Density Ratio Estimation-based Bayesian Optimization with Semi-Supervised Learning
Authors: Jungtaek Kim
Venue: ICML 2025 poster (2025)
Keywords: Bayesian Optimization
Density Ratio Estimation-based Bayesian Optimization
Bayesian Optimization with Semi-Supervised Learning
Primary area: optimization->everything_else

Abstract: Bayesian optimization has attracted huge attention from diverse research areas in science and engineering, since it is capable of efficiently finding a global optimum of an expensive-to-evaluate black-box function. In general, a probabilistic regression model is widely used as a surrogate function to model an explicit distribution over function evaluations given an input to estimate and a training dataset. Beyond the probabilistic regression-based methods, density ratio estimation-based Bayesian optimization has been suggested in order to estimate a density ratio of the groups relatively close and relatively far to a global optimum. Developing this line of research further, supervised classifiers are employed to estimate a class probability for the two groups instead of a density ratio. However, the supervised classifiers used in this strategy are prone to be overconfident for known knowledge on global solution candidates. Supposing that we have access to unlabeled points, e.g., predefined fixed-size pools, we propose density ratio estimation-based Bayesian optimization with semi-supervised learning to solve this challenge. Finally, we show the empirical results of our methods and several baseline methods in two distinct scenarios with unlabeled point sampling and a fixed-size pool, and analyze the validity of our methods in diverse experiments.

Density Ratio Estimation-based Bayesian Optimization with Semi-Supervised Learning
Jungtaek Kim
Abstract
structure optimization (Kim et al., 2024), and language model fine-tuning (Jang et al., 2024), since it is capable of efficiently finding a global optimum of an expensive-to- evaluate black-box function. As studied in previous litera- ture on Bayesian optimization (Snoek et al., 2012; Martinez- Cantin al., 2018; Springenberg et al., 2016; Hutter et al., 2011), probabilistic regression model, which can estimate distribution of function evaluations over inputs, is widely used a surrogate function; A Gaussian process (Ras- mussen & Williams, 2006) is a predominant choice for the surrogate function. An analogy between probabilistic regres- sion models in Bayesian optimization is that they rely on an explicit function over function evaluations p(y x, D) given an input estimate, denoted as x, and a training dataset D.
Bayesian optimization has attracted huge atten- tion from diverse research areas in science and engineering, since it is capable of efficiently find- ing global optimum of an expensive-to to-evaluae black-box function. In general, probabilistic re- gression model is widely used as surrogate func- tion to model an explicit distribution over function evaluations given an input to estimate and a train- ing dataset. Beyond the probabilistic regression- based methods, density ratio estimation-bai based Bayesian optimization has been suggested in or- der to estimate a density ratio of the groups rel- atively close and relatively far to a global opti- mum. Developing this line of research further, supervised classifiers are employed to estimate a class probability for the two groups instead of a density ratio. However, the supervised classifiers used in this strategy are prone to be overconfident for known knowledge dates. Supposing that we have access to unlabeled points, e.g., predefined fixed-size pools, we pro- pose density ratio estimation-base Bayesian op- timization with semi-super learning to solve this challenge. Finally, we show the empirical results of our methods and several baseline meth- ods in two distinct scenarios with unlabeled point sampling and a fixed-size pool, and analyze the validity of our methods in diverse experiments.
Beyond the probabilistic regression-based Bayesian opti- mization, density ratio estimation (DRE))-base Bayesian optimization has been studied recently (Bergstra et al., 2011; Tiao et al., 2021). Furthermore, likelihood-free Bayesian optimization, which is equivalent to DRE-bbased Bayesian optimization with a particular utility function, has been pro- posed by Song et al. (2022). Bergstra al. (201 attempt to model two densities p(x y S y², D) and p(x y y¹, D), where y' is a threshold for dividing inputs to two groups that are relatively close and relatively far to global solution, in order to estimate Ç-relative density ratio (Yamada et al., 2011). On the other hand, instead of modeling two densities separately, Tiao et al. (2021); Song et al. (2022) estimate a density ratio using class-probability estimation (Qin, 1998). As discussed in the previous work, this line of research pro- vides a new understanding of Bayesian optimization, which allows us to solve Bayesian optimization using binary classi- fication. Moreover, it can reduce the amount of computation required for building surrogate functions.
global solution candi-
1. Introduction
Bayesian optimization (Brochu et al., 2010; Garnett, 2023) has attracted immense attention from various research areas such as hyperparameter optimization (Bergstra et al., 2011), battery lifetime optimization (Attia et al., 2020), chemical reaction optimization (Shields et al., 2021), ) nanophotonic
However, the supervised classifiers utilized in the DRE- based Bayesian optimization are prone be overconfi- dent for known knowledge on global solution candidates or region that have been already exploited. In this pa- per, supposing that we have access to unlabeled points, we propose a novel DRE-based method with semi-supervised learning, which is named DRE-BO-SSL, to solve this over- exploitation problem. Although our direct competitors, i.c. BORE (Tiao et al., 2021) and LFBO (Song et al., 2022).
Uniiersiity of Wisconsin Madison, Madison, WI 53706, USA. Correspondence to: Jungtack Kim <jungtack kim @ eis..
Proceedings of the nnnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Density Ratio Estimation-basse Bayesian Optimization with Semi-Supervised Learning
(a) MLP, BORE, Iterations to
Clas Fuadle
83
(b) MLP, LFBO, Iterations to
(c) DRE-BO-SSI, Label propagation, Iterations 5
(d) DRE-BO BO-SSL, Label spreading, Iterations to
Figure 1. Comparisons of BORE and LFBO by multi-layer perceptrons, and DRE-BO-SSE with label propagation and label spreading for Branin function. row shows Iterations to with five initial points. (blue), x (red), and (green) indicate data points of I y", data points of > v", and query points, respectively. Moreover, (olive) stands for unlabeled points and its transparency represents class probability predicted by semi-supervised classifier. A query point chosen by maximizing output the classifier. More results shown in Figure 8.
show their strengths through theoretical and empirical anal- yses, algorithm betters ability to consider wider region that satisfies p(x S y², D) 2 p(x y >> yt, D), than the competitors, shown in Figure 1. By this in- tuitive example Figure 1, we presume that DRE--OO- SSL appropriately balances exploration and exploitation rather than the existing methods. Compared a supervised classifier, e.g., random forests (Breiman, 2001), gradient boosting (Friedman, 2001), and multi-layer perceptrons, our semi-supervised classifiers, i.e., label propagation (Zhu & Ghahramani, 2002) and label spreading (Zhou et al., 2003), are less confident in terms of the regions of global solution candidates using unlabeled data points; see Figures and 8 and Sections and for detailed examples and analyses.
account distinct scenarios with unlabeled point sam- pling and with predefined fixed-size pool. For the first scenario, we randomly sample unlabeled data points from the truncated multivariate normal distribution using mini- max tilting method (Botev, 2017), to allow for the possibility of adopting a cluster assumption (Seeger, 2000). Finally, we demonstrate that our method shows superior performance compared the exiting methods in diverse experiments in- cluding synthetic benchmarks, Tabular Benchmarks (Klein & Hutter, 2019), NATS- (Dong et al., 2021), and 64D minimum multi-digit MNIST search. Note that Tabular Benchmarks, NATS-Bendn and 64D minimum multi-digit MNIST search have access to fixed-size pools. To validate our methods, we provide thorough analyses on the compo- nents of DRE-BO-SSI Section 6 and the appendices.
To make use of semi-supervised classifiers, take into
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Density Ratio Estimation-based Bayesian Optimization with Semi-Supervised Learning
To sum up, our contributions are itemized as follows:
cannot change dramatically; more detailed analysis on Ç is provided in Sections 6 and I.
we identify the over-exploitation problem of supervised classifiers in DRE-based Bayesian optimization;
2. Background and Related Work
we propose DRE-based Bayesian optimization with semi-supervised learning, named DRE-BO-SSL for two distinct scenarios with unlabeled point sampling and a predefined fixed-size pool;
Bayesian Optimization. It is a prineipled and efficient ap- proach to finding a global solution of a challenging objective, e.g., expensive-to-evalee et al., 2010; Garnett, 2023). To focus regression model a surrogate function, we omit the details of Bayesian optimization here; see the references by Brochu et al. (2010); Garnett (2023) for details. In Bayesian optimization, Gaussian process regression (Rasmussen & Williams, 2006) is widely used as a surrogate function (Srini- vas et al., 2010; Snoek et al., 2012) because of its flexibil- ity with minimum assumptions on model and smoothness. While Gaussian process is a probable choice, Bayesian optimization with diverse surrogates such as Student-/. pro- cess regression -Martinez-Cantin et al., 2018), Bayesian neural networks (Springenberg et al., 2016), and tree-based models (Hutter et al., 2011; Kim & Choi, 2022) has been proposed. An analogy between such models is that they model p(y x, D) explicitly, so that it can be used to define acquisition function with the statistics of p(y x, D). Note that we solve the problem of minimizing the objective functions in this work.
black-box functions (Brochu probabilistic
we demonstrate the effectiveness of our method in various experiments including NATS- and 64D minimum multi-digit MNIST search.
1.1. Over-Exploitation Problem
As illustrated in Figures and 8, the supervised classifiers used DRE-based Bayesian optimization suffer from the over-exploitation problem. Interestingly, many deep learn- ing models also share similar problem, which is called an overconfidence problem (Guo et al., 2017; Müller et al., 2019), due to various reasons, but primarily due to overpa- rameterization. It is noteworthy that the definition of the over-exploitation problem in DRE-based Bayesian optimiza- tion is different from the overconfidence problem in general classification. The definition in DRE-based Bayesian op- timization does not imply that single data point has high probability for particular class, but it implies that few data points have high probabilities for class of inter- est. More precisely, our definition indicates the problem of overconfidence over known knowledge on global solution candidates. For example, the definition in general classifi- cation includes a case that 50% of data points are biased to one class and the remainder is biased to another class. On the contrary, this definition does not include such a case and it only includes case that a small number of data points or the small region of a search space, which is colored in yellow in Figures and 8, are biased to some particular class, i.c., Class in bRE-dased Bayesian optimization.
