Distributionally Robust Active Learning for Gaussian Process Regression
Authors: Shion Takeno
Yoshito Okura
Yu Inatsu
Aoyama Tatsuya
Tomonari Tanaka
Akahane Satoshi
Hiroyuki Hanada
Noriaki Hashimoto
Taro Murayama
Hanju Lee
Shinya Kojima
Ichiro Takeuchi
Venue: ICML 2025 poster (2025)
Keywords: Gaussian process
Active learning
distributionally robust
experimental design
Primary area: general_machine_learning->online_learning_active_learning_and_bandits

Abstract: Gaussian process regression (GPR) or kernel ridge regression is a widely used and powerful tool for nonlinear prediction. Therefore, active learning (AL) for GPR, which actively collects data labels to achieve an accurate prediction with fewer data labels, is an important problem. However, existing AL methods do not theoretically guarantee prediction accuracy for target distribution. Furthermore, as discussed in the distributionally robust learning literature, specifying the target distribution is often difficult. Thus, this paper proposes two AL methods that effectively reduce the worst-case expected error for GPR, which is the worst-case expectation in target distribution candidates. We show an upper bound of the worst-case expected squared error, which suggests that the error will be arbitrarily small by a finite number of data labels under mild conditions. Finally, we demonstrate the effectiveness of the proposed methods through synthetic and real-world datasets.

Distributionally Robust Active Learning for Gaussian Process Regression
Shion Takeno' Okura Yoshito Yu Inatsu
Aoyama Tatsuya' Tomonari Tanaka' 1 Akahane Satoshi!
Hiroyuki Hanada
Noriaki Hashimoto
Taro Murayama
Hanju Lee* Shinya Kojima* Ichiro Takeuchi 13
Abstract
Gaussian process regression (GPR) model (Rasmussen & Williams, 2005) is often used as a base statistical model for AL algorithms due to its flexible prediction capability (Seo et al., 2000; Yu et al., 2006; Guestrin et al., 2005; Krause et al., 2008b; Hoang et al., 2014; Hübotter et al., 2024). Standard AL methods for the GPR are based on information gain (Guestrin al., 2005; Krause et al., 2008b; Kirsch et al., 2021; Kirsch & Gal, 2022; Bickford Smith al., 2023; Hübotter et al., 2024). Most information gain-based approaches are heuristics without theoretical guarantees. A notable exception is the work by (Guestrin et al., 2005; Krause et al., 2008b), which shows that the US for the GPR model is optimal to maximize the information gain from the obtained data labels regarding the GP prior. Furthermore, from the analysis of kemeliieed bandits (e.g.. Srinivas et al., 2010; Salgia et al., 2024), we can see that the US and RS guarantee the convergence of the maximum of posterior variance (See Proposition 2.3 for details). Another com- monly used AF is variance reduction (Seo et al., 2000; Yu et al., 2006; Shoham & Avron, 2023; Hübotter et al., 2024), which can be computed efficiently in the GPR. However, these AFs do not incorporate the importance of the unla- belled dataset, that is, the prior information regarding the target distribution. In addition, to our knowledge, except for the worst-case analysis in Proposition 2.3, there are no theoretical guarantees for the target prediction error.
Gaussian process regression (GPR) or kernel ridge regression is a widely used and powerful tool for nonlinear prediction. Therefore, active learning (AL) for GPR, which actively collects data labels to achieve an accurate prediction with fewer data labels, is an important problem. How- ever, existing AL methods do not theoretically guarantee prediction accuracy for target distribu- tion. Furthermore, as discussed in the distribu- tionally robust learning literature, specifying the target distribution is often difficult. Thus, this paper proposes two AL methods that effectively reduce the worst-case expected error for GPR, which is the worst-case expectation in target dis- tribution candidates. We show an upper bound of the worst-case expected squared error, which suggests that the error will be arbitrarily small by a finite number of data labels under mild condi- tions. Finally, we demonstrate the effectiveness of the proposed methods through synthetic and real-world datasets.
1. Introduction
Active learning (AL) (Settles, 2009) is a framework for achieving high prediction performance with fewer data when labeling new data is expensive. For this purpose, AL algo- rithms actively acquire the label of data that improves the prediction performance of some statistical model based on acquisition fumctions (AFs). Many types of AFs have been proposed, such as uncertainty sampling (US), random sam- pling (RS), variance reduction, and information gain, as summarized in (Settles, 2009).
Several studies have tackled the development of the target distribution-aware AL (Kirsch et al., 2021; Kirsch & Gal, 2022; Bickford Smith al., 2023). In particular, as ex- tension of the distributionally robust learning (Chen et al., 2020), Frogner et al. (2021) proposed distributionally ro- bust AL (DRAL), which aims to minimize the worst-case error in the set of target distributions to obtain a robust model. However, since these studies employed the heuristic AL methods based on, e.g., information gain and expected model change (Settles, 2009), the theoretical guarantee has not been shown.
'Department of Mechanical Enginecring, Nagoya University, Aichi, Japan 'Dpparmeett of Computer Science, Nagoya Insti- tute of Technology. Aichi, Japan 'RIKEN AIP, Tokyo, Japan "DENSO CORPORATION, Aichi, Japan. Correspondence to: Shion Takeno <takeno s. .mllab.nnt @ comi, Ichiro Takeuchi <takeuchi www.roiichhhooooo.o nó @ f.mail maail.aggoy nagooa- up.
This paper develops DRAL framework for the GPR model. We aim to minimize the worst-case expected error, where the worst-case scenario and the expectation are taken re- garding the target distribution candidates and chosen target distributions, respectively. Note that our formulation gener- alizes target distribution aware AL since it includes the case
Proceedings of the 42nd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Distributionally Robust Active Learning for Gaussian Process Regression
in which the unique target distribution can be specified. We perform the theoretical analysis under two conditions called Bayesian and frequentist assumptions (Srinivas et al., 2010), in which we leverage several useful lemmas in kernelized bandit literature (Srinivas et al., 2010; Vakili et al., 2021a;b; Kusakawa et al., 2022).
k(2;, I IRttt is the identity matrix, and y, = (Um):-..: :,m.,) ) E R'. Finally, for later use, let the pos- terior variance at x² when 2; x be of(x | x) == of(x" 2 = æ). Note that the posterior variance calcula- tion does not require yr- Furthermone, it is known that fe (x())) is equivalent to the kernel ridge regression estimator with regularization parameter À = o² (Kanagawa et al., 2018).
Our contributions are summarized follows:
Maximum Information Gain (MIG): Further, we define MIG (Srinivas et al., 2010; Vakili et al., 2021b);
1. We show several properties of the worst-case squared error for the GPR model, which suggests that the error can be bounded from above using the posterior vari- ance even if the input domain is continuous. Along the way proving the error properties, we show the Lipschitz constant of the posterior mean of GPs in Lemmas 2.9 and 3.3, which may be of independent interest.
Definition 2.1 (Maximum information gain). Let f ~ GP k) over X C [0. Let A = {a.)== C X. Let fa (f(a.)); ;-1 ca = (G) where Vi,G o(0, and ya = fA + EA E R.. Then, MIG YT is defined as follows:
max fA), ACX:A|=TT
2T
2. We propose two DRAL methods for the GPR model, inspired by the RS and the greedy algorithm. Our proposed methods are designed to guarantee the con- vergence of the (expected) posterior variance.
where / is the Shannon mutual information.
It is known that MIG is sublinear for commonly used kernel functions, for example, " = (d for lin- ear kernels, ir ((log ((logT)d++) 1) for squared exponen- tial (SE) kernels k x¹) and 'r =O(T (log T) T)ara for Matérn-v kernels - Ju (221 12-=--= where f 0 are the lengthscale and smoothness parameter, re- spectively, and r'()) and J, are Gamma and modified Bessel functions, respectively (Srinivas et al., 2010; Vakili et al., 2021b).
3. We show the probabilistic upper bounds of the error incurred by the proposed algorithm, which suggests that under mild conditions, the error can be arbitrarily small by finite number of data labels.
exp (-I|æ - x' (B/(()))
AMat = 21 " T(o)
Finally, we demonstrate the effectiveness of the proposed methods via synthetic and real-world regression problems.
2. Background
Lipschitz Consatant of (x)): We will use the following useful result from Theorem E.4 in (Kusakawa et al., 2022):
This section provides the known properties of the GPR.
Lemma 2.2 (Lipschitz constant for posterior standard de- viation). Let k(x, x') IRd x Rd + R be linear, SE, or Matérn-v kernel and k(æ, x) S 1. Moreover, assume that a noise variance o² is positive. Then, for any 2 1 and Dy, the posterior standard deviation (x())) satisfies that
2.1. GPR model
The GPR model (Rasmussen & Williams, 2005) a kernl- based regression model. Let us consider that we have already obtained the training dataset of input-output pair D. {[x;, Va,) ) where Vi, x E X C Rdd Va, E R, and d is an input dimension. The GPR model assumes that, without loss of generality. f follows zero-mean GP. that is, f ~ (0 k), where h: : X' x X' + RR is a predefined positive semidefinite kernel function. In addition, the i- th observation Ya, assumed to be contaminated by i.i.d. Gaussian noise e, ~ _N(0, o2) as Ya. = fxx) + Eq. Then, the posterior distribution of f becomes again GP, whose mean and variance are analytically derived as follows:
Vz, x' ER3, lo,(x) - az(2')] la 2'112,
where Tig is a positive constant given by
if (((x,,") )') is the linear kernel, if k x!) is the SE kernel, if km,, x!) is the Matérn kernel,
y2
La =
V2
where V >>>
(e(x)= = (x) T K + 02)1) "'y.
2.2. Uncertainty Sampling and Random Sampling
"e (xx)) = k(ar, .) - (z)) (K + =2I))) kx))))
For the GPR model, the US selects the most uncertain input as 1-th input:
where ke(x) = (k(z, æI)...., k(2, xz)) T K E Rixt is the kernel matrix whose -eemmet is
E R²
33 arg max 02-1
xex
2

