Explaining Hyperparameter Optimization via Partial Dependence Plots
Authors: Julia Moosbauer
Julia Herbinger
Giuseppe Casalicchio
Marius Lindauer
Bernd Bischl
Venue: NeurIPS 2021 Poster (2021)
Keywords: Automated machine learning
hyperparameter optimization
interpretable machine learning
partial dependence plots

Abstract: Automated hyperparameter optimization (HPO) can support practitioners to obtain peak performance in machine learning models.
However, there is often a lack of valuable insights into the effects of different hyperparameters on the final model performance.
This lack of explainability makes it difficult to trust and understand the automated HPO process and its results.
We suggest using interpretable machine learning (IML) to gain insights from the experimental data obtained during HPO with Bayesian optimization (BO).
BO tends to focus on promising regions with potential high-performance configurations and thus induces a sampling bias.
Hence, many IML techniques, such as the partial dependence plot (PDP), carry the risk of generating biased interpretations.
By leveraging the posterior uncertainty of the BO surrogate model, we introduce a variant of the PDP with estimated confidence bands.
We propose to partition the hyperparameter space to obtain more confident and reliable PDPs in relevant sub-regions.
In an experimental study, we provide quantitative evidence for the increased quality of the PDPs within sub-regions.

Explaining Hyperparameter Optimization via Partial Dependence Plots
Julia Moosbauer', Julia Herbinger', Giuseppe Casalicchio, Marius Lindauer, Bernd Bischl Department of Statistics, Setwii-Maximilianng Institute of Information Processing. Leibniz University Hannover, Hannover, Germany {julia maliaa.,oobbaee, , julia herbinggr,,,, giuseppe.. caa.eeppe.aallccchii.... bernd. bischh uni umi-muenchen.c lindauer@tnt. mni-hannover.com de
Munich, Munich, Germany
Abstract
Automated hyperparameter optimization (HPO) can support practitioners to obtain peak performance in machine learning models. However, there is often lack of valuable insights into the effects of different hyperparameters on the final model performance. This lack of explainability makes it difficult trust and understand the automated HPO process and its results. We suggest using interpretable machine learning (IML) to gain insights from the experimental data obtained during HPO with Bayesian optimization (BO). BO tends to focus on promising regions with potential seigh-erformance configurations and thus induces a sampling bias. Hence, many IML techniques, such as the partial dependence plot (PDP). carry the risk of generating biased interpretations. By leveraging the posterior uncertainty of the BO surrogate model, we introduce variant of the PDP with estimated confidence bands. We propose partition the hyperparameter space to obtain more confident and reliable PDPs in relevant sub-regions. In an experimental study, we provide quantitative evidence for the increased quality the PDPs within sub-regions.
Introduction
Most machine learning (ML) algorithms are highly configurable. Their hyperparameters must be chosen carefully, as their choice often impacts the model performance. Even for experts, it can be challenging to find well-performing hyperparameter configurations. Automated machine learning (AutoML) systems and methods for automated HPO have been shown to yield considerable efficiency compared manual tuning by human experts (Snock al., 2012). However, these approaches mainly return a well-performing configuration and leave users without insights into decisions of the optimization process. Questions about the importance of hyperparameters or their effects on the resulting performance often remain unanswered. Not all data scientists trust the outcome of an AutoML. system due the lack of transparency (Drozdal et al., 2020]. Consequently, they might not deploy AutoML model, despite all performance gains. Providing insights into the search process may help increase trust and facilitate interactive and exploratory processes: A data scientist could monitor the AutoML process and make changes to it (e.g., restricting or expanding the search space) already during optimization to anticipate unintended results.
Transparency, trust, and understanding of the inner workings of an AutoML system can be increased by interpreting the internal surrogate model of an AutoML approach. For example, BO trains surrogate model to approximate the relationship between hyperparameter configurations and model performance. is used to guide the optimization process towards the most promising regions of the hyperparameter space. Hence, surrogate models implicitly contain information about the influence of
'These authors contributed equally to this work.
35th Conference on Neural Information Processing Systems (NcurIPS 2021).

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hyperparameters. If the interpretation of the surroaate matches with a data scientist's 's expectation, confidence in the correct functioning of the system may be increased. If these do not match, provides an opportunity to look either for bugs in the code or for new theoretical insights.
We propose analyze surrogate models with methods from IML to provide insights into the results of HPO. In the context of BO, typical choices for surrogate models are flexible, probabilistic black- box models, such as Gaussian processes (GP) or random forests. Interpreting the effect of single hyperparameters on the performance of the model to be tuned is analogous to interpreting the feature effect of the black-box surrogate model. We focus on the PDP [Friedman 2001] which is a widely-used method to visualize the average marginal effect of single features on a black-box model's prediction. When applied to surrogate models, they provide information on how a specific hyperparameter influences the estimated model performance. However, applying PDPs out of the box to surrogate models might lead to misleading conclusions. Efficient optimizers such as BO tend to focus on exploiting promising regions of the hyperparameter space while leaving other regions less explored. Therefore, a sampling bias in input space introduced, which in turn can lead to poor fit and biased interpretations in underexplored regions of the space.
Contributions: We study the problem of sampling bias in experimental data produced by AutoML systems and the resulting bias of the surrogate model and assess its implications on PDPs. We then derive an uncertainty measure for PDPs of probabilistic surrogate models. In addition, we propose a method that splits the hyperparameter space into interpretable sub-regions of varying uncertainty to obtain sub-regions with more reliable and confident PDP estimates. In the context of BO, we provide evidence for the usefulness of our proposed methods on synthetic function and in an experimental study in which we optimize the architecture and hyperparameters of deep neural network. Our Supplementary Material provides (A) more background related to uncertainty estimates, (B) notes on how our methods are applied to hierarchical hyperparameter spaces, (C) details on the experimental setup and more detailed results, (D) a link to the source code.
