876, 2019.
Thsan Mohd Yassin, Mohd Nasir Taib, Norasmadi Abdul Rahin, Mohd Khairul Mohd Salleh, and Husna Zainol Abidin. Particle swarm optimization for narx structure selection application on de motor model. In IEEE Symposium on Industrial Electronics and Applications (ISIEA 2010), 2010.
Zhiliang Ying. Asymptotic properties of maximum likelihood estimator with data from Gaussian process.
Journal of Multivariate analysis, 36(2):280 296, 1991.
Hon Sum Alec Yu, Dingling Yao, Christoph Zimmer, Mare Toussaint, and Duy Nguyen- Tuong. Active learning Gaussian process state space model. Machine Learning and Knowledge Discovery in Databases. Research Track: European Conference, (ECML. PKDD), 2021.
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Lijun Zhang. Online learning in changing environments. International Joint Conference on Artificial Intelligence (IJCAT), 2021.
Peng Zhao, Yu- Yu-Xiang Wang, and Zhi-Hua Zhou. Non-stationary online learning with memory and non- stochastic control. International Conference on Artificial Intelligence and Statistics (ATSTATS), 2022.
Christoph Zimmer, Mona Meister, and Duy Nguyen-Tuong. Safe active learning for time-series modeling with Gaussian processes, Advances in neural information processing systems (NewrIPS), 31, 2018
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A On P-elementary covariance marginalizable covariance functions
This section provides more information about P-elementary covariance marginalizable covariance functions. In particular, we provide cookbook of which GP covariance constructions result in P-elementary covariance marginalizable covariance functions.
A.1 Motivating example
Recall Definition 4.3 a covariance function P-elementary covariance marginalizable w.r.t. a finite measure Tr
μ on R if the integral / 2, (r,, x2) dp(x,) exists as a P-elementary function ø1 and z2. This condition was used in Theorem 4.4 and hence allowed for closed form computations of IMSPE and T-IMSPE.
Let see this definition in action for the special case of the squared exponential covariance function with lengthscale of one and signal variance of one and Gaussian measure, already considered in Subsection 2.4 The rest of this appendix section vastly generalizes this example.
Example A.1. According to Formula 4 IMSPE selects the next measurement point x, E Rixd such that the averaged posterior variance
J Rd
k(xrlo0 x) du((r))
is minimal for any chosen measure j Rxxd Consider specifically d the prior Gaussian process (P) with mean and covariance function
k(x1, = exp (- (51 (_ x22)
2
and j the standard Gaussian distribution given by the (Lebesgue, denoted by A) densily
dp
x
exp
dx
22T
2
Assume furthermore one existing measurement point at x = 1.
Now let us consider which point E R to choose next, according to IMSPE.
[ (1,0 1) dlp((r))
= / ((p)) xxr)
k(xx, k(1,xa))
k(x,, 1) k(1, 1)
he xr)] ((11,x7))
k(ar; 1)]
du(xz)
/ 1- [k(2,,))) k(zr,1)) k(1,2x))
k(Z,,1 1)
-1 Tk(o, [(():z7)) x(1,z,)
dx(xx))
1
-k(l, x.,)
[K( du(x,))
[k(r)))))) k(x,,1]] 1)]
lo x2)
x(1,xx))
k(1, xx))
1
-k((1
[k(,,, xr) k(1,27))
()(x))))) k(xr;)) 1)]
du(xr)
)21,
+(1,
=1 x(1,2x))
(k(o,, k(ar, 1)2 2k(1, vr,
(..)) ) 1)) dp(xx)
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This integral is computable, if we can compute the above products (or squares) of covariance functions. Computing this is closed form is precisely the condition of & being P-elementary covariance marginalizable w.r.b. μ.. Luckily, this is possible for Gaussians, see also Example A.4 for a more general statement of this fact.
1
= (1 )2))
1// exp
-xx)² (Ir
2
du(xz)) + / cxp
(- (xr - 1)3) -1))) 2
2 dj((z,)
2
32
(1-x,)
x2,)²
(e, -x.)
(x,, - 1)2
2
exp
exp
exp
du(x+r)
R
2
2
2
1
=1
(1
A(1,T.)22
/2
1/1 exp
(I.,- x2.)²
2
12 d)(x)))
(I,, (_(x,
1)2
2
x
exp
+
exp
exp (-2
d)(xr)
2
2
(1 - x2a)2
(3rr x2,)²
(Jrr 1)2
x2
2 exp
exp
exp
exp
dd(x))
R
2
2
1
=1-
(1
((1,,x=))
S exp
(((x, z2.²
d\(x)) +
exp
(-(x, 1)²
d)(xr)
2
(1 (1-xaa)² /
(x,
x2
(x.-
1)2
x²
2 exx
exp
d)(xz))
2
R
2
2
2
Again, we stress that these integrals are computable in closed form.
1
2
x
2m
2/2
exp (- (-o (ōx 8x, + 5)
=1 (1- 1(1, x. 2
exp
exp
v/
3
3
v3
1
1
x
1
1
2
(f 8x, + 5)
1 (1- (1-exp(- - )2)
exp
exp
exp
v3
V
3
v3
The resulting acquisition function shows which values for x. are most suitable to yield information concentrated around zero with variance 1 (as specified by the measure 14) for the given Gaussian process. We plot this function in Figure 2 and discuss it in Subsection 2.4
This example has shown on the special case that we need to integrate certain products of covariance functions in closed form to compute IMSPE and T-IMSPE in closed form. This was the definition of P-elementary covariance marginalizable covariance functions, which will be formally used in the proof of Theorem 4.4 in Appendix B similar to the example above.
A.2 Examples of P-elementary covariance marginalizable covariance functions
We now generalize this definition of a covariance function k being P-elementary covariance marginalizable, such that it takes two covariance functions info consideration. This generalization is necessary for taking sums of two covariance functions, see Proposition A.10 Our proofs below are given in this slightly more general framework.
Definition A.2. Let μ be finite measure on Rdd We call pair (k1, of two covariance functions P-elementary cross-covariance marginalizable w.n.t. z if the integral ʃ ky (x1, 2, (2 12) dp(x,) exists as P-elementary function in Ii and 22.
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Obviously, a covariance function & is P-elementary covariance marginalizable w.r.t. f if (k:, k) is P-elementary cross-oovariacee marginalizable. Hence, we will show some of the following results for P-elementary cross- covariance marginalizability, which implies these results for P-elementary covariance marginalizability,
The definition of P-elementary cross-covariance marginalizability is non-trivial, as many relevant covariance functions are elementary Joosss-covar marginalizable w.r.t. relevant measures. While we state the following examples with P being PyTorch, these results should obviously hold for any reasonable programming framework.
Example A.3. Constant covariance functions (2(,) = ca are PyTorch-elementary cross-covariance marginalizable w.r.t. any finite measure μ.. The integral
I ki (1),, 22) du(a,)= - 6202 (pla))
= = (D E
is real number since is finite measure. Hence, this real number can be represented by PyTorch.
Example A.4. Consider the automatic relevance detection squared exponential (SE) covariance function
1
2 34
ks 0² exp
exp
2
E
Then, & and k are PyTorch-elementary cross-covariance marginalizable w.r.t. any continuous uniform distribution on a multidimensional interval or any Gaussian distribution. This holds, since any finite product of Gaussian functions is again Gaussian, leading integrals involving the error function erf or more squared exponential functions. To prevent digging even deeper into gory details of indices, here we demonstrate this in the one-dimensional case for a uniform mcasure c [a.b]
o (I1, F, KKS, (2r,, 12) dp(()))
1 (x) xx,)²
1 (22 22,3
c. c. exp
o2 exp
2
d)(x;)
l7
2
lg
1 (x) xr
(I22 - x,,)
=coio
2
cxp
2 1
exp
2
d)(x,)
a
(1(1 )
}²)
=coio²
exp
28762
d)(z,)
219.112 + 26321
Glg(I1 (I] x2)²
=coio²
exp
-(G + 6
Ir
2(1 + Pg)
17 7+G G
d)(z,)
Vil122 exp
expp
(- (01-12) 2(77
erf
((bb2))+(3(b-2
erf
(la-a1)
4122 /1l22/28/+22
/22//24+222
=co
/28 28
The same computation for a Gaussian measure follows from an additional completion to the square and the multidimensional case follows from the one-dimensional case and Fubini's theorem on the order of integration. We refer to Appendix for more dctails on this covariance function for T-IMSPE and IMSPE.