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Distributionally Robust Active Learning for Gaussian Process Regression
The RS randomly selects a 1-th input by fixed probability distribution p(æ:) over Xt:
This assumption holds for stationary and four times differ- entiable kernels (Theorem 5 of Ghosal & Roy, 2006), such as SE kernel and Matérn-y/ kernels with v > 2 (Section 4 of Srinivas et al., 2010). These assumptions are commonly used (Srinivas et al., 2010; Kandasamy et al., 2018; Paria et al., 2020; Takeno et al., 2023; 2024).
Lt p(x).
For both algorithms,, the upper bound of the maximum vari- ance is known:
As with Lemma 5.1 in (Srinivas et al., 2010), the credible interval can be obtained as follows:
Proposition 2.3. Assume X' is compact subset of Rd. If we run the US, the following inequality holds:
Lemma 2.6. Suppose that X is finite and Assumption 2.4 holds. Pick 8 (0,1) and t N. Then, for any given Dr,
max (2) S CITT T
#Et
Pr (1(m)) Ma(x)) - /8 ( Vn E Ex 1 2 1 - 6,
where C1 = 11/1loog1 + 2- Furthermore, if we run the RS, the following inequality holds with probability at least 1 8, where E (0, 1), under several conditions:
where Ba = 2 1og((2118))
2.3.2. FREQUENTIST ASSUMPTION
max (x) 00(0) 02 #EX
(0(0)))
We assume that f is an element of the reproducing kernel Hilbert space (RKHS) specified by the kernel k as with (Srinivas et al., 2010; Chowdhury & Gopalan, 2017; Vakili et al., 2021a; 2022; Li & Scarlett, 2022):
Proof. For the US see, e.g., Eq. (16) in (Vakili et al., 2021a) and Lemma 5.4 in (Srinivas et al., 2010). For the RS, see Theorem 3.1 in (Salgia et al., 2024).
Assumption 2.7. Let I be an element of RKHS H4 speci- fied by the kernel A used in the GPR model. Furthermore, the RKHS norm of f is bounded as |Iʃ ll, S B < co for some B > O, where || l|xe denotes the RKHS norm of H.. In addition, the i-th observation ya: is contaminated by inde- pendent sub-Gaussian noises {Ei as ya: = fxx,) + Es. That is, for all E N, for all y) E IR, and for some R O, the moment generating function of e satisfies E[exp(ne;) S exp( y r'k²) Finally, the kernel function is normalized as k(x, x 1 for all x, x' e X.
When Tr is sublinear, the above upper bounds suggest that the maximum variance will be arbitrarily small within the finite time horizons.
2.3. Regularity Assumptions and Predictive Guarantees
Here, we provide the details of Bayesian and frequentist assumptions and predictive guarantees for both assumptions.
Furthermore, for continuous X, we assume the follow- ing smoothness condition as with (Chowdhury & Gopalan, 2017; Vakili et al., 2021a; 2022):
1. BAYESIAN ASSUMPTION
We consider the following assumption:
Assumption 2.8. The kernel function k satisfies the follow- ing condition on the derivatives. There exists a constant L such that,
Assumption 2.4. The function is a sample path f GP(O, k) and the i-th observation yo, is contaminated by i..d. Gaussian noise 6 (O, o²) Ya,, f(x,) + c.. In addition, the kernel function is normalized as k(x x²) S for all x' E X.
@2k(u,
1/2 < Lu.
sup sup #e jE
du,ou, 1A 10
Furthermore, for continuous X, we assume the following smoothness condition:
This assumption provides the Lipschitz constant of f: Lemma 2.9 (Lemma 5.1 in (De Freitas et al., 2012)). Sup- pose that Assumption 2.8 holds. Then, any 9 E Hy is Lipschitz continuous with respect to [[gll Lx-
Assumption 2.5. Let X' C [0, rdd be compact set, where > O. Assume that the kernel i satisfies the following condition on the derivatives of sample path f. There exist
the constants a, b > 0 such that.
We rely on the confidence bounds for non-adaptive sampling methods, which is a direct consequence of Theorem 1 in (Vakili al., 2021a) and the union bound:
0f(u)
2 (
Pr sup x
L.) S aexp
Oug
72
Lemma 2.10. Suppose that X is finite and Assumption 2.7 holds. Pick E 10, and N. Assume that (x:)xe(4) is
for all j E [d].
3

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Distributionally Robust Active Learning for Gaussian Process Regression
independent of (e))) Then, the following holds:
Furthermore, need the Lipschitz constant of pr,. In the frequentist setting, the Lipschitz constant for prr can be derived as { vt log t) by Lemma 4 in (Vakili et al., 2021a) and Lemma 2.9. To obtain slightly tighter upper bound, we show the following lemma:
Pr (1f(x)) - me(x)] - oz()), E x) 2 1--6,
2 -(B [ R 2 2log((
where 3g-
Lemma 3.2. Fix 8 E 1) and t. E (T]. Suppose that Assumptions 2.7 and 2.8 hold. Then, pel.) is Lipschitz continuous with the Lipschitz constant,
3. Problem Statement and Its Property
This section provides details on our problem setup and its properties.
R B+
d 3
L
27 + 2 2og
o
3.1. Problem Statement
with probability at least 1 - 6.
We aim to minimize the worst-case expected errors regard- ing the GP prediction (ar after T'th function evaluations:
We show the proof in Appendix A.I. Since the MIG YT is sublinear for the kernels on which we mainly focus, the upper bound (LL is tighter than OlL /log t).
FT-
max E pEp p [[((=)) [
(2)
In the Bayesian setting, the upper bound of the Lipschitz constant for Pr has not been shown our knowledge. Therefore, we show the following lemma:
where P is a set of target distributions over the input space X called ambiguity set (Chen et al., 2020). We assume that maxpep Ep(==) [g(x")] exists for any continuous function g: X + 12 This paper concentrates on the setting where the training input space from which we can obtain labels includes the test input space.
[T]. Suppose that Assumptions 2.4 and 2.5 hold and the kernel has mixed @k((.z) partial derivative 0z,02z for all j E [d]. Set a and b as in Lemma 2.5. Assume that (es) is independent of (c))cctt) and f. Then, t4 and re(ar) = f(x) (x( satisfies the following:
Lemma 3.3. Fix 8 E (0,1) and
Our problem setup can be seen as the generalization of the target distribution- AL and the AL for the worst- case error maX (f(x) pr (x))?. This is because our problem is equivalent to the target distribution aware AL if we set [P] = and to the worst-case error minimization if P includes {p E Px Ex E X, p(x) = 1}. where Px is the set of the distributions over X.
(p)(u) Oug (r.(u)
I
Pr sup
> L
< 2a exp
12
1
ΦE
u=
1
Pr
sup x
>
L
< 2a exp
Oug
1==
62
3.2. High Probability Bound of Error
for all j E [d].
If the input space X' is finite, we can obtain the upper bound of Eq. (2) as the direct consequence of Lemmas 2.6 and 2.10:
See Appendix A.2 for the proof, in which we leverage Slepian's inequality (Proposition A.2.6 in van der Vaart & Wellner, 1996) and the fact that the derivative of the sample path follows GP jointly when the kernel is differentiable.
Lemma 3.1. Fix 8 E (0 and T E N. Suppose that Assumption 2.4 holds and 3s is set as in Lemma 2.6, or Assumption 2.7 holds and 38 is set as in Lemma 2.10. Then, the following holds with probability at least - 6:
By leveraging the above results, even X' is continuous, we can obtain the following upper bound of Eq. (2):
Lemma 3.4. Suppose that Assumptions 2.7 and 2.8 hold. Fix E (0, 1) and T E N. Then, the following holds with probability at least 5:
Eq S 36 max Ep(ar) [o7( PEP
max max log( F)}
Er S 2,8.T max E pp(ar) [of(2")] +0 pEP
For continuous X. the confidence parameter 3s x log |x'| diverges if we apply Lemmas 2.6 and 2.10 directly. There-
T
fore, in this case, the Lipschitz property is often lever- aged (Chowdhury & Gopalan, 2017; Vakili et al., 2021a). The Lipschitz constant of if can be directly derived from the Assumption 2.5, or Assumption 2.8 and Lemma 2.9 (Srini- vas et al., 2010; De Freitas et al., 2012).
where 38,T = B+ 2d (Tdr + 1) + 2 log (2)
Lemma 3.5. Suppose that Assumptions 2.4 and 2.5 hold.
Fix 10,1) and T E N. Then, the following holds with

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Distributionally Robust Active Learning for Gaussian Process Regression
probability least 6:
Algorithm Proposed DRAL methods
log(F)) T
Require: Domain X, GP prior μ and k, ambiguity set P
Fr 2,,,T max E pp(=)) [x)] +0 PEp
1: Do f 0
2: for = 1......, do
3: Update 1--1(-) according to Eq. (1)
where B8.r = log +) 2 (2g(2/)).
4: Compute H, according to Eq. (3) or Eq. (4)
See Appendices A.3 and A.4 for the proof.
5: end for
Observe V1. .i......95 3r
Consequently, we can minimize Eq. (2) by minimiz- ing maxpep E Bp(=)) [x*]] In this perspective, the US and RS are theoretically guaranteed because of maxpep Ep(t*) [ar)) S max (ar) and Propo- sition 2.3. However, the US and RS do not incorporate the information of P. Therefore, the practical effectiveness of the US and RS is limited.
Update pr(() and (.) according to Eq. (1)
8: return pr(+) and (.)
where (( arg maxpep 1 (xx) and we as- sume that generate the sample from pr- By using the worst-casc distribution Pr for each iteration, this algorithm incorporates the information of P.
3.3. Other Performance Mesuares
Second, we consider the greedy algorithm since its practical efficiency has often been reported (c.g., Bian et al., 2017). However, in our setup, the algorithm that greedily decreases the expected posterior variance should be
Although we mainly discuss the squared error, other mea- sures can also be bounded from above:
Lemma 3.6. The worst-case expected absolute error for any T E N bounded from above as follows:
arg min max pp x)),
max Ep(ar) l1f(x=))= - (x |] VEr, PEP
xEx PEp
where FT defined as in Eq. (2).
which requires huge computational time in general due to min-max optimization. Thus, we consider an approximately greedy algorithm as follows:
Lemma 3.7. The worst-case expectation of entropy for any T E N is bounded from above follows:
max maxEp(a>) Ep(e-) [H [f(x)) |Dr]] S log (2teBr) PEP
arg min E (m))) x));
#EX
where Er maxpEp Ep(my) (x*)) and H (x)
where Pr (x = arg pax E case distribution defined by (x, (x)))e(t-1]- On the other hand, the theoretical guarantee for this algorithm is challenging for us. Hence, inspired by the fact that the US has theoretical guarantee, we set the constraint so that the chosen input is uncertain than (p. ) (x*)]:
(xx)) )} is the worst-
Dr] = log (log(Viite (2)))T())) is Shannon entropy.
See Appendices A.5 and A.6 for the proof. Therefore, mini- mizing maxpep Ey(ar) jo (x*)) also provides the conver- gence of the absolute error and the entropy!.
4. Proposed Methods and Analysis
arg min (m 1006(0 æ6A.
(np x)).
(4)
We aim design algorithms that enjoy both similar con- vergence guarantee the US and RS and practical effective- ness, incorporating the information of P. In particular, we consider two algorithms inspired by the greedy algorithm and the RS and show theoretical guarantees. Algorithm shows the pseudo-code of the proposed algorithms.
where 2; := xx E X of
Ep (p (xx)
Note that X > holds due to the definition.
Necessity of Constraints: We considered that the con- straint regarding (x)) makes the analysis easy since the US that maximize o,- (æx) achieves the error convergence, as shown in Proposition 2.3. Therefore, we employ the constraint on Xx We set the threshold of the constraint |,? sake of the analysis. On the other hand, our experimental results suggest that the greedy (ap- proximated) expected eгтor reduction algorithm without the constraint shows superior performance. Therefore, remov- ing or alleviating the constraint can be important future work.
4.1. Algorithms
First, we consider the RS-based algorithm. The algorithm straightforward as follows:
xt ~ (x),
For the absolute error, we can design algorithms that directly reduce de not of, and achieves similar thcoretical guarantee.
5

