Z ERO - SHOT T RANSFER L EARNING FOR G RAY - BOX H YPER - PARAMETER O PTIMIZATION
Anonymous authors
Paper under double-blind review
A BSTRACT
Zero-shot hyper-parameter optimization refers to the process of selecting hyperparameter conﬁgurations that are expected to perform well for a given dataset upfront, without access to any observations of the losses of the target response.
Existing zero-shot approaches are posed as initialization strategies for Bayesian
Optimization and they often rely on engineered meta-features to measure dataset similarity, operating under the assumption that the responses of similar datasets behaves similarly with respect to the same hyper-parameters.
Solutions for zeroshot HPO are embarrassingly parallelizable and thus can reduce vastly the required wallclock time of learning a single model.
We propose a very simple HPO model called Gr ay-box Zero( O )- S hot I nitialization (GROSI) as a conditional parametric surrogate that learns a universal response model by exploiting the relationship between the hyper-parameters and the dataset meta-features directly.
In contrast to existing HPO solutions, we achieve transfer of knowledge without engineered metafeatures, but rather through a shared model that is trained simultaneously across all datasets.
We design and optimize a novel loss function that allows us to regress from the dataset/hyper-parameter pair unto the response.
Experiments ondatasets demonstrate the strong performance of GROSI, compared to conventional initialization strategies.
We also show that by ﬁne-tuning GROSI to the target dataset, we can outperform state-of-the-art sequential HPO algorithms.
I NTRODUCTION
Within the research community, the concentration of efforts towards solving the problem of hyperparameter optimization (HPO) has been mainly through sequential model-based optimization (SMBO), i.e. iteratively ﬁtting a probabilistic response model, typically a Gaussian process (Rasmussen (2003)), to a history of observations of losses of the target response, and suggesting the next hyper-parameters via a policy, acquisition function, that balances exploration and exploitation by leveraging the uncertainty in the posterior distribution (Jones et al.
(1998); Wistuba et al.
Snoek et al.
However, even when solutions are deﬁned in conjunction with transfer learning techniques (Bardenet et al.
; Wistuba et al.
; Feurer et al. (2015)), the performance of
SMBO is heavily affected by the choice of the initial hyper-parameters.
Furthermore, SMBO is sequential by design and additional acceleration by parallelization is not possible.
In this paper, we present the problem of zero-shot hyper-parameter optimization as a meta-learning objective that exploits dataset information as part of the surrogate model.
Instead of treating HPO as a black-box function, operating blindly on the response of the hyper-parameters alone, we treat it as a gray-box function (Whitley et al.
(2016)), by capturing the relationship among the dataset meta-features and hyper-parameters to approximate the response model.
In this paper, we propose a novel formulation of HPO as a conditional gray-box function optimization problem, Section 4, that allows us to regress from the dataset/hyper-parameter pair directly onto the response.
Driven by the assumption that similar datasets should have similar response approximations, we introduce an additional data-driven similarity regularization objective to penalize the difference between the predicted response of similar datasets.
In Section 5, we perform an extensive battery of experiments that highlight the capacity of our universal model to serve as a solution for: (1) zero-shot
HPO as a stand-alone task, (2) zero-shot as an initialization strategy for Bayesian Optimization (BO), (3) transferable sequential model-based optimization.
A summary of our contributions is:
• a formulation of the zero-shot hyper-parameter optimization problem in which our response model predicts upfront the full set of hyper-parameter conﬁgurations to try, without access to observations of losses of the target response; • a novel multi-task optimization objective that models the inherent similarity between datasets and their respective responses; • three new meta-datasets with different search spaces and cardinalities to facilitate the experiments and serve as a benchmark for future work; • an empirical demonstration of the performance of our approach through a battery of experiments that address the aforementioned research aspects, and a comparison against state-of-the-art HPO solutions for transfer-learning.
R ELATED W ORK
The straightforward zero-shot approaches for HPO consist of random search (Bergstra & Bengio (2012)), or simply selecting hyper-parameters that perform well on general tasks (Brazdil et al.
Some recent work has also shown that simply selecting random hyper-parameters from a restricted search space signiﬁcantly outperforms existing solutions, and improves the performance of conventional SMBO approaches (Perrone et al.
The restricted search space is created by eliminating regions that are further away from the best hyper-parameters of the training tasks.
Another prominent direction for zero-shot HPO depends heavily on engineered meta-features, i.e. dataset characteristics (Vanschoren (2018)), to measure the similarity of datasets.
Following the assumption that the responses of similar datasets behave similarly to the hyper-parameters, it has been shown that even the simplest of meta-features (Bardenet et al.