Example A.5. Using the Risch-Algorithm as implemented in Maple, polynomial covariance functions, half-integer Matérn covariance functions, and Wiener process covariance are PyTorch-elemennary covariance marginalizable w.r.t. both continuous uniform distribution on an interval or any Gaussian distribution. Additionally, these three covariance functions, together with constant and squared exponential covariance functions, all pairwise PyTorch--lementary cross-covariance marginalizable.
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Table 1: These tables shows pairs of standard covariance functions and whether they are Pylerch-elementary pross-covariane marginalizable w.r.t. any continuous uniform distribution on multidimensional interval (left table) or any Gaussian distribution (right table). The reasons for this summary are given in the examples A.6 and A.7 The negative results stemm from the fact, that Risch's algorithm in Maple failed to compute the necessary integrals in Maple in closed form. A checkmark on the diagonal means that a covariance function is syJorrh-elementary covariance marginalizable. For the negative results in brackets for the cosine covariance function, see the comment in Example A.7 For the extension from cosine covariance functions to random Fourier features (RFF) see Example A.11
uniform distribution
SE
Wiener
cosimu
RFF
Matérn
RQ
periodic
Gaussian distribution
polynoni
St:
Wiener
costie
RFF
Matérn
RQ periodic
polynomial
polynomial
SE
Wiener process
Wiener process
cosine RFF Matérn RQ Periodic
cosine RFF Matérn RQ Periodic
The proof of this is easy to understand high level, although detailed technical proof ends in gory fight against indices, case distinctions, and factor: First,, using Fubini's theorem, everything reduces to the one-dimensional case, Second, case distinction in the minimum-functtin in the Wiener process covariance function is easily dealt with by splitting the one-dinensional integrals at its positions of non-differentiabiliyy this results in integrals where only one case appears. Afterwards, the Wiener process is just a polynomial covariance. Now, the integrands are just products of polynomials and exponential functions with most quadratic exponents. These integrals can be solved by potentially ugly combinations of partial integration, completion squares in exponents, and usage of the Gaussian error function, similar to Example A.4
Example A.6. Neither the rational quadratic (RQ) nor the periodic covariance function seem to be eyeeenetylym covariance marginalizable w.r.t. non-degenerate continuous uniform distributions on multidimensional interval non-degenerate Gaussian distributions, let alone PyTorch-clementary cross- covariance marginalizable w.r.t. any of the standard covariance functions.
Example A.7. Consider the PyTorch-elementary (cross)ccovariance marginalizability of the cosine covariance function w.r.t. uniform distribution marginalizable, as the integral basically reduces to a square of cosines. Similarly, Risch's 's algorithm in Maple is able to verify that the cosine covariance function is Py TyToch-eeementary cross-covariance marginalizable together with constant, linear, polynomial, and Wiener process covariance functions. More interestingly, it seems that the pair of squared exponential and cosine covariance function is not PyToorch-eeementary covs-covv marginalizable w.r.t. both continuous uniform distribution on an interval or any Gaussian distribution. While Maple can find an antiderivative for the defining integral in closed form, this antiderivative involves complex error functions]] which are not implemented in current versions of Porch. Perhaps, this pair will be PyTorch-clementary cross-covariance marginalizable in a future version of PyTorch.
interval. It is easily recognized PyTorch-elementary covariance
We recommend implementations of the Risch-Algorithm as implemented in Maple, Maxima, or Mathematica to solve these integrals.
At least neither the authors nor-and much more important- the implementation of the Risch algorithm in Maple were able to compute these integrals in closed form.
TThe following Python code shows, that the error function in Py'forch does not support complex arguments in a recent version.
>> torch.
12.0.1'
>> torch. (torch. (ww.rccttttoorec.oo.......oo (cmpleeee1))) Traceback (nost recent call last):
"<stdin>", 1, in Kmodule>
RuntineError: "erf_vnl _ml__al__pp"" implemented 'ComplexFooat!
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Now consider the cosine covariance function and its PyTorch-clemente (cross))covariance marginalizable w.r.t. a Gaussian distribution. Even showing that it is Py PyTorch-elementryy covariance marginalizable needs some manual intervention to simplify the results of Risch's 's algorithm in Mapld] As the squared exponential covariance function is a Gaussian, this trick works to show that the pair of cosine covariance function and squared exponential covariance function are PyToech-elementary cross-covariance marginalizable. The pair of cosine covariance and polynomial covariance is easily seen as PyTorch-clementtry cross-covariance marginalizable using repeated partial integration. Similar to above, we get complex error functions when trying to compute the integrals for the eyteenh-elementary cross-covariance marginalizability of the cosine covariance when paired with either half-integer Matérn covariance or Wieder process covariance, rendering these two pairs not Pytorch-elementary cross-covariance marginalizability in current version of PyTorch.
We summarize the results of the previous examples A.4 to reproduce most of these integrals without the help of a computer algebra system.
and A.7 in Table I We do not recommend
A.3 A cookbook of constructing P-elementary covariance marginalizable covariance functions
Knowing the base covariance functions, which are P-elementary (cross-covvrriacce marginalizable, we now consider the standard constructions of composite covariance functions. We construct a cookbook that states which of these constructions results in P-elementary (cross- crross-)ovvriancee marginalizable covariance functions. As a first example, scaling covariance functions does not change their P-elementary cross- covarione marginalizability.
Proposition A.8 (Scaling covariance functions). If poir (k1, k2) of two covoriance functions is P-elementary cross-covariance marginalizable w.r.t. some measure μ then so is (oik2, o22k2 for any 02 > 0.
Proof. The proof is obvious from the lincarity of integrals.
Multiplying covariance functions on independent inputs keeps P-elementary cross-covariance marginalizabilitt, Proposition A.9 (Covariance functions with independent inputs). Consider direct sum decomposition Rd = Rdi @ Rd with μI finite measure on Rdi and #2 finite measure on IRd2 Assume that covariance function k factors over this decomposition, i.e. there are two covariance functions ky IR @ Rd: + R, i= 1,2, with k(x, = (x x11) (2()),12) for x = x22) E Rdi @ Red and x' = x2) E Edd @ Rdz. Tf kı resp- l2 are P-elementory cross-covarian marginalizable w.r.t. μi resp. u2, then so is k w.r.t. Pr @ P2.
Proof. This follows directly from Fubini's theorem on the order of integration.
One special case of this previous result is that the measure μ might be chosen as a product measure of a bounded uniform measure in some coordinate directions and Gaussian mcasure in different coordinate directions.
The class of P-elementary covariance marginalizable covariance functions is not closed under addition, for example the sum of a cosine covariance function and various other covariance functions is not Pytorch- clementary covariance marginalizable w.r.t. uniform or Ganssian measures. However, the addition of pairs of P-elementary covariance marginalizable covariance functions sometimes results in P-elementary covariance marginalizable covariance functions.
Proposition A.10 (Sums of covariance functions). Assume that a pair (k1,k2) of two covoriance functions is P P-elementary croes-covariance marginalizzbbe and both kn and k2 are P-elementary covariance marginalizable w.r.t. some measure μ. Then, kj + 12 is P-elementary covariance marginalizable.
Forr simplicity we consider a special case where parameters are set to The following Maple code computes of the relevant integral for the PyTorch-elementtay covariance marginalizability the cosine covariance function, subtracts the intended results, and shows that this difference zero. A direct computation seems itposssible. e simplify(
int Int (x-x))=coo (exp(x))xp(xxxx))) (2x "ewppppe==")xxxxxiiittttttt irnfifittt sew--iiiinttty...fiiieey. infinity)
(x-y)- -ece-yy-cosxxxeee (PP)(
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Proof. Consider
/ (k1(21107) + (1 + k2(22, xr)) du(x))
= f h1 2, (2 x.) + (2) ) (T)
12 (22, 2r) ho (11, h22 1r) dp(x))
f Aa (2), xr )a (12, x,) du(x,) + (x (2 dp(x,)
+ / (2(21, x4 (1 xT) dp(x,)) +
x.) du(x,)
All four integrals exist as P-elementary functions by assumption. Since the class of P-elementary functions is closed under addition, the proposition holds.
Example A.11. By this proposition, all said for the cosine covariance function in Example A.7 also holds for covariance functions from random Fourier features (Hensman al. 2018), as these are just specific linear combinations of cosine covariance functions.
As direct corollary, if three more covariance functions are all P-elementary covariance marginalizable and pairwise P-elementary cross-covariance marginalizable w.r.t. some finite mcasure μ., then so is their sum. Looking at. Table 1 we hence see that e.g- the sum of a polynomial covariance function, a Wiener process, and a squared exponential covariance function is P-elementary covariance marginalizable.
Adding measures keeps P-elementary cross-covariance marginalizabilit.