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Distributionally Robust Active Learning for Gaussian Process Regression
Computational Complexity: The computation of the GP has O(T9) computational complexity, which can be allevi- ated by scalable GP learning approaches (Liu et al., 2020). However, more careful proofs incorporating an approxi- mation error in GP learning are required to derive similar theoretical analyses as ours, as with (Vakili et al., 2022). On the other hand, the computational complexity of the maximization maxpeP (p(=)) 1 )], the expectation (p.(=-) , (x=)] and the sampling from pt (a) depends on the ambiguity set P and may increase in proportion to d. Although our experiments focus on the set of discrete probability distributions with a moderate size of [ the above computations may be complicated if P is the set of continuous probability distributions or |.X| is huge. Exten- sions such more computationally intractable ambiguity sets P. e.g., the ball defined by Wasserstein distance and Kullback- divergence (Hu & Hong, 2013; Frogner et al., 2021), is crucial future work.
Proof. We can obtain the result by combining Lemmas 3.1, 3.4, and 3.5, Theorems 4.1 and 4.2, and the union bound. Note that assume |xl > T.
Thus, the error incurred by the proposed algorithms con- verges to 0 with high probability for discrete and continuous imput domains, at least with linear, SE, and Matérn kernels.
5. Related Work
As discussed in Section 1, many AL algorithms have been developed (Settles, 2009). The AL. algorithms for classifi- cation problems are heavily discussed compared with the regression problem (for example, Houlsby et al., 201 1; Zhao et al., 2021; Bickford Smith et al., 2023). In particular, theoretical properties for binary classification problems are well-investigated (Hanneke al., 2014). On the other hand, the theoretical analysis of AL for the regression problem relatively limited.
4.2. Analysis
AL is often referred to as optimal experimental design (OED) (Lindley, 1956; Cohn, 1993; Chaloner & Verdinelli, 1995; Cohn et al., 1996; Ryan & Morgan, 2007). The OED frameworks aim to reduce the uncertainty of target parameters or statistical models. For this purpose, many measures for the optimality have been proposed, such as A-optimality (average), D-optimality (determinant), and V- optimality (variance) (Pukelsheim, 2006; Allen-Zhu et al., 2017). The OED methods for various models, such as the linear model (e.g., Allen-Zhu et al., 2017), neural net- work (e.g.. Cohn, 1993), and GPs (e.g.. Yu et al., 2006), have been proposed. Our analysis concentrates on the V- optimality of the GPR (Seo et al., 2000; Yu et al., 2006; Shoham & Avron, 2023) and its DR variant, for which, to our knowledge, thcoretical guarantee has not been shown.
Here, we show the error convergence by Eqs. (3) and (4):
Theorem 4.1. Fix 8 E (0, 1). Assume that X' C Rdd is a compuct subset. If we run Algorithm with Eq. (3), the
following holds with probability at least 1- 8:
2CTTT
(1)( T
max pEp
(07
+ +0
T
where C₁ = 1/1 1
Theorem 4.2. Assume that X C IR is compact subset. If
run Algorithm 1 with Eq. (4), the following holds:
Cirr
max Ep(ar) 1077 PEP
where C1 = 1/ log( 11 0
See Appendix B for the proof, in which Lemma 3 in (Kirschner& & Krause, 2018) is used to show Theorem 4.1.
In OED or AL, subset selection algorithms (Das & Kempe, 2008) are often leveraged. The subset selection is general problem whose goal is to find the subset that maximizes some set function. Therefore, the AL can be seen as the sub- set selection of æ1...., 3₁ X. In this literature, the sub- modular property of the set function, for which the greedy algorithm can be optimal, is commonly investigated (Das & Kempe, 2008; Krause et al., 2008b; Guestrin et al., 2005; Bian et al.. 2017). The criteria for the AL, such as the D- optimality of the GPR (Krause et al., 2008b; Guestrin et al., 2005), sometimes satisfy the submodular property. Further- more, Das & Kempe (2008) have shown sufficient condi- tions for the greedy algorithms be optimal in Theorem 3.4 (an assumption can be rephrased as k( x²) < 4T in our problem) and Section 8. However, even if we consider mini- mizing Ep(=-) (x7())) with [P] = 1, these conditions and the submodularity do not hold in general. Therefore, the DR maximization of submodular function (c.g.. Krause et al., 2008a; Staib et al., 2019) also cannot be applied directly.
Consequently, our proposed methods achieve almost the same convergence as those of the US and RS shown in Proposition 2.3. Furthermore, by combining Lemmas 3.1, 3.4, and 3.5, we can see that the upper bound of Er:
Corollary 4.3. Fix 8 E (0, 1) and T E N. Then, we run Algorithm 1, the following hold with probability at least 1-8:
1. When Assumption 2.4 or Assumptions 2.7 holds.
ET = (logo//o1///1/// (log(1X1/5))
T
2. When Assumptions 2.4 and 2.5 or Assumptions 2.7 and
2.8 hold,
Er = 0 log 10g((T/))T))
T
6

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Distributionally Robust Active Learning for Gaussian Process Regression
US - RS
variance reduction
EPIG -
DR variance reduction +
DR random
CDR variance reduction
Expected Squared Error
0.06
n=0.0
Expected Squared Emor Expected Squared Error
0.150
0.000
Expected Squared Error
0.6
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n- 0.1
0.125
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1:25-
0.04
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0.03
000075
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0.75-
0.02-
0.2
0.50
0.025
0.1
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0.00
0.000
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Iterations
Iterations
Iterations
Iterations
Expected Squared Emror
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Iterations
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Iterations
Figure I. Result of the emor Er in the synthetic data experiments with = 0, 0.001, 0.01, 0.1. The horizontal and vertical axes show the number of iterations and Er, respectively. The error bar shows mean and standard errors for 20 random trials regarding the random initial point (and the algorithm's 's randomness). The top and bottom columns represent the results of the SE and Matém kernels, respectively.
Several studies have discussed the target distribution-awan AL. At least from (Sugiyama, 2005), the effectiveness of AL incorporating the information of target distribution for mis- specified models has been discussed. Transductive AL (Seo et al., 2000; Yu et al., 2006; Shoham Avron, 2023) can be interpreted as the expected error minimization when the uni- formly random target distribution p(x) 1/|xX| is specified. Kirsch et al. (2021); Kirsch& & Gal (2022); Bickford Smith et al. (2023) extended this setting so that an arbitrary target distribution can be considered. These existing methods are heuristic algorithms that do not guarantce the convergence of the error. On the other hand, Hübotter al. (2024) show transductive AL methods with theoretical guarantees. How- ever, for the VTR algorithm in (Hübotter al., 2024), their analysis assumes assumption of submodularity, which may not hold as discussed immediately after A Assumption 3.2 of (Hübotter et al., 2024). Furthermore, their analysis for the VTR algorithm results in m (x) = (/r/////TI /T), which is vacuous in the usual regime that T < |X| (see Theorems 3.3 and C.13 and Section C.6.2 of Hübotter al. (2024)).
parameter so that the worst-case expected loss is minimized, where the worst-case is taken regarding the target distribu- tion candidates called an ambiguity set. Therefore, DRAL is an intuitive extension of DRL to AL.
Another related literature is core-set selection (Sener & Savarese, 2018), which selects the subset of the training dataset to maintain prediction accuracy while reducing the computational cost. Our proposed methods can be applied to the core-set selection for the GPR. However, its effective- ness may be limited since the information on training labels is not leveraged.
Other highly relevant literature is kernelized bandits, also called Bayesian optimization (BO) (Kushner, 1964; Srinivas et al., 2010; Shahriari et al., 2016). BO aims for efficient black-box optimization using the GPR model. For this purpose, several properties of GPs, such as the confidence intervals and the MIG, have been analyzed (Srinivas et al., 2010; Vakili et al., 202 la;b). Our analyses heavily depended on the existing results in this field.
In addition, level set estimation (LSE) (Gotovos et al., 2013; Bogunovic et al., 2016; Inatsu et al., 2024) is an AL frame- work using the GPR model, which aims to classify the test in- put set by whether or not a black-box function value exceeds given threshold. In particular, Inatsu et al. (2021) consider the variant of LSE, which aims to classify by whether or not the DR measure defined by the black-box function exceeds given threshold. Thus, our problem setup differs from the problem of (Inatsu et al., 2021).
Frogner et al. (2021) further extended to DRAL using the AF called expected model change (Settles, 2009) for the non- Bayesian model. In addition, Liu et al. (2015) considers DRAL for non-Bayesian classification models. However, these methods are heuristic greedy algorithms and are not theoretically guaranteed for the prediction error.
The DRAL is inspired by the DR learning (DRL) (Chen & Paschalidis, 2018; Chen et al., 2020). DRL considers learning robust statistical model by optimizing the model
7