Reproducibility and Open Science: The implementation of the proposed methods as well as reproducible scripts for the experimental analysis are provided in public git-repository
Background and Related Work
Recent research has begun to question whether the evaluation of an AutoML system should be purely based on the generated models' predictive performance without considering interpretability (Hutter jet al. 2014a, Plisterer et al. 2019 Freitas 2019, Xanthoooulos et al., 2020). Interpreting Au systems can be categorized as (1) interpreting the resulting ML model on the underlying dataset, or (2) interpreting the HPO process itself. In this paper, we focus on the latter.
L : A- + IR be a black-box cost function, mapping hyperparameter configuration A = (Ai,..., Aa) to the model errof obtained by learning algorithm with configuration X. The hyperparameter space may be mixed, containing categorical and continuous hyperparameters. The goal of HPO is to find X E arg mİnxEA c(A). Throughout the paper, we assume that a surrogate model é A - given as approximation to c. If the surrogate is assumed to be a GP, č(A) is random variable following Gaussian posterior distribution. In particular, for any finite indexed family of hyperparameter configurations (X(1) A (8)) AA, the vector of estimated performance values is Gaussian with
posterior mean vi = (m (A (0) i= ) and covariance K (i (A(0, A(D)
Hyperparameter Importance. Understanding which hyperparameters influence model performance can provide valuable insights into the tuning strategy [Probst et al. 2019). To quantify relevance of hyperparameters, models that inherently quantify feature relevance - c.g., GPs with ARD kernel Neil 1996 can be used as surrogate models. Hutter et al. [014a] quantified the importance of hyperparameters based on a random forest fitted on data generated by BO, for which the importance of both the main and the interaction effects of hyperparameters was calculated by functional
ANOVA approach. Similarly, Sharma et al. 2019 quantified the hyperparameter importance of
Thhere exist various implementations Greenwell 2017
Pedregosa et al. 2011
). extensions Greenwell
ct al. 2018 Goldstein et al. 2015 and applications |Fiedman and Mculman 2003 Cutler al. 2007
https://giithe //github.. com/ eede 1ma/paper___. 2021_ xautom]
Typically, the model error is estimated via cross- validation hold-out testing.

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residual neural networks. These works highlight how useful it is to quantify the importance of hyperparameters. However, importance scores do not show how a specific hyperparameter affects the model performance according to the surrogate model. Therefore, we propose to visualize the assumed marginal effect of hyperparameter. A madel-agnostic interpretation method that can be used for this purpose is the PDP.
PDPs for Hyperparameters. Let S C. {1, .... df denote an index set of features, and let C {1, 2, d} \ S be its complement. The partial dependence (PD) function [Friedman 2001 of
c: A - IR for hyperparameter(s) S is defined a
(s(As) == Exx [c(A)] S c(As, Ac) dl((A)..
(I)
When analyzing the PDP of hyperparameters, model performance uniformly across the hyperparameter space. In line with prior work [Hutter et al. 2014a], we therefore assume P to be the uniform distribution over Ac- Computing es (xs) exactly is usually not possible because c is unknown and expensive to evaluate in the context of HPO. Thus, the posterior mean in of the probabilistic surrogate model é(\) is commonly used as a proxy for c. Furthermore, the integral may not be analytically tractable for arbitrary surrogate models é. Hence, the integral is approximated by Monte Carlo integration, i.e.,
are usually interested in how their values Ag impact
ēs (As) = 1 1=1 mn (As, A0
Σ
R
(2)
for a sample (A
P(A)). in (AS, represents the marginal effect of Äs for one
i=1....77
specific instance i. Individual conditional expectation (ICE) curves [Goldstein et al 2015 visualize against Ag for
the marginal effect of the i-th observation by plotting the value of m (As, (1)
set of grid points A E As. g E {1,.... G}. Analogously, the PDP visualizes (s(As) against the
grid points. Following from Eq. 4 the PDP visualizes the average over all ICE curves. In HPO, the marginal predicted performance
related concept. Instead of approximating the integral via Monte Carlo, the integral over è is computed exactly. Hutter et al. [2014a] propose an efficient approach to compute this integral for random forest surrogate models.
Uncertainty Quantification in PDPs. Quantifying the uncertainty of PDPs provides additional information about the reliability of the mean estimator. Hutter et al. [2014a] quantified the model uncertainty specifically for random forests as surrogates in BO by calculating the standard deviation of the marginal predictions of the individual trees. However, this procedure is not applicable to general probabilistic surrogate models, such as the commonly used GP. There are approaches that quantify the uncertainty for ML models that do not provide uncertainty estimates out-of-the-box. (Catri and Bailey (016) suggested a bootstrap approach for tree ensembles to quantify the uncertainties of effects based on PDPs. Another approach quantify the uncertainty of PDPs is to leverage the ICE curves. For example, Greenwell [017] implemented a method that marginalizes over the mean + standard deviation of the ICE curves for each grid point. However, this approach quantifies the underlying uncertainty of the data at hand rather than the model uncertainty, as explained in Appendix 1. A model-agnostic estimate based uncertainty estimates for probabilistic models is missing far.
Subgroup PDPs. Recently, a new research direction concentrates on finding more reliable PDP estimates within subgroups of observations. Molnar et al.) [2020) focused on problems in PDP estimation with correlated features. To that end, they apply transtormation trees to find homogeneous subgroups and then visualize PDP for each subgroup. Grömping (0200) looked at the same problem and also uses subgroup PDPs, where ICE curves are grouped regarding a correlated feature. Britton 2019] applied a clustering approach to group ICE curves to find interactions between features. However, none of these approaches aim at finding subgroups where reliable PDP estimates have low uncertainty. Additionally, to the best of our knowledge, nothing similar exists for analyzing experimental data created by HPO.
"To keep notation simple, we denote e(A) as function of two arguments (As, Aa) to differentiate compo- nents in the index set S from those in the complement. The integral shall be understood multiple integral of e where Aj. j E C. are integrated out.
