Neural Exploratory Landscape Analysis

## Abstract

Abstract Recent research in Meta-Black-Box Optimization (MetaBBO) have shown that meta-trained neural networks can effectively guide the design of black-box optimizers, significantly reducing the need for expert tuning and delivering robust performance across complex problem distributions. Despite their success, a paradox remains: MetaBBO still rely on human-crafted Exploratory Landscape Analysis features to inform the meta-level agent about the low-level optimization progress. To address the gap, this paper proposes Neural Exploratory Landscape Analysis (NeurELA), a novel framework that dynamically profiles landscape features through a two-stage, attention-based neural network, executed in an entirely end-to-end fashion. NeurELA is pre-trained over a variety of MetaBBO algorithms using a multi-task neuroevolution strategy. Extensive experiments show that NeurELA achieves consistently superior performance when integrated into different and even unseen MetaBBO tasks and can be efficiently fine-tuned for further performance boost. This advancement marks a pivotal step in making MetaBBO algorithms more autonomous and broadly applicable. The source code of NeurELA can be accessed at https://anonymous.4open.science/r/Neur-ELA-303C .

## 1 Introduction

Black-Box Optimization (BBO) is a discipline in mathematical optimization characterized by the absence of both the closed-form expression of the target objective function and its derivative during the optimization process.
Traditional BBO methods, such as Genetic Algorithm [], Evolutionary Strategy [], and Bayesian Optimization [], have been extensively studied in the past half century. They are applied to address a broad range of black-box scenarios, including automated machine learning [], bioinformatics [], prompt tuning in large language models []. However, these methods often require manual tuning with deep expertise to ensure the optimal performance on unseen tasks. Recent advances in Meta-Black-Box Optimization (MetaBBO) [, , , , ] leverage meta-learning [] to streamline the traditional labour-intensive tuning process across different problem distributions. This effort not only reduces the need for expert-level knowledge, but also enhances generalization towards unseen problems. Generally, MetaBBO employs a bi-level optimization paradigm, wherein the meta-level control policy (often implemented as a neural network) is optimized according to the feedback from the low-level optimization process [], as depicted on the left side of Figure 1.

One of the core challenges to ensure the generalization performance of MetaBBO algorithms is to design an information-rich profiling system for the low-level optimization status (highlighted in yellow box on the left side of the Figure 1). Firstly, the optimization status must include the identification features to differentiate various optimization problems. This design is crucial to enable the meta-level neural policy to be aware of what is the target optimization problem. Secondly, the optimization status should align with the low-level optimization dynamics, which helps to inform the meta-level neural policy about where the low-level optimization progress has reached.

Figure: Figure 1: Left: The general workflow of MetaBBO algorithms, which follows a bi-level optimization paradigm. The landscape analyser timely profiles the low-level optimization progress, informing the meta-level neural policy to dynamically output desired configuration for the low-level BBO method. Right: The average optimization performance (larger is better) of integrating our pre-trained landscape analyser into diverse MetaBBO tasks (red line). The blue line denotes the original performance of the MetaBBO algorithms, while the green line denotes the performance of integrating the traditional exploratory landscape analysis features into the MetaBBO algorithms.
Refer to caption: extracted/5881667/pics/metabbo.png

Existing MetaBBO algorithms widely adopt Exploratory Landscape Analysis (ELA) [, ] to profile the low-level optimization status. While the approaches provide a comprehensive understanding of problem and algorithm characteristics, they still face several limitations. 1) Many features derived from ELA are highly correlated with each other [] and exhibit strong sensitivity to random sampling strategies []. This necessitates an additional step of feature selection for the given MetaBBO scenario, requiring a certain level of expert knowledge. 2) The landscape features given by ELA are more like an “offline” portray of the target optimization problem. This is somewhat incompatible with the dynamic nature of MetaBBO paradigm, which requires both the “offline” problem properties and the “online” optimization progress. 3) Computing some of these features, particularly those related to convexity and local optimum properties, consume additional function evaluations (FEs). The consumption of the computational resource, initially reserved for the low-level optimization, ultimately degrades the final performance of the MetaBBO algorithms.

To address the above limitations,
we propose Neural Exploratory Landscape Analysis (NeurELA), a novel, learnable framework designed to dynamically profile optimization status for the MetaBBO algorithms in an end-to-end manner. Our NeurELA encounters certain challenges: on the one hand, we need to carefully design a universal neural network-based landscape analyser that is compatible with a variety of MetaBBO algorithms. On the other hand, it is crucial to enable effective learning within this analyser to ensure it robustly generalizes not only to unseen optimization problems but also to previously unencountered MetaBBO algorithms. We address the challenges as follows.

1) Our research begins by designing a comprehensive operating space, $\Omega$, within which our landscape analyser $\Lambda$ functions. This space encompasses a collection of MetaBBO tasks, each designated as $\mathbb{T}:=\{\mathbb{A},\mathbb{D}\}$. Here, $\mathbb{A}$ specifies a particular MetaBBO algorithm, and $\mathbb{D}$ represents the associated set of optimization problems to meta-train and test the algorithm. The primary objective is to optimize $\Lambda$ such that it enhances the performance of various $\mathbb{A}$ across their respective problem sets in $\Omega$. Our introduction of $\Lambda