(2013)) improve the performance of single task BO algorithms (Feurer et al.
The target response is initialized with the top-performing hyper-parameters of the dataset nearest neighbor in the meta-feature space.
The shortcomings of using engineered meta-features are that they are hard to deﬁne (Leite & Brazdil (2005)), and are often selected through trial-and-error or expert domain knowledge.
As a remedy, replacing engineered meta-features with learned meta-features (Jomaa et al.
(2019)) compensates for such limitations, by producing expressive meta-features agnostic to any meta-task, such as HPO.
Zero-shot HPO is also posed as an optimization problem that aims to minimize the meta-loss over a collection of datasets (Wistuba et al.
(2015a))
by replacing the discrete minimum function with a differentiable softmin function as an approximation.
The initial conﬁgurations boost the single task BO without any meta-features.
In (Wistuba et al.
(2015b))
, hyper-parameter combinations are assigned a static ranking based on the cumulative average normalized error, and dataset similarity is estimated based on the relative ranking of these combinations.
Winkelmolen et al.
(2020) introduce a Bayesian Optimization solution for zero-shot HPO by iteratively ﬁtting a surrogate model over the observed responses of different tasks, and selecting the next hyper-parameters and datasets that minimize the aggregated observed loss.
Aside from zero-shot HPO, transfer learning is employed by learning better response models (Wistuba et al.
based on the similarity of the response.
Feurer et al.
(2018) propose an ensemble model for BO by building the target response model as a weighted sum of the predictions of base models as well as the target model.
In addition to the transferable response models, Volpp et al.
(2019) design a transferable acquisition function as a policy for hyper-parameter optimization deﬁned in a reinforcement learning framework.
As a replacement to the standard Gaussian process, Perrone et al.
(2018) train a multi-task adaptive Bayesian linear regression model with a shared feature extractor that provides context information for each independent task.
In contrast to the literature, we formulate the problem of zero-shot HPO as a gray-box function optimization problem, by designing a universal response model deﬁned over the combined domain of datasets and hyper-parameters.
We rely on the embeddings to estimate the similarities across datasets and design a novel multi-task optimization objective to regress directly on the response.
This allows us to delineate from the complexity paired with Bayesian uncertainty, as well as the trouble of engineering similarity measures.
H YPER - PARAMETER O PTIMIZATION
Consider a dataset D
= { (cid:0) x
(Train) , y (Train) (cid:1) , (cid:0) x
(Val) , y (Val) (cid:1) , (cid:0) x
(Test) , y (Test) (cid:1) } for a supervised learning task, with training, validation and test splits of predictors x ∈ X and targets y ∈ Y .
We aim at training a parametric approximation of the target using ˆ y :
=
f
( θ, λ ) :
X → Y , where θ ∈ Θ denotes the parameters and λ ∈ Λ its hyper-parameters, by minimizing a loss function L : Y × Y → R as:
[Equation]
We hereafter denote the validation error as the response (cid:96) ( λ ) :
= L (cid:0) y
(Val) , f (cid:0) x
(Val) ; θ ∗ , λ (cid:1)(cid:1) .
Unfortunately, a direct optimization of the response (cid:96) ( λ ) in terms of λ is not trivial, because θ ∗ is the result of the minimization problem and its gradients with respect to λ are not easy to compute.
Instead, in order to learn the optimal hyper-parameters λ we train a probabilistic surrogate ˆ (cid:96) ( λ ; β ) : Λ × B → R parameterized by β ∈ B , with B as the space of response model parameters, that minimizes the log-likelihood of approximating the response (cid:96) ( λ ) over a set of K evaluations
[Equation]
( λ K )) } .
We denote P as the probability of estimating the response given a surrogate model.
Given the surrogate, the next hyper-parameter to be evaluated λ (next) is computed by maximizing an acquisition function
A (e.g. EI (Mo ˇ ckus (1975)) as:
[Equation]
max
[Equation]
M ETA -L EARNING OF C ONDITIONAL G RAY -B OX S URROGATES
Let us deﬁne a collection of T datasets as (cid:8) D (1) , . . .
, D ( T ) (cid:9) and let (cid:96) ( t ) ( λ ) measure the response of the hyper-parameter λ on the t -th dataset D ( t ) .
Furthermore, assume we have previously evaluated
K ( t ) many hyper-parameters λ ( t ) k , k ∈ { 1 , . .
, K ( t ) } on that particular dataset.
We condition the surrogate ˆ (cid:96) to capture the characteristics of the t -th dataset, by taking as input the meta-features representation of the dataset as φ ( t ) .