"Grid points are typically chosen as an equidistant grid or sampled from P(As). The granularity G is chosen by the user. For categorical features, the granularity typically corresponds to the number of categories.
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Figure I: Illustration the sampling bias when optimizing the 2D Styblinski Tang function with BO and the Lower Confidence Bound (LCB) acquisition function a(>) r)()) r 8()) for r = 0.1 (left) and r = 2 (middle) vs. data sampled uniformly at random (right).
Figure 2: The two horizontal cuts (left) yield two ICE curves (right) showing the mean prediction and uncertainty band against A for é with T = 0.1 on the 2D Styblinski-Taie function. The upper ICE curve deviates more from the true effect (black) and shows higher uncertainty.
3 Biased Sampling in HPO
Visualizing the marginal effect of hyperparameters of surrogate models via PDPs can be misleading. We show that this problem is due to the sequential nature of BO, which generates dependent instances (i.e., hyperparameter configurations) and thereby introduces sampling and resulting model bias. To save computational resources in contrast to grid search or random search, efficient optimizers like BO tend to exploit promising regions of the hyperparameter space while other regions are less explored (see Figure Consequently, predictions of surrogate models are usually more accurate with less uncertainty in well-explored regions and less accurate with high uncertainty under-explored regions. This model bias also affects the PD estimate (see Figure ICE curves may be biased and less confident they are computed in poorly-learned regions where the model has not seen much data before. Under the assumption of uniformly distributed hyperparameters, poorly-learned regions are incorporated the PD estimate with the same weight as well-learned regions. ICE curves belonging to regions with high uncertainty may obfuscate well-learned effects of ICE curves belonging to other regions when they are aggregated PD estimate. PDPs visualizing only the mean estimator of Eq. do not provide insights into the reliability of the PD estimate and how it is affected by the described model bias.
PDP. Hence, the model bias may also lead to a less reliable
4 Quantifying Uncertainty in PDPs
Pointwise uncertainty estimates of probabilis- tie model provide insights into the reliability of the prediction é(A) for specific configuration A. This uncertainty directly correlates with how explored the region around À is. Hence, includ- ing the model's 's uncertainty structure into the PD estimate enables users to understand in which regions the PDP is more reliable and which parts of the PDP must be cautiously interpreted We now extend the PDP of Eq. (2) to probabilistic surrogate models (e.g., GP). Let As be (Pc)) sample that is used to compute the Monte
High Sunpling Blas
Modum Sampling Bias
Law Sampping Blas
0-
-50
5.0
5.500
5.50
25 50
Figure 3: PDPs (blue) with confidence bands for surrogates trained on data created by BO and LCB with r 0.1 (left), i.i.d. dataset (right) vs. the true PD (black).
fixed grid point and { (6)
(middle) and uniform
iml.....
Carlo estimate of Eq.
2). The vector of predicted performances at the grid point As is and with (posterior) mean in (As) : (m (As. > i=l...
(As) (e As. 1(0
i=l....
"Note that we aim at representing model uncertainty in PD estimate, and not the variability of the mean prediction (see Appendix A.I for detailed justification).

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a (posterior) covariance R (As) == (k
( (()ss))))) (ss)) i.j=l...,." Thus, êg (As)
Ei=1 (As, X is random variable itself. The expected value of ê3 (As) corresponds to the PD of the posterior mean function vit at As, i.e.:
1
E i=1 n
(As:
- n Σ m (As. (3)
ms (As)
Ea css (As)]
Ea
The variance of êg (As) is
(As) = (As)) = Ve 2 T e)s
-
-
(As) 1.
(4)
For the above estimate, it is important that the kernel is correctly specified such that the covariance structure is modeled properly by the surrogate model. Eq. treating the pairwise covariances as unknown, i.c.:
44) can be approximated empirically by
1 TI Σ K (As),
S (As)
(5)
In Appendix A.2, we show empirically that this approximation is less sensitive to kernel misspecifi- cations. Please note that the variance estimate and the mean estimate can also be applied other probabilistic models, such as GAMLSS transformation trees, or a random forest. An example for PDPs with uncertainty estimates is shown in Figure for different degrees of sampling bias.
5 Regional PDPs via Confidence Splitting
As discussed in Section 5 (efficient) optimization may imply that the sampling is biased, which in turn produce misleading interpretations when IML is naively applied. We now aim to identify sub- regions A' C A of the hyperparameter space in which the PD can be estimated with high confidence, and separate those from sub-regions in which it cannot be estimated reliably. In particular, we identify sub-regions in which poorly-leamed effects do not obfuscate the well-learned effects along each grid point, thereby allowing the user to draw conclusions with higher confidence. By partitioning the entire hyperparameter space through a tree-based approach into disjoint and interpretable sub-regions, a more detailed understanding of the sampling process and hyperparameter effects achieved. Users can either study the hyperparameter effect of a (confident) sub-region individually or understand the pplooation-expploitatiio sampling HPO by considering the complete tree structure. The result of this procedure for a single split is shown in Figure 5
The PD estimate on the entire hyperparameter space A is computed by sampling the Monte Carlo estimate (A )EN ~ P(Ac), N == {1.2. n). We now introduce the PD estimate on sub- region N' A simply as (A DaE only using A' {i E N MeAae Since we are interested in the marginal effect of the hyperparameter(s) S at each Àg E As. we will usually visualize the PD for the whole range As. Thus, all obtained sub-regions should of the form A' As A/c with Ao C Ac. This corresponds to an average of ICE curves in the set i E N'. The pseudo-code to partition a hyperparameter (sub-)space A and corresponding sample (A
(2 JEEN Ac, / C (1,..., n. into two child regions is shown in A Algorithm [ This splitting is recursively applied in CART like procedure [Beiman et al. 19845) to expand a full tree structure, with the usual stopping criteria (e.g., maximum number of splits, minimum size of region, or a minimum improvement in each node). In each leaf node, the sub-regional PDP and its corresponding uncertainty estimate are computed by aggregating over all contained ICE curves.