Proposition A.12 (Linear combinations of mcasures). If pair k2) of two covariance functions is P-elementory cross-covariance marginalizable w.r.l. both the measures μn and μ2, then is (k1, k2) w.r.l. Q1l1 + 02j2 for 01, 02 E I.
Proof. The proof is obvious from the linearity of integrals.
This fact allows to use IMSPE and T-IMSPE in domains that are no hyperrectangles, as long as the domain can be reasonably approximated by hyperrectangles. We can even subtract measures this way, e.g. to cut out parts of an area. One could even use negative measures as obtained from this proposition to avoid getting information in certain areas.
B Proofs of closed form computability of IMSPE and T-IMSPE
We provide our proof of Theorem 4.4 about the computability of IMSPE and T-IMSPE for P-elementary covariance marginalizable covariance functions and its corollaries. We do not claim novelty to the idea of this proof, since similar proofs exist for the special case of polynomial covariance functions since (Sacks et al. 1989bb. The discussion additional special cases are spread over the subsequent literature, see Subsoction 3.1 e.g. Lemma 3.1 (Binois et al.] 2019). In comparison previous proofs, we provide longer and more explicit proof.
Proof of Theorem 4-4 Consider the 1 x 1-matrix k(xr(11;; x)) in the integral
f [1 dd(zz))
from the definition of IMSPE, We can make the integrand more explicit as k(1,|1, xx) k(Ir, k(a,, (z. x. (.).))))(( Ix) (I, - x. 2.),
k(x,))) k(1,),)) 1- k(,,)x))
=(xr, (k(2,,a) k(zr, k(z,,2) k(IO,II)) k(I,,IT))
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The Schur complement
S. = k(x,, K(a,,, (k((,,
the scalar o² k(r,,, I.), and the vector
= k(z,, 2)((, =
(k(x, (Ma x(xx
allows to describe the above matrix inverse in closed form.
k(x,,)) x..) 2: I.).
Th(a,, x) -1 + k:x,, - N. 15- k( xx) -k(2, (((2,,))=k(1, x )S²
55 k(x,, 27 (xx,,))
S.¹
Ailas, x) 1 + 5T L L
-5- 1b
S.1
()k(I,1))1 L -L
ST1
L
Having a closed form of the inverse, we can write
k(Ir11,0,,)
x(x,x)
k(I, xx)
k(I, xr) k( xrr)
= =,( - [k(,,,)) k(rr x.)]
k(x,,))
k(F.;II.).
= k,),, [k(a,,)) k(zr, xx)] S-1 x) + L -L"
k(x,, xr.) k( x
-L
S,,((,, + L -LT
k(x,, xr)) k(r,,, xrr)
= of S. [Mx,,)) x(T
-1
-,,((I,))
1Kx,, xr)) + L (I, - L ko xT)) -Lk(Z, x, (K(,,
= 0 8- [KZ,,)) ((Tr,2)))
0. S- (S.k(z, 2)k(I, ((x, + +)),, LL(x,
k(r,, x)) K(x., - -x (
k( (K),,,
0. 8. 25, kz2,2,)
+
k(x,,,,)
- k(rr, (k),,, k(,, + (,,
Writing k(2,)) = CTC as Cholesky decomposition, C = k k(z,,))/OT, C. == O,k(,,z,), OT =
x)TT,)/CT,, C. = C(,,((, and the correlation k(z,, 2.) = k(E,, this simplies to
k A(x (2,
(V)(E,z))) of - -s MM (C Cik(E,,)
- CV, k(,,
= - S. (S.C C. C. cC. C. - cC +2)
o. c.. C. S C. + 5- C, S-1 cC.C, C. - S-12
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Now
k(
(1,, Ixx, x)) du(z))
(T)
| of du(x;)
C C.C.dr(
C.C.C.C. C. C. du(ar)
(1.1)
(1:2)
(1.3)
+
/ S. C.
ST ST C. du(()))) ST (
(1.4)
(1.5)
(1.6)
Here, 2² and S. are scalars, C. is a vector of linear combinations of k(r,, C. is vector of lincar combinations of k(x,, I). This means that all entries are either constant or a constant times a product of 21) k(x,, All linear combinations and all constants can be computed via numerical linear algebra, mostly using the Cholesky decomposition. In particular, these integrals, considered as functions in w, are P-olementary, since l is P-clementary covariance marginalizable.
T-IMSPE way of applying IMSPE to specific GPs. Hence, the claim for T-IMSPE follows from that of IMSPE.
Proof of Corollary 4.5
Initially, we compute O(n²) Cholesky decomposition of the data covariance matrix
k(x,x)) For each Fay We make O(n²) evaluations of closed form functions and small finite number of forward and backward substitutions of Cholesky factors, each computable in O(n²).
The statistics literature, see e.g. Gramacy 2020), also concerned with avoiding the Cholesky decomposition in O(n²), which is possible by rank-onc-update formulas of the covariance matrices. These computational improvements are only possible when hyperparameters are not retrained between iterations.
C Detailed explicit formulas of T-IMSPE and IMSPE for squared exponential covariance functions
We give very explicit formulas for T-IMSPE and IMPSE when using squared exponential covariance function.
Therefore, consider the 6 integrals in M on page 27 independently. covariance function
We assume a squared exponential
1 ((x1)) (x2))
h R x Rdd R :21
(2) + II
0² exp
2
h=1
1
((x) ) (22))))
o² exp
2
h=1
with automatic relevance determination. For simplicity, we assume p to be probability distribution, i.e. ((Rd) 1, and later will also specifically concentrate Gaussian measures and uniform measures.
C.1 Integral (I.1)
f foddu(ar) o² 1.),, This term is independent of x., k is stationary.
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or for μ N(m, diag(s)) this integral evaluates to Published in Transactions on Machine Learning Research (04/2025)
C.2 Integral (1.2)
/ C.d(())))))
/
dp(x,)
(1) (M( ()) 1, k( x.,) dp(x,)
(k(x, -)= f (z) xx) du(xx))
xk(x,z)- ".... / II
((xx)) (xx)))² ((xz)) - (2x))a²)) du(xr))
exp
(
A=1
(Ixx) (x:)n)² + ((zz)) - (2r)))² da(xrr)
(k(1,z)) 11.
II
exp
This should be integrable easily, e.g. for the continuous uniform distribution on [a1, b1] x... x [as, bal this integral evaluates to
(k(r,x)) )ij 00 7
erf [20 (2,-(z (x.))
(bb, (1
[(xx)) - (xx))a)2
28n
erl
26,,
la
exp
469
20n
2bb
1=1
-12 (2,(1))) 3- (--2(((m))) -m,) - (f (
(k(2, x) 2((z) 44.
d
e
exp
2,, +2x
VEg + 2s
1-=1
hm
This term is independent. of a..
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C.3 Integral (1.3)
/ C.C.
-- C.C.c....ptr,
-
-- / MZr.)) ((t.::))((000MM:::)))))))ttt x,)xx))
S. /
Marr (KR") k(0, (r)(.))
S,
M(a,,a)) Kiyy - k(z,, (x)(a,)
S-
Kitig
((Ir) k(zj, x, (({,)
bj=1
(I, x;,² + (xz 2z,)2) plxr))
S.
KiKij
I1 h=1
exp
{==
1 (I, x;,3 + (Iz 22,)2
S. 04
KiKi
exp
mlar)
(2
{,==
h=1
for == (CT (V), = k(xx 2(((,,) x.). This should be integrable easily, e.g. for the continuous
uniform distribution on [aibb]] x x [aa, ba] this integral evaluates to
S. o KiKi VT"
II ln
11=1
h=1
erl 2an Pan-(x,))
-wede erf
2bh (2,-(2z)n-(e4 (xx))
(Ix,)) (Ir))))
2/
28
exp
463
2ah 2bh
or for = N(m, diag(s)) this integral evaluates to
E. (,a(4
2- ((,,((())) -m,,) (,r (mx))))
Σ 11=1
n
S.
E
exp
2P 2P.02 +2x4)
/ V2h 2s
This term depends s. (via xs.
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C.4 Integral (1.4)
/ 5, C.C.o((
-
h( Ixp C
d
((rx)) - (1r)))
0²
exp
2
lg
h=1
1 ((xx)) )n - (xr)a)²)
S-10² 0²
k(ar.z) 20,
exp
l
h=1
1 ((x.)) -(x))N)2) ((TT,
S 0
Kij
exp
2
lg
1 ((xx)n - -(xr))) + ((xr)) - (2x)))2) (1x))))))))))))) mlxrr)
Σ
R4 S.S 04
exp
2
h=1
1 ((x+)) (x+))) + ((x,)s - (Zi)a)) m(xer)
Σ Kg S-1 04
exp
for == (VM,))M2, = k(xx k((,, This should be integrable casily, e.g. for μ the continuous
uniform distribution on (11, b1] x x [aa, ba] this integral evaluates to
d II ln
k4 S.t 04
h=1
erf 2aa (aas-(x,)1, (1p,n erf
(bh -cx )a (w.)a
II exp
((x=)) - (22)))2)
24
2h
46
20h 2bp
or for μ N(m, diag(s)) this integral evaluates to
[ -m,)² ((x))))))) mm
12- )
((x)))))-(ss) (xxs)a))
exp
Ky S- 0
II 4n
27 (Fj, +2.))