Therefore, a dataset-aware surrogate can be trained using meta-learning over a cumulative objective function O ( β ) as:
[Equation]
4. T HE M ETA - FEATURE E XTRACTOR
Introducing engineered meta-features has had a signiﬁcant impact on hyper-parameter optimization.
However, learning meta-features across datasets of varying schema in a task-agnostic setting provides more representative characteristcs than to rely on hard-to-tune empirical estimates.
The meta-feature extractor is a set-based function (Zaheer et al.
(2017)) that presents itself as an extended derivation of the Kolmogorov-Arnold representation theorem (Krkov´a (1992)), which states that a multi-variate function φ can be deﬁned as an aggregation of univariate functions over single variables, Appendix B.
Each supervised (tabular) dataset D ( t ) :=
(cid:0) x ( t ) , y ( t ) (cid:1) consists of instances x ( t ) ∈ X ∈ R N × M and targets y ( t ) ∈ Y ∈ R N × C such that N , M and C represent the number of instances, predictors and targets respectively.
The dataset can be further represented as a set of smaller components, set of sets , D ( t ) =
(cid:110)(cid:16) x ( t ) i,m , y ( t ) i,c (cid:17) | m ∈ { 1 , . .
, M } , i ∈ { 1 , . . .
, N } , c ∈ { 1 , . . .
, C } (cid:111) .
A tabular dataset composed of columns (predictors, targets) and rows (instances) is reduced to single predictortarget pairs instead of an instance-target pair.
Based on this representation, a meta-feature extractor parameterized as a neural network (Jomaa et al.
(2019)), is formulated in Equation 4.
For simplicity of notation, we drop the superscript ( t ) unless needed.
[Equation]
with f : R 2 → R K f , g : R K f
→ R K g and h :
R K g
→ R K represented by neural networks with K f , K g and K output units, respectively.
The objective of this formulation is to capture the correlation between each variable (predictor) and its assigned target.
In contrast to existing parametric approaches (Edwards & Storkey (2016)) which are bound to ﬁxed-datset schemas making them ill-suited for real world applications, this meta-feature extractor representation is agnostic to dataset schema, mainly due to its permutation invariance property.
It is important to note that φ is permutation invariant, i.e. unaffected by any permutation on the input, which allows us to obtain the same output for the same multi-variate data regardless of the order of input.
4. T HE A UXILIARY D ATASET I DENTIFICATION T ASK
The meta-features extracted by optimizing Equation 3 alone lack any semantic meaning, and thus are not really considered meta-features, as much as they are simple latent represenations, as they might result in dissimilar datasets being colocated in the meta-feature space, i.e. not versatile enough for proper distinction between the datasets.
The dataset identiﬁcation task, introduced previously as dataset similarity learning (Jomaa et al.
(2019)), ensures that the meta-features of similar datasets are colocated in the meta-feature space, providing more expressive and distinct meta-features for every dataset.
Let p D a joint distribution over dataset pairs such that ( D ( t ) , D ( q ) , s ) ∈
T × T × { 0 , 1 } with s being a binary dataset similarity indicator.
We deﬁne a classiﬁcation model
ˆ s : T × T → R + that provides an unnormalized probability estimate for s being 1 , as follows:
[Equation]
where Z : R k × R k
→ R + represents any distance metric, and γ a tuneable hyper-parameter.
For simplicity, we use the Euclidean distance to measure the similarity between the extracted metafeatures, i.e. Z (cid:0) φ ( t ) , φ ( q ) (cid:1) =
(cid:107) φ ( t ) − φ ( q ) (cid:107) , and set γ = 1 .
The classiﬁcation model is trained by optimizing the negative log likelihood:
[Equation]
with p D + as the distribution of similar datasets, p D + = { ( D ( t ) , D ( q ) , s ) ∼ p D | s = 1 } , and p D − as the distribution of dissimilar datasets, p D − = { ( D ( t ) , D ( q ) , s ) ∼ p D | s = 0 } .
Similar datasets are deﬁned as multi-ﬁdelity subsets (batches) of each dataset.
4. D ATA - DRIVEN S IMILARITY R EGULARIZATION
Our surrogate differs from prior practices, because we do not consider the response to be entirely black-box.
Instead, since we know the features and the target values of a dataset even before evaluating any hyper-parameter, we model a gray-box surrogate by exploiting the dataset characteristics φ when approximating the response (cid:96) .
As a result, if the surrogate faces a new dataset that is similar to one of the T datasets from the collection it was optimized (i.e. similar meta-features φ extracted directly from the dataset), it will estimate a similar response.