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Distributionally Robust Active Learning for Gaussian Process Regression
US - RS-
variance reduction
EPIG
- DR variance reduction
+
DR random
CDR variance reduction
Expected Seuared Eroo
8-00
Expected so BrT
0-0000
n-001
Expected Squared Eiror
0.1
8.2
18
100
200
300
400
500
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300 400 lterations
700
100
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100
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lterations
lier Lione
[eer Alerations
Expected Squares Ererr
0.0
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n- 0.001
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Expected Squared Error 7 s.
0.3
5.D.
L.
4.5
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Itarations n 0.001
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Expected Squared Erer
0.8.
Expected Squared Ener
Expetted Squared Litor
3.0
0..
5.--
0.0
150
300
300
Iterations
Iterations
Iterations
Iterationn
Figure 2. Result of the expected squared error Er in the woaldworld data experiments with = 0, 0.001, 0.01, 0.1. The horizontal and vertical axes show the number of iterations and Er, respectively. The error bar shows mean and standard emrors for 20 random trials regarding the random initial point (and the algorithm's 's randomness). The top, middle, and bottom rows represent the results of the King County house sales dataset, red wine quality dataset, and auto MPG dataset, respectively.
6. Experiments
Lo norm. Note that r = 0 matches the case that the unique target distribution Pret is specified. Since we consider the case of discrete X', maximization over P can be written as linear programming for which we used CVXPY (Diamond & Boyd, 2016; Agrawal et al., 2018).
In this section, we demonstrate the effectiveness of the pro- posed methods via synthetic and real-world datasets. We employ RS, US, variance reduction (Yu et al., 2006), and ex- pected predictive information gain (EPIG) (Bickford Smith et al., 2023) as the baseline. Note that we do not employ the method of (Frogner et al., 2021) since adapting their method to GPR models is not apparent, and they focus on the ambiguity sets defined by Wasserstein distance over a continuous input domain. We show the implementation de- tails of EPIG in Appendix C.2. Furthermore, as the ablation study, we performed the method, referred to as DR variance =))] that is, the unconstrained version of Eq. (4). We referred to the proposed methods as DR random and constrained DR
The aim of our experiments is to show that the proposed al- gorithms are consistently well-performing for any size of P. i.e., 17, in contrast to the baselines, which can deteriorate for some 7). The same parameter 1) used for the computation of the performance measure Ey and the actual algorithm. Therefore, we expect that our proposed algorithms show good performance among all 17 0, 0.001, 0.01, 0.1. On the other hand, if 1 is large, then P starts to include more distributions. Therefore, the proposed algorithms behave similarly to the US when 7) is large. Hence, the proposed algorithms have comparable performance to the usual AL methods when 17 is large, though the proposed algorithms are superior to those if is small. On the other hand, EPIG is designed for the case of 17 = O, that is, the test distribution is explicitly specified. However, EPIG aims to decrease the entropy Ep(ze) ( (o)g()=))=))] not Ep(a) (o? (x)). There- fore, EPIG not suitable for decreasing Fr and can be inefficient in our experiments even for the case of ry =
reduction, that greedily minimizes (x. [o
(a* [x](x
variance reduction (CDR variance reduction). We evaluate the performance by the error Er defined in Eq. (2). Fur- thermore, for the synthetic dataset, we show the result of maxpep Ep(a)) [of( DD] in Appendix C.I.
In these experiments, by some 7) 2 we define the ambi- guity set follows:
= lp E Px Prof - pllco < <n}
where Px and Pref are sets of all distributions over X' and some reference distribution, respectively, and || Il denotes
8

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Distributionally Robust Active Learning for Gaussian Process Regression
6.1. Synthetic Data Experiemnts
that reduce the target variance and incorporate information about target distribution candidates for practical effective- ness. Then, we proved the theoretical eπror convergence of the proposed methods, whose practical effectiveness is demonstrated via synthetic and real-world datasets.
We set = (-1, 133, where |X| 11 1331. The target function f is the sample path from GPs, where we use SE and Matérn-y kernels with v 5/2. We use the fixed hyperparameters of the kernel function in the GPR model, which is used to generate ʃ. and fix o² = 10-4 The first imput #1 is selected uniformly at random, and T' is set to 400. Furthermore, we set Pref = (.2...))
Limitation and Future Work: We can consider several future research directions. First, since we do not show the optimality of the convergence rate, developing a (near) opti- mal algorithm for Er is vital. For this goal, the approximate submodularity (Bian et al., 2017) may be relevant from the empirical superiority of DR variance reduction. Second, since the expectation over p(x*) may be intractable, an analysis incorporating the approximation error or develop- ing an efficient algorithm without expectation computation may be crucial (DR random does not require the expectation but is often inefficient). Third, although our analyses only require the existence of the maximum over P, our exper- iments are limited to the discrete distribution set defined by the Lo ball. Thus, more general experiments regarding, e.g., the continuous probability distributions and the ambi- guity sets defined by Kullback-Leibler divergence (Hu & Hong, 2013) and Wasserstein distance (Frogner et al., 2021), are interesting from the practical perspective.
Figure I shows the result. We can see that DR and CDR variance reductions show superior performance consistently for all the kernel functions and 1), although the DR ran- dom is often inferior to those due to the randomness. This result suggests that the DR and CDR variance reductions effectively incorporate the information of P. Furthermore, although the constraint by X, is required for the theoret- ical analysis in CDR variance reduction, we can confirm that it does not sacrifice the practical effectiveness. On the other hand, the usual AL methods, such as US and variance reduction, deteriorate when 17 is small since they do not incorporate the information of P. When is large, since our problem approaches the worst-case error minimization, the US and variance reduction result in relatively good results. On the other hand, the EPIG designed for the case 17 = is inferior for all 1 since the EPIG is based on the entropy O (log(a, (z))), not the squared error.
Impact Statement
6.2. World Dataset Experiments
This paper focuses on the theoretical and algorithmic as- pects of the machine learning method. We consider that some potential societal consequences of this paper need not necessarily be highlighted here.
We use the King County house sales?, the red wine qual- ity (Cortez& & Reis, 2009), and the auto MPG datasets (Quin- lan, 1993) (See Appendix C.3 for details). For all experi- ments, we used SE kernels, where the hyperparameters and 0² are adaptively determined by the marginal likelihood maximization (Rasmussen & Williams, 2005) per 10 itera- tions. The first input is selected uniformly at random. Fur- thermore, we normalize the inputs and outputs of all datasets before the experiments and set Pref (O, 0.3I
Acknowkedgements
This work was partially supported by JST ACT-X Grant Number (JPMJAX23CD and JPMJAX24C3), JST PRESTO Grant Number JPMJPR24J6, JST CREST Grant Num- bers (JPMJCR21D3 including AIP challenge program and JPMJCR22N2), JST Moonshot R&D Grant Num- ber JPMIMS2033-05, JSPS KAKENHI Grant Number (JP20H00601, #923K169433 JP23K 19967, JP24K1 and JP24K20847), NEDO (JPNP20006), and RIKEN Cen- ter for Advanced Intelligence Project.
Figure 2 shows the result. We can confirm the same ten- dency that DR and CDR variance reductions show superior performance consistently, as the synthetic data experiments shown in Figure 1. Note that the fluctuations come from the hyperparameter estimation.
7. Conclusion
References
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A. Proofs for Section 3
A.1. Proof of Lemma 3.2
From the definition pe, we obtain
mc( S 0/ ()y
S (.)x + v, (-)e:
f(z)) where (x)) = (x)( + f. (f.
f(a) _ and et = (4) Therefore, the Lipschitz constant of μr is bounded from above by the Lipsethiz constants of o. (-)5, and vi (-)Ez.
For the first term of (-) we follow the proof of Lemma of (Vakili al., 2021a). Recall the RKHS-based definition of kernel ridge estimator:
H4, arg min
E (Vm. u(xx))² + llllll
Me. 2=1
Therefore, we can derive
min HEHA
Σ (f(x;) ((x;))² + + 1M1l 2f t ()f.ll.
5 Σ ((oc)))(xx)))) + ffllsl 11
(:fE
Ifflle
Hence, we obtain -all ()ff. & If I B. By combining Lemma 2.9, 27 (.),, f BLg Lipschitz continuous.
For the second term of ()),, we leverage the confidence bounds of kernel ridge estimator (Theorem 3.11 in Yadkor, 2013). Let g: X' 1] IR g(x, 0) g 1) and fix j [d]. Then, the zero function g belongs to the RKHS with any kernel function T.. Thus, we design the following kernel function Te:
A ((x,0), (z, 0)) k(ar, 2);
T ((xx1 (2, (z,1)) 8 k2 x) Ür,02z ak( 2)
k ((x,0) (z, 1))
Øzj
for all I, z E X. Note that since the kernel function, k has partial derivatives due to Assumption 2.8, the derivative of the kernel and the kernel itself are the kernels again as discussed in, e.g., Sec. 9.4 in (Rasmussen & Williams, 2005) and Sec. 2.2 in (Adler, 1981). Thus, we can interpret U. (+)e; as the kernel ridge estimator for g(x, 1), where F,(x) =
Ok? (w) (K, 02 o2) In addition, ||g||zg 0. Therefore, from Theorem 3.11 in (Abbasi-Yadkori, 2013) and 02,
(g(a 1) 0. (x)) = 1 (a) e.). we obtain
R / + 2log (1/6), Var x) Pr [o] (x) S 00(())
5,
where 8 10 to this kernel ridge estimation. Note that since the kernel matrix K, is defined by k(æ,, z). the MIG is the usual one defined by X and In addition, due to the monotonic decreasing property of the posterior variance, (x))) S Lks we obtain
and and (x) To ((x, 1), (x, akT (w) dz,
K, + 02 02 -10k(()))))
(m)
dz, is the posterior variance that corresponds
LR V2 + 2og (1/6), Vx E x) 2 21-
Pr T.
(x)ee
a
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Distributionally Robust Active Learning for Gaussian Process Regression
and thus,
LaR 1% + 2 log (1/0) 2 21- 6.
Pr
sup LEx
(x)e I
Consequently, by using the union bound for all j E [d], we derive
L R
Pr sup sup 1. (x) S V2 + 2 log (d/ō)
2 1 8,
jc(d] mEX
which shows that v (2))e is LRR // + 2 2og (d/ö) Lipschitz continuous.
Combining the Lipschitz constants of v. (-),, and v/ (-),,, obtain the result.
A.2. Proof of Lemma 3.3
First, we fix (ar)) without loss of generality since
(p)
L ((x))ee)))
Pr
sup
E
Pr
sup LEE
Oug
dug
.EN
That is, the upper bound the conditional probability given any (at))et directly suggests the upper bound of the target probability on the left-hand side. Note that from the assumption (mx))ett) is independent of (e:)se[) and f, the observations y, follows Gaussian distribution even if (Ir))e(t) is fixed.
We leverage Slepian's inequality shown Proposition A.2.6 in (van der Vaart & Wellner, 1996):
Lemma A.1 (Slepian, Fernique, Marcus, and Shepp). Let X and Y be separable, mean-zero Gaussian processes indexed by a common index set T such that
E E x,) S EI - y2)
for all s,t E T. Then,
Pr sup Xx 21 IET
Pr
sup Y; > 2 IET
for all A >
The separability (Definition 5.22 in Handel, 2016) holds commonly. As discussed in Remark 5.23 in (Handel, 2016), for example, the sample path is almost surely continuous, then the separability holds. Furthermore, the sample path defined by the commonly used kernel functions, such as linear, SE, and Matérn-/ kernels with v 2 1, continuous almost surely (Costa al., 2024). In addition, if the kernel function is continuous, the posterior mean function is also continuous, almost surely.
First, provide the proof of the result regarding (z)).. Since y, (ax))e(t) (O K; + o²I,), can that Hs (x:))et CP(O. ky, (xx=)). =)). where kn, (x, x) k, T (K + o2,, ke(z). Furthermore, it is known that if the kernel has mixed partial derivative d'k(a,s) f and its derivative 0f(x)/0zz jointly follow GPs (Rasmussen & Williams, Br,ü2z 2005; Adler, 1981). Specifically, the derivative distributed
0f(z) gP dz,
(. (,,((2,))= = pk(u,0)
ou,dv, 1==0,===
for j E [d]. Note that since the prior mean of / zero, the prior mean of p(t) is also zero. As with f, the derivative of is: distributed as
(1) 0))(2z)
8 (u, v) Ou,
P
(2:))eft
GP ky (2,2))=
Ox,
lu=av==
14