Dantoo Estimation. Whereas knowing data dis- tribution p(x) is important, it is difficult directly estimate p(x) (Sugiyama et al., 2012). For specific machine learning problems such as importance sampling (Kloek & van Dijk, 1978) and mutual information estimation (Bishop, 2006), we can bypass direct density estimation and estimate a density ratio. More recently, Rhodes et al. (2020) tackle a density- chasm problem using telescopic density-ratio estimation, which replaces an original problem with set of logistic regression problems. Since the work by Rhodes et al. (2020) suffers from the issue on distribution shift, the recent work by Srivastava et al. (2023) proposes a density ratio estima- tion method with multinomial logistic regression, which is capable of mitigating the distribution shift using multi- class classification. In Bayesian optimization, Bergstra et al. (2011) have proposed a strategy with tree-structured Parzen estimators to estimate density ratio alternative to probabilistic regression-based acquisition functions. In ad- dition, the existing work by Tiao et al. (2021); Song et al. (2022) suggests methods with binary classifiers in order to estimate class probabilities as a density ratio; see Section for the details of this literature.
By the aforementioned definition, at early iterations of Bayesian optimization, supervised classifier tends to over- fit to a small size of D, due to relatively large model capacity. This consequence makes a Bayesian optimization algorithm highly focus on exploitation. Similar to our ob- servation, the imbalance of exploration and exploitation in the DRE-based approaches is also discussed in the recent work by Oliveira et al. (2022); Pan et al. (2024). Moreover, the consequence mentioned above is different from the char- acteristies of probabilistic regression- baased Bayesian opti- mization because the regression- based methods are capable of exploring unseen regions by dealing with uncertainties. Besides, even though a threshold ratio Ç might be able to mitigate this problem, an overconfident supervised classi- fier is likely to keep getting stuck in local optimum as yt
Semi- Semi-SSpervised Learning. It is learning scheme with both labeled and unlabeled data (Zhu, 2005; Chapelle et al., 2006; Bengio et al., 2006). To cooperate with labeled and unlabeled data, this strategy generally utilizes geometry
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Density Ratio Estimation-basee Bayesian Optimization with Semi-Supervised Learning
of data points connectivity between points, and assigns pseudo-labels to unlabeled data points, which referred as transductive learning (Gammerman et al., 1998). As a semi- supervised learning method on a similarity graph. Zhu & Ghahramani (2002) propose a label propagation algorithm which iteratively propagates the labels of unlabeled data points using labeled data. Zhou et al. (2003) compute the labels of labeled and unlabeled data points by weighted iterative algorithm with initial labels. Belkin & Niyogi (2002) predict pseudo-labels by finding a linear combination of few smallest eigenvectors of the graph Laplacian.
Therefore, a class probability over for Class is consid- ered an acquisition function; it is simply derived by Tiao et al. (2021) as the following:
A(x 5. Dz) -
(4)
Eventually, the class probability is estimated by various off- the-shelf classifiers such as random forests and multi-layer perceptrons.
Song et al. (2022) have suggested general framework of likelihood free Bayesian optimization, called LFBO:
A( 1 (,)
3. DRE-based Bayesian Optimization
arg max ED, (((())))))(((())))000 (5) 5:X
Unlike probabilistic regression- basedd Bayesian optimiza- tion, DRE-based Bayesian optimization employs a den- sity ratio-based acquisition function, defined with a den- sity p(x | y S y², Dt), where x is d-dimensional vec- tor, y is its function evaluation, y¹ is threshold, and D. = [(x,,:)){ Ho) is a dataset of + 1 pairs of data point and evaluation. In particular, the work by Bergstra et al. (201 1) defines an acquisition function based on (-relative density ratio (Yamada et al., 2011):
which is versatile for any non-negative utility function w'y; where ʃ is a strictly convex function, f' is the derivative of f. and f is the convex conjugate of f. By the properties of LFBO, it is equivalent to an expected utility-based acquisition function. Along with the general framework, Song et al. (2022) prove that BORE is equiv- alent to the probability of improvement (Kushner, 1964) and LFBO with u(y:: y²) = Ily S v²) is also equivalent to the probability of improvement, where is an indica- tor function. Moreover, they show that their method with w(y: y²) = max(y² - y. 0), which can be implemented as weighted binary classification, equivalent the expected improvement (Jones et al., 1998).
A(x (c)
p y S De)
<p(x y S y1 D2) (1 ()p(x y yt, D.) (1)
where < ply - v*) E 10.1) is a threshold ratio. We need to find a maximizer of (1), by optimizing the fol- lowing composite function: h(p(x y S p.)p/((
4. DRE-based Bayesian Optimization with Semi-Supervised Learning
y > y De)). where h(x)= = (< + (1 - ー Since h is strictly increasing function, we can directly maxi- mize p(x < y,, D:)p/(((x y > y², Dr). In the pre- vious work by Bergstra et al. (2011), two tree-structured Parzen estimators are used to estimate the respective densi- ties, p(x y S y², Dr) and p(x y > y², DD).
We introduce DRE-based Bayesian optimization with semi- supervised learning, named DRE-BO-SSE by following the previous studies in DRE-based and likelihood-free Bayesian optimization (Bergstra et al., 201 1; Tiao et al., 2021; Song et al., 2022), which were discussed in Section 3.
While the work by Bergstra et al. (2011) utilizes two distinct tree-structured Parzen estimators, Tiao et al. (2021) propose method to directly estimate (1) using class- dlss-prrobaility estimation (Qin, 1998; Sugiyama et al., 2012), which is called BORE. Since it can be formulated as a problem of binary classification in which Class 1 is group of the top < of Dr and Class 0 is a group of the bottom 1- Ç of D, in terms of function evaluations, the acquisition function defined in (1) induces the following:
Similar to the standard Bayesian optimization and exist- ing baE-based Bayesian optimization, DRE-BO-SS! iter- ates the undermentioned steps as presented in Algorithm 1. Firstly, a threshold y! is calculated by considering V1., with Ç. Secondly, labels C, of data points in Dr are assigned to one of two classes; group of S y is assigned to Class I and a group of > y, is assigned to Class O. If we are given unlabeled data points X.,. the corresponding points X,, are used, but if not available it samples n. unlabeled data points X, from by utilizing a strategy described in Section 4.2, Then, it estimates pseudo-labels C, of semi-supervised learning model by following the procedure presented in Sec- tion 4.1 and Algorithm 2. Using C, it chooses the next query point X,+1:
p(x 2 =)
A(x| | ç, D.) = [p(x 2 = 1) + (1 - <)p(x 2 = 0)
(2)
By the Bayes' theorem, (2) becomes the following:
A(x (.D.) p(z = x
33)
X(+1 = arg max TC (x: (, Xw),
(6)
+ +
XEX
4

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Density Ratio Estimation-basel Bayesian Optimization with Semi-Supervised Learning
Algorithm 1 DRE-BO-SSL
are defined as X [ X E RRottta)dd Note that in our problem c 2, because address the problem with two classes.
Input:
Iteration budget T, search space X, a black-box objective f. a threshold ratio Ç, semi-supervised clas- sifier mc. and unlaheled data points X, if available and the number of unlabeled data points Tuu, otherwise.
As shown in Algorithm 2, we first initialize labels to propa- gate C E R (mm=-nu)x2, it is initialized as the following:
Output: Best candidate x
1:
Initialize Do = {xo: }o]} by randomly selecting X from X' and evaluating X, by f.
C [Co, C1,... Ont 1, Crys Cmy+ Cnnthl...-.Cntnmn.- (7)
2: for = 0 to T do
where Co, C1, C are one-hot representations
3: Calculate a threshold y! using Ç.
[CiJ../Cil2..> [Ci]mss and Coyy Cny+l..... Cny+nu-1 are zero vectors. Denote that [C]; is ith row of C. Then, we compute a similarity between two data points x, Xj E X. that we compare all pairs in X. For example, one of pop- ular similarity functions, i.e., radial basis function kernel, can be used:
4: Assign labels C, of data points in D, with vt.
5: if ,,, are not available then
6: Sample ne unlabeled data points x. from X.
7: end if
8: Estimate pseudo-labels C₂ by following the proce-
dure in Algorithm 2.
Wiy exp (-91|x; - x, 11)
(8)
9: Choose the next query point X++1 by maximizing
Te, (x: S. De, X.) for x E X.
where 3 is free parameter given. As discussed in the work by Zhu & Ghahramani (2002), we can learn 3 in (8) by minimizing an entropy for propagated labels C:
10:
Evaluate X by I and update D++1.
11: end for
12: Determine the best candidate xt, considering Vo:T-
m
H[²)= = Σ CHI log(C)e.
(9)
where TT, (x: Ç, Dr, X..) predicts a class probability over x for Class 1; see (14).
See Figure 4 for the results of learning 3 and Section F for analysis on the effects of B. By (8). we compute a transition probability from x, x; by Pij = Way/
We adopt a multi-started local optimization technique, e.g., L-BFGS-B (Byrd et al., 1995), to solve (6). However, a flat landscape of Tc, (x: <, Dr, X.) over may occur, so that optimization performance can be degraded. To tackle this issue, a simple heuristic of randomly selecting query point among points with identical highest class probabilities complements our method. Since the multi-started technique is utilized and the output of TT. is bounded in [0, 1], a flat landscape is easily recognized by comparing the outcomes of the multi-started strategy.
that similarities W E (x == +u Wkj- Note and transition probabilities P E (x) +m.) are defined, where [Wly = Wiy and Pla = Pij. Moreover, by the definition of Pij. P = D W, where D is a degree matrix whose diagonal entry is defined as the following:
D
Σ 3=1
[W
(10)
With initial Ò and P. we repeat the following steps: (i) computing the next C: (ii) normalizing C row-wise, until change of Ć converges to a tolerance or the number of iterations propagated reaches to maximum iterations T. In particular, label propagation (Zhu & Ghahramani, 2002) updates Č and constrains the labels of labeled data to C₁:
4.1. Label Propagation and Label Spreading
Here we describe sumi-super learning techniques (Zhu, 2005; Chapelle al., 2006; Bengio et al., 2006) to build DRE-BO-SSL. We cover transductive setting (Gam- merman et al., 1998), which is to label unlabeled data by utilizing given labeled data, and then an inductive setting, which is to predict any point using pseudo-labels of unla- beled and labeled data.