Yet, if we know apriori that two datasets are similar by means of the distance of their meta-features, we can explicitly regularize the surrogate to produce similar response estimations for such similar datasets, as:
[Equation]
Overall we train the surrogate model to estimate the collection of response evaluations and explicitly capture the dataset similarity by solving the following problem, Equation 8, end-to-end , where α ∈ R controls the amount of similarity regularization, and δ ∈ R controls the impact of the dataset identiﬁcation task:
[Equation]
N ETWORK A RCHITECTURE
Our model architecture is divided into two modules, ˆ l :=
φ ◦ ψ , the meta-feature extractor φ , and the regression head ψ .
The meta-feature extractor φ : R 2 → R K h is composed of three functions,
Equation 4, namely f , g and h .
The regression head is also composed of two functions, i.e. ψ : ψ 1 ◦ ψ 2 .
We deﬁne by ψ 1 : R K h × Λ → R K ψ 1 as the function that takes as input the meta-feature/hyperparameter pair, and by ψ 2 : R K ψ 1 → R the function that approximates the response.
Finally, let
Dense(n) deﬁne one fully connected layer with n neurons, and ResidualBlock(n,m) be m × Dense(n) with residual connections (Zagoruyko & Komodakis (2016)).
We select the architecture presented in Table 1 based on the best observed average performance on the held-out validation sets across all meta-datasets, Appendix E.
[Table]
E XPERIMENTS
Our experiments are designed to answer three research questions 1 :
• Q1 :
Can we learn a universal response model that provides useful hyper-parameter initializations from unseen datasets without access to pervious observations of hyper-parameters for the dataset itself?
• Q2 :
Do the proposed suggestions serve as a good initialization strategy for existing SMBO algorithms?
• Q3 :
Aside from zero-shot HPO, does the performance of our method improve by reﬁtting the response model to the observations of the hyper-parameters for the target dataset and how well does our approach compare to state-of-the-art methods in transfer learning for HPO?
5. T RAINING P ROTOCOL
In Algorithm 1 we describe the pseudo-code for optimizing our response model via standard metalearning optimization routines.
We use stochastic gradient descent to optimize the internal model, and Adam optimizer (Kingma & Ba (2015)) to optimize the outer loop.
We set the number of inner iterations to v = 5 , and use a learning rate of 0 .
001 for both optimizers.
We use a batch size oftasks sampled randomly with each iteration.
The code is implemented in Tensoﬂow (Abadi et al.
Ther performance of the various optimizers is assessed by measuring the regret, which represents the distance between an observed response and the optimal response on a response surface.
For hyper-parameter optimization, the meta-datasets are provided beforehand, consequently the optimal response is known.
Since we normalize the response surfaces between (0 , 1) , we observe the normalized regret.
The reported results represent the average over 5-fold cross-validation split for each meta-dataset, with 80 meta-train, 16 meta-valid, and 24 meta-test sets.
5. M ETA - DATASET
We create three meta-datasets by using 120 datasets chosen from the UCI repository (Asuncion & Newman (2007)).
We then create the meta-instances by training a feedforwad neural network and report the validation accuracy.
Each dataset is provided with a predeﬁned split 60% train, 15% validation and 25% test instances.
We train each conﬁguration for 50 epochs with a learning rate of 0 .
The hyper-parameter search space is described in Table 2.
[Table]
The layout hyper-parameter (Jomaa et al.
(2019)) corresponds to the overall shape of the neural network
[Table]
and provides information regarding the number of neurons in each layer.
For example, all the layers in the neural network with a (cid:3) layout share the same number of neurons .
We introduce an additional layout, (cid:52) , where the number of neurons in each layer is successively halved until it reaches the corresponding number of neurons in the centeral layer, then doubles successively.
We also use dropout (Srivastava et al.
and batch normalization (Ioffe & Szegedy (2015)) as regularization strategies, and stochastic gradient descent (GD), ADAM (Kingma & Ba (2015)) and
RMSProp (Tieleman & Hinton (2012)) as optimizers.
SeLU (Klambauer et al.
(2017)) represents the self-normalizing activation unit.
The search space consists of all possible combinations of the hyper-parameters.
After removing redundant conﬁgurations, the resulting meta-datasets have 256 , 288 and 324 unique conﬁgurations respectively.
For the purposes of our algorithm, we need access to the datasets used to generate the meta-features 3 .
Further details are available in Appendix C.