Vtg. + 287
R=1
This term depends on
C.5 Integral (I.5)
f SS (S-1CTT Cop(zr). This is the same integral as the last one.
C.6 Integral (I.6)
f - S 2 = / -S. 1-1-- )z dd((xx)
1 [(xx)) (2x)))2 2
=/
- S
exp
dp(xr)
h=1
d
((xr); - (Ia))))) du(xr)
= -- 0
exp
lg
h=
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This is easily integrable, e.g. for j the continuous uniform distribution (01, [1] ... x [ad, bal this integral erf( as - (2x)1)
evaluates to
d V II
erf(
6b (a. In
-S-
c
En
2a, 2bb
A=1
or for Mmm, diag(s)) this integral evaluates to
En exp(
[ma - (22- 2a12
" 7222
17
28 2
This term depends on Zx.
C.7 All terms in [ together
For the continuous uniform distribution Ja, 6] have:
/
k(xr [e, x) du(xx))
o²
Σ
(k(2, 11) 00 VT
II Ln
6/=1
h=1
((Iz)) - (xi))))
erf
220h-1 (20n-(2,)n-
26
erf
2bh (an (n,)n
26h
exp
48
20h 2bp
1=1
Σ
5. o Ki VT
cn
1/=1
(12))
(2s))a)
erf
2an -1x)) (2)
26
erl
2b. 10r,-(a,11 -(xc)
26
exp
462
2ah 2bp
d
+
K; S 0t /
8n
1=1
h=1
erf 2as (xx)) -1(.) orf (2h 22b )a (xx.)
((xx)) - (xx))a)
24
28,,
exp
46
20h 2bp
d
erf( Bh (m-))a
erf( bn (x-)2
ST 4
Pn
en
2ar 2bh
h=1
o2
(k(z,x))1) -1) + +S +S.'K(K)
0 VT"
Cn
1-=1
((I))s - (ss))a²
orf
2an - (2,)), (1rs)
26p
erf
22h (2,))
26a.
exp
182
20h
2bp
d
+
Ri
S-1 0
n
d
Cx
:=1
h=1
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(xx.)) - (xx)a)²
erf
2np (20)=(0))1 (18h)) erf
26p
2hp (x-)) (m,
2
exp
4
2ah 26,
1=1
ers( an (o-]a) -erf( on- ((2))
S-
π
En
11
2ap - 2bp
For multivariate normal μ (mm, where diag (11 .1....,, sa) ....,
Rdd we have:
f k(zz| x) du(xr)
((((()) 10mmmmmp m. 2-- ((2cc) -mm m
n
(k(x, 7))
exp
20 (2, +2o
/27 28
h=1
-B,()) mma )2 12-18 ((0.)) -ma, -- (()))))))))))))
d
exp
21,, 1 2x
Kijjj
VCg + 28
1-j=1
((x.), m,,) )2 ((x;)) -mm 3 -3-
(0x(((()))))))))) ((x,)) (xr)),)
+ 2
K4 S.1 04
II Pn
exp
263 (Eg +2x
Ve8 + 25%
d Ln exp(
(ma (2.)
F. -2
S
25
o²
(k(2,z))
))5 + 11),,
+ -0
II ln
-12((2,) -m, - m. > of (10 (2 (1,, +207))
exp
II k=1
17 + 28
((.)) mm 2- -f ((x:))) m. ) ) -- ((xx (uss)))
exp
27, (25, +28
+ 2
Kig S 0*
/ 1 + 28
1=1
d En exp(
(ma (2.)a
24
S.
25
h=1
D On Baselines
Here, we comment on our choice for entropy as the only baselines in the experimental Section We chose entropy (only) baseline acquisition function, since it is the state of the art for active learning and safe active learning in machine learning. This is different to Bayesian optimization, where multiple acquisition functions are commonly used. In Appendix D.1 we give reasons why several classical methods are no suitable basclincs. Marginalized β-diversity and marginalized mutual information are acquisition functions we developed during the paper and do not appear in the literature. They have no closed form, and their numerical approximations are very sub-par our preliminary ablation experiments in Appendix D.2 Furthermore, we comment the superiority of entropy over IMSPE for GPs with trainable hyperparameters remarked in Subsoction 2.4 which excludes IMSPE baseline.
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D.1
Classical baselines
D-optimal designs non-active version of entropy, as described in Section 3.1 IMSE, ALC, IV, and IVAR are just other names of IMSPE, as commented Scction 3.1 A-optimal designs and V-optimal designs are non-active versions of IMSPE in different interpretations, and hence not suitable for a comparison.
ALM is just another name for the entropy, as described in Section 2.3
D.2 Comparison of IMSPE to marginalizing 8-diversity and marginalizing mutual information
Maximizing the mutual information I(r.; v, | xr) (Krause et al. 2008 between a new datapoint I, w.r.t. some points of interest 2, conditioned on previously existing data a classic choice to place sensors or conduct measurements in active learning settings. is originally restricted discrete sensor positions, where it leads to an NP hard decision problem. Despite this restriction, applicability of mutual information has increased as discussed in Subsection
5 DJ Iy For active learning purposes we can replace the mutual information by the squared 3-diversity
z) == exp(I(xx:; (Leinster 2021 van Dam 2019 Chiu et al. 2014), as is is just a composition with a monotonous and injective map. Instead of maximizing the (squared) 3-diversity, we can also minimize the inverse squared f-diversity D3 (I.;0r/a)) Consider Ix, J, E Rixd and Raxd
2 x) exp(I(x_; [x))=²
e x) - H(., x,, x)) -2 1
exp (5 log((.r) + 2 log (det (k(2 (r.,, +x),))))))
exp (1$ log (2x)) log (det (k(x,|=))))
exp (log (det (2, x))))
exp (log (det (k(2,, [z))))
det (k(,11/
det (x)))))x))
determinant of Ixl-matrix
The denominator is constant w.r.b. n. and can be disregarded when minimizing Dj (xrixxz)))))) w.r.t. a.. We can write the numerator in terms of prior covariances:
k(x,|x., x) = k(z,, - k(ar, (x., (a..,))) ())
Minimizing this acquisition function obviously leads to choosing v. R Ir, which is obviously a usually bad choice, since any real form ob optimization is ignored in favor of just choosing the reference point. Similar phenomena appear for multiple reference points 2,, where each reference point leads to a local minima in the loss landscape of xr -> minimize the above criteria when averaged over xy;
To prevent these attracting local minima in the loss landscape, one
/ D 2 (Ix;z, dp(xx)) = / k(zrlz)
k(x, ((())(o,))) du(x,)
xx,
average inverse squared 3-diversity
f ( ((), dp(x,)
IMSPE
/ - -[r,s, dp(x,)
average negative mutual information 8hhe interpretation of 3-diversity that of the diversity between two groups. In case, this aim to minimize the diversity between z+ and ær, when conditioned x. We were unable find anything close resembling closed formula for any other power of the B-diversity.
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Here, we assume some (finite or probability) measure j..
The average negative mutual information and the IMSPE are connected via a Jensen gap.
f 1( [x) du(z,)
/ (2 log (det (k(z+l=))) -2 log (det (())) dp(a))
/ (2 log (M(x,
1
log (zx,,/ x))) dp(xx)
2
1 / log (1(2,1/222 x)) du(xx)) / log (k(z)(z)) dp(zx)
2
Jensen
1
log (ʃ (((z)een x) du(xx)) duu(a))))
1 / log (k(x, du(x,)
2
2
constant in st.
- 1 2
(f dp(())) (equal up to constant)
log
Here, the latter line is just one half the log of the acquisition function IMSPE. Hence, the mutual information and IMSPE are closely connected. IMSPE has the advantage for continuous domains, as one can marginalize the reference points in closed form.