C++1 + P
(11)
[C++]] + [Ciles
(12)
Suppose that each data point is defined on a d-dimensional compact space X IRd. We consider nu labeled points X, E xd their corresponding labels C₁ Re and The unlabeled points X,, E Rd where is the number of classes. X and C₁ are query points that have already been evaluated and their class labels; we define X, = [xo..... .x, for D.- = {(x,, = For the sake of brevity, the concatenated data points of X, and X,,
for i E [mx] at Iteration t. On the other hand, label spread- ing (Zhou et al., 2003) propagates Č by allowing a change of the labels of labeled data with a clamping factor a E (0, 1):
Č++1 + oD 1/2 1/2,, + (1 (1-a)) C)Cu,
(13)
where Co is initial propagated labels, which are defined in Line of Algorithm 2. Note that D D-1/2WD - 11/2 can
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Density Ratio Estimation-basee Bayesian Optimization with Semi-Supervised Learning
be pre-computed before the loop defined from Lines 4 7 of Algorithm 2.
the truncated multivariate normal distribution, we adopt the minimax tilting method (Botev, 2017). Compared to the method by Genz (1992), it yields high acceptance rate and accurate sampling. In this paper Σ is set as an identity matrix, and I and u are determined by a search space. We will provide more detailed discussion on point sampling in Sections 6 and G.
By the nature of transductive setting, it only predicts the labels of particular data using the known data (Gammerman et al., 1998), which implies that it cannot estimate a categor- ical distribution of unseen data, To enable it, given unseen x, we define an inductive model with C:
W TOR j= W [C]
5. Experiments
2
(14)
We compare baseline methods with DRE-BO-SSI. in the fol- lowing optimization problems: synthetic benchmarks for a scenario with unlabeled point sampling, and synthetic bench- marks, Tabular Benchmarks (Klein Hutter, 2019), NATS- Bench (Dong et al., 2021), and minimum multi-digit MNIST search for scenario with a fixed-size pool. Note that Tab- ular Benchmarks, NATS-Bench, and minimum multi-digtt MNIST search are defined with a fixed number of possible solution candidates, which implies that they are considered as combinatorial optimization problems. By following the previous work by Tiao et al. (2021); Song et al. (2022), we set a threshold ratio as Ç = 0.33 for all experiments; the effects of a threshold ratio Ç are analyzed in Sections 6 and I. To solve (6), we use L-BFGS-B (Byrd et al., 1995) with 1,000 different initializations. All experiments are repeated 20 times with 20 fixed random seeds, where 5 initial points are given to each experiment. The sample mean and the standard error of the sample mean are reported. Other miss- ing details including the details of the competitors of our methods are presented in Section E.
for i E [2]. where w E Ruetn is similarities of x and X
by (8). [C denotes ith column of C. This inductive model is better than or equivalent other classifiers without unla-
beled data in certain circumstances; its analysis is provided in Section D.
4.2. Unlabeled Point Sampling
As described above, if unlabeled points are not available, we need to generate unlabeled points under a transductive learn- ing scheme. However, it is challenging problem unless we know a landscape of Tee adequately. Many studies by Seeger (2000); Rigollet (2007); Singh al. (2008); Ben-David et al. (2008); Carmon et al. (2019); Wei al. (2020); Zhang et al. (2022) investigate how unlabeled data can affect a model and whether unlabeled data helps improve the model or not.
In order to make use of the theoretical findings of previous literature, we define a cluster assumption:
Assumption& 4.1 (Cluster assumption in the work by Seeger (2000)). Two points X; E X' should belong to the same label if there is a path between x; and Xj which passes only through regions of relatively high P(X). where P is distribution over a random variable X X.
As the competitors of our method, we test the following baseline methods:
Gaussian process, EI and UCB: It is a Bayesian op- timization strategy, which is defined with Gaussian process regression with the Matérn 5/2 kernel (Ras- mussen & Williams, 2006), where expected improve- ment (Jones al., 1998) or Gaussian process upper confidence bound (Srinivas et al., 2010) is used as an acquisition function;
By Assumption 1. the idea of clustering on the Fuclidean space or spectral clustering on a graph can be naturally ap- plied in semi-supervised learning (Seeger, 2000; Joachims, 2003), which is not the scope of this work.
To build DRE-BO BO-SSL associated with Assumption 4.1, we sample unlabeled data points from the truncated multivariate normal distribution so that each sample is in a compact X:
Random forest, BORE and LFBO: These are BORE and LFBO that employ random forests (Breiman, 2001) with 1,000 decision trees, where minimum samples to split are set to 2 for these baselines;
f(z) = exp(- z)I((1 < Az <Su u)
(15)
AZ u)
Gradient boosting, BORE and LFBO: These methods are BORE and LFBO with gradient boosting classi- fiers (Friedman, 2001) with 100 decision trees, where a learning rate for the classifier is set to 0.3;
where 1, auue E Eed are lower and upper bounds, Σ AAT is covariance matrix, I is indicator function, and Z ~ (0, I2) is random variable. It is challenging to calculate a denominator of (15), P(1 < A T Z S u). and simulate from f() because an integration of the denominator and an accept-reject sampling strategy from fz) are cumbersome in this multi-dimmesional case. To effectively sample from
XGBoost, BORE and LFBO: Similar to gradient boosting, BORE and LFBO with XGBoost (Chen & Guestrin, 2016) have 100 decision trees as base learn- ers with a learning rate of 0.3;
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Density Ratio Estimation-basee Bayesian Optimization with Semi-Supervised Learning
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MLP, BORE and LFBO: These methods are built with two-layer fully-connected networks; the detail of the multi-layer perceptron is described as follows.
5.1. A Scenario with Unlabeled Point Sampling
Synthetic Benchmarks. We run several synthetic func- tions for our methods and the baseline methods. For unlabeled point sampling. we sample 100 points from N(x,, (x), .N(x,,, I2), where [n./nı] or [na/m]] + 1 points are sampled from each truncated dis- tribution, so that n. points are sampled in total. As shown in Figure 2, our methods outperform the baseline methods incorporating labeled data with unlabeled points. Interest- ingly, out methods beat Gaussian process-based Bayesian optimization. It implies that ours can fairly balance explo- ration and exploitation. Furthermore, we present the results
The architecture of the multi-layer perceptron is set as the following: (i) first layer: fully-connected, input dimension- ality d, output dimensionality 32, ReLU; (ii) second layer: fully-connected, input dimensionality 32, output dimen- sionality 1, Logistic where d is the dimensionality of the problem we solve.
Note that most configurations for the baselines follow the configurations described in the work by Song et al. (2022).
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Density Ratio Estimation-basse Bayesian Optimization with Semi-Supervised Learning
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Figure 5. Results with 20 repeated experiments on Tabular Benchmarks for scenario with fixed-size pools. LP and LS stand for label propagation and label spreading, respectively.
Eoupetoo gerioss II
Sitllowd HOTU
Gareraa pereee K
Nillowe BRIIE
Dirpsiin grotnes EI
Xitllowd DOTE
Guoestis DCB
XtiDoott LEBO
Gazetzz UCB
NGllooe LFG3
Eorautio LCS
Stillowt, LI BO
BORE
BORE
BORE
DORE
Bazthii feori LFB0
MILP. L2B0
Repless Jarod, LEBO
MUP. LFB3)
Ruzben LF3O
MLP..
Goolial Dasiita, BORZ
DRE DO-SS. (LP)
Guoile bonling, BORE
DRE BO-SAL C.P..
Goolial looocaa BORE
DRE DRE--0--5EL (LP)
Anrmnayy
Grodant lookria, LEGO
DRE-DOSSEE (L3)
Accuracy
Crabizt benitaz, LFBO
DRE BO-SEL (LS)
Avenray
Govdral louani,, LFBO
DEE-BOOSS (LS)
Iteration
Iseration
Iteration
(a) CIFAR-10
(b) CIFAR-100
(c) ImageNet-16-120
Figure 6. Results with 20 repeated experiments NATS- for scenario with fixed-size pools.
of learning 33 in Figure 4, where B is adaptively selected by (9) every iteration.
this paper, for each dataset. Our algorithms show superior performance compared to other approaches. Similar the synthetic functions, we argue that the adoption of a prede- fined pool leverages its performance. In some cases, the Gaussian process-based strategy is better than our methods.
5.2. Scenarios with Fixed-Size Pools
Synthetic Benchmarks. Several synthetic benchmark functions are tested for our methods and the baseline meth- ods. To generate fixed-size pool for each benchmark, we uniformly sample 1000 points from a bounded search space before an optimization round is started. As presented in Fig- ure 3, our methods perform better than the baseline methods. It implies that the use of unlabeled data helps improve op- timization performance. Also, the results of learning 3 are demonstrated in Figure 4. Learned / is likely to converge to some value as iterations proceed according to the results.
NATS-Bench (Dong et al., 2021), which is the up-to-date version of NAS-Bennhh2D5 (Dong & Yang, 2019), is used to test our methods and the baseline meth- ods. NATS-Bench is a neural architecture search benchmark with three popular datasets: CIFAR-10, CIFAR-100 and ImageNet- 120, and it has 32,768 architectures, i.e., a fxee-size pool in this paper, for each dataset. Similar to the experiments mentioned above, our methods work well in three datasets, compared to the existing methods; Figure 6 for the results.
NATS-Bench.