---

Distributionally Robust Active Learning for Gaussian Process Regression
for all j [d] In addition, the covariance is given as
Cov (())) Ok( u)
Ouj luzz
Then, that the posterior variance of the derivative can be obtained in the same way the usual GP calculation, as follows:
Var (F()(z)) | Di) k(xxx)) - (x,
Cov (z)))) Di) k(xx)) 2) kye (2,, z).
On the other hand, we can obtain that
E -() = k(xx)) + 1(2z22) 2) 22(22 x),
(14 - m1 (x) (=))² (ar))seto) = k,, x)) + kn, (2,z) - 2k (x,, 2),
for all x, = E X. Then, we obtain
(z) - (2)) - E [11 (x)-)))))))) (2x))ete)
= x)x + =(2,,) 2k(x 2) - (1n,(x,,) - (2, 2) - 2k (0e, z))
= Var (z)) (z) D.) + Var (z)) Di) - 2Cov (z), (1) D.) 2 O.
Consequently, by applying Lemma A.1, we obtain
Pr sup n4 (ax) 2 A x
(ot))Et1) S Pr
(10) 2 (10(
sup
med
(5) (z) and f( (x: follow centered GPs, we obtain
Since 14
Pr
sup m
(
(wi)) < Pr
sup u (x) 2 (2i)eett)
s 22Pr
sup mE
f(0) (zz) 2 2)
Hence, from Lemma 2.5, we obtain the desired result. We can obtain the result regarding (x)) (x))
(a:) in almost the same proof. We can see that
(0) n ( (x)) ~ CP k(æ, 2) - (x,, z)).
Then,
pr (x) - (F()(z)) -1)))))))(1000 (1) (m))) (mi)eel)) (1a0)~0010 - -R (p)mm))
-
= Kin (2, x)- + ky, (z, x) 2k (x, x) 2 0.
The remaining proof is the same as the case of p1l
A.3. Proof of Lemma 3.4
As with the existing studies (e.g., Srinivas et al., 2010), we consider the discretization of input space. Let X' C X' be finite set with each dimension equally divided into [rdr], where > O. Therefore, [X = [rrr]d and sup=e || - [<]]] < 1. where [x] is the nearest input in X, that is, [x]] = arg minge | x: l|12- Note that we leverage X purely for the analysis, and X not related to the algorithm.
15

---

Distributionally Robust Active Learning for Gaussian Process Regression
From Assumption 2.8 and Lemma 2.9, we see that is L Lipschitz continuous. Furthermore, from Lemma 3.2, f is
2m - 2oo (3) Lipschitz continuous with probability at least - 5. Combining the above, we see that /2
Lk
B +
pe is Li
2B
1 2/27 27: + 2 log (4)
Lipschitz continuous with probability at least 1 - 8.
From the above arguments, by combining Lemma 2.10 and the union bound, the following events hold simultaneously with probability least 1 d:
1. f(x) - PT (x) is Lies (T's Lipschitz continuous, where Lies (T) = Li 2B + !n BV 2TT + log (2/)
2. The confidence bounds on X' hold; that is,
Va E 1f(x)) - (1r(x))] S (3f2r((2))
where 85,7 = B + R 2d (rdr + 1)+ + 2 log (:) 2
Then, we can obtain the upper bound as follows:
ET = max pp(=")
(f(x*) )- - ur(x*))²
PEP
max E p(=) pEP
(f((=*)) ur((=")) + Lies (T)]l()) [==]
(: The above event 1)
f((x0)) - (r((=(]) Lree (T)
< max E p(=*)) pEP
The definition of X)
31/22 (oTT((t) [Tx")) + Liee (T)
< max pp((")) pEP
(: The above event 2)
27 1/22 82220T(00' oT(x") 3//22 Lo + Lies (T))
< max Bp(ar) pEP
(: Leemma 2.2)
pp( [o7(2")]+ +2 12/2 Lo + Lrea(T))
2
< 23,r max E PEP
(a 22 +b)
If we set T = T, noting that Lies 0 and g1/2 (log(TT66) we obtain the following: 8,7
J (x") + O max (rr, log log(3/))}
ET < 23,T max E
pEP Bp(=))
7
Although by setting T = (T), we can make the second term small arbitrarily, B8,r = θ (T)))
and Ep(ar) [x*)))] 0(1/T)) (Lemma 4.2 in Takeno et al., 2024). Therefore, since the first term is Q (og(T]/) max Ior. (T/8) if yr is sublinear, do not set more large value for simplicity.
and
T2
0 T
A.4. Proof of Lemma 3.5
As with the existing studies (e.g., Srinivas et al., 2010), consider the discretization of input space. Let X X' be finite set with each dimension equally divided into [rdr], where >> O. Therefore, |X| [rrr] and suPæEx || [a) < +
where [e] is the nearest input in X, that is, [æx] arg min | x ||- Note that leverage X' purely for the analysis. and Y not related to the algorithm.
In addition, from Lemma 3.3, the following inequality holds with probability least 1 8:
Or.(u)
S bV/oo(2ad/8)
sup sup jdd =
Ou,
==
16

---

Distributionally Robust Active Learning for Gaussian Process Regression
which implies that Lrea. the Lipschitz constant of re(a)) = f(x) --r(rrr)...)))))))))))))>))))- can be bounded from above.
Then, by combining the above argument, Lemma 2.6, and the union bound, the following events hold simultaneously with probability at least 1 8:
1. f(x) (Ar (x) is Lres Lipschitz continuous, where Lres - by/log( 4ad/8).
2. The confidence bounds on X' hold; that is,
Va E 1f(xz)) pr(x))] S f or(x),
where 35,7 = 2dl log(Tdr + 1) + 220((
Hence, we can obtain the upper bound as follows:
ET = max Ep(x') PEP
(f(x*) - )
max ep(n*) pEP
(f((=*)) - uT ((x")) + Lres [=*]]11)²]]
(: The above event 1)
Lyos 2
max Bp(m*) pEP
[((=')) -ur((=)])
(: The definition of X)
91/2 02.for(( or([x=]] Liees
S. max Bp(x)) pEp
(: The above event 2)
8/.f2 L, Lres
2
81/2 or(xx) +
< max Bp(ar) pEp
6,r
(: Lemmm 2.2)
g1/2 Lo Lres
< 235,r max E pp(m)) pEp
(x)) +
(a (a{42)) +b)
Then, by setting T, we see that
[of(x*)] O (log(T/6)
Er < 23,,T max maxEp((r)
pEP
7
Although by setting 7 = (T()), we can make the second term small arbitrarily, we do not do so since the first term is dominant compared with O (8/(T/8) T
term.
A.5. Proof of Lemma 3.6
Since the maximum (x("") arg maxpep Ep(ar) [If(x=*) - (x(")))] exists, we obtain
Bp(ar) 11f(x*) "r((*))] Exr( [1f(x=) (=())))]] PEP
(pp((r) [(f(x*) Ho (x)))
max Ep(ar) [(f(x=)) - pr((x)))
PEP
where we used Jensen's inequality.
17

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Distributionally Robust Active Learning for Gaussian Process Regression
A.6. Proof of Lemma 3.7
Since the maximum (x=) = arg maxpep Ep(ar) [[] D:]l exists, we obtain
max 1 PEP
pp(a) [M[(() D:]] B pr (logla))llo22eeelll (x))
- log(2ncB Typ (x (x)())))
(: Jensen's inequality)
_ log
2nee max 1 Bp(ce) lp(ce)of (x)1 pEP
B. Proofs for Section 4
B.1. Proof of Theorem 4.1
From the definition, (x) is monotonically decreasing along with D:-1 C De. Therefore, for all < T and ,I,
max E (p
æTl
S max E
p(mm)) [1....,...] xt].
pEp
pEP
Note that æ Dr are random variables due to the randomness of the algorithm. Hence, we obtain
1
max Ep Ep(=)) (o} æ1..... xT| S
max EE p(a= pppa-))o0 102- 21. .1...... x1-1]
T
pEp
1
(p.(a.)lo2111 ( ,1,..,, 11-)
Definition of pc)
7
Then, apply the following lemma (Lemma 3 in Kirschner & Krause, 2018):
Lemma B.1. Let Y, be any non-negative stochastic process adapted to filtration (Fi), and define m. E|y, F. Further assume that Y, < be for a fixed, non-decreasing sequence (bt) Then, if by 2 1, with probability at least for any 2 1, holds that,
T Y, + 4br log + Sby log(4br) +
1
Σ my Σ
The random variable Ep,,()) [a? ()) F1...., xy) satisfies the condition this lemma by setting b = for all t E [T].
Therefore, with probability at least
2 Σ o , + 4log 1 + (lg((4) +
max E
1o x1 Ii....,, xrl <
pEp
T
1=1
< T
20 + +4log g
+ 8log(4)) +1
Here, we use Et (z.) Ci (Lemma 5.2 in Srinivas et al., 2010).
B.2, Proof Theorem 4.2
From the definition, (x) is monotonically decreasing along with D.-1 C Dr. Therefore, for all S T and 01,...,,37
max E
pEP p((= (p(pa)((ff (of @1 #i...., xrl S max Bp(a=))o?(xx") $1...., [.y].
pEP
18

---

Distributionally Robust Active Learning for Gaussian Process Regression
US
RS
variance reduction
EPIG +
DR variance reduction +
DR random
CDR variance reduction
0.0.0
0.001
n=0.11
0=0.1
1.0
Expected Loss
0.8-
expected Lose
0.8
Expected Loss
0.8
Expected Loss
0.8
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0.2-
0.0
100
200
300
100
300
400
100
200
300
100
300
400
Iterations
Iterations
Iterations
Iterations
0.0
n= 0.001
f
0.01
n=
1.0
1.0
1.0
Expected Loss
Expected Loss
0.8-
Expected Loss
0.8-
Expected Loss
0.8
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0.0
0.0
100
200
300
400
0
100
200
300
400
100
200
300
400
100
200
300
400
Iterations
Iterations
Iterations
Iterations
Figure 3. Result of maxper (()) [oi (a=" "] in the synthetic data experiments with r7 0, 0.001, 0.01, 0.1. The horizontal and vertical axes show the number iterations and maxpep (pp()))) [a?(x=)) )] respectively. error bar shows mean and standard errors for 20 random trials regarding the random initial point (and the algorithm's randomness). The top and bottom represent the results of the GPR model with SE and Matérn kernels, respectively.
Hence, we obtain
T
max E p(1=)) 107(x=) x1,,................. pEp
æT! S
max E
pp "002_1 x1, x1-1]
T
pEp
t=1
1
2
1 rt
(: Definition of X.)
T
t=1
C17T
( Lemma 5.2 (Srinivas et al., 2010))
T
C. Other Experimental Settings and Results
C.1. Results for Variance
Figure 3 shows the result of maxpep Ep(x") (o] (o(())) which suggests that the proposed methods effectively minimize maxpep Ep(=)) (x²)).
C.2. Details on Implementation EPIG
EPIG defined as follows (Bickford Smith et al., 2023):
Dt = arg max (p(p-)) [H yyk Dz-1, H Hyy [y
DD=
xc
Although Bickford Smith et al. (2023) have discussed the efficient computation for EPIG, in the regression problem, the EPIG can be computed analytically except for the expectation over p(x") as follows:
of a*)
K² arg max (p())) log =EX
of
(x)
of (x
where (x, x²) is the posterior covariance between x and x². Since we focus on the discrete input domain in the experiments, the expectation over p(æ*) can also be computed analytically. We used the above equation for the implementation.
19