The criterion to evaluate a specific partitioning is based on the idea of grouping ICE curves with similar uncertainty structure. To be more exact, evaluate the impurity of PD estimate sub-region A' with the help of the associated set of observations N' = {i E N eAe, also referred
as nodes, as follows: For each grid point As, we use the L2 loss in L (\s to evaluate how the
$ 'Generalized additive models for location, scale and shape
"Clsssification and regression trees
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Entire Hyperperameter Space
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0.15
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Figure ICE curves of 8 of As for the left (green) and right (blue) sub-region after the first split. The darker lines represent the respective PDPs. The orange vertical line marks the value Äs of the optimal configuration.
Figure 5: Example of two estimated PDPs (blue line) and 95% confidence bands after one parti- tioning step. The orange vertical line is the value of As from the optimal configuration, the black curve is the true PD estimate ey (As).
uncertainty varies across all ICE estimates i N" using Si (As) WT EiEr 8 (As, (As:10)) and aggregate the loss L (As,N") over all grid points in (N²):
C (Ass.N)= = = (5² (As: S (As))² and Rez(N") = E g=1 C(A ,N'. (6)
Hence, we measure the pointwise Ly-distance between ICE curves of the variance function
Algorithm 1: Tree-based Partitioning
8 (As, , (0) and its PD estimate 22 (As)
input: NV
within sub-region N'. This seems reasonable, as ICE curves well-explored regions of search space should, on average, have lower uncertainty than those in less less-cxplored regions. However, since only split according hy- perparameters in C' but not in S, the partition- ing does not cut off less explored regions w.r.t. Ag. Thus, the chosen split criterion groups ICE curves of the uncertainty estimate such that receive sub-regions associated with low costs c (and thus high relevance for user) to be more confident in well-explored regions of As and less confident in under-explored regions. Fig- ure shows that ICE curves of the uncertainty measure with high uncertainty over the entire range of Às are grouped together (right sub-region). Those with low uncertainty close to the optimal configuration Äs and increasing uncertainties for less suitable configurations are grouped together by curve similarities in the left sub-region. The respective PDPs are illustrated in Figure 5 where the confidence band in the left sub-region decreased compared the confidence band of the global PDP especially for grid points close to the optimal value of As. Hence, by grouping observations with similar ICE curves of the variance function, resulting sub-regional PDPs with confidence bands provide the user with the information of which sub-regions of Ag are well-explored and lead to more reliable PDP estimates. Furthermore, the user will know which ranges of As can be interpreted reliably and which ones need to be regarded with caution.
for j E Codo
for Every split t on hyperparameter Xj do
Agpl {i E N N
NP -i N 25
I 1) Ri, (NPitt Riz
end for end for Choose (j', E arg mr
Return N7 and Noie for (j,1) jr, ,6,z)
To sum up, the splitting procedure provides interpretable, disjoint sub-regions of the hyperparameter space. Based on the defined impurity measure, PDPs with high reliability can be identified and analyzed. In particular, the method provides more confident and reliable estimates in the sub-region containing the optimal configuration. Which PDPs are most interesting to explore depends on the question the user would like to answer. the main interest lies understanding the optimization and exploring the sampling process, a user might want to keep the number of sub-regions relatively low by performing only few partitioning steps. Subsequently, one would investigate the overall structure of the sub-regions and the individual sub-regional PDPs. If users more interested in interpreting

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hyperparameter effects only in the most relevant sub-region(s), they may want to split deeper and only look at sub-regions that are more confident than the global PDP.
Due to the nature of the splitting procedure, the PDP estimate on the entire hyperparameter space is a weighted average of the respective sub-regional PDPs. Hence, the global PDP estimate is decomposed into several sub-regional PDP estimates. Furthermore, note that the proposed method does not assume numeric hyperparameter space, since the uncertainty estimates as well as ICE and PDP estimates can also be calculated for categorical features. Thus, it is applicable to problems with mixed spaces as long as a probabilistic surrogate model and particularly its uncertainty estimates are available. In Appendix B we describe how our method is applied to hierarchical hyperparameter spaces.
Since the proposed method is an instance of the CART algorithm, finding the optimal split for cate- gorical variable with levels generally involves checking 29 subsets. This becomes computationally infeasible for high values of q. It remains an open question for future work if this can be sped by an optimal procedure as in regression with 1.2 loss [Fisher] 1958) and binary classification [Breiman al. 1984a or by clever heuristic for multiclass classification Wrrigtt and König [2019].
6 Experimental Analysis
In this section, we validate the effectiveness of the introduced methods. We formulate two main hypotheses: First, experimental data affected by the sampling bias lead to biased surrogate models and thus to unreliable and misleading PDPs. Second, the proposed partitioning allows us to identify an interpretable sub-region (around the optimal configuration) that yields a more reliable and confident PDP estimate. In first experiment, we apply our methods to BO runs on synthetic function. In this controlled setup, we investigate the validity of our hypotheses with regards to problems of different dimensionality and different degrees of sampling bias. In a second experiment, we evaluate our PDP partitioning in the context of HPO for neural networks on a variety of tabular datasets.
We assess the sampling bias of the optimization design points by comparing their empirical distribu- tion a uniform distribution via Maximum Mcan Discrepancy (MMD) [Gretton al. 2012 Molnar iet al. 2020]. which is covered in more detail in the Appendix C.I. We measure the reliability of a PDP, i.e., the degree to which user can rely on the estimate of the PD estimate, by comparing it to the true PD es (As) as defined in Eq. ① More specifically, for every grid point X (9) we compute the negative log-likelihood (NLL) of cs (As) under the distribution of ês (As) pointwise for every grid point A The confidence of a PDP is illustrated by the width of its confidence bands vis (As) + 91-a/2 85 (As), with 91-a/2 denoting the (1 a/2)-quantile of standard normal distribution. We measure the confidence by assessing 8g (As) pointwise for every grid point. In
particular, we consider the mean confidence (MC) across all grid points 1 Н. (X(²) as well as
the confidence at the grid point closest to ÅS abbreviated by OC, with Ā being the best configuration evaluated by the optimizer. To evaluate the performance of the confidence splitting, we report the above metries on the sub-region that contains the best configuration evaluated by the optimizer, assuming that this region is of particular interest for user of HPO. PDPs are computed with regards to single features for G = 20 equidistant grid points and n = 1000 Monte Carlo samples.