Tabular Benchmarks. Comparisons of our methods and the existing methods are carried out in these hyperparameter optimization benchmarks (Klein & Hutter, 2019), as in Fig- ure 5. We can benchmark variety of machine learning models, which are defined with specific hyperparameters and trained on one of four datasets: naval propulsion, protein structure, Parkinson's 's telemonitoring, and slice localization. There exist 62,208 models, which are used as pool in
64D Minimum Multi-Digit MNIST Search. This task, which is proposed in this work, is to find minimum multi- digit number, where fixed number of multi-digit MNIST images are given. As visualized in Figure 10, three random images in the MNIST dataset (LeCun et al., 1998) are con- catenated. Eventually, "000" and "999" are global minimum and global maximum, respectively. Since inputs are images
8

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Density Ratio Estimation-base Bayesian Optimization with Semi-Supervised Learning
C (= 0.01 and = 0.8. While it has depend on opti- mization problems, = 0.33 and Ç 0.5 generally appropriate choices according to this analysis. The details of this study can be found in Section and Figure 16.
Germice process
Xithooe BORE
Grusina powcoss UCB
XcC LFDO
Ranisss Jeci, BORE
ML., SCRE
Ranloc: LP00
MLP. 2.BB
Mininam Value
10-2
Squditz,
DRS BI--SL, (1.)
10-7
Flat Landscape of Class Probabilities over Inputs. Re- gardless of the use of either supervised semi-supervised classifier, flat landscape of class probabilities can occur in the framework of DRE-based Bayesian optimization. To overcome the issue of optimizing a flat landscape, we add a simple heuristic of selecting a random query point from points with identical highest class probabilities if the land- scape is flat, as described in Section 4. Since off- aff-the-shelf local optimization methods struggle to optimize a flat land- scape, this simple heuristic is effective.
10"
10-0
400
500
Iteration
Figure 7. Results with 20 repeated experiments on 64D minimum multi-digit MNIST search.
and each concatenated image consists of three different digit images, this high-dimensional optimization problem is chal- lenging. The size of a fixed-size pool 80,000, As shown in Figure 7, our methods show satisfactory performance compared to other baseline methods.
General Framework of DRE-BO-SSL. As a future di- rection, we expect that general framework of DRE-BO- SSL can be defined, which is similar to likelihood approach by Song et al. (2022). However, it is difficult to define an utility function involved in both labeled and unlabeled data. For example, if we assume that an utility function is u(y y¹) = max(y) - y, 0), y for an unlabeled data point is unknown. Although it depends on the form of utility function, we need to define y of an unlabeled data point by utilizing the information we have if the utility function is related to
Missing details of these experiments are shown in Section E.
6. Discussion
We discuss interesting topics on our methods and the proper- ties of DRE-BO-SSLL Moreover, we provide more thorough discussion and limitations in the appendices.
7. Conclusion
Effects of the Number of Points and Sampling Distri- butions for Unlabeled Points. We choose distribution for unlabeled point sampling as the truncated multivariate normal distribution in order to satisfy the cluster assumption. To analyze our algorithm thoroughly, we demonstrate the ef- fects the number of sampled points and sampling distribu- tions for unlabeled points in Section G, varying the number of unlabeled points and utilizing diverse sampling distribu- tions, i.e., uniform distributions, Halton sequences (Halton, 1960), and Sobol' sequences (Sobol', 1967).
In this paper we have proposed a DRE-based Bayesian optimization framework with semi-supervised learning, named DRE-BO-SSL. Unlike the existing work such BORE and LFBO, methods make use of semi- supervised classifiers where unlabeled data points are sam- pled or given. The superior results by our methods and the thorough analyses on the characteristics of DRE-BO- SSL exhibit the validity of our proposed algorithms.
A Acknowledgements
Effects of Pool Sampling. Because the computational complexity of label propagation and label spreading de- pends on a pool size, we need to reduce a pool size appro- priately in order to speed up the algorithms. Therefore, we uniformly sample a subset of the fixed-size set, which is used as unlabeled points. Figure 14 reports elapsed times over subset sizes for pool sampling. Larger subsets make the framework slower expected. More detailed analysis on the effects of pool sampling is demonstrated in Section H.
This research was supported in part by the University of Pittsburgh Center for Research Computing through the re- sources provided. Specifically, this work used the H2P cluster, which supported by National Science Foundation award number OAC-21 17681.
Impact Statement
Our work does not have any direct negative societal impacts because it proposes a new Bayesian optimization method to optimize a black-box function. However, in the sense of optimization, this line of research can be used in optimizing any objectives including unethical optimization tasks.
Effects of Threshold Ratios. We study the effects of a threshold ratio Ç in order to understand how we can choose S. As shown in Figure 16, both small Ç and large Ç gen- erally represent worse performance; see the results with

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Density Ratio Estimation-basee Bayesian Optimization with Semi-Supervised Learning
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A. Additional Comparisons of BORE and LFBO
(a) Random forests, BORE, Iterations 5
28
53
Clacs
(b) Random forests, LFBO, Iterations to 5
(c) Gradient boosting, BORE, Iterations
Clac de
(d) Gradient boosting. LFBO, Iterations 1 5
Clas Waa
(e) XGBoost, BORE, Iterations 1 to 5
25
XGBoost, LFBO. Iterations to
Figure 8. Comparisons of BORE and LFBO by random forests, gradient boosting, and XGBoost for the Branin function. It follows the configurations described Figure 1.
In addition to Figure I, we include additional comparisons of BORE and LFBO by random forests (Breiman, 2001), gradient boosting (Friedman, 2001), and XGBoost (Chen & Guestrin, 2016) for the Branin function, For Figures and we use
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Density Ratio Estimation-basee Bayesian Optimization with Semi-Supervised Learning
Ç 0.33 and 0.5.
B. Additional Related Work
Although many recent Bayesian optimization methods using regression models (Eriksson et al., 2019; Balandat et al., 2020; Cowen-Rivers et al., 2022; Ament et al., 2023) show strong performance on various benchmark functions, we do not compare our methods to them, as such comparisons are beyond the scope of this work. The goal of this work to demonstrate the effectiveness DRE-based Bayesian optimization with semm-suuereised learning the settings of DRE-based Bayesian optimization. Picheny et al. (2019) propose novel Bayesian optimization framework that relies variable ordering in latent space to handle ill-conditioned discontinuous objectives. Regarding the choice of the truncated multivariate normal distribution, it may be related to the boundary issue of Bayesian optimization; see the Ph.D. thesis of Swersky (2017), Such similar issue is discussed in the existing work by Oh et al. (2018); it proposes Bayesian optimization approach with cylindrical kernels concentrating on a region proximal to the center of the search space. Moreover, the prior work by Hvarfner et al. (2024) also reports the similar consequence in Bayesian optimization. From the perspective of DRE-based Bayesian optimization, Gaussian process classification (Rasmussen & Williams, 2006) might be applicable for defining binary classifiers; it left for future work.
C. Details of DRE-BO-SSL
Algorithm 2 Labeling Unlabeled Data
Input: Labeled data points X,, their labels C₁ unlabeled data points X., maximum iterations т, and tolerance e. Additionally, clamping factor for label spreading.
Output: Propagated labels C.
1: Initialize propagated labels Č X.
2: Compute similarities W and degree matrix D.
3: Compute transition probabilities P with W and D.
4: repeat
5: Propagate Č with P and the previous Č, and additionally a for label spreading.
6: Normalize Č row-wise.
7:
until change of C converging
reaching
Algorithm 2 describes procedure to label unlabeled data points; see the main article for the details of DRE-BO-SSL.
D. Analysis of DRE-BO-SSL
Under the cluster assumption, i.e., Assumption 4.1, margin y is defined as minimum distance between two decision boundaries.
Definition D.1. Let a compact connected decision set be C, X for i {0,1) 1 and boundary subset, i.e., set of boundary
points, of compact connected set S be as. A margin y is defined
(2I n C2 0) - 1) x1 eac X0x,xx000
min
||x] Xell2-
(16)
Using Definition D. we claim that a semi-supervised classifier in DRE-BO-SSL can mitigate the over-exploitation problem presented Section 1.1, because it can expand a decision set C1 for Class 1 by reducing y with unlabeled data. However, we need verify if a large decision set derived from the characteristics of non-parametric classifiers, since semi-supervised classifier we use is converted the Nadaraya- -atsonn non- non-parametric model (Nadaraya, 1964; Watson, 1964) without unlabeled data. As shown in Figure 12, there is strong relationship between the performance of semi-supervised classifiers without unlabeled data, i.e., the Madaraya-Watson estimator, and one of semi-supervised classifiers with unlabeled data. It implies that configuration selection for semi-supervised classifier is dependent on class of objective function, and the presence of unlabeled data is likely to be effective for alleviating the over-exploitation problem.
Singh et al. (2008) provide a sample error bound of supervised and semi suni-supervieed learners, related to 7/ ni, nu. and d. This work proves that a semi-supervised learner can be better than any supervised learners, by assuming that ny > n
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Density Ratio Estimation-basel Bayesian Optimization with Semi-Supervised Learning
and access to perfect knowledge of decision sets. However, these theoretical results cannot be directly applied in sequential problem because this work assumes that both labeled and unlabeled data points are independent and identically distributed. Nevertheless, these results hint a theoretical guarantee better performance of semi-supervised classifiers with unlabeled data points. Further analysis for the circumstances of Bayesian optimization is left for future work.
E. Experimental Details
Here we present the missing details of the experiments shown in the main part.
To earry out the experiments in work, dozens of commercial Intel and AMD CPUs such as Intel Xeon Gold 6126 and AMD EPYC 7302. For the experiments minimum multi-digit MNIST search, the NVIDIA GeForce RTX 3090 GPU is used.
To minimize (9) for finding an adequate 3 of label propagation and label spreading, we use L-BFGS-B with a single initialization, which is implemented SciPy (Virtanen al., 2020), for the experiments in Figures 2, 3, 5, 6, and 7. For the other empirical analyses, we set J3 as a specific fixed value; see the corresponding sections for the details.
To reduce the computational complexity of DRE-BO-SSL for scenario with fixed-size pool, we randomly sample 2,000 unlabeled points from the predefined pool for all experiments excluding synthetic benchmark functions. More thorough analysis can be found in Section H.