---

Distributionally Robust Active Learning for Gaussian Process Regression
C.3. Details Real-World Datasets
The King County house sales dataset is dataset used to predict house prices in King County by 7-dimensional features, such the area and the number of rooms. This dataset has been used for testing the regression (Park et al., 2020), and similar dataset has also been used for the AL studies (Park & Kim, 2020). Although this dataset includes 20000 data points, we used random sample of 1000 data points for simplicity.
Red wine quality dataset (Cortez & Reis, 2009) is dataset used to predict the quality of wines from the wine ingredients expressed by Il-dimensional features. This dataset includes 1600 data points and has been used for the regression problem (Cortez al., 2009).
Auto MPG dataset (Quinlan, 1993) is dataset used predict automobile fuel efficiency from 6-dimensional features, such as the weight of the automobile and engine horsepower. Auto MPG dataset has been used for the AL research (Park & Kim, 2020). This dataset includes 399 data points.
20


Reviews:
[{"review_id": "JOsObcGdkE", "reviewer": "Reviewer_Ai57", "summary": "This paper examines active learning in the context of Gaussian Process Regression (GPR). In particular, it seeks to minimize the worst-case error with respect to the (unknown) test-time marginal distribution, by considering the worst-case distribution in an ambiguity set. First the paper upper bounds this worst-case error as a function of the worst-case predictive variance. This motivates two selection algorithms (DR and CDR) which aim to minimize this worst-case variance, hence minimizing an upper bound on worst-case error. Similar theory is derived to show that this strategy also upper bounds worst-case absolute error and worst-case entropy. The empirical results evaluate the achieved value of worst-case error in comparison to number of sampling iterations on synthetic and real-world data.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We are grateful for the reviewer's careful reading and insightful and detailed comments, based on which we will revise the paper as much as possible. We will concentrate on and answer several seemingly important questions below.\n\n>in line 384 there is a comment about the derivative of log(a), in the context of EPIG. This is intriguing but I didn't totally follow the logic here. Why do we care about this derivative when is small? \n\nEPIG aims to decrease the entropy $\\mathbb{E}\\_{p(x^*)} [ \\log (\\sigma^2_t (x^*))]$ instead of posterior variance $\\mathbb{E}\\_{p(x^*)} [ \\sigma^2_t (x^*)]$. \nThen, if we can decrease $\\sigma^2_t (x) = 1$ to $\\sigma^2_{t+1} (x) = 0.9$, the difference of the posterior variance is $0.1$ and the difference of the entropy is $|\\log 1 - \\log (0.9)| \\approx 0.1$.\nTherefore, the amounts of decrease in the posterior variance and the entropy are almost the same.\nOn the other hand, if we can decrease $\\sigma^2_t (x) = 0.1$ to $\\sigma^2_{t+1} (x) = 0.01$, the difference of the posterior variance is $0.09$ and the difference of the entropy is $|\\log (0.1) - \\log (0.01)| \\approx 2.3$.\nTherefore, the amount of decrease in the entropy is larger than that of the posterior variance in this case.\nThis is because the gradient of logarithmic function $1 / a$ is large around $a = 0$.\n\nHence, compared with our proposed algorithms, the EPIG focuses on additionally decreasing the posterior variance in the area where the posterior variance is already small.\nThis is the reason why EPIG deteriorates when the performance measure ($E_T$) is based on the posterior variance.\n\n>when the ambiguity set is small, the proposed methods do better, and when it is large, the baselines (in particular, US) does just as well. To me the first observation doesn't make sense, and the second observation makes it unclear what the utility of the proposed algorithms are, if they don't outperform baselines in the setting they are designed for. In particular, I am surprised how well US does in the large ambiguity set case.\n\nIn our experiments, the same ambiguity sets $\\mathcal{P}$ is used for the computation of $E_T$ and for the actual algorithm. \nTherefore, our proposed methods are expected to perform well for all $\\eta$.\n\nOn the other hand, if $\\eta$ is large, $\\mathcal{P}$ becomes a large set, and our proposed methods approach the US. \nFor example, $\\eta = 1$, the ambiguity set $\\mathcal{P}$ defined by $L_{\\rm inf}$ ball contains all the distributions over $\\mathcal{X}$.\nTherefore, $p_t(x) = {\\rm argmax}\\_{p \\in \\mathcal{P}} \\mathbb{E}\\_{p(x^*)} [\\sigma^2_t (x^*)] $ becomes the distribution that have the probability 1 at ${\\rm argmax}\\_{x \\in \\mathcal{X}} \\sigma^2_t(x)$ and 0 otherwise.\nConsequently, DR random and DR variance reduction results in the US.\n\nInversely, if $\\eta$ is small, the proposed methods can utilize the information of $\\mathcal{P}$ compared with the US.\nFurthermore, since EPIG does not show superior performance for $E_T$ (see the above answer), our proposed methods perform better than baseline methods.\n\nConsequently, we believe that the benefit of the proposed algorithms is the versatility for the size of $\\mathcal{P}$.\nThe proposed algorithms approach the classical AL (the US) and the test distribution-aware AL for large $\\mathcal{P}$ and small $\\mathcal{P}$, respectively.\nTherefore, the proposed algorithms perform well for arbitrary sizes of $\\mathcal{P}$.\n\n>However, I think more discussion is needed about Frogner et al. 2021 who also propose distributionally robust active learning.\n\nFrogner et al., 2021 focus on the DRAL for the ambiguity set defined by Wasserstein distance over continuous distribution.\nTherefore, their proposed algorithm does not match our experimental problem setup.\nIn addition, the used heuristic AL algorithm is an expected model change, whose AF is the norm of the gradient with respect to the model parameter of the loss function.\nHowever, since the GPR (kernel ridge regression) is a nonparametric model, even the definition of the AF is not straightforward.\nFor the above two reasons, we did not employ the method in Frogner et al., 2021 as the baseline.\n\n>Liu, A., Reyzin, L., & Ziebart, B. (2015, February). Shift-pessimistic active learning using robust bias-aware prediction. In Proceedings of the AAAI Conference on Artificial Intelligence (Vol. 29, No. 1).\n\nWe thank you for pointing out the vital related work, for which we will add the discussion.\n\n>the discussion right before equation (4) could be improved. Can you better motivate your choice of constraint set here?\n\nAt least the US can satisfy the theoretical convergence, the constraint that enforces the choice of the input with large variance may be beneficial.\nInspired by this thought, our constraint is derived from the proof.\nOn the other hand, more intuitive motivation may be obtained by removing or alleviating the constraints.\nThis is also an interesting future direction.", "claims_and_evidence": "The theoretical claims in this paper are well supported. The upper bound between worst-case error and worst-case variance is a nice connection, and is well-supported by theory. They also show a sample complexity analysis of their sampling approaches over the sampling horizon.\n\nTo me, the potentially weak aspect of this paper is the empirical results. There are several issues that I see:\n- first, it is hard to interpret these results visually. 20 random trials per method seems low, especially for these small datasets, and more trials would help reduce the variance of the results. It is difficult to tell if there are any significant differences between the proposed methods and baselines. In one or two cases perhaps the uncertainty sampling (US) baseline does worse, but in other cases the DR methods don't do as well as the baselines. At a high level, the proposed methods do qualitatively seem to perform better in most cases, but these results could be strengthened by additional analysis such as significance testing between area under the performance curve values, and showing the number of times each method \"wins\" over other methods according to a significance test\n\n- along these lines, there is something I am very confused by in the experiments: when $\\eta = 0$, this is equivalent to the simple transductive case where the test distribution is known exactly. I am confused why in this setting, the proposed DR methods seem to have the biggest gain over baselines. On the other hand, as $\\eta$ is increased (which in theory should be where DR active learning shines), there is hardly any improvement over US. So it's not clear to me what the empirical advantage of this proposed approach is", "methods_and_evaluation_criteria": "Yes, and in particular the worst-case error is measured as a performance benchmark. The authors should clarify on their y-axes that \"expected squared error\" means worst-case error E_T. I assume this does in fact mean squared error with respect to the worst-case p.m.f. over the actual (discrete) points for each dataset test-split. Is that correct? Can the authors clarify exactly what is being shown on the y-axis, for both the synthetic and real-world experiments?", "theoretical_claims": "I skimmed the proofs and they seem reasonable. The theoretical claims seem well-supported by tools in prior literature", "experimental_designs_or_analyses": "The experimental design seems reasonable, and I have included comments above", "relation_to_broader_scientific_literature": "The authors present a thorough related work section. However, I think more discussion is needed about Frogner et al. 2021 who also propose distributionally robust active learning. This previous work is briefly mentioned, and the authors say that no theory was provided there, but on the empirical side there is no experimental comparison against that method (that I can see). Can the authors elaborate more about this prior work? What was the setting there - GPR regression or something else? Was there an empirical selection algorithm that could also be tested here? If so, why wasn't it evaluated as a baseline? What theory is shown in this prior work? From what I can tell that seems to be the most relevant prior work in the literature so it's important to clearly distinguish what the contribution is here, in comparison.", "essential_references_not_discussed": "The paper\n\nLiu, A., Reyzin, L., & Ziebart, B. (2015, February). Shift-pessimistic active learning using robust bias-aware prediction. In Proceedings of the AAAI Conference on Artificial Intelligence (Vol. 29, No. 1).\n\nis highly relevant here and should be discussed, since it also essentially designs an active learning algorithm for the worst case. It is for classification, so couldn't be compared here, but should be mentioned.", "other_strengths_and_weaknesses": "I thought all aspects of this paper were excellent, until I got to Section 6 (Experiments). I think the authors need to address what to me seems like a discrepancy above: when the ambiguity set is small, the proposed methods do better, and when it is large, the baselines (in particular, US) does just as well. To me the first observation doesn't make sense, and the second observation makes it unclear what the utility of the proposed algorithms are, if they don't outperform baselines in the setting they are designed for. In particular, I am surprised how well US does in the large ambiguity set case.\n\nIf the authors could better justify the empirical aspects of this work, I would increase my rating. Otherwise, the theoretical contributions, related work, introduction, background etc were all very interesting and very well written.", "other_comments_or_suggestions": "- in section 4, what is the \"greedy algorithm\" (line 260)? This should be defined. I assume it is greedy expected error reduction\n- the discussion right before equation (4) could be improved. Can you better motivate your choice of constraint set here?\n- this paper would be strengthened with lower bounds on worst-case expected error. the authors do acknowledge this in the limitations by mentioning how it would be good to know how optimal this approach is\n- substitute \"rows\" for \"columns\" in the figure captions\n- in line 384 there is a comment about the derivative of log(a), in the context of EPIG. This is intriguing but I didn't totally follow the logic here. Why do we care about this derivative when $a$ is small? Can the authors elaborate on the point they were going for here?