5. B ASELINES
We introduce two sets of baselines to evaluate against the different aspects of our approach:
Z ERO -S HOT H YPER - PARAMETER O PTIMIZATION
• Random search (Bergstra & Bengio (2012)) is the simplest approach where the hyperparameters are selected randomly.
• Average Best represents the top hyper-parameters that had on average the highest score on the meta-train datasets.
• NN- (cid:104) METAFEATURE (cid:105) (Feurer et al. (2015)) refers to the process of selecting the top performing hyper-parameters of the nearest neighboring dataset based on their metafeatures.
We use two sets of well-established engineered meta-features, which we refer to as MF1 (Feurer et al.
and MF2 (Wistuba et al.
(2016)), as well as learned metafeatures (Jomaa et al.
(2019)), which we denote by D2V .
The similarity is measured by the Euclidean distance.
Ellipsoid (Perrone et al.
(2019)) is also a random search approach, however the hyperparameters are sampled from a hyper-ellipsoid search space that is restricted to encompass as many optimal hyper-parameters from the training dataset as possible.
S EQUENTIAL - MODEL BASED O PTIMIZATION FOR T RANSFER L EARNING
• GP (Rasmussen (2003)) is standard Gaussian process response model with a Matern 3/2 and automatic relevance determination.
This approach is trained independently on each dataset.
• TST-R
(Wistuba et al.
(2016)) is a two-stage approach where the parameters of the target response model are adjusted via a kernel-weighted average based on the similarity of the hyper-parameter response between the target dataset and the training datasets.
We also evaluate the variant of this approach that relies on meta-features, by replacing the engineered meta-features with learned meta-features, TST-D2V .
• RGPE (Feurer et al.
(2018)) is an ensemble model that estimates the target response model as a weighted combination of the training datasets’ response models and the target itself.
The weights are assigned based on a ranking loss of the respective model.
• ABLR (Perrone et al.
(2018)) is a multi-task ensemble of adaptive Bayesian linear regression models with all the tasks sharing a common feature extractor.
• TAF-R (Wistuba et al.
(2018)) learns a transferable acquisition function, unlike the aforementioned algorithms that focus on a transferable response model, that selects the next hyper-parameter based on a weighted combination of the expected improvement of the target task, and predicted improvement on the source tasks.
• MetaBO (Volpp et al.
(2019)) is another transferable acquisition function, optimized as a policy in a reinforcement learning framework.
This approach, however, demands a pre-computed target response model as part of the state representation.
In our approach, we learn a universal response model based on the underlying assumption that the response is not only dependent on the hyper-parameters, as is assumed in black-box optimization techniques, but also on the dataset itself, presenting the problem as a gray-box function optimization.
5. R ESULTS AND D ISCUSSION Q1: Z ERO - SHOT HPO AS A STAND - ALONE PROBLEM
In Table 3 we report the ﬁnal normalized regret achieved by the different zero-shot approaches for the ﬁrst 20 hyper-parameters (Feurer et al.
Our method consistently provides better hyper-parameters for such a small budget 4 .
The use of engineered meta-features to represent datasets for HPO solutions is not reliable, as the results achieved by NN- (cid:104) MF1 (cid:105) and NN- (cid:104) MF2 (cid:105) are no better than random.
On the other hand, using the meta-features extracted from the dataset directly,
NN- (cid:104) D2V (cid:105) serves as a better approximation.
Furthermore, random sampling from the restricted hyper-ellipsoid also outperforms the use of initialization strategies based on meta-features.
We obtain the zero-shot hyper-parameters via Algorithm 2.
The D2V meta-features are obtained via Algorithm 5.
[Table]
Q2: Z ERO - SHOT HPO AS AN INITIALIZATION STRATEGY FOR SINGLE - TASK SEQUENTIAL HPO METHODS
We use the aforementioned initialization strategies to warm-start single task GP with a Matern 3/ kernel and automatic relevance determination as the response model.
The quality of our suggested hyper-parameters are reﬂected in the improved performance of the response model at the early stages compared to metafeature-based initialization and random search, Figure 1.
The pseudo-code for sequential model-based optimization is provided by Algorithm 3.
[Figure]
Q3: S EQUENTIAL G RAY - BOX F UNCTION O PTIMIZATION
The proposed universal response model provides useful hyper-parameters upfront without access to any observations of losses of the target responses.
However, by iteratively reﬁtting the model to the history of observations, the response prediction is improved, as depicted in Figure 5, and summarized in Table 4 for an additional 75 trials after warm-starting with 20 conﬁgurations.
We reﬁt our algorithm by optimizing Equation 3, on the history of observed losses on the target dataset,
Algorithm 4.