To compare baseline methods with our methods, we assess optimization algorithms using a simple regret:
simple reg.
min (xx) - f(x"),
(17)
where x" is a global optimum.
Our proposed methods and baseline methods are implemented with scikit-learn (Pedregosa et al., 2011), PyTorch (Paszke al., 2019), NumPy (Harris et al., 2020), SciPy (Virtanen al., 2020), XGBoost (Chen & Guestrin, 2016), and BayesO (Kim & Choi, 2023). Scikit-learn and SciPy are under the BSD 3-Clause license, NumPy is under the modified BSD license, XGBoost is under the Apache 2.0 license, and BayesO is under the MIT license. On the other hand, PyTorch is under its own license; refer to its repository for the license.
E.1. Details of Synthetic Benchmarks
Here describe the details of synthetic benchmarks.
Beale Function. This function is defined as follows:
f(x) (1.5 + 2 + (2.25 +1++110 x1222) + (2.625 21 + 21
(18)
where [21, x2] E [-4.5, 4.5)a.
Branin Function. It is defined as follows:
f(x) (xe2 (5.1/4m ) (5/x)m1 - -6)=+10 10 (1/8x)) cos(#1) 10,
(19)
where [21, m2
E [[-5, 10), [0, 15]]-
Bukinó Function. This benchmark is given by the following:
f(x) = 100 / [x2 - 0.01Iq1 + 0.01|21 [||+++++-1 + 10),
(20)
where x [21, 22] E [[-15, -5. 1-3.3]
Six-Hump Camel Function.
It is given by the following:
f(x) (4 - 2.1 =f/3) + (-4 4zj)az.
(21)
where x = [x1, x2] E [[-3, 3), [-
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Density Ratio Estimation-baset Bayesian Optimization with Semi-Supervised Learning
E.2. Details of Tabular Benchmarks
Table I. Search space for Tabular Benchmarks. "tanh" and "relu" represent hyperbolic tangent and ReLU, respectively,
Hyperparameter
Possible Values
number units The number units for 2nd layer
layer
{16, 32, 64, 128, 256, 512) [16, 32, 64, 128, 256, 512) {0.0, 0.3, 0.6
Dropout rate for 1st layer Dropout rate for 2nd layer Activation function for Ist layer Activation function for 2nd layer Initial learning rate Learning rate scheduling Batch size
{0.0, 0.3, 0.6]
("tanh", "relu")
("tanh". "relu") ., x 10 1003.11. .1x 10 10-2. 2.5 x 10 x.x 10
(5 10 10 1x 10
("cosine", "constant") {8, 16. 32, 64)
The search space of Tabular Benchmarks (Klein & Hutter, 2019) is described in Table 1. To handle categorical and discrete variables, we treat them as integer variables by following the previous literature by Garrido-Merchán Hernández- (2020). More precisely, the value of each integer variable corresponds the index of the original categorical or discrete variable. When evaluating the variables, the integer variables transformed into the original variables.
E.3. Details of NATS-Bench
Inpus image
Conv.
Cell
X
Residual block -
Cell
x
Residual block -
Cell
xN
avg. pooling
Output
Figure 9, Neural network architecture in ANTS-Bench. Orange blocks optimized.
Table 2. Search space for NATS-Beneh There exist 8 32,7668 models.
Hyperparameter
Possible Values
Output channels of 1tt convolutional layer
{8, 16, 24, 32, 40, 48, 56, 64)
Output channels of stt cell stage
{8. 16, 24, 32, 40, 48, 56, 64
Output channels of 1st residual block
{8. 16, 24, 40, 48, 56, 64
Output channels of 2nd cell stage
{8. 16, 24, 32, 40,
56. 64
Output channels of 2nd residual block
{8. 16. 24, 32, 40, 48, 56, 64)
We describe the search space for NATS- -ennh (Dong et al., 2021) in Figure 9 and Table 2.
E.4. Details of Minimum Multi-Digit MNIST Search
Since multi-digit MNIST, which is composed of images of size (28, 84) and shown in Figure 10. is high-dimensional, some of the methods used in this work, e.g., methods with random forests, gradient boosting, and XGBoost, struggle to process such data. Therefore, we embed an original image to a lower-dimensional vector using an auxiliary convolutional neural network. The convolutional neural network is trained to classify a three-digit image to one of labels from "000" to "999." with the following architecture:
First layer: convolutional, input channel 1, output channel 8, kernel size 3 x 3, padding 1, ReLU, max-pooling x 2;
Second layer: convolutional, input channel 8, output channel 16, kernel size 3 x 3, padding 1, ReLU, max-pooling
2 x 2;
Third layer: convolutional, input channel 16, output channel 32, kernel size 3 3, padding 1, ReLU, max-pooling
2 2;
Fourth layer: fully-connected, input dimensionality 960, output dimensionality 128, ReLU;
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Density Ratio Estimation-ban bbased Bayesian Optimization with Semi-Supervised Learning
1
144
4
0
7
6
0 5 6
300 0 2
Figure 10. Examples on 64D minimum multi-digit MNIST search.
Fifth layer: fully-connected, input dimensionality 128, output dimensionality 64, ReLU;
Sixth layer: fully-connected, input dimensionality 64, output dimensionality 1000, Softmax.
The Adam optimizer (Kingma & Ba, 2015) with learning rate 1 x 10-3 is used to train the network for 100 epochs. To train and test the model fairly, we create a training dataset of 440,000 three-digit images, a validation dataset of 40,000 three-digit images, and test dataset of 80,000 three-digit images using training dataset of 55,000 single-digit images, a validation dataset 5,000 single-digit images, and a test dataset 10,000 single-digit images in the original MNIST dataset (LeCun al., 1998). For example, supposing that test dataset has 1,000 single-digit images per class is not true for the MNIST dataset, but it is assumed for explanation-ante "753" can be created. We therefore randomly sample fixed number of three-digit images from vast number of possible combinations. In addition, early stopping technique is utilized by comparing the current validation loss the average of validation losses for the recent five epochs. Eventually, our network achieves 99. 99.6% in the training dataset, 97.0% in the validation dataset, and 96.9% in the test dataset.
would like to generate three-digit image "753," 10 combinations for
To construct a fixed-size pool, we use 80,000 embeddings of dimensionality 64, which are derived from the outputs the fifth layer without ReLU, by passing the dataset of three-digit images through the network.
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Density Ratio Estimation-basee Bayesian Optimization with Semi-Supervised Learning
F. Discussion on Free Parameter in Label Propagation and Label Spreading
Sirgle Begne
Baae
Sugle Raan
Saplle
Strele
30m
Beration
Boratoo
Ieznntion
(a) Beale, I.P
(b) Branin,
(c) Bukin6, L.P
(d) Six-hump camel, LP
1010
1010
00301
10
1000
20005
1001C
Staple Dopoet
Staple Bogert
D0
120
Destion
licratio
Etentiin
(c) Beale, LS
(f) Branin, LS
Bukin6, LS
(h) Six-hump camel, LS
Figure 11. Effects of a free parameter 3 in label propagation, denoted as LP, and label spreading, denoted as LS. All experiments are repeated times.
In the Bayesian optimization process of PRE-BO-SSLL a free parameter / in label propagation and label spreading is learned every iteration by minimizing (9); see Figure 4 for the results on learned 3. Furthermore, to show the effects of 3, we empirically analyze 3 as depicted in Figure 11. We sample 1,000 unlabeled points and use all of them as unlabeled points without pool sampling. For the cases of four benchmark functions, higher /3 tends to show better performance than lower B. These results considerably correspond with the results in Figure
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Density Ratio Estimation-baset Bayesian Optimization with Semi-Supervised Learning
G. Discussion on Unlabeled Point Sampling
No ta
Sc Landattoc
X. Candaniave
Xp inseaae.h
Ganokea-100
Gemiies-1000 1930
Graviaz
Genvias 1D
Gaaskes- 10000
Cacoian-
Cacoiia 10000
Ge
Geoelas ao LIC
Cacmtes
Stnple Ttggs
Sample Tteget
Simple Begne
Smplle Rogert
100
200
200
100
.00
.013
20
100
130
-00
100
Teeratio
Derrationn
Errodion
Deeattm
(a) Beale, LP
(b) Branin, LP
(c) Bukin6, LP
(d) Six-hump camel, LP
Cluscas-1)
Gazoine]]
Lanaraz
Gr
Stull Boant
Stuple Boget
Stuple Baget
Sangle
Derectinn
Teration
Ieeation
Eneeation
(e) Beale, LS
(f) Branin, LS
(g) Bukin6, LS
(h) Six-hump camel, LS
Figure 12. Effects of the number of unlabeled points for unlabeled point sampling. I.P and LS stand for label propagation and label spreading, respectively. We repeat all experiments 20 times.
05
Gassdin
Sdvd
Gassdon
Savt Baaee
Ganodin Uritiess
Sihal libro
Saxelar Uritiers
Sihal
Dniters
Iiaoo
Stmple 00
Staple Regnet
Staple Tespert
Stuple Rogeet
100
300
lanalion
Dinniion
Firnation
(a) Bcale, LP
(b) Branin, LP
(c) Bukin6, LP
(d) Six-hump camel, LP
Laneras Uritien
Cirarazz Thilbenn
Balro
Fabis
Uhillen
Saupl: Jeges
Siuple Regoot
Sampl:
Sippl:
121
100
lonation
(e) Beale, LS
(1) Branin, LS
(g) Bukin6, LS
(h) Six- Six-hump camel, LS
Figure 13. Effects sampling strategies for unlabeled point sampling. LP and LS stand for label propagation and label spreading. respectively. We repeat all experiments 20 times.
We design two studies to analyze the effects of the number of unlabeled points n. and sampling strategies in process of unlabeled point sampling, where unlabeled points are not provided and 3 = 0.5 is given.