\n- figures would be easier to interpret on a log-y scale. in fact this might be a necessity - it is very hard to read the figures when all of the methods perform so similarly\n- the \"Impacts statements\" section is actually an opportunity to comment more about the worst-case aspect. Why exactly do we care about the worst-case? Perhaps there could be society impacts of this (performance on minority groups, fairness aspects, etc). Now that I think about it, I think the paper could better motivate overall why we care about worst-case performance (either in the introduction or conclusion).\n- the plots in the appendix would also benefit from a log y-scale\n- can the authors include a simple figuring showing which points are actually selected by their method, in comparison to say US, on a standard GPR toy problem? I'd be curious to see qualitatively which points are selected for worst-case error mitigation", "questions_for_authors": "Please see my comments above, especially about the experimental section, and please address what I perceive as contradictions in the experimental results regarding the size of the ambiguity set and how surprisingly well US performs when the ambiguity set is large, and how surprisingly well DR/CDR performs when the ambiguity set is small. I would have expected the *opposite* result, where US performs well when $\\eta = 0$ and DR/CDR performs better when $\\eta$ is larger. In fact, this observation is even more concerning in the appendix plots: US seems to do just as good of a job as DR/CDR in reducing the actual max variance. What benefit does DR/CDR really have then, since they are designed to minimize this worst-case variance, if US does just as good of a job?\n\n## update after rebuttal\nThank you for your response. Your clarifications have addressed my concerns. Please make any needed edits in the next iteration of the paper. I have increased my score from Weak Accept to an Accept", "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": "", "extra_scores": {}}, {"review_id": "lsFEhuHSGW", "reviewer": "Reviewer_iagm", "summary": "The authors investigate the problem of distributionally robust active learning for Gaussian process regression, addressing limitations of existing approaches that primarily rely on heuristic and information gain-based methods. They provide a rigorous theoretical analysis with guarantees on the posterior mean and implement their framework using greedy and random search acquisition functions. The proposed method is evaluated on both experimental and synthetic datasets, with the objective of minimizing worst-case prediction errors. Performance is benchmarked against several baseline models, including uncertainty sampling (US), random sampling (RS), variance reduction, expected predictive information gain (EPIG), and  distributionally robust approaches both constrained and un-constrained.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the careful reading and constructive comments.\n\n>How can the analysis we extended to other surrogates for active learning other than GPR?\n\nFor the extension to other surrogate models, such as Bayesian neural networks (BNN) and deep kernel learning (DKL), there is a problem that the posterior variance does not necessarily decrease.\nTherefore, the extension is not easy as with that the theoretical analysis of bandit with BNN and DKL models is still often difficult.\nWe believe that the analyses for those models are crucial tasks in the community.\n\nOn the other hand, our analysis is expected to be extendable for the GPR with neural tangent kernel by combining the analyses of neural bandits, for example, in [1, 2].\n\n[1] Parnian Kassraie, Andreas Krause, Neural Contextual Bandits without Regret, Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, PMLR 151:240-278, 2022.\n\n[2] Taehyun Hwang, Kyuwook Chai, Min-Hwan Oh, Combinatorial Neural Bandits, Proceedings of the 40th International Conference on Machine Learning, PMLR 202:14203-14236, 2023.", "claims_and_evidence": "Are the claims made in the submission supported by clear and convincing evidence? \n - Yes", "methods_and_evaluation_criteria": "Do proposed methods and/or evaluation criteria (e.g., benchmark datasets) make sense for the problem or application at hand?\n- Yes", "theoretical_claims": "Did you check the correctness of any proofs for theoretical claims?\n- Yes but not throughly.", "experimental_designs_or_analyses": "Did you check the soundness/validity of any experimental designs or analyses? \n- Yes", "relation_to_broader_scientific_literature": "I am not very familiar with related works in the Target distribution aware AL and AL for worst case error so cant comment on this.", "essential_references_not_discussed": "I am not very familiar with related works in the Target distribution aware AL and Al for worst case error so cant comment on this.", "other_strengths_and_weaknesses": "Strengths:\n- Evaluates the proposed method on a diverse set of datasets, including both synthetic and experimental data, while providing comprehensive baseline model comparisons.\n- Clearly presents related work, effectively distinguishing prior contributions from the novel aspects introduced in this study.\n\nWeakness:\n- The scope of discussion is restricted to only Gaussian process regression (GPR) surrogates in active learning. However, active learning can incorporate alternative surrogate models, such as deep kernel learning and Bayesian neural networks (BNNs).", "other_comments_or_suggestions": "Please see Question's.", "questions_for_authors": "How can the analysis we extended to other surrogates for active learning other than GPR?\n- Bayesian Neural Nets\n- Deep kernel's (Stochastic Variational Deep Kernel Learning - Wilson et al 2016)", "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": "", "extra_scores": {}}, {"review_id": "6d3r4a9adS", "reviewer": "Reviewer_CodQ", "summary": "The paper introduces a framework for Distributionally Robust Active Learning (DRAL) in Gaussian Process Regression (GPR), which focuses on minimizing the worst-case expected error over potential target distributions. The authors derive upper bounds on the worst-case expected squared error and showing that the worst-case expected error can be reduced to arbitrarily small values with a finite number of labeled data points. Experimental results on both synthetic and real-world datasets demonstrate the effectiveness of the proposed DRAL methods compared to existing AL strategies.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the careful reading and constructive comments.\n\n>While the paper claims that the methods are efficient, it lacks a detailed computational complexity analysis or runtime comparisons. \n>The paper does not discuss how well DRAL scales with increasing dataset size or feature dimensionality.\n>It is not clear whether the scalable GP learning approaches can be combined with the proposed methods\n\nExcept for the computation of $\\max\\_{p \\in \\mathcal{P}} \\mathbb{E}\\_{p(x^*)}[ \\sigma^2\\_t(x^*) ]$, the doemstic computational complexity is $O(T^3)$, which comes from the computation of kernel matrix inversion.\nFurthermore, although the computational complexity of our algorithms does not depend on the feature dimension directly, the computation of the maximization and the expectation in $\\max_{p \\in \\mathcal{P}} \\mathbb{E}_{p(x^*)}[\\sigma^2_t(x^*)]$ can become difficult in proportion to the feature dimension.\n\nAs the reviewer suggested, the computational complexity $O(T^3)$ may be avoided by the scalable GP learning approaches.\nIn such cases, more careful proofs incorporating the approximation in GP learning, like [1], are required for rigorous analysis.\n\nRegarding the computation of $\\max_{p \\in \\mathcal{P}}$, we assume that this computation is feasible in this paper.\nOn the other hand, several studies in the DR learning literature have considered the efficient computation of $\\max_{p \\in \\mathcal{P}}$ for more complex ambiguity sets $\\mathcal{P}$ defined by, for example, Wasserstein distance and KL divergence [2, 3].\nThe extension for those ambiguity sets is an important future direction, as described in Section~7.\n\nFinally, regarding the computation of the expectation $\\mathbb{E}_{p(x^*)}[\\sigma^2_t(x^*)]$, our proposed method DR random does not require this computation, although DR variance reduction requires this expectation computation.\nWe consider that developing an algorithm that does not require the expectation computation and achieves performance as good as DR variance reduction is interesting as future work, as described in Section~7.\n\n[1] Sattar Vakili, Jonathan Scarlett, Da-Shan Shiu, Alberto Bernacchia, Improved Convergence Rates for Sparse Approximation Methods in Kernel-Based Learning, Proceedings of the 39th International Conference on Machine Learning, PMLR 162:21960-21983, 2022.\n\n[2] Hu, Z. and Hong, L. J. Kullback-Leibler divergence constrained distributionally robust optimization. Available at Optimization Online, 1(2):9, 2013.\n\n[3] Frogner, C., Claici, S., Chien, E., and Solomon, J. Incorporating unlabeled data into distributionally robust learning. Journal of Machine Learning Research, 22(56):\n1–46, 2021.\n\n>The evaluation focuses only on GPR and does not compare against deep learning-based AL techniques, such as transformer-based AL methods.\n\nAlthough comparisons against deep learning-based AL methods are important for practical validation, we focused on the performance of the AL algorithm rather than the base regression models.\nWe believe that, for the extension to the deep learning models, our experimental validation implies that our AL algorithm can be a better basis than other AL algorithms for the GPR.\n\n>The authors do not provide experimental validation showing how quickly posterior variance converges in practical settings.\n\nWe show the convergence of $\\max\\_{p \\in \\mathcal{P}} \\mathbb{E}\\_{p(x^*)}[ \\sigma^2\\_t(x^*) ]$ for the synthetic problems in Figure 3 in Appendix C.\nIf the hyperparameter estimation for the kernel is stable, the convergence of $\\max\\_{p \\in \\mathcal{P}} \\mathbb{E}\\_{p(x^*)}[ \\sigma^2\\_t(x^*) ]$ shows similar results also in the other datasets.", "claims_and_evidence": "* The authors provide theoretical derivations, including upper bounds on the worst-case expected squared error and posterior variance convergence.\n* Probabilistic upper bounds (Lemma 3.4, Theorem 4.1) are established, showing that posterior variance can be reduced arbitrarily small with a finite number of labeled data points.\n* The authors conduct experiments on synthetic data and real-world datasets (King County house sales, red wine quality, auto MPG). The results show lower expected squared errors for the proposed DRAL methods compared to uncertainty sampling (US), random sampling (RS), and variance reduction.\n\n\n* While the paper claims that the methods are efficient, it lacks a detailed computational complexity analysis or runtime comparisons. It is not clear whether the scalable GP learning approaches can be combined with the proposed methods.\n* The evaluation focuses only on GPR and does not compare against deep learning-based AL techniques, such as transformer-based AL methods.", "methods_and_evaluation_criteria": "* The evaluation focuses on expected squared error as the primary performance metric, which is appropriate for regression tasks.\n* The authors validated DRAL on both synthetic and real-world datasets; however, these datasets used are relatively small-scale and contain structured numerical data.