We evaluate two policies for selecting the next hyper-parameter after reﬁtting, (1) greedily selecting the hyper-parameter with highest predicted response, GROSI(+1), and (2) selecting the next hyper-parameter randomly from the top 5 hyper-parameters with the highest predicted response, GROSI(+5).
In contrast to the baselines that select hyper-parameters through an acquisiton function which capitalizes on the uncertainty of the posterior samples, we incorporate uncertainty by selecting the next hyper-parameter from the top-5 hyper-parameters uniformly at random, and thus introduce a small trade-off between exploration and exploitation.
Furthermore, our method outperforms the state-of-the-art in transfer learning approaches for HPO.
Across all three meta-datasets, we achieve the best ﬁnal performance, Table 4, as well as better intermediary results 5 .
The baselines are warm-started with 20 randomly selected hyper-parameters (Feurer et al.
For better readability, the uncertainty quantiﬁcation can be found in Figure 4.
[Figure]
[Table]
It is worth mentioning that the neural acquisition function trained by MetaBO, the only baseline that does not integrate uncertainty in its policy, demonstrates poor performance, which can be attributed to the large search space.
5. A BLATION S TUDY
We perform several ablation experiments to analyze the contribution of each objective to the overall performance.
The results are detailed in Table
[Table] [Table]
Treating zero-shot HPO as a simple regression model by optimizing Equation 3 alone is suboptimal and does not scale across all meta-datasets.
We notice that adding the auxiliary dataset identiﬁcation task, Equation 6 brings on signiﬁcant improvement, similarly with the similarity driven regularization,
Equation 7.
This reinforces the notion that the responses of similar datasets behave similarly with regards to the hyper-parameters.
Both losses help generate more expressive meta-features, the former more directly, by optimizing the inter- and intra-dataset similarities, and the latter indirectly by penalizing the difference in the predicted response.
We also initialize the meta-feature extractor, φ , by pretraining it independently, Algorithm 5.
There too, we notice that the competitive performance compared to training from scratch, nevertheless, the performance on the Optimization Md by optimizing all losses jointly on a pretrained meta-feature extractor can lead to underﬁtting on the regression task.
A small sensitivity analysis can be found in Appendix F.1.
C ONCLUSION
In this paper, we formulate HPO as a gray-box function optimization problem that incorporates an important domain of the response function, the dataset itself.
We design a novel universal response model for zero-shot HPO that provides good initial hyper-parameters for unseen datasets in the abscence of associated observations of hyper-parameters.
We propose and optimize a novel multi-task objective in order to estimate the response while learning expressive dataset meta-features.
We also reinforce the assumption that similar dataset behave similarly with respect to hyper-parameters by introducing a novel similarity-driven regularization technique.
As part of future work, we will investigate the impact of our approach within the reinforcement learning framework.
A D ETAILED P ROBLEM S ETTING
By a learning task we denote a pair ( p, (cid:96) ) of an unknown distribution p of pairs ( x
, y ) ∈ R M + L , with M, L ∈ N , and a loss (cid:96) :
R L × R L → R .
A function ˆ y : R M → R L is called a model for task p and
[Equation]
its (expected) loss .
Let a be a learning algorithm that yields for every sample D of pairs ( x, y ) from a task ( p, (cid:96) ) and hyper-parameters λ ∈ R P a model ˆ y for the task.
We call
[Equation]
the loss (or the response) of hyper-parameters λ .
We say validation loss for the loss estimated on fresh validation data.
Sequential single-task hyper-parameter optimization problem.
Given an initial number K of pairs ( λ k , l k ) of hyper-parameters and their (validation) losses and a budget B ∈ N of trials, ﬁnd sequentially B many hyper-parameters λ K +1 , . . .
, λ K + B , such that their smallest loss
[Equation]
λ k
[Equation]
is minimal among all such sequences.
To compute the next guess λ k +1 , the hyper-parameters λ 1 , . .
. , λ k tried so far and their (validation) losses l k :=
(cid:96) ( λ k ) can be used.
Zero-shot cross-task hyper-parameter optimization problem.
Let p task be an unknown distribution of supervised learning tasks.
Given a sample of triples (( p, (cid:96) ) , λ, l ) of learning tasks ( p, (cid:96) ) , hyper-parameters λ and their losses l , ﬁnd for a fresh task ( p, (cid:96) ) and a budget B ∈ N — without any observations of losses of hyper-parameters on this task — a set { λ 1 , . . .
, λ B } of hyper-parameters, such that their smallest loss min k ∈ 1: B (cid:96) ( λ k ) is minimal among all such sets.
Sequential cross-task hyper-parameter optimization problem.