For the first study, we conduct five settings, no unlabeled data, which implies that transductive learning is not used, and n. = 10, 100, 1000, 10000. Interestingly, the tendency of the number of unlabeled points are unclear as presented
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Density Ratio Estimation-basel Bayesian Optimization with Semi-Supervised Learning
in Figure 12. It implies that setting for the number of unlabeled data points depend the characteristics of benchmark functions, which is common in Bayesian optimization and black-box optimization. Besides, is different across benchmarks and iterations and it lets optimization results sensitive to nl. Therefore, we cannot determine suitable setting without access to black-box function of interest.
As another elaborate study, we test the effects of sampling strategies. Four strategies, the truncated multivariate normal distributions, uniform distributions, Halton sequences (Halton, 1960), and Sobol' sequences (Sobol', 1967), are compared. As depicted in Figure 13, the normal distribution is better than the other sampling methods in four cases and shows robust performance in most of the cases, but it always the best. Similar the previous study on the effects of nuy we presume that it is also affected by y, which is hard to define in practice.
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Density Ratio Estimation-bass Bayesian Optimization with Semi-Supervised Learning
H. Discussion on Pool Sampling
200
300
-79
120
Tiue Eanh?
730
800
480
200
Sibbel 8om
Scbwe Stor
Sulwri Stor
Sulent Stor
(a) Beale,
(b) Branin,
(c) Bukin6,
(d) Six-hump camel, LP
3.0
300
240
244
Tiae (v6.
-00
Time (pe
X4 201
10:0 106 280 R00
A0 1MS I500 200 cMM Smm
20 1MB 340 X.X0 Di 880 Subert
30
135 160 3.0 3:5-0
Sabert Ste
(e) Beale, LS
(I) Branin, LS
(g) Bukin6, LS
(h) Six-hump camel, LS
Figure 14. Results with 20 repeated experiments on elapsed times varying subset sizes via pool sampling. LP and L. stand for label propagation and label spreading, respectively.
1381
Stogle Dozn
Stagle Doase
Stagle Boaan
Stagle Dount
Reratton
Boraton
Incnstion
Beraboo
(a) Beale, LP
(b) Branin,
(c) Bukin6,
(d) Six-hump camel, LP
100
2000
660
2006
200
200
1030
3000
10
1100
Stuple Hogret
Siugle Rogre
Staple
12'4
Torratio
Beale. LS
(1) Branin, LS
Bukin6, LS
(h) Six-hump camel, LS
Figure 15. Effects of pool sampling for a case with fixed-size pools. We repeat all experiments 20 times, and LP and L$ stand for label propagation and label spreading. respectively.
To see the impact of an additional hyperparameter, i.c., the size of a subset of the original pool, which is introduced to speed up semi-supervised learning algorithms, we demonstrate numerical analysis on pool sampling where the size of predefined pool is 4,000 and = 0.5 is given. Based on Figures 14 and 15, we can accelerate our framework without significant performance loss.
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Density Ratio Estimation-basee Bayesian Optimization with Semi-Supervised Learning
I. Discussion on Threshold Ratio
[:5 1006
0..0
066 1000
00
18-
1006
Strple Boane
Stuple Dogne
Strppe Dozne
Stuple Boure
by
200
(a) Beale, LP
(b) Branin, LP
(c) Bukin6, LP
(d) Six-hump camel, LP
12
0.99
Stupk Dourst
Stugk Doguet
Sluok
Daretoon
Iecatton
Irenstion
Beale, LS
Branin, LS
Bukin6, LS
(h) Six- -uxphumppo camel, LS
Figure 16. Effects of threshold ratio with 3 0.5. We repeat all experiments 20 times, and LP and LS stand for label propagation and label spreading, respectively.
Figure 16 demonstrates the effects of threshold ratio Ç, where we use 3 = 0.5. It follows the setting described in Section F. As presented in Figure 16, a smaller Ç, for example, < = 0.01 or Ç = 0.1 tends to show worse performance than larger Ç. which implies that the over-exploitation not due to conservative y². Interestingly, the results with = 0.8 also generally under-perform. We presume that is basically due to over-exploration.
J. Limitationn
As discussed above, our algorithms slow down if a pool size is significantly large. As presented in Figure 14, elapsed times are certainly dependent on subset sizes. To tackle this issue, we suggest a method to randomly select subset of the pool, but a more sophisticated subsampling method can be devised for our framework. In particular, we can leverage the impacts of the subset of the pool by utilizing the geometric information of unlabeled data points.
23


Reviews:
[{"review_id": "geM0QRzk5l", "reviewer": "Reviewer_GQCC", "summary": "This paper proposes an algorithm for black-box optimisation which is inspired by Bayesian Optimisation by density-Ratio Estimation (BORE, Tiao et al., 2021), a BO algorithm that transforms the usual BO regression problem into classification, and is combined with semi-supervised learning to better balance the exploration-exploitation trade-off. The paper starts by showing that BORE and likelihood-free Bayesian optimisation (LFBO) algorithms in general tend to overexploit due to the sole reliance on a small amount labelled training data. It is then shown that a semi-supervised learning approach allows an algorithm to better explore the search space by making use unlabelled query points and their inherent uncertainty to promote exploration. Two semi-supervised learning algorithms for BO are then proposed based on label propagation (Zhu & Ghahramani, 2002) and label spreading (Zhou et al., 2003) techniques. Empirical results demonstrate a few performance when compared to BORE, LFBO and traditional BO baselines.", "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 thank the reviewer for your constructive feedback to improve our work. We answer your questions and concerns in the following.\n\n> One my main concerns with this paper in its current form is that, to me, it seems that semi-supervised learning would only make sense if the data had a meaningful underlying distribution. (skipped)\n\n> It's not very clear, from a theoretical standpoint, what the extra machinery derived with the unlabelled data is doing for (skipped)\n\nFirst of all, our algorithm with fixed-size pools does not encounter this issue. As described in *Section 1*:\n\n*To make use of semi-supervised classifiers, we take into account two distinct scenarios with unlabeled point sampling and with a predefined fixed-size pool,*\n\nwe assume two scenarios. More importantly, the real-world applications such as Tabular Benchmarks, NATS-Bench, and 64D minimum multi-digit MNIST search have access to fixed-size pools. Thus, the \"data distribution\" determined by the algorithm is defined based on the *geometry information* of possible query point candidates, where the candidates are from the true distribution. This information helps understand the search spaces of the problems we are tackling. In this context, the entropy (*Eq. 6*) is computed using the propagated labels of the unlabeled points from fixed-size pools, rather than from randomly-sampled points.\n\nFor the concern that the semi-supervised learning problem with randomly-sampled points is ill-posed, unlabeled points help shape the model’s understanding of the search space. This guides exploration by highlighting unexplored regions, even without labels. From an empirical perspective, we think that the unlabeled data can promote more informed and robust exploration. In addition, we would argue that our framework with semi-supervised learning can be understood from a regularization standpoint; eventually it can mitigate the over-exploitation problem.\n\nWe will clarify it in the revised version.\n\n> I think this type of assumption is usually applicable to the case of i.i.d. data, not making sense in BO settings (skipped)\n\nWe agree with your point. We have already discussed this issue in *Section D*. We would like to highlight our discussion here:\n\n*However, these theoretical results cannot be directly applied in our sequential problem because this work assumes that both labeled and unlabeled data points are independent and identically distributed. Nevertheless, these results can hint a theoretical guarantee on better performance of semi-supervised classifiers with unlabeled data points. Further analysis for the circumstances of Bayesian optimization is left for future work.*\n\nThis topic will be left for future work.\n\n> It is not explicitly defined whether the algorithm is defined for maximisation or minimisation problems\n\nThe algorithm is defined for *minimization problems* in our work. We will clarify it in the revision.\n\n> In-text citations should be explicitly mentioned.\n\nThank you for pointing these out. We will fix them in the final version.\n\n> In line 192, \"as an acquisition function; it is simply denoted as...\", the equation that follows is not simply a definition (skipped)\n\nIt is a good point. We will clarify this in the final version by mentioning that it is the derivation of Tiao et al. (2021).\n\n> Implementation details (skipped) can be moved to the appendix.\n\nWe will move it to the supplementary material.\n\n> Propagated labels are introduced in Eq. 6 before a proper definition.\n\nThank you for pointing this out. *Eq. 6* and its corresponding description will be moved to the end of *Section 4.1*.\n\n> Line 74: \"in order that it is possible that a cluster assumption (Seeger, 2000) is assumed\" should be revised.\n\nIt will be revised like this:\n\n*to allow for the possibility of adopting a cluster assumption (Seeger, 2000).*\n\n> What is $f'$ denoting in Eq. 3? The derivative of $f$?\n\nYes, it is the derivative of $f$. We will clarify it in the revision.\n\n> Why would a simple uniform query distribution on the domain not be suitable for this work? I'm trying to understand why the truncated normal distribution was used (skipped)\n\nThis question is closely related to the boundary issue of Bayesian optimization; see Section 4.4.1 of the Ph.D. thesis of Kevin Swersky [1]. The similar issue is discussed in the previous work [2]. This work [2] proposes a Bayesian optimization approach with cylindrical kernels concentrating on a region near the center of the search region. Also, the recent work [3] also mentions this boundary issue in Bayesian optimization.\n\nWe will clarify it in the final manuscript by extending our discussion presented in *Section G* and *Figure 12*.\n\n[1] Swersky, K. Improving Bayesian optimization for machine learning using expert priors. University of Toronto, 2017.\n\n[2] Oh, C., et al. BOCK: Bayesian optimization with cylindrical kernels. ICML, 2018.