\n* The paper does not discuss how well DRAL scales with increasing dataset size or feature dimensionality.", "theoretical_claims": "* The authors establish that posterior variance decreases monotonically with increasing labeled data. The proof assumes that the GP model is well-specified which can be common in practice and this convergence guarantee may not hold. The authors do not provide experimental validation showing how quickly posterior variance converges in practical settings.", "experimental_designs_or_analyses": "Yes, the numerical results look solid.", "relation_to_broader_scientific_literature": "N/A", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "N/A", "other_comments_or_suggestions": "Typo: Line 010 right: gaion -> gain", "questions_for_authors": "N/A", "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": "", "extra_scores": {}}, {"review_id": "F4na6H4s3L", "reviewer": "Reviewer_Y8KU", "summary": "The paper explores a distributionally robust active learning framework designed to minimize worst-case posterior variance over an ambiguity set of data distributions. It builds on bounds for posterior variance under uncertainty sampling and links these results to regret minimization. The analysis focuses on probabilistic bounds and sampling strategies, with an emphasis on uncertainty-based selection.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the careful reading and constructive comments.\nHowever, we believe that several reviewer's comments come from misunderstandings, for which we would like to elaborate our claims.\n\nFirst of all, our proposed methods are stated in Section 4 and thus not US and RS.\nTherefore, we do not claim the optimality of the US.\nRather, we claimed that the US could deteriorate in our problem setup, though its theoretical convergence (which may not be tight) can be shown by Proposition 2.3.\nOur main Theorems 4.1 and 4.2 (obtained by Lemmas 3.4 and 3.5) show that if our proposed methods run, the error $E_T$ must converge to zero if $\\gamma_T$ is sublinear.\n\nSecond, our theoretical claims do not depend on the definition of ambiguity sets, though we assume the existence of $\\max_{p \\in \\mathcal{P}}$.\nTherefore, the theoretical claims hold for general ambiguity sets defined by, for example, Wasserstein distance and KL distance.\nOn the other hand, the practical performance and feasibility of the algorithm may be problematic.\nDealing with these practical problems is an important future work, as described in Section 7.\n\n\n>the posterior variance of f should be f_var-sigma^2(given f and epsilon are independent).\n\nAs shown in, for example, Eq. (2.24) of [1], the posterior variance of $f(x)$ given $\\mathcal{D}_t$ is $\\sigma^2_t(x)$. \nNote that the noise variance $\\sigma^2$ is affected to $\\sigma^2_t(x)$ through the computation of $(K + \\sigma^2 I)^{-1}$ in Eq. (1).\n\n [1] C. E. Rasmussen & C. K. I. Williams, Gaussian Processes for Machine Learning, the MIT Press, 2006.\n\n\n>it assumes without justification that minimizing an upper bound of E_T is equivalent to minimizing E_T itself.\n\nWe believe that when an objective function is computationally intractable, minimizing the tractable upper bound (or maximizing the lower bound) is a very common approach in the machine learning community.\nThis is because, for example, converging the upper bound to zero immediately implies that the objective function converges to zero.\nFor instance, the evidence lower bound maximization for the variational Bayes learning is generally accepted.\n\n\n>A critical issue in the paper is the complete absence of any discussion regarding the structure, form, or assumptions on the ambiguity set P.\n\nAs stated above, our theoretical analyses only assume the existence of $\\max_{p \\in \\mathcal{P}}$ (and practical feasibility).\nTherefore, the ambiguity sets can be set almost arbitrarily depending on the specific problems.\n\n\n> The inequality relies on Lemma 5.4, which only establishes that total regret is bounded by the information gain but does not directly bound the max posterior variance. Moreover, Lemma 5.4 assumes a GP-UCB sampling strategy, whereas Proposition 2.3 concerns uncertainty sampling, making it unclear whether the bound remains valid in this setting.\n\nIn the derivation of Lemma 5.4 in [2], the fact $\\sum_{t=1}^T \\sigma^2_t(x_t) \\leq 2 \\gamma_T / \\log(1 + \\sigma^{-2})$ is shown for any sampling strategy.\nNote that the GP-UCB sampling strategy is required to show $r_t \\leq 2 \\beta_t^{1/2} \\sigma_t(x_t)$ in [2].\n\n[2] Srinivas, N., Krause, A., Kakade, S., and Seeger, M. Gaussian process optimization in the bandit setting: No regret and experimental design. In Proceedings of the 27th International Conference on Machine Learning, pp. 1015–1022. Omnipress, 2010.\n\n>Even if the inequality in Proposition 2.3 holds, the proof does not establish whether the max posterior variance actually converges to a constant or vanishes as T increases\n\nAs described immediately after Proposition 2.3, if $\\gamma_t$ is sublinear, Proposition 2.3 implies that the maximum posterior variance converges to 0.\nSimilarly, Theorems 4.1 and 4.2 show that the error $E_T$ converges to 0 if $\\gamma_t$ is sublinear when our proposed algorithms are run.\nIn addition, as described after Definition 2.1, $\\gamma_t$ is sublinear for standard kernels such as SE and Matern kernels.\n\n>First, Lemma 3.5 provides an upper bound on E_T in terms of max of expectation under p of sigma_T. Then, they use the inequality max of expectation (under p) of sigma_T is less <= max of sigma_T, suggesting that minimizing the latter should also minimize the former.\n\nOur proposed methods aim to directly decrease $\\max_{p \\in \\mathcal{P}} \\mathbb{E}_{p(x^*)}[\\sigma^2_t (x^*)]$. Please see Section 4.", "claims_and_evidence": "1. Equation (1) is described as the posterior variance of f(x) but it actually corresponds to the posterior variance of the (noisy) observations, y(x). Since the noise variance is explicitly included in the model, the posterior variance of f should be f_var-sigma^2(given f and epsilon are independent). Additionally, the paper introduces an inconsistent notation for the noise term epsilon_i,  implying that noise depends on the data points x_i, but later it was treated as a global data invariant parameter.\n2. Lemma 3.5 is a core part of the paper's proposed method, yet it has a fundamental issue: it assumes without justification that minimizing an upper bound of E_T is equivalent to minimizing E_T itself. This assumption is incorrect unless the bound is proved to be somewhat tight, which is not discussed in the paper. Without that, we can not ensure that minimizing the bound effectively minimizes the original function. At the very least, the authors should quantify the bound’s gap, prove its asymptotic tightness, or demonstrate that the minimizers of both functions align.\n3.A critical issue in the paper is the complete absence of any discussion regarding the structure, form, or assumptions on the ambiguity set P. In standard distributionally robust optimization (DRO), defining P  is crucial, as it determines the tractability of the optimization, the validity of robustness guarantees, and the tightness of theoretical bounds. Typical DRO formulations either constrain P via distance metrics (e.g., Wasserstein distance or KL divergence), define it through known priors, or impose structural assumptions over some the distribution family. However, this paper does not provide any explicit definition or constraints on P, making its distributional robustness claim unclear. I am particularly confused about how one can discuss a DRO method without making any assumptions about P, as the entire foundation of DRO relies that. This omission is particularly problematic in Section 4, where the results are stated for all distributions in P, yet there is no indication of what P actually represents. Does the paper implicitly assume that the bound in Proposition 2.3 applies to all distributions in P?. In my understanding. Proposition 2.3 only bounds the posterior variance under uncertainty sampling, which does not necessarily extend to arbitrary distributions in P.", "methods_and_evaluation_criteria": "The paper evaluates distributional robustness by defining the ambiguity set P as the set of distributions within an L_{\\inf}neighborhood of a specific reference Gaussian distribution with the size of the neighborhood controlled by eta. However, this specification raises concerns about whether the evaluation fairly assesses distributional robustness. DRO evaluations typically test robustness across a broad range of potential distribution shifts(including realworld data distribution shift), rather than restricting P  to a small perturbation around a fixed Gaussian. Since the method is only evaluated under this particular form, it remains unclear whether the proposed approach truly minimizes worst-case errors across diverse distributions or if its performance gains are specific to the chosen experimental setup.", "theoretical_claims": "1. Proposition 2.3 lacks a complete proof, as it only cites previous results without detailing the crucial derivation steps. The inequality relies on Lemma 5.4, which only establishes that total regret is bounded by the information gain but does not directly bound the max posterior variance. Moreover, Lemma 5.4 assumes a GP-UCB sampling strategy, whereas Proposition 2.3 concerns uncertainty sampling, making it unclear whether the bound remains valid in this setting. Even if the inequality in Proposition 2.3 holds, the proof does not establish whether the max posterior variance actually converges to a constant or vanishes as T increases. A more rigorous derivation is needed to justify the claim.\n2.The argument that Proposition 2.3 theoretically guarantees the effectiveness of uncertainty sampling  and random sampling  is stated too vaguely to me. The authors do not explicitly outline how Proposition 2.3 supports their claim, and when I tried to infer their reasoning as follows, a potential issue emerges: First, Lemma 3.5 provides an upper bound on E_T in terms of max of expectation under p of sigma_T. Then, they use the inequality max of expectation (under p) of sigma_T is less <= max of sigma_T, suggesting that minimizing the latter should also minimize the former. Finally, they use Proposition 2.3, which bounds max of sigma_ under US, concluding that US or RS and RS are theoretically justified. Now, even if we momentarily accept this reasoning and assume that both bounds involved are tight—meaning minimizing max of sigma_T effectively minimizes E_T—a fundamental issue remains: the variance sigma_T  appearing in Proposition 2.3 is specific to a Gaussian process trained on data selected by US, whereas the quantity that should be minimized in Lemma 3.5 seems to be a general variance term that depends on the sampling strategy to be optimized(not necessarily US). The correct approach to minimizing max of sigma_T  should be to optimize the data selection process so that the Gaussian process variance is reduced globally. However, Proposition 2.3 only states that if US is used, then the corresponding Gaussian process variance decreases at a certain rate. This does not imply that US is the optimal or even an effective choice for minimizing max of sigma_T in a general sense.", "experimental_designs_or_analyses": "See Evaluation Criteria.", "relation_to_broader_scientific_literature": "I am not sure.", "essential_references_not_discussed": "NA", "other_strengths_and_weaknesses": "NA", "other_comments_or_suggestions": "NA", "questions_for_authors": "NA", "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": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper is in active learning in the context of GP Regression. The authors propose an approach that seeks to minimize the worst-case error with respect to the (unknown) test-time marginal distribution, by considering the worst-case distribution in an ambiguity set. This submission was reviewed by 4 reviewers in the field, and the paper ended up with an average overall recommendation of 3.25 (min: 2, max: 4). The average score increased from 3.0 -> 3.25 during the discussion.

The reviewers appreciated many aspects of the work (especially, presentation of the methods, the derivations). On the other hand, they also have concerns regarding the presentation of the experiments and proper presentation of the computational efficiency. The paper, however, has clear merits and could be a good addition to the program of the conference.

Please note: The reviewers also point out a number of minor issues that should be fixed in the camera-ready version.
Decision: Accept (poster)