6.1 BO on a Synthetic Function
We consider the d-dimensional Styblinski-Tamm function C : [-5,5]
R, A
Ei= (X + 16X3 + 52:) for d E f3,5, 8). Since the PD is the same for each dimension i, we only present the effects of A₁. We performed BO with a GP surrogate model with a Matérn-3/2
kernel and the LCB acquisition function a () = m + (()) with different values 7 E {0.1, 1,5]
to control the sampling bias. We compute the global PDP with confidence bands estimated according to Eq. (5) for the GP surrogate model é that was fitted in the last iteration of BO. We ran Algorithm 1
and computed the PDP in the sub-region containing the optimal configuration. All computations were repeated 30 times. Further details on the setup are given in Appendix C.2.1.
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Table 1: The table shows the relative improvement of the MC and the NLL via Algorithm | with 1 and splits, compared to the global PDP along with the sampling bias for a 7 = 0.1 (high), T 2 (medium), and = 5 (low). Results are averaged across 30 replications.
40-
4.5
8
4.0-
10
8.5
5 MC (%)
5 NLL (%)
d
MMD
Pag
Pap
3
Map 3
Figure 6: The figure presents the MC (left) and the NLL (right) for d E (3,5,8) for a high (T 0.1), medium (T= 1), and low (T = 5) sampling bias across 30 replications. With a lower sampling bias, we obtain narrower confi- dence bands and lower NLL.
low (0.18)
7.65
13.64
5.89
10.92
medium (0.51)
12.86
36.92
4.78
7.70
high (0.56)
34.84
2.77
-1.62
low (0.15)
6.63
15.45
2.82
6.05
medium (0 (0.45)
19.67
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11.99
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3.58
9.67
0.84
2.40
medium (0.42)
8.86
23.03
1.51
high (0.56)
6.59
19.84
1.53
4.29
As presented Figure g the PDPs for surrogate models trained on less biased data (measured by the MMD) yield lower values of the NLI., as well as lower values for the MC. Table I shows that a single tree-based split reduces the MC by up to almost 20%, and up to 37% when performing partitioning steps. Additionally, the NLL improves with increasing number of partitioning steps in most cases. The results on the synthetic functions support our second hypothesis that the tree-based partitioning improves the reliability in terms of the NLL and the confidence of the PD estimates. The improvement of the MC is higher for a medium to high sampling bias, compared to scenarios that are less affected by sampling bias. We observe that (particularly for high sampling bias) there are some outlier cases which the NLL worsens. More detailed results are shown in Appendix C.3.1.
6.2 HPO on Deep Learning
In second experiment, we investigate HPO in the context of surrogate benchmark (Eggensperger ct al. 2015] based on the LCBench data [Zimmer et al. 2021]. For each of the 35 different OpenML Vanschoren et al. 2013] classification tasks, LCBench provides access to evaluations of deep neural network on 2000 configurations randomly drawn from the configuration space defined by Auto-PyTorch Tabular (see Table 5 in Appendix C.2). For each task, we trained random forest as an empirical performance model that predicts the balanced validation error of the neural network for given configuration. These empirical performance models serve cheap evaluate objective functions, which efficiently approximate the result of the real-world experiment of running a deep LCBench instance. BO then acts this empirical performance model
learning configuration as its objective 10
For each task, ran BO to obtain the optimal architecture and hyperparameter configuration. Again, we used a GP with a Matérn-3/2 kernel and LCB with 7 = 1. Each BO run was allotted a budget of 200 objective function evaluations. We computed the PDPs and their confidences, which are estimated according to Eq. ( based on the surrogate model è after the final iteration. We performed tree-based partitioning with up to 6 splits based on uniformly distributed dataset of size 1000. All computations were statistically repeated 30 times. Further details are provided in Appendix C.2.2.
For the real-world data example, we focus on answering the second hypothesis, i.e., whether the tree- based Algorithm D improves the reliability of the PD estimates. We compare the PDP sub-regions after 6 splits with the global PDP. We computed the relative improvement of the confidence (MC and OC) and the NLL of the sub-regional PDP compared to the respective estimates for the global PDP. As shown in Table 2. the MC of the PDPs is on average reduced by 30% to 52% depending on the hyperparameter. At the optimal configuration Às the improvement even increases to 50% 62%. Thus, PDP estimates for all hyperparameters are on average independent of the underlying dataset cllarly more confident in the relevant sub-regions when compared to the global PD estimates, especially around the optimal configuration Äs. In addition to the MC, the NLL simultaneously improves. In Appendix C.3.2, we provide details regarding the evaluated metrics on the level of the dataset and demonstrate that our split criterion outperforms other impurity measures regarding MC
Please note that the random forest is only used as surrogate in order to construct an efficient benchmark objective, and not as surrogate in the BO algorithm, where use GP.
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and OC, Furthermore, we emphasize in Appendix C.3.2 the significance of our results by providing comparison naive baseline method.
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betch a20 6.1788
wegft decay
Figure 7: PDP (blue) and confidence band (grey) of the GP for hyperparameter max. number of units (batch size) on the left (right) side. The black line shows the PDP of the meta surrogate model representing the true PDP estimate. The orange vertical line marks the optimal configuration Às. The relative improvements from the global PDP to the sub-regiona PDP after 6 splits are for max. number of units (batch size): MC 61.6% (28.4%), OC 63.5% (62.2%), 8 NLL 48.6% (30.1%).