\n\n[3] Hvarfner, C., et al. Vanilla Bayesian Optimization Performs Great in High Dimensions. ICML, 2024.", "claims_and_evidence": "One of the main claims of the paper is that BO algorithms based on density-ratio estimation (DRE) are inherently biased towards over-exploitation. Arguments for this claim are mainly presented via illustrations and intuitive arguments with some mild theoretical justification, though no formal proof is provided. In addition, it is claimed that semi-supervised learning helps remediating this problem by reducing over-confidence in classifiers. Illustrations, intuitive arguments and empirical performance improvements are presented as evidence for such claim.", "methods_and_evaluation_criteria": "One my main concerns with this paper in its current form is that, to me, it seems that semi-supervised learning would only make sense if the data had a meaningful underlying distribution. However, in BO, the \"data distribution\" is basically determined by the algorithm itself, at least in continuous search spaces. The algorithm gets around some of these issues by randomly sampling unlabelled points from the domain according to a user-defined query-point distribution. However, imposing a distribution over the query points could be problematic in itself due to both theoretical (e.g., it imposes additional structure on the problem which may or may not be beneficial) and practical (e.g., how can one define a suitable data distribution for arbitrary domains) reasons. Despite the paper discussing a few of these aspects, there is not much emphasis on the intrinsic fact that the data in BO does not intrinsically follow a given underlying distribution. So the semi-supervised learning problem \"as currently defined\" seems ill posed.", "theoretical_claims": "No theoretical results are provided with the paper.", "experimental_designs_or_analyses": "The experimental evaluation is quite thorough, involving both synthetic and real-data problems, a reasonable number of representative baselines, and ablation studies.", "relation_to_broader_scientific_literature": "The paper potentially advances research in Bayesian optimisation algorithms, which are an essential component of automated machine learning, hyper-parameter tuning and adaptive experimental design frameworks. Specifically, the paper focuses on likelihood-free methods for BO, which avoid defining an explicit prior distribution over the objective function, relying instead on a direct estimator for the algorithm's acquisition function. The algorithms presented in the paper potentially mitigate the over-exploitation problem in traditional approaches in this area by incorporating traditional semi-supervised learning techniques.", "essential_references_not_discussed": "Nothing else significantly relevant that I could think of.", "other_strengths_and_weaknesses": "* Assumption 4.1 (cluster assumption): I think this type of assumption is usually applicable to the case of i.i.d. data, not making sense in BO settings where the data is not i.i.d., being actively selected and dependent on all the previous algorithmic choices.", "other_comments_or_suggestions": "* It is not explicitly defined whether the algorithm is defined for maximisation or minimisation problems, though it is implicitly defined for minimisation due to the density on the numerator in Eq. 1.\n* In-text citations should be explicitly mentioned. For example, \"While the work (Bergstra et al., 2011) utilizes...\" (line 177-178) would be better stated as \"While the work of Bergstra et al. (2011) utilizes...\", or \"While Bergstra et al. (2011) utilizes...\". A similar issue is found in line 254.\n* In line 192, \"as an acquisition function; it is simply *denoted* as...\", the equation that follows is not simply a definition, but the result of a mathematical derivation in Tiao et al. (2021). So saying it's simply denoted as its result does not make sense.\n* Implementation details, such as the use of multi-start L-BFGS-B for the optimisation of the acquisition function, can be moved to the appendix.\n* Propagated labels are introduced in Eq. 6 before a proper definition.\n* Line 74: \"in order that it is possible that a cluster assumption (Seeger, 2000) is assumed\" should be revised.", "questions_for_authors": "* What is $f'$ denoting in Eq. 3? The derivative of $f$?\n* It might seem that what the algorithm is mainly using randomly sampled data points from the search space to tune the RBF hyper-parameter $\\beta$ in Eq. 5 via maximum entropy. It's not very clear, from a theoretical standpoint, what the extra machinery derived with the unlabelled data is doing for, e.g., the model's epistemic uncertainty or other factors that might promote exploration.\n* Why would a simple uniform query distribution on the domain not be suitable for this work? I'm trying to understand why the truncated normal distribution was used, a distribution which might, e.g., concentrate the unlabelled data around the centre of the domain.", "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": "4GU0rBy3kx", "reviewer": "Reviewer_xCG2", "summary": "This paper observes the overconfidence, or to say, over-exploit behavior of supervised learning methods in DRE-BO. With unlabeled points available, semi-supervised learning is combined with DRE-BO to tackle the issue. The proposed method, DRE-BO-SSL, are empirically evaluated at two cases including unlabeled point sampling and a fixed-size pool and shows promising performance.\n\n## update after rebuttal\nMy questions are answered by the authors. A revised version with those clarifications will improve this paper.", "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 thank the reviewer for your constructive feedback to improve our work. We answer your questions and concerns in the following.\n\n> I am sort of confused about the relationship between experiments and the motivation. DRE-BO-SSL is motivated by the overconfidence of DRE-BO-SL, giving the impression that the performance of DRE-BO-SL is limited because of the lack of sufficient exploration, which makes those SL-based methods keep improving current best solution while miss real global optima in uncertain area. (skipped)\n\n> Like I mentioned before, the motivation is not well-treated in experiments. A better regret curve does not mean the method explores the search space better.\n\n> The evaluation criteria on optimization performance are sufficient, while the evaluations on improvements towards exploration are missing.\n\nAn empirical regret analysis on the benchmarks used in our work supports our motivation, as improved performance in terms of regret indicates that the Bayesian optimization algorithm effectively balances exploration and exploitation, ultimately leading to lower regret values. If the algorithm fails to explore effectively, improvement in regret values cannot be achieved. Therefore, our performance evaluation aligns with our initial motivation.\n\nA similar discussion can be found in Section 10.1 of the Roman Garnett's book [1]. We would highlight one paragraph in the reference [1].\n\n*Regret is an unavoidable consequence of decision making under uncertainty. Without foreknowledge of the global optimum, we must of course spend some time searching for it, and even what may be optimal actions in the face of uncertainty may seem disappointing in retrospect. However, such actions are necessary in order to learn about the environment and inform future decisions. This reasoning gives rise to the classic tension between exploration and exploitation in policy design: although exploration may not yield immediate progress, it enables future success, and if we are careful, reduces future regret. Of course, exploration alone is not sufficient to realize a compelling optimization strategy, as we must also exploit what we have learned and adapt our behavior accordingly. An ideal algorithm thus explores efficiently enough that its regret can at least be limited in the long run.*\n\n[1] Garnett, R. Bayesian Optimization. Cambridge University Press, 2023.\n\n> Unclear about the choice of hyperparameters. Although the authors reported results with different values of hyperparameters, those results are inconsistent over benchmarks. For example, Figure 11 could not tell which number of unlabeled points should be chosen Figure 12 does not give a clear guidance about which sampling strategies should be used. Figure 14 seems to me that choosing a subset with size 10 is enough, which is counter-intuitive.\n\nWe think that this is a natural characteristic of black-box optimization, in which the hyperparameters depend on specific optimization problems. If a particular method or configuration consistently dominates alternatives, black-box optimization is not challenging. In this work, no trials are conducted before evaluating a black-box objective, so optimal hyperparameters are not used. It is important to note that our methods with default configurations outperform the baseline approaches despite *not using optimal hyperparameters*.\n\n> For the illustration example in Figure 1, it indeed shows that BORE and LFBO focus on area around the best current, and DRE-BO-SSL could explore well. However, it makes sense to spend several iterations to exploit a promising area and then explore uncertain domain. From this example it is unclear to me that if BORE and LFBO will get stuck at that local optimum or will explore other areas later. Is it more informative if the authors could plot similar figures at iterations 10, 20, 30, etc. to see their performance?\n\nResulting performance comparison between the baseline methods and our methods (*Figures 1 and 8*) is shown in *Figure 2b*. Ultimately, our methods outperform the baselines by effectively balancing exploration and exploitation.\n\nWe will clarify this in the final manuscript.\n\n> For Tabular benchmarks and NATS-Bench, the decision variables are integers as mentioned in Appendix E. The optimization of Eq. (4) in discrete search space is not mentioned in the paper.\n\nFollowing the previous work [2], we transform continuous variables to discrete variables before evaluating them. We have briefly mentioned in *Section E.3*, but we will more clarify this in the final version.\n\n[2] Garrido-Merchan, E. C. and Hernandez-Lobato, D. Dealing with categorical and integer-valued variables in Bayesian optimization with Gaussian processes. Neurocomputing, 2020.\n\n> Could the authors please check all citations to make sure they are properly ordered?\n\nThank you for pointing this out. We will fix them in the final paper.", "claims_and_evidence": "I am sort of confused about the relationship between experiments and the motivation. DRE-BO-SSL is motivated by the overconfidence of DRE-BO-SL, giving the impression that the performance of DRE-BO-SL is limited because of the lack of sufficient exploration, which makes those SL-based methods keep improving current best solution while miss real global optima in uncertain area. However, the experiments are showing that DRE-BO-SSL has better optimization performance compared to other baselines, which is definitely good to the BO community but somehow mismatches the initial motivation.", "methods_and_evaluation_criteria": "Extending the SL part to SSL in DRE-BO seems to be quite natural to me. The evaluation criteria on optimization performance are sufficient, while the evaluations on improvements towards exploration are missing.", "theoretical_claims": "This paper contributes in a methodology and empirical way and there are no theoretical claims to check.", "experimental_designs_or_analyses": "This paper conducts thorough experiments, applying two SLL techniques, LP and LS to scenarios with/without unlabeled points on various benchmarks. The effects of different hyperparameters are also discussed and tested in supplementary materials.", "relation_to_broader_scientific_literature": "This paper introduces semi-supervised learning to DRE-based BO, which could be seen as a combination of two literatures.", "essential_references_not_discussed": "From my perspective, this paper already included essential references.", "other_strengths_and_weaknesses": "**Strengths**\n\n- This paper combines DRE-BO with SSL, which is novel.\n\n- This paper is well-written and easy to follow.\n\n- Experiments are thoroughly conducted.