Given both, (i) a sample of triples (( p, (cid:96) ) , λ, l ) of learning tasks ( p, (cid:96) ) , hyper-parameters λ and their losses l , and (ii) a fresh task ( p, (cid:96) ) and a budget B ∈ N , ﬁnd sequentially B many hyper-parameters λ 1 , . . .
, λ B , such that their smallest loss min k ∈ 1: B (cid:96) ( λ k ) is minimal among all such sequences.
To compute the next guess λ k +1 , the hyper-parameters λ 1 , . .
. , λ k tried so far and their (validation) losses l k :=
(cid:96) ( λ k ) as well as all data on other tasks can be used.
B T HE M ETA - FEATURE E XTRACTOR
The meta-feature extractor is a set-based function, and is represented as an extended derivation of the Kolmogorov-Arnold representation theorem (Krkov ´ a (1992)), which states that a multi-variate function φ can be deﬁned as an aggregation of univariate functions over single variables:
[Equation]
It is important to note that φ is permutation invariant, i.e. unaffected by any permutation on the input, which allows us to obtain the same output for the same multi-variate data regardless of the order of input.
As a simple variant of this formulation (Zaheer et al.
, we can replace the set of functions h m , with single function h , and g m,j with a function g .
In this paper, we incorporate the meta-feature extractor as part of the response model, effectively learning a conditional response
C M ETA - DATASETS C
L AYOUT H YPER - PARAMETER
Below are some examples of the number of neurons per layer for networks with different layout hyper-parameters given 4 neurons and 5 layers:
• Layout (cid:3) :
[4,4,4,4,4] • Layout (cid:67) : [4,8,16,32,64] • Layout (cid:66) :
[64,32,16,8,4] • Layout (cid:5) : [4,8,16,8,4] • Layout (cid:52) :
The search space consists of all possible combinations of the hyper-parameters.
After removing redundant conﬁgurations, e.g. (cid:52) layout with 1 layer is similar to a (cid:3) layout with 1 layer, the resulting meta-datasets have 256 , 288 and 324 unique conﬁgurations respectively.
C. H YPER - PARAMETER E NCODING
Below is description of the encodings applied to our hyper-parameters.
We also like to note that the scalar values are normalized between (0 , 1) .
[Table]
C. T HE UCI D ATASETS
Table 7 is an overview of the UCI datasets used to generate the meta-datasets.
[Table]
3 mushroom  2 monks2 optical  10 arrhythmia  13 primary-tumor  15 breast-tissue  6 conn-bench-vowel-deterding  11 statlog-australian-credit  2 soybean  18 tic-
[Table]
-toe
[Table]
merluccius states 2f  3 lymphography  4 chess-krvkp  2 monks2 audiology-std  18 waveform
[Table]
-segmentation  7 fertility
[Table]
-display  10 lenses  3 heart-va  5 wine-quality-red
6 pittsburg-bridges-SPAN  3 parkinsons  2 oocytes trisopterus nucleus 2f  2 wine-quality-white
[Table]
german-credit  2 pima  2 acute-inﬂammation
2 pittsburg-bridges-
[Table]
-OR-D  2 car  4 low-res-spect  9 horse-colic
2 musk2 heart-switzerland  5 pittsburg-bridges-REL-L  3 oocytes trisopterus states 5b  3 breast-cancer  2 congressional-voting  2 spambase
[Table]
-nephritis  2 iris  3 credit-approval  2 thyroid  3 hill-valley  2 mammographic  2 oocytes merluccius nucleus 4d  2 ilpd-indian-liver  2 seeds  3 blood  2 ozone  2 waveform-noise  3 magic  2 statlog-heart  2 statlog-image  7 pittsburg-bridges-TYPE  6 cylinder-bands  2 echocardiogram  2 lung-cancer  3 ﬂags  8 dermatology  6 letter  26 cardiotocography-10clases  10 zoo  7 heart-cleveland  5 ecoli  8 haberman-survival  2 yeast  10 balance-scale  3 hayes-roth  3 wine  3 libras  15 miniboone  2 breast-cancer-wisc-prog  2 hepatitis  2 glass
6 post-operative
D A LGORITHMS
We deﬁne p T as the task distribution that represents pairs of datasets and hyper-parameters, i.e.
T ( t,k ) =
( D ( t ) , λ ( t ) k ) ∈ T × Λ , and p D be the distribution of the datasets as deﬁned in Section 4.2.
Algorithm 1 provides the overall optimization framework for GROSI, our approach.