To further study our suggested method, we now
highlight few individual experiments. We chose one iteration of the shuttle dataset. On the two left plots of Figure 7 we that the true PDP estimate for max. number of unitis is de- creasing, while the globally estimated PDP trend increasing and thus misleading. Although the confidence band already indicates that the PDP cannot be reliably interpreted on the entire hy- perparameter space, it remains challenging to draw any conclusions from it. After perform- ing 6 splits, we receive confident and reliable PD estimate on an interpretable sub-region. The same plots are depicted for the hyperparameter batch size on the right part of Figure 7 This example illustrates that the confidence band might not always shrink uniformly over the entire range of Às during the partitioning, but often particularly around the optimal configuration Äs.
Table 2: Relative improvement of MC, OC, and
NLL on hyperparameter level. The table shows the respective mean (standard deviation) of the average relative improvement over 30 replications for each dataset and 6 splits.
Hyperparameter
MC (%)
OC (%)
NLL (%)
Batch size
40.8 (14.9)
61.9 (13.5)
19.8 (19.5)
Leaming
503.(13.7)
(1.6)
17.9 (20.5)
Max. dropout
49.7 (15.4)
(11.9
(18.))
Max. units
51.1
58.6(12.7)
24.6 (22.0)
Momentum
(14.5)4.5)
(12.7
19.7 (21.7)
Number of layers
30.6
50.9
13.8 (32.5)
Weight decay
36.3
(1.0
11.9 (19.7)
Discussion and Conclusion
In this paper, we showed that partial dependence estimates for surrogate models fitted on experimental data generated by efficient hyperparameter optimization can be unreliable due to an underlying sampling bias. We extended PDPs by an uncertainty estimate to provide users with more information regarding the reliability of the mean estimator. Furthermore, we introduced a brse-based partitioning approach for PDPs, where we leverage the uncertainty estimator to decompose the hyperparameter space into interpretable, disjoint sub-regions. We showed with two experimental studies that we generate, on average, more confident and more reliable regional PDP estimates in the sub-region containing the optimal configuration compared to the global PDP.
One of the main limitations of PDPs is that they bear the risk of providing misleading results if applied correlated data in the presence of interactions, especially for nonparametric models [römping 2020]. However, existing alternatives that visualize the global marginal effect of a feature such as accumulated local effect (ALE) plots (Apley and Zhu 2020 do also not provide a fully satisfying solution to this problem GGömming 2020). As solution to this problem. Grömping 2020] suggests stratified PDPs by conditioning on a correlated and potentially interacting feature to group ICE curves. This idea is in the spirit of our introduced tree-based partitioning algorithm. However, in the context of BO we might assume the distribution in Eq. ① to be uniform and therefore no correlations are
present. Instead of correlated features, we are faced with sampling bias (see Section where

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we observe regions of varying uncertainty. Hence, instead of stratifying with respect to correlated features and aggregating ICE curves in regions with less correlated features, we stratify with respect to uncertainty and aggregate ICE curves in regions with low uncertainty variation. Nonetheless, it might be interesting to compare our approach with approaches based on the considerations made by Grömping 2020 or potentially improved ALE curves.
Another limitation when using single-feature PDPs as in our examples is that hyperparameter interactions are not visible. However, two-way interactions can be visualized by plotting two- dimensional PDPs within sub-regions. Another possibility detect interactions is to look at ICE curves within the sub-regions. If the shape of ICE curves within a sub-region is very heterogeneous, it indicates that the hyperparameter under consideration interacts with one of the other hyperparameters. Hence, having the additional possibility to look at ICE curves of individual observations within sub-region is an advantage compared to other global feature effect plots such as ALE plots (Apley and Zhu 2020), they are not defined on an observational level. While we mainly discussed GP surrogate models on a numerical hyperparameter space in our examples, our methods are applicable to wide variety of distributional regression models and also for mixed and hierarchical hyperparameter spaces, We also considered in Appendix C.3.2 different impurity measures. While the one introduced in this paper performed best in our experimental settings, this impurity measure as well as other components are exchangeable within the proposed algorithm. In the future, we will study our method more complex, hierarchical configuration spaces for neural architecture search.
The proposed interpretation method based on surrogate and consequently does provide insights about what the AutoMI system has learned, which in turn allows plausibility checks and may increase trust in the system. To what extent this allows conclusions on the true underlying hyperparameter effects depends on the quality of the surrogate. How to efficiently perform model diagnostics to ensure a high surrogate quality before applying interpretability techniques subject to future research.
While we focused on providing better explanations without generating any additional experimental data, it might be interesting investigate in future work how confidence and reliability of IML methods can be increased most efficiently when a user is allowed to conduct additional experiments.
Overall, believe that increasing interpretability of AutoML will pave the way for centnned AutoMI. Our vision is that users will be able to better understand the reasoning and the sampling process of AutoML systems and thus either trust and accept the results of the AutoMI system interact with it in a feedback loop based on the gained insights and their preferences. How users can then best interact with AutoML (beyond simple changes of the configuration space) will be left open for future research.
Acknowledgments and Disclosure of Funding
This work has been partially supported by the German Federal Ministry of Education and Research (BMBF) under Grant No. 011S 1836A. The authors of this work take full responsibilities for its content.
10

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References
D. W. Apley J. Zhu. Visualizing the effects of predictor variables in black box supervised learning models. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 82(4): 1086,,,000,,,,, 2020.
B. Bischl, J. Richter, J. Bossek, D. Horn, J. Thomas, and M. Lang- mirmbo: A modular framework for model-based optimization of expensive black-box functions, 2018.
L. Breiman, J. H. Friedman, R. A. Olshen, and C. J. Stone. Classification and Regression Trees. Wadsworth, 1984a. ISBN 0-534-98053-8.
L. Breiman, J. H. Friedman, R. A. Olshen, and C. J. Stone. Classification and Regression Trees. Wadsworth, 1984b.
M. Britton. VINE: Visualizing statistical interactions in black box models. CoRR, abs/ 190400004,,,,, 2019.
Cafri A. Bailey. Understanding variable effects from black box prediction: Quantifying effects tree ensembles using partial dependence. Journal of Data Science, 14(1):67-95, 2016.