The comparison to the inverse squared 8-divergence D xx) is more complicated. While Dz (a.; Tr
x) = M(a.la..)) and (x) are equal up to a factor that is constant in v.. Sadly, this factor is not constant in k(r.(m_.=))
k(x,[z)
Tr, over which we marginalize. Hence, f /xxp//. 2)) du(a,)) and f (r, Ix) dμ(x,) = f k(z,2z) du(x,) are differently weighed integrals. In the latter case, one weighs reference points F, higher k(z, 2)) = 0, i.e. reference points near the data points. We do not think that this is suitable in practice. In addition to closed form intograls and very suboptimal initial experiments, we did not pursue the ß-diversity any further.
We elaborate on preliminary empirical comparison between the marginalizations suggested above, and not only between (T-)IMSPE and entropy. First, we were neither able to compute f J 1-1(2,;;, xx) du(z,) nor "D722(xx: My x) du(x,,) in closed form for any choice of covariance function we tried. Furthermore, numerical or MC approximations to these integrals suffer from a problem: the appearance of local optima al all points where the integral is numerically or stochastically evaluated, In preliminary experiments, this led to the same suboptimal behavior for finitely many reference points z,- More preciscly, for the minimization of both -I( 24 and 2 (xx; I), one chooses V, is local optimum close to one of the 2,. In fact, any data point in x will repel I. and any point in will attract x.. This leads to many local optima, that are not optimal for active learning. The many local optima are a problem for the optimization in safe active learning, as many more starts the optimization are necessary, leading to drastically increased computation time. Further increasing the number of reference points xr leads to more, but less pronounced, local extrema; this did not improve the situation. In Figure we see how optimal the acquisition functions entropy, IMSPE, and a numerical approximation of IMSPE are for placing a single data points in one to twelve dimension.
We cannot change the zr during a single optimization rum, as this leads to unstable optimization. Even choosing the x, different in every active learning step resulted in suboptimal exploration, as mcasurements are not prioritized by being near the safety boundary, but by being close to one of the Iy..
E Further results of the experiments
E.1 Further results for the seasonal change experiment
For the example about seasonal change from Subsection 5.1 the average amount of safe area that recognizod safe area is also similar for entropy (0.401 + 0.048) and T-IMSPE (0.394 + 0.039).
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log-odds average posterior variance
10
9
8
7
6
5
4
3
2
entropy
1
(mumerirally approximated integral 1024 points) IMSPE (uumerically appruximated integral with 256 points) IMSPE (aumerically approximated integral with points)
0
IMSPE (closrd form)
7
10
12
dimensions
Figure 7: For various dimensions 1 d 12, we consider the average posterior variance over the interval [-1, of GP after adding a two datapoints. The first datapoints is at the origin, and the second datapoint it optimized with 5 different acquisition functions. For a better visual representation, we print the average variance dag which bounded between 0 and 1, in log-odds, i.e. as log (i-z better. The prior GP is has squared exponential covariance functions with all hyperparameters set to one. The entropy acquisition functions (red) is very suboptimal, since the second point is always placed at a position with maximal variance after adding the origin, which is the boundary of the domain. While the numerical approximations to the intergral in the definition IMSPE (shades light. blue) come close to the symbolically computed IMSPE (dark blue), they rarely find the same optimum.. For example one dimension, the optimally places points are at (entropy). -0.633 (IMSPE approximated with 64 points), -0.598 (LMSPE approximated with 256 points), -0.536 (IMSPE approximated with 1024 points), which are at best near the real optimum at -0.532 (closed form IMSPE).
o where smaller still
uuy
For the example about scasonal change from Subscction 5.1 we conducted an ablation study about the effect dwindling superiority of T-IMSPE over entropy, once the seasonal effect weakens. We consider learning system with seasonal changes of various changes, again by rotating the domain in the function
McCormick (2)
sin(x1 + x2) + (xn x22 11501 + 2.5x2 + 1,
by the formula
1
ssul, (t, 21, x2) 4 - -1+ McCormick
10
aal
al
CO8
sin
sin
sin
2
2
10
2
10
1
at
at
sin
sin
COS
sin
x22
2
2
10
10
where E 2,, 4,5) determines the strength (or speed) of the seasonal change. Note that in Subsection 5.1 used = 5. The value 0 corresponds to seasonal change. We keep all other parameters as in the main experiments.
In Figure 8 we see the results of this experiment. Entropy is superior for a E (0, 1}, whereas for a 2 2 we have that T-IMSPE results in lower RMSE values. This is in accord with both the literature as discussed in Subsection 2.4 and the results this paper: entropy the superior acquisition function when temporal changes are non-existant or weak, whereas T-IMSPE is the superior acquisition function for applications with temporal changes.
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T-IMSPE (ours)
1.5
Entropy
RMSE safe area
0.5
0
2
4
amount of scasonal change a
Figure 8: Box plots of the RMSE values in the safe area of the experiments from active learning with seasonal changes in Appendix E.1 We compare the results of 25 runs between T-IMSPE (blue) and entropy (red) for various values of the strength of the seasonal change We see that T-IMSPE is superior to entropy for larger values of whereas for smaller values of a we see the superiority of the entropy.
It remains an open practical question how to detect, whether a system is sufficiently enough time changing, where we currently have no clear answer. Computation time is comparable between T-IMSPF, entropy and IMSPE, so this is cannot guide a decision. We checked the length scales in direction of time as criterion, and they were not sufficiently correlated with time variance. Perhaps one could use model selection for Gaussian processes: a GP that constant time direction more suitable model the data, then the system might not be sufficiently time variant; this is speculative. It remains to guess for a potential user of T-IMSPE, entropy, IMSPE, which acquisition function more suitable in situation. If a system noticably time variant, suggest T-MSSPE. Note that IMSPE behaves exactly as T-IMSPE if system is constant in time; hence, it might be safe choice to always use (T-)IMSPE. In the borderline of mild or no time variance, all methods show comparable performance and our preliminary experiments point to a strong dependence on the dataset; in this overlap, the choice is not too relevant.
E.2 Further results for the drift experiment
Consider the experiment about drift in examples discussed in Subsection 5.2 Due to the drift, any active learning scheme won't bring down the RMSE significantly. Instead, the goal is to hold the RMSE an acceptable level despite the drift. In the main paper, we have shown the RMSE gain of T-IMSPE over entropy Figure also shows the raw RMSE values.
in Figure
E.3 Further results for the rail pressure experiment
Subsection 5.3 considered the rail pressure experiment for dynamic systems. Here, we show additional results. Also, the amount of safe area that is recognized as safe area is drastically larger for T-IMSPE, see Figure 10 Table 2 shows some additional numerical results. In addition, this lables serves as an ablation for the maximal distance in an elipsiodal norm, allowed by the safe active learning algorithm in this experiment. This axis-parallel cllipse with semi-axes 80.3 and 2.17 is actually a circle in scaled space (see Appendix F.1 for the formula for scaling) of radius 0.1. Increasing the radius 0.15 0.2, i.c. allowing more dynamic behaviour, massively decreases the quality of the results in safe active learning.
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T-IMSPE (ours)
Entropy
IMSPE
6
5
RMSE in safe area.
3
160
260
%6%
%00
%609
2609
%60
%608
%606
%601
Figure 9: Box plots of the RMSE values in the safe area of the experiments from active learning under drift in Subsection 5.2 We compare the results of 500 runs between T-IMSPE (blue), entropy (red), and IMSPE (green) on average during the runs (left) and then ascending at specific time steps. In contrast to the other use cases, the drift prevents a major decrease of the RMSE over time. These are the raw values of Figure from the main paper, where the difference of experiments with the same seeds were considered.
Recall dynamic safe eval
60 %
entropy
50 %
T-IMSPE (ours)
40 %
30 %
20 %
10 %
steps
250
500
750
1000
Figure 10: This diagram show the increase in recall for classifying the safe validation data correctly into safe in the rail pressure model from Subsection 5.3 Entropy (red) show a slow incline, whereas our approach T-IMSPE rises faster, resulting in much higher recalls. The dashed line is the mean of 100 runs, the area shows the 2a area and the solid lines show the same exemplary runs as Figure
F Technical Details on experiments
Our implementation done in the PyTorch environment Paszke et al. 2017).