4: while do 5:
Set
Sample
Evaluate
After optimizing our objective via Algorithm D we apply Algorithm 2 to observe the results presented in Tables 3 and 5.
For sequential model-based optimization, a surrogate ˆ l is ﬁtted to the observed responses of the unknown function.
Several initialization strategies exist to exepedite the transfer of information across tasks, Section 5.4.
In Algorithm 3, we present the generic pseudo-code for SMBO, that requires an acquisition function, a , to sample the next iterate from the domain.
In Section 5.4, we propose to initialize our response model on the target dataset, then iterativly tune it to that particular dataset.
Initially, we select top K conﬁgurations based on Algorithm 2, our zero-shot approach.
Then, via Algorithm 4, we sample uniformly at random from the top X ranking conﬁgurations.
If X = 1 , then this represents the greedy policy.
Meta-feature learning from datasets with varying schema was initially proposed in (Jomaa et al.
For our approach, we introduce a set-based meta-feature extractor module to handle datasets of varying schema as well, however, we optimize Equation 8 and use the dataset identiﬁcation task as an auxiliary objective.
However, to pre-train the meta-feature extractor for the Ablation study,
Section 5.5, as well as in order to extract meta-features for the NN-D2V, and TST-D2V, we follow
Algorithm 5, with p D + as the distribution of similar datasets, p D + = { ( D ( t ) , D ( q ) , s ) ∼ p D | s = 1 } , and p D − as the distribution of dissimilar datasets, p D − = { ( D ( t ) , D ( q ) , s ) ∼ p D | s = 0 } .
Similar datasets are deﬁned as multi-ﬁdelity subsets (batches) of each dataset.
E E XPERIMENTAL D ETAILS
Our model architecture is divided into two modules, ˆ l :=
φ ◦ ψ , the meta-feature extractor φ , and the regression head ψ .
The meta-feature extractor φ : R 2 → R K h is composed of three functions,
Equation 4, namely f , g and h .
The regression head is also composed of two functions,
i
[Table]
e. ψ : ψ 1 ◦ ψ 2 .
We deﬁne by ψ 1 : R K h × Λ → R K ψ 1 as the function that takes as input the meta-feature/hyperparameter pair, and by ψ 2 : R K ψ 1 → R the function that approximates the response.
Finally, let
Dense(n) deﬁne one fully connected layer with n neurons, and ResidualBlock(n,m) be m × Dense(n) with residual connections (Zagoruyko & Komodakis (2016)).
In order to select a single universal response model
[Table]
we evaluate the validation performance on the three network architectures described in Table 8.
We select the architecture that has the best average performance between the three across the three meta-datasets, Table 9, which turns out to be
Architecture 3.
The architectures assign different number of trainable variables for the meta-feature extractor and the coupled regression head.
[Table] [Table]
F A DDITIONAL E XPERIMENTAL R ESULTS F. H YPER - PARAMETER S ENSITIVITY A NALYSIS
We optimize our response model by minimizing Equation 8, which includes the dataset identiﬁcation task, Equation
[Table]
and the similarity-driven regularizion task, Equation 7, with auxiliary weights δ and α assigned to both respectively.
We report below the performance of our universal response model for different auxiliary weights.
The results conﬁrm the importance of emphasizing the auxiliary dataset identiﬁcation task in conjunction with the similarity-driven regularization loss, which reenforces the intuition that similar datasets behave similarly with respsect to hyper-parrameter response.
The reported results throughout the paper are based on δ = 1 and α = 0 .
[Table]
F. A DDITIONAL R ESULTS Q1: Z ERO - SHOT HPO AS A STAND - ALONE PROBLEM
In Table
[Table]
we report the average time required by each zero-shot approach evaluate the optimal hyper-parameter.
The results do not reﬂect the time required to suggest these hyper-parameters
[Table]
but the training time required to optimize the underlying model given the suggested hyper-parameters.
Although the hyper-parameters suggested by our algorithm require slightly more time to observe, we still achieve better response, with less regret, Table 3.
[Table]
As a plausibility argument for the usefulness of our zero-shot strategy
[Table]
we depict in Figure 3 the top 10 suggested hyper-parameters by our approach
[Table]
as well as two initialization strategies on the actual response surface.
Our picks can be seen colocated near the different optima in the search space whereas hyper-parameters of other strategies are dispersed
[Table]
Q3: S EQUENTIAL G RAY - BOX F UNCTION O PTIMIZATION
We reﬁt the universal response model to the observations of the response on the target dataset by optimizing Equation 3.
We depict the improvement achieved over the zero-shot approach in Figure
[Table]
[Figure]
[Figure] [Figure]