\n\n**Weaknesses**\n\n- Like I mentioned before, the motivation is not well-treated in experiments. A better regret curve does not mean the method explores the search space better.\n\n- Unclear about the choice of hyperparameters. Although the authors reported results with different values of hyperparameters, those results are inconsistent over benchmarks. For example, Figure 11 could not tell which number of unlabeled points should be chosen Figure 12 does not give a clear guidance about which sampling strategies should be used. Figure 14 seems to me that choosing a subset with size 10 is enough, which is counter-intuitive.", "other_comments_or_suggestions": "- For the illustration example in Figure 1, it indeed shows that BORE and LFBO focus on area around the best current, and DRE-BO-SSL could explore well. However, it makes sense to spend several iterations to exploit a promising area and then explore uncertain domain. From this example it is unclear to me that if BORE and LFBO will get stuck at that local optimum or will explore other areas later. Is it more informative if the authors could plot similar figures at iterations 10, 20, 30, etc. to see their performance?\n\n- For Tabular benchmarks and NATS-Bench, the decision variables are integers as mentioned in Appendix E. The optimization of Eq. (4) in discrete search space is not mentioned in the paper.\n\n- Could the authors please check all citations to make sure they are properly ordered? e.g., line 14, line 157, etc.", "questions_for_authors": "I asked all questions above and my major concerns are stated in weaknesses.", "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": "72MhAQUlvh", "reviewer": "Reviewer_Y5ev", "summary": "In this article, the authors consider using a classifier to perform Bayesian optimization rather than regression models. To avoid issues with overconfident predictors, the proposition is to rely on unobserved inputs (unlabeled data), propagate the labels to this set before selecting the next query point among them. This selection is based on maximizing the probability to be in the class with lowest values. Unlabeled data can be generated by sampling. A benchmark is provided against some related methods.\n\n## update after rebuttal\n\nI thank the authors for their response. Their answered my comments,  with a focus on DRE-based BO approaches. Yet a more general discussion of regression vs classification plus comparisons to the state of the art BO would be beneficial to enlarge the discussion.", "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 thank the reviewer for your constructive feedback to improve our work. We answer your questions and concerns in the following.\n\n> Given observation values, the regression problem is supposedly simpler than a classification (skipped)\n\nWe think that this question is regarding the general advantages of DRE-based Bayesian optimization (BO). We want to address it from three different perspectives.\n\nFirstly, one of the most popular surrogate models in regression-based BO is a Gaussian process (GP) [1]. The computational complexity of GP regression is $\\mathcal{O}(n^3)$, where $n$ is #query points. As a result, DRE-based BO can be more efficient in terms of computational complexities than GP-based BO.\n\nSecondly, classification can be viewed as a form of bounded regression. For example, using a logistic function, the output of a classification model is bounded in $[0, 1]$. Importantly, bounded regression usually makes model training easier than using the regression-based formulation.\n\nLastly, the studies [2, 3] show that DRE-based BO is equivalent to regression-based BO under specific conditions; it has been discussed in *Section 3*. Consequently, this DRE-based strategy is built on the aforementioned advantages while preserving the abilities of the regression-based method.\n\nWe will clarify these in the final paper.\n\n[1] Garnett, R. Bayesian Optimization. Cambridge University Press, 2023.\n\n[2] Tiao, L. C., et al. BORE: Bayesian optimization\nby density-ratio estimation. ICML, 2021.\n\n[3] Song, J., et al. A general recipe for likelihood-free Bayesian optimization. ICML, 2022.\n\n> The choice of synthetic test functions chosen is not explained.\n> Can you state mathematically the class of optimization problem considered early on?\n\nFollowing the criteria described in [4], Beale, Branin, and Six-Hump Camel belong to 4) multi-modal with adequate global structure, which is often considered as the main target problem of BO [4, 5]. Bukin 6 seems to belong to 5) multi-modal with weak global structure.\n\nWe have provided the mathematical forms of these benchmarks in *Section E.2*. Also, the details of the real-world benchmarks tested in this work are described in *Sections E.3, E.4, and E.5*.\n\n[4] Diouane, Y., et al. TREGO: a trust-region framework for efficient global optimization. J. Glob. Optim., 2023.\n\n[5] Jones, D. R., et al. Efficient global optimization of expensive black-box functions. J. Glob. Optim., 1998.\n\n> It is hard to conclude anything from Figure 5 without more repetitions.\n\nWe would argue that most BO algorithms can generally solve Tabular Benchmarks without significant difficulty. However, our methods still show competitive or better performance compared to the baselines.\n\n> Considering ranks only is studied (skipped) GPs for classification are available as well\n\nWe will cite them by adding sentences like these:\n\n*Ordinal Bayesian optimization [6] proposes a novel Bayesian optimization framework that relies on variable ordering in a latent space to handle ill-conditioned or discontinuous objectives.*\n\n*GP classification [7] can be used in DRE-based Bayesian optimization.*\n\n[6] Picheny, V., et al. Ordinal Bayesian optimisation. arXiv preprint arXiv:1912.02493, 2019.\n\n[7] Rasmussen, C. E. and Williams, C. K. I. Gaussian Processes for Machine Learning. MIT Press, 2006.\n\n> The empirical comparison does not include state of the art BO methods\n\nWe appreciate your constructive suggestions. However, because our work focuses on addressing the over-exploitation problem in DRE-based BO, the main competitors of our methods are other DRE-based BO approaches. We think that adding such BO methods does not change the core message of our work. Nevertheless, we will cite and discuss those methods in the final version.\n\n> Figure 1 and 8: problem with captions\n\nCould you please specify the problem of the captions of *Figures 1 and 8*? We will fix this problem in the final version if you let us know.\n\n> Can you provide running times for the benchmark experiments?\n\nWe provide running times for the benchmark experiments. Because our methods are defined with similarity matrices and transition probabilities $\\mathbf{W}, \\mathbf{P} \\in \\mathbb{R}^{(n_l + n_u) \\times (n_l + n_u)}$ where $n_l$ and $n_u$ are #labeled points and #unlabeled points; see *Eqs. 7, 8, and 9*, ours are relatively slower compared to the baselines. However, as discussed in our paper, our methods can mitigate the over-exploitation problem using both labeled and unlabeled points.\n\nFurthermore, enhancing the efficiency of DRE-based BO with semi-supervised learning is left for future work.\n\n| Method | Time |\n| --- | --- |\n| RF, BORE | 351.2 $\\pm$ 51.7 |\n| RF, LFBO | 392.0 $\\pm$ 65.0 |\n| GB, BORE | 17.5 $\\pm$ 2.0 |\n| GB, LFBO | 20.4 $\\pm$ 3.2 |\n| XGBoost, BORE | 59.2 $\\pm$ 27.7 |\n| XGBoost, LFBO | 58.9 $\\pm$ 24.9 |\n| MLP, BORE | 197.6 $\\pm$ 78.1 |\n| MLP, LFBO | 239.6 $\\pm$ 89.6 |\n| DRE-BO-SSL (LP) | 3850.5 $\\pm$ 2314.8 |\n| DRE-BO-SSL (LS) | 820.8 $\\pm$ 193.0 |", "claims_and_evidence": "Given observation values, the regression problem is supposedly simpler than a classification one obtained by removing function values above or below a threshold. Why this would work better in some cases should be better explained.", "methods_and_evaluation_criteria": "The choice of synthetic test functions chosen is not explained. Detailed results based on features of the considered function would be welcome, see for instance in Diouane, Y., Picheny, V., Riche, R. L., & Perrotolo, A. S. D. (2023). TREGO: a trust-region framework for efficient global optimization. Journal of Global Optimization, 86(1), 1-23.", "theoretical_claims": "Only a preliminary analysis is given.", "experimental_designs_or_analyses": "The analysis is mostly valid. It is hard to conclude anything from Figure 5 without more repetitions.", "relation_to_broader_scientific_literature": "The paper connects BO with label propagation and unsupervised learning.", "essential_references_not_discussed": "Considering ranks only is studied, for instance by Picheny, V., Vakili, S., & Artemev, A. (2019). Ordinal bayesian optimisation. arXiv preprint arXiv:1912.02493.\n\nGPs for classification are available as well, see Williams, C. K., & Rasmussen, C. E. (2006). Gaussian processes for machine learning (Vol. 2, No. 3, p. 4). Cambridge, MA: MIT press.", "other_strengths_and_weaknesses": "The empirical comparison does not include state of the art BO methods, such as:\n- Eriksson, D., Pearce, M., Gardner, J., Turner, R. D., & Poloczek, M. (2019). Scalable global optimization via local Bayesian optimization. Advances in neural information processing systems, 32.\n- Cowen-Rivers, A. I., Lyu, W., Tutunov, R., Wang, Z., Grosnit, A., Griffiths, R. R., ... & Bou-Ammar, H. (2022). Hebo: Pushing the limits of sample-efficient hyper-parameter optimisation. Journal of Artificial Intelligence Research, 74, 1269-1349.\n- Ament, S., Daulton, S., Eriksson, D., Balandat, M., & Bakshy, E. (2023). Unexpected improvements to expected improvement for bayesian optimization. Advances in Neural Information Processing Systems, 36, 20577-20612.", "other_comments_or_suggestions": "Figure 1 and 8: problem with captions\n\n## update after rebuttal\nSorry, I did not catch the small difference between subcaptions", "questions_for_authors": "- Can you state mathematically the class of optimization problem considered early on?\n- Can you provide running times for the benchmark experiments?", "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 proposes a semi-supervised extension of density ratio estimation-based Bayesian Optimization (DRE-BO), aiming to address over-exploitation in supervised classifier-based approaches by incorporating unlabeled points through label propagation and spreading techniques. The approach is evaluated on both synthetic and real-world black-box optimization tasks.

The reviewers agree the paper is well-written and that the use of semi-supervised learning within DRE-BO is novel and empirically promising. Key concerns raised include theoretical justifications for the proposed setting (particularly the validity of semi-supervised learning in BO where data distributions are induced by the algorithm) and limited exploration analysis beyond regret curves. Additional concerns involve hyperparameter sensitivity and missing comparisons to broader BO baselines.

The authors provided a thorough and thoughtful rebuttal, clarifying the setup (especially regarding fixed-size pools), empirical justifications for their choices, and additional insights on regret-based exploration evaluation. Reviewers acknowledged these clarifications and maintained weak accept scores, suggesting the paper makes a useful methodological contribution despite some limitations. Therefore, I recommend acceptance.
Decision: Accept (poster)