All our GPs use the squared exponential covariance function with a separate length scale l as hyperparameter for each input (automatic relevance determinantion). Additionally, the signal variance of and noise
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Table 2: We show various additional results on dynamic railpressure experiment from Subsection 5.3 We compare entropy (first three rows) to T-IMSPE (soconod three rows) with three different maximal distances (0.1, 0.15, and 0.2) for each step in scaled space. The experiment in the paper used the clear superior distance of 0.1. The first row shows the computation time in seconds an NVIDIA RTX3080 GPU of all 1000 safe active learning steps. Here, entropy clearly takes longer, even though in theory it should have fast evaluation time by small constant factor. Debug outputs indicates that entropy leads to significantly more failed optimization runs, where the safety and other constraints could not be kept. Both approaches choose over 90 % of points as being safe, with T-IMSPE achicving 99 % safety. Our safety criterion of two standard deviations of the GP model prediction stay below the safely bound would predict that- -igoorig model error independently chosen points are about 97.7 % safe. The last row shows the recall of our safety criterion safe trajectories, i.c. the percentage of safe trajectorios in the test data, which are recognized as being safe. In these recall values, T-MMSPE is more than twice good as entropy. See also Figure [0] where the recall values are plotted over time, instead of the average as here. Results are the mean of all five seeds, with standard deviation in brackets.
strategy
max. distance
comp. time (s)
safe points (%)
static recall (%)
0.1
13093(+334)
97.5(#0.4)
18.3(16.4)
Entropy
0.15
(+032(+663)
94.0(+0.3)
16.5(+2..)
0.2
15604(+1122)
90.3(++0.2)
9.2(+2.3)
0.1
(792(1212)
99.0(=0.4)
38.5(+3.4)
T-IMSPE (ours)
0.15
3915(+179
97.1
32.8(+3.6)
0.2
4305(+271)
93.3(+1.2)
25.6(+7.2)
variance of as as additional hyperparameters, as in (Rasmussen & Williams, 2006 The GP has constant mean function m as hyperparameter. Por the initial training of the GP hyperparameters, we use the SQ implementation from PyGranso (Liang et al.] 2022 Curtis et al 2017 After each new measurement, we retrain all hyperparameters with 30 steps of ADAM. When training hyperparameters instead of minimizing the negative log likelihood, we minimizing the negative log a-posteriori. This is the same GP for both acquisition functions entropy and T-IMSPE.
The choice of a GP with squared exponential covariance function for all experiments might be unintiutive. Specific covariance functions might have improved the results of the examples, e-g. periodic cosine covariance functions for the experiment with seasonal chance in Section 5.1 However, there are several reasons for experiments with a standard covariance function such as the squared exponential covariance.
1. We did not want to assume any specific time-varying structure (other than dynamic structure in the third experiment) in our covariance function.
We prefer a consistent approach over several experiments.
The GP used in the seasonal change experiment in Section 5.1 does not directly model periodic behavior. Despite this, T-IMSPE still steers the data acquisition such that even a GP with a uninformative covariance functions learns the periodic behavior, see Figure 4 We see it an advantage to T-IMSPE that periodic structures are automatically captured with standard Gaussian process.
4. We conducted preliminary experiments with periodic covariance function in the scasonal change experiment in Section 5.1 These experiments used the numerical approximations to T-IMSPE, since there is no closed form version for the periodic covariance in T-IMSPE, see Table [ The results were as follows: Safe active learning behaved very suboptimal with all acquisition functions when we trained the hyperparameter for the period length of the periodic covariance function, since the hyperparameter for the period length was hard to learn in Gaussian process with multiple inputs and few measurement points. This lead to many local optima, most of which showed very suboptimal
Witthhout constraints, this basically Newton optimization.
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behavior, and very overfit models. When we manually set the period to the correct value (something that seems to be unreasonable in many practical examples) and used numerical approximation to T-IMSPE, entropy was superior to the numerically approximated T-IMPSE. This is consistent (and one of many reasons) for our observation in Appendix that numerically evaluated integrals perform suboptimally.
We choose the following priors for the experiments. For the seasonal change experiment from Subsection 5.1 and for the drift experiments from Subsection 5.2
the priors are
softplus (4) N/5, 12)
softplus (4x) ~ .NO,
softplus softplus"(ar) ~ N(1,13)
softplus (an)
- 1²) ~ (o.)
m m~,((10,0.0)
for softplus(x) = log(1 [0g(1+ + exp(x)), temporal length scale l and spatial length scale ey For the rail pressure experiments from Subsection 5.3 the priors are
softplus -(() ~ (00.5.0..)
softplus
-(a,) ~ -V(0,,,00))
softplus (attplus"(o,)-ooa ~ NN(33,,01
For the rail pressure example, it is particularly important to have conservative priors, since otherwise safety constraints are kept much less often. The priors for the mean are particularly strict. This prevents small mean functions and hence exploration into unsafe area which are deemed safe due to unsuitable extrapolation. Taking the mean as trainable hyperparameter with prior instead of fixing it allows for flexible models once enough data is collected. While using GPy'Trrch (Gardner et al. 2018), reimplemented the priors to avoid inconsistencies in GPyTorch.
The optimization for safe active learning is done again using the SQP implementation from PyGranso with random restarts. In case of the first or second unsuccessful optimization, where no point keeping the constraints could be found, we start an additional optimizations with new starting points, such that at most restarts are performed. In the optimizer, all tolerances are set 10-4 and maximal 200 iterations are allowed. The most important constraints, the safety constraint, is taken ( [x. y) / (x. (x) < e, where e the constraint and Msate (x. (x,y) resp. //sate (x.[z) are the mean resp. standard deviation of the posterior GP x.. This corresponds to 0.977.
We adapted entropy (mildly) to the time-variant case: maximize the variance by only allowing the current time step, similar to the usage of the acquisition function in (Fiducioso et al. 2019). However, to the best of the authors' knowledge, IMSPE is the only acquisition function that allows to specify that knowledge should be collected to add information for future time steps.
The finite measure for T-IMSPE in the experiment with seasonal changes and for the drift experiments are chosen as 10) where to the current time point, is the allowed spatial domain, and is the indicator function. In other words, we want to reduce the variance equally over D and up to 10 time steps into the future. For the rail pressure experiment, we chose p measure for T-IMSPE, to search for trajectories that reduce the variance over the space of all trajectories. The exponent 4 in D is due to having time steps in our NX model.
As in the seasonal change example, the drift example starts with 8 initial measurements at times 0,..., positioned at the inital points of a Sobol sequence the safe area. Afterwards, 100 further mcasurements at times 8...., 107 are conducted according the two respective safe active learning criteria entropy and T-IMSPE.
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1.50
P2
B2
0.70
-0.3
6:25
t
0.0
T1
6:2
1:0
00.55
-09
2
-0.35
-0.5
-0.30
.0.5
0.73
-0.73
-0.55
2
2
4
4
2
-0.15
2
2
-0.75
-
- 18.23-
-0.
0.10
0.26
0.76
0.5
22
2.0
3:0
31
-0.5
-0.6
-0.76
-0.70
2
22
2
.0.7
-05
-02
4
20
-0.05
0.75
0.50
0.5
4.0
-0.0
T2
41
0.75
0.23
5:0
-0.5
322
31
2
2
-0.5
-0.75
-0.75
-0.78
-0.75
2
2
4
4
2
2
Figure 11: Plots of the function from
7
for the seasonal change experiment for various values of t.
In the rail pressure example, we also model real world safety procedure, mirroring a similar procedure in (Zimeer et al. 2018 Tebbe et al. 2024]. Once a measurement is not safe, we jump back to the area that was initially declared as safe. Note that the jump back itself is not necessarily safe itself, due to potentially being a long trajectory. However, once inside the area that was initially declared as safe, the behavior will quickly stabilize, which is not necessarily the case outside of the initially safe domain. The NX structure was chosen similar to (Zimmer et al. 2018 Tebbe et al. 2024), i.e. based on expert knowledge. If, in a different application, this expert knowledge is not available,
use techniques for NX structure optimization to find a suitable value, e.g. from (Yassin et all 2010). The domain is D [1000, 4000] [0, 60), the safety constraint is to keep the rail pressure below 18, and the initially known safe domain for steady state behavior is [2093, 2414] x [14.36, 23.0]. Bigger jumps result in uncontrollable dynamic behavior, hence we restrict (N4+11, ux+1) to an ellipse around (r4, Uş) with axis-paralle semi-axes 80.3 and 2.17. We choose 256 initial measurements the known safe domain, while keeping distances between points short enough that the initial measurements are all safe. After an unsafe measurements with rail pressure above 18, we will return to the safe domain to stabilize the dynamic behavior.
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4
T2
4
Tp
0.0
1.0
T1
11
2
2
2
4
2
4
4
4
2.0
41
00
00
0.70
aa
60.2
Figure 12: Plots of the function from S for the drift experiment for various values of t.
F.1 Formulas for the experiments
The formula of the seasonal change experiment in Subsection 5.1 uses the classical McCormick function [MCorriick 1976)
McCormick (x1, + sin(11 + x2) + (I1 x22²² -1.5x1 + 2.5xz + 1,
in the formula
1
ssul: (t, 11, x22 + -1 11 McCormick
10
1
t
1
cos
sin
sin
sin
x2,
(7)
2
2
2
1
2
sin
sin
2
sin
x2
See Figure 11 for a visual representation of this function. We consider this function in the spatial domain
[-4,4], where the area [-0.5, [-1,1] is deemed save initially. Measurements (t, 21))2) are deemed safe if 21, < 0.