D. R. Cutler, T. C. Edwards K. Beard, Cutler, T. Hess, Gibson, and J. Lawler. Random forests for classification in ecology. Ecology. 88(11):2783-277) 2007.
Drozdal, Weisz, Wang, G. Dass, B. Yao, C. Zhao, M. Muller, L. Ju, and Su. Trust AutoML: Exploring information needs for establishing trust automated machine learning systems. Proceedings of the 25th International Conference on Intelligent User Interfaces, pages 297-307, 2020.
K. Eggensperger, Hutter, Hoos, K. Leyton-Brown. Efficient benchmarking hyperparameter optimizers via surrogates. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 29, pages 1114-1120, 2015.
W. D. Fisher. On grouping for maximum homogeneity. Journal of the American statistical Asxociation, 53(244): 789-798, 1958.
A. Freitas. Automated machine learning studying the trade-off hetween predictive accuracy and inter- pretability. In Third IFIP International Cross-Domain Conference for Machine I Learning and Knowledge Extraction (CD-MAKE 2019), volume 11713, pages 48-66. Springer, August 2019.
J. H. Friedman. Greedy function approximation: A gradient boosting machine. Annals of Statistics, pages 1189 1232, 2001.
J. H. Friedman and J. Mculman. Multiple additive regression trees with application in epidemiology. Statistics in medicine, 22(9):1365-13 2003.
A. Goldstein, A. Kapelner, Bleich, and E. Pitkin. Peeking inside the black hox: Visualizing statistical learning with plots of individual conditional expectation. Journal of Computational and Graphical Statistics, 24(1): 44 65, 2015.
B. M. Greenwell. pdp: An R Package for Constructing Partial Dependence Plots. The R Journal, 9(1):421-45 2017.
B. M. Greenwell, B. C. Bochmke, and A. J. McCarthy, A simple and effective model-based variable importance measure. arXiv preprint arXiv: 04755, 2018.
A. Gretton, K. M. Borgwardt, M. Rasch, B. Schölkopf, and A. Smola. kernel two-sample test. Journal of Machine Learning Research, 13(25))723-773, 2012.
U. Grömping. Model-agnostic effects plots for interpreting machine learning models. Report 1/2020, Reports Mathematics, Physics and Chemistry. Department 11, Beuth University of Applied Sciences Berlin, 2020.
Hutter, H. Hoos, and K. Leyton- Brown. An efficient approach for assessing hyperparameter importance. In Proceedings of the 31th International Conference on Machine Learning, ICML, volume 32, pages 754-762. JMLR.org Logg 2014a.
Hutter, L. Xu, H. Hoos, and K. Leyton-Brown. Algorithm runtime prediction: Methods & evaluation. Artificial Intelligence, 206:79-111, 2014b.
Levesque, Durand, Gagné, and Sabourin. Bayesian optimization conditional hyperparameter spaces. In 2017 International Joint Conference on Neural Nenworks, IJCN 2017, Anchorage. AK, USA, May 14-19, 2017, pages 286 293. JEEE, 2017. doi: 00.119/JJCNN.2017..966888 URL httph::: org/10 1109/ IJCNN 2017. 7965867

---

C. Molnar, G. König, B. Bischl. and G. Casalicchio. Model-agnostic feature importance and effects with dependent features- A conditional subgroup approach. CoRR, abs/2006.04628, 2020.
R. M. Neil, Bayesian Learning for Neural Networks. Springer-Verlag Berlin, Heidelberg, 1996. ISBN 0387947248.
F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss,
V. Dubourg, J. Vanderplas, A. Passos, Cournapeau, M. Brucher, M. Perrot, and Duchesnay. Scikit-learn Machine learning in Python. Journal of Machine Learning Research, 12:2825 2830, 2011.
E. Pfisterer, J. Thomas, and B. Bischl. Towards human centered AutoML. CoRR, abs/1911 2019.
P. Probst, A. Boulesteix, and B. Bischl. Tunability: Importance of hyperparameters of machine learning algorithms. Journal of Machine Learning Research, 20:53:1-53:32, 2019.
A. Sharma, N. van Rijn, F. Hutter, and A. Müller. Hyperparameter importance for image classification by residual neural networks. In Discovery Science 22nd International Conference, DS, volume 11828 of Lecture Notes in Computer Science, pages 112-126. Springer, 2019.
J. Snock, H. Larochelle, and Adams. Practical Bayesian optimization of machine learning algorithms. In Advances Neural Information Processing Systems 25, pages 1166,,968,,2 2012.
K. Swersky, D. Duvenaud, J. Snoek, F. Hutter, and M. A. Osborne. Raiders of the lost architecture: Kernels for bayesian optimization in conditional parameter spaces. arXiv: Machine Learning, 2014.
J. Vanschoren, N. Rijn, Bischl, and L. Torgo. Openml: networked science in machine learning. SIGKDD Explor, 15(2):99-60. 60, 2013.
M. N. Wright and König. Splitting on categorical predictors in random forests. Peerl, 7:c6339, 2019.
I. Xanthopoulos, Tsamardinos, V. Christophides, E. Simon, and A. Salinger. Putting the human back in the AutoML loop. Proceedings of the Workshops of the EDBT//CCT 2020 Joint Conference, volume 2578 of CEUR Workshop Proceedings. CEUR-WS orrg. 2020.
L. Zimmer, M. Lindauer, and F. Hutter. Auto-PyTorch Tabular: Multi-fidelity metaleaming for efficient and robust Lu, IEFE TPAMI, 2021. Preprint via Early Access.