41

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Published in Transactions on Machine Learning Research (04/2025)
4
T2
0.0
2N
T1
2
-0.
2
4
4
4040
32
372
1235
t
-4
2
4
4
2
4
-111
Figure 13: Plots of the function from
8
for the drift experiment for various values of
42

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Published in Transactions on Machine Learning Research (04/2025)
The formula of the drift experiment in Subsection 5.2 is given by
1
((2 2+ sin ( (+1
drft: (2, > 1000
(8)
(2)s - 25 + 1.
10
for the classical Rosenbrock function Rosenbrock 1960)
Rosenbrock (I1, - (8 |x] x2l + (1 21 )))
This function is visualized in Figure 12 and Figure 13 We consider this function in the spatial domain 4]2, where the area [-0.5, 0.5) x 1-1, is decmed save initially. Mcasurements at (t, (1, 22) are deemed safe if drft(t, z1, x2) < O. Note that the safe area is decreasing as f increases.
43


Reviews:
[{"review_id": "yZkRC6KYXg", "reviewer": "Reviewer_RkUu", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "# Strengths\n\n(S1) The problem considered in this manuscript---active learning for data efficiency *and* safety---is highly relevant and the suggested approach seems to carry some originality.\n\n(S2) The authors show that the proposed integrals can be computed in closed form, which is of practical benefit.\n\n(S3) An interesting aspect of the paper is the notion of \"elementary functions that can be computed in closed form in PyTorch\", etc. (Sec. 4.3).  This appears to be very relevant.  I am wondering though whether such arguments and formalizations have been considered in other areas of computer science like automatic differentiation, etc.  There seems to be no discussion of related work on these aspects; it might be worth adding.\n\n(S4) Numerical experiments illustrate the superior quantitative performance over state of the art criteria for active sampling.\n\n\n# Weaknesses\n\n(W1) My main difficulty is that I have trouble pin-pointing the exact technical problem that is being solved. Essentially, I am missing a precise problem formulation.  It seems that Sec. 2 somehow introduces the problem that is being considered, without spelling this out explicitly.  This would help for clarification and transparency of what is actually the technical problem that is considered in this work. Furthermore, Sec. 4, first line talks about \"time-varying models\" without making these precise.  Sec. 4.1 seems to combine a problem statement (what is the time-dependent model/problem) with a solution. It would probably be clearer if this was disentangled.  \n\nI would like to ask the authors to consider adding a specific (sub)-section at an appropriate place early in the paper, where the technical system is formalized (including the \"type\" of system, cf. (W2), and the time dependence that is considered) and where the problem that shall be solved in this work is precisely stated. Owing to the lack of a precise problem formulation, I have difficulty identifying the core technical problem that was solved in this work.\n\n\n\n(W2) *One* core aspect of (W1) is that it is not clear to me what type of system the authors are considering. Is it a dynamical system, x_t+1 = f(x_t, ...) (i.e., one with a state/memory), or a static system, y = f(x)? Note: both can be time varying (i.e., have explicit dependence of f on time t).\n\n* This should be stated as part of a precise problem formulation (W1)\n* What system is considered has important implications on how sampling can be done. For example, in the Introduction, I was puzzled by the sentence \"where measurements can be taken at almost arbitrary positions in a continuous domain\" At first, after reading title and abstract, I was assuming that the authors would do active learning for dynamical systems, where samples (usually state/action combinations) can *not* be chosen freely, as the system has to \"evolve\" to reach a certain state x_t (see, e.g., Buisson-Fenet et al 2020).  If they do consider static systems, it might be possible to freely choose the input. \n\n* I suggest to discuss the conceptual difference where sampling in dynamical systems is different as one cannot freely choose the input, but has to \"go through the physics/dynamics\" instead.  A potential place where the authors could elaborate on this might be Sec. 3.2 (related work).  At least, this discussion should reflect the different nature of the sampling and contrast other works with what is done in this work.\n\n\n\n(W3) The list of contributions should be improved:\n\n* \"We provide several theoretical contribution\" is too vague for a list of contributions; it should be spelled out (or at least an indication given) what results are obtained.\n* In the list of contributions, I would have found it insightful to read what \"real world examples\" are being considered.\n* It is not clear to me how \"We distinctly delineate the theoretical scope of closed-form integrals\" is related to the discussion so far in the introduction; hence, the contributions appears out of context (at this stage).\n\n\n\n(W4) Safe active learning (SAL) setting (Sec. 2.2): It would be helpful to clarify a bit better how the proposed setting is different from the state of the art.  For example, training of safe GPs (g_safe = GP(...)) is not new AFAIK, so references should be added.  Then, is the formulation (1) a new one, or also existing work?  It did not become clear to me what is new and what is background here, partly due to missing references.  It would be good if this was clarified.\n\nFurthermore, is the considered SAL setting one where there exists a TRUE safety function g_safe, but we don't know it (which would be more of a frequentist setting), or is there no such function (more Bayesian).  In particular, I was wondering about (1), whether the safety constraints refers to the true one, or to our information about it given the current data.  This I would expect to have consequences on the algorithm and the type of guarantees that one may be able to give. In my opinion, it would be critical to clarify the setting.  Both settings exist in the literature, but they can have consequences regarding the type of bounds, guarantees, etc. that are available (see, e.g., [R1, R2])\n\n\n\n(W5) The role of hyperparameter optimization is not clear in the context of the proposed SAL framework. On p. 4, \"GP hyperparaemter training is well established and fast\", this sentence appears a bit out of context.  Furthermore, hyperparameter optimization can have a significant effect on safety, at least in other safe learning settings (see e.g. [R2]), so it would be good to discuss 1) if it is also the case here, and 2) if hyperparameter optimization is used at all (and of what type).\n\n\n\n(W6) Sec. 2.3: IMO, this is too short as this seems to be a main basis for what is proposed herein.  First, reference(s) should be given.  Second, I would appreciate understanding the key idea as to why IMSPE is chosen, and preferred to entropy, for example.  Details can be deferred to the appendix (as is currently the case), but the main ideas should be in the main paper.\n\n\n\n(W7) Related work: The paper states that it considers active learning for *time-varying* systems.  There is recent work on BO with time-varying objectives, which is not reviewed in related work, however.  I was wondering whether this is relevant.  In particular, there is also recent work on safety *and* time-varying problems [R3, R4], which should probably be discussed in this context. Orthogonal to this, also see (S3) regarding related work.\n\n\n\n(W8) The technical developments in Sec. 4 require clarification.\n\n* It should be discussed what parts of the model in (3) are novel, and which parts are known.  For example, using GPs that have time as an argument is rather common in BO with time variations / spatio-temporal GPs.\n* I did not understand the idea of Sec. 4.2.  Does (4) still include the time dependency?  It seems to be removed, but this method is still titled T-IMSPE (with the \"T\" in the acronym).\n* Sec. 4.2, first/second line: is y_t = f(x_t, ...) on purpose?  I thought this was the dynamics with x_t+1 = f(x_t, ...)\n* Sec. 4.2, line 4:  What does the notation $x_{i,t}$ (double indices) mean?  Has this been introduced?\n\n\n\n(W9) In Sec. 4.4, the authors present a \"cookbook\" for GP kernels and those fulfilling Def. 4.3, however referring to the appendix for all the details.  As this is an interesting aspect and result of this work, it might be worth to present the kernels / results that compose the cookbook in 1-2 tables in the main paper, while leaving the details in the appendix.  This could also include the references to where (some of) the results have been known in prior work.\n\n\n\n(W10) Presentation and discussion of experimental results.\n\n* The description of the experiment in Sec. 5.1 is insufficient. From the description, one cannot understand what the system is about, why it is challenging / relevant in the context of this work, etc.  The main paper should be self-contained without the need to go to the supplementary material for main aspects. Just referencing an equation in the appendix is not a good structure IMHO and appears like these results were just pushed to the appendix without reconsidering the structure (maybe to make the page limit...).  Similarly for Sec. 5.2.\n* Experiments in Sec. 5.1:  It seems that the evaluation (test data) is restricted to the safe area only.  I understand that model quality should be evaluated there.  However, isn't it also of interest how often samples in the unsafe areas were taken and how this compares between the methods?  IMO, this could (and maybe should) also be discussed and maybe shown with appropriate plots (not just one number in the text).\n* The results (e.g., Figure 2) illustrate that the proposed criterion (T-IMSPE) is quantitatively superior to the other criteria.  However, it wasn't so clear how relevant this improvement is from the application point of view.  Based on the presentation, I get the impression that the improvement is there, but I'm not so sure that it is super relevant. I wonder if some discussion / clarification can be added in this regard.\n\n\n\n(W11) In parts, I found the writing and polishing level of this manuscript to be below standard. Examples:\n\n* Figure 1 does not seem to be referenced from anywhere in the text.\n* For the pseudo-code (p. 4), I suggest to use a proper algorithm environment.\n* What exactly is/are the algorithm(s) that is being run in the experiments?  Please make this explicit, e.g., referencing to pseudo code + equation reference to the criterion.\n* At times, I got the impression that components of the work were moved to the appendix, without taking much care of how this affects the flow and understanding in the main paper.  At least for me, I found the number of references to the appendix a bit disturbing and would have liked the main paper to be more self-contained (e.g., by making sure that all main statements/results/insights are in the main paper). As an example, in the intro of Sec. 5, the main thoughts behind the experimental setups should be briefly discussed in the main paper.  Another example, see comments (W9) + (W10).\n* The paper has a super extensive appendix. While this *might* be needed for the results in the paper, I recommend to make an effort to re-evaluate if the appendix can be shortened.  \n* Bottom of page 8, is there a duplication of text?  The last two paragraphs appear very similar...\n* Please do a careful pass for typos, such as punctuation.  *Examples*: \n  * Page 1, forth line from the bottom -> period instead of semicolon?\n  * end of Sec. 2.1 missing a period.\n\n\n# Minor comments and suggestions\n\n* Introduction: \"Many systems pose challenges for experimentation due to their high costs, whether in actual implementation or simulation,\" -- I'm not quite sure if I can follow the authors argumentation.  They argue that experiments on hardware or simulation are often expensive; then they say that data-driven simulation is better.  So, simulation replaces simulation in a sense.  I believe the authors are getting at surrogate models, etc., but I suggest to 1) clarify what is meant here, 2) consider if this is the intended motivation here (because there might be other motivations as well for looking into active learning for dynamical systems).\n* \"DoE is not even applicable in the face of safety constraints\" -- I don't understand this statement.  I would consider Bayesian Optimization (BO) a method for DoE.  And Safe BO is a kind of DoE, where one exactly considers safety constraints when deciding on the next configuration.  Hence, this statement needs to be clarified or modified, and it seems that Safe BO should be more properly discussed here.\n* Please explain what NX structures mean (in the Intro or in Sec. 2.1).\n* p. 2, \"we start from the acquisition function IMSPE ...\" -> References should be added for the different acquisitions functions, IMO.\n* p. 3, Remark 2.1: \"Therefore, these points are not statistically independent random draws of a Gaussian\" -- I don't understand this sentence.  In particular, why does the non-iid property follow from the previous sentence (points being far away...)\n* p. 3, Remark 2.1: What does \"slightly lower\" mean quantitatively for being \"acceptable\"?  The statement is not technically precise.\n* Check writing \"Pytorch\" vs. \"PyTorch\" for consistency\n\n\n\n# References (of this review)\n\n[R1] Fiedler et al., Practical and rigorous uncertainty bounds for gaussian process regression, AAAI 2021\n\n[R2] Fiedler et al., On Safety in Safe Bayesian Optimization, TMLR 2024\n\n[R3] Jialin Li, et al., Safe Time-Varying Optimization based on Gaussian Processes with Spatio-Temporal Kernel, NeurIPS 2024\n\n[R4] Holzapfel et al., Event-triggered safe Bayesian optimization on quadcopters, L4DC 2024", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The manuscript proposes an active learning framework that aims at both data efficiency and safety during learning. A system is considered that is time-varying, which includes relevant aspects in practice like drifts and seasonal changes. The T-IMSPE criterion is proposed for active sampling, which minimizes posterior variance over a future horizon. The proposed criterion and corresponding active learning method is compared to established methods in numerical experiments, as well as an example with real-world data.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Please address and/or respond to the above-mentioned weaknesses.", "broader_impact_concerns": "No concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "J9xeEwDsp4", "reviewer": "Reviewer_EHv6", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strength**:\n- This paper extends the IMSPE acquisition function to account for future states, enabling better data collection in systems with temporal dependencies.\n- I like that the paper provides guidelines on which GP kernels, domains, and measures are suitable for T-IMSPE, and explores its theoretical boundaries.\n- The proofs are technically solid\n- Interesting toy problems and real-world scenario experiments are evaluated, which demonstrates the applicability of T-IMSPE.\n\n**Weakness**:\n- the connections and differentiations of safe learning methods with SAL are poorly explained. Most safe learning tries to acquire dynamics model data safely, and the purpose is fundamentally different the formulation presented in this paper. Additionally, there are more literature on the safe reinforcement learning with Gaussian Process in recent years, the author should comment their connections with those works. \n- Though the theoretical proofs are sufficient, the author dives deeply into the mathematical aspects of T-IMSPE without providing sufficient intuition or simplified examples to guide non-specialist readers. This approach may alienate practitioners and those outside the immediate research niche, limiting the paper’s broader impact.\n- The experiments are interesting but the impact is limited. I don't think those are well established benchmarks. The authors should at least validate the applicability of T-IMSPE in a broader set of safe exploration tasks, such as safe reinforcement learning benchmark tasks. This will answer the scalability to high dimensional robotics tasks. Good platforms include:\n     - Safety Gymnasium: A Unified Safe Reinforcement Learning Benchmark\n     - Guard: A Safe Reinforcement Learning Benchmark\n- Figures and diagrams, while useful, are dense and lack adequate annotation for clarity. For example, Figure 1 requires significant prior knowledge to interpret, which limits its instructional value.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper introduces T-IMSPE (Time-aware Integrated Mean Squared Prediction Error), a novel acquisition function designed for safe active learning in time-varying systems. The method aims to minimize posterior variance across both current and future states, addressing the challenges posed by system dynamics such as drift and seasonal changes.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "- more reference and connections with safe RL with Gaussian Process in recent literature should be discussed\n- more intuitive explanation, comments should be provided to the theoretical proof and theories themselves \n- safe robotics benchmark experiments are required", "broader_impact_concerns": "No concern", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "OvF8PZ0or9", "reviewer": "Reviewer_3k2z", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "### Strengths\n1. The paper is well written and easily understood.\n2. Adding a time dimension to IMSPE is done intuitively, also allowing easy closed-form computation.\n3. The experiment section is nicely done (kudos on having p-values), and clearly shows better performance compared with entropy and IMSPE.\n\n### Weaknesses & questions\n1. The T-IMSPE is not applicable to all kernels. This is all the more reason to investigate sparse GPs/variational inference, where an inexact posterior is computable and a closed-form is not necessary.\n2. From what I understood, tuning the kernel parameters and computing the posterior (i.e., training) is done first, and then the T-IMSPE method is to be used, correct? This is why it is claimed that \"T-IMSPE \\[has\\] the same computational complexity as computing the entropy.\"\n3. I am a bit confused as to why the authors restricted test data to the safe area. They mention that \"model quality is only of interest there,\" but wouldn't it also be important to see if the methods are sampling \"unsafe\" points?\n4. In Section 5 you reference Appendix F and \"the attached code,\" but I did not see attached/linked code anywhere.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The submission introduces T-IMSPE, a new acquisition function for safe active learning in time-varying systems that minimizes posterior variance over both current and future states. The authors show the closed-form computability of T-IMSPE, and thus its $O(n^3)$ and $O(n^2)$ time-complexity in training and testing, respectively. Finally, empirical results demonstrate that T-IMSPE outperforms the traditional method of entropy as a measure for information in modeling quality while adhering to safety constraints.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. It would be interesting to investigate the performance (both time and MSE) of the T-IMSPE method with sparse GPs (variational inference) compared with the exact closed-form computation of the posterior. Sparse GPs should allow for drastically improving the computational complexity.\n2. From what I understood, T-IMSPE is the objective that is used in the safe active learning and instead of $I(\\cdot)$ in equation (1), correct? I think it would be better if this is explicitly mentioned, because in the current version of the paper, equation (1) and the pseudocode are never referenced after Section 2.2.\n3. The work assumes correct model specification. What happens if the correct model (e.g., kernel parameters or even kernel type) is not given? How well does the method perform compared with the entropy method?\n\n### Minor comments\n1. Typo on page 3: \"This **yield** the probability...\" should be \"This **yields** the probability...\"\n2. Typo on page 5: \"In both cases, T-IMSPE **work**...\" should be \"In both cases, T-IMSPE **works**.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: See above.
Decision: Accept (poster)