12


Reviews:
[{"review_id": "iQXdUjRq3hP", "reviewer": "Reviewer_J9Xw", "summary": "The paper suggests a modification of partial dependence plot for assessing the influence of hyperparameter on the fitting of the model in the context of Bayesian optimization. Bayesian optimization provides a surrogate function that describes the influence of hyperpatameter on the cost function with associated uncertainty. The surrogate function can be used as a proxy of the actual cost function to generate the partial dependence plot. However, this approach usually does not capture the associated uncertainty. Moreover, given Bayesian optimization explores the high-performing hyperparameter space more to find a suitable solution, it samples the hyperparameter space in a biased manner by design, and thus, it is more uncertain in region where less hyperparameters are sampled. Estimating the uncertainty of the partial dependence plot from these regions can bias the partial dependence plot. The authors resolve this issue by partitioning the hyperparameter space in regions based on the certainty of the surrogate function, and show that estimating partial dependence plot from region that has low uncertainty provides a better estimate, and the partial dependence plot becomes more interpretable.", "questions": "", "limitations": "", "rating": 7, "confidence": 4, "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": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "- The paper tackles an interesting problem. Explaining the contribution of hyperparamter choices in complex machine learning models helps the end-users select a hyperparameter value more informatively. Partial dependence plot is an effective way of achieving this. The proposed approach additionally provides uncertainty around the partial dependence plot, and breaks the parameter space down to estimate this uncertainty better.\n- The paper is well written and explores issues with directly estimating the partial dependence plot from surrogate function, and the proposed  modifications well.\n- The paper demonstrates extensively using simulated and real data that the estimated confidence interval drops by a large margin when splitting the hyperparameter space in regions over using the global estimate. This along with graphical representation of the region splits provide additional information on the effect of hyperparameter.\n- The authors mention that the real-world data were modelled with deep neural network but line 290 it is mentioned that they use random forest. Table 2 and Figure 7, howeve\nr, discusses deep neural network.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "The authors are clear about the limitations of their work, i.e., proposed heuristics. The work, to my understanding, does not have any negative societal impact.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "6 hours"}, "extra_text": {}}, {"review_id": "bFTHXON8Kr4", "reviewer": "Reviewer_kioz", "summary": "The paper presents several ideas to help understand and interpret the effects of different hyperparameters on the model chosen by automated machine learning. \n\nThe contributions include proposing an uncertainty measure for PDPs of probabilistic surrogate models including Bayesian optimization and others. To further increase interpretability, the paper suggests a procedure of dividing the hyperparameter space into subspaces/sub-regions, so that the hyperparameter effect of sub-regions can be analyzed individually. \n\nThe experiment results of the d-dimensional Styblinski-Tang function show the PDPs for surrogate models trained on less biased data produces lower values of the NLL and MC. , as well as lower values for the MC. Splitting into sub-regions helps to reduce MC generally. For high sampling bias, MC seems decrease when splitting 1 or 3 times, however, NLL can become worse in some cases. \n\nIn the other experiment, the paper studies HPO for 35 different OpenML classification tasks. The study found that the MC of the PDPs reduces significantly after 6 splits when comparing the PDP in sub-regions with the global PDP.\n ", "questions": "", "limitations": "", "rating": 5, "confidence": 3, "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": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Using Partial Dependence Plots (PDPs) to interpret the effects of hyperparameters is not a new idea. The paper proposes a new uncertainty measure and also the idea of divide the hyperparameter space into sub-regions by constructing a tree structure. The difference with existing work is clear, and incremental, in my opinion. The paper cites many previous studies. \n\nThe paper claims the new method produces \"more confident and more reliable regional PDP estimates in the sub-region containing the optimal configuration compared to the global PDP \". The experiment results confirm that claim by testing on a Synthetic Function and a set of OpenML classification tasks. The baselines in the paper use the same method proposed by the paper. It will be interesting to have results of comparing with other methods, for instance,  Greenwell [2017] or Grömping [2020].  The paper lists the limitation of the proposed approach. \n\nThe paper is well written, overall. In Algorithm 1: Tree-based Partitioning, it is not clear to me how to choose the split t. Hyperparameters can be categorical or continuous, so the optimization problem to computing t can be difficult.\n\nI believe interpretability is an import topic for ML and AutoML. Being able to explain why a model is chosen by AutoML helps to convince people to adopt the model. The conclusion from the study is not surprising for BO based AutoML. I don't feel the proposed method produces significant results.  \n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Yes. ", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "6"}, "extra_text": {}}, {"review_id": "33-Z83gr-e", "reviewer": "Reviewer_APrH", "summary": "This paper focuses on the topic of explaining the hyperparameter optimization (HPO) process usually used in AutoML via partial dependence plots (PDP). In particular, the authors argue that naive methods suffer from sampling bias of HPO, and demonstrate how to quantify uncertainty in the resulting PDP and propose a further improvement by looking at PDP in sub-regions. ", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "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": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Originality: As far as I know the method proposed is original. There are lots of work on interpretable machine learning directly applied on models, but explaining the process of hyperparameter optimization is a relatively less studies while important area.\n\nQuality: The claims made in the paper are supported by the experiment studies. The authors are also honest about discussing both pros and cons of the proposed method.\n\nClarity: Overall it has good clarity. However I think the author needs more discussion in the motivation of the proposed method. See later comments. I do feel that the intro and background are a bit too long. The authors can consider moving the mathematical introduction of PDP to late sections.\n\nSignificance: I believe this topic is of importance to practitioners and data scientists.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "1. One major disadvantage of PDP is that it can provides misleading results for correlated features, which usually happen in practice including in HPO setting. It is definitely worth some discussion in the last section.\n\n2. The proposed method can be applied to explain ml models directly. It's worth to see how it can interpret using one or two examples. ", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "3"}, "extra_text": {}}]

Author rebuttal:

Meta-review: We thank the authors for the detailed clarifications they provided during the rebuttal. Model understanding is an important topic and understanding the impact/role of hyperparameters is still an open question. The reviewers all agreed that this paper makes interesting contributions even though the authors built on top of earlier work that uses partial dependency plots to interpret the role of hyperparameters. The reviewers found that the experiments supported the claims and that the discussion was balanced, covering both pros and cons of the proposed approach.
Decision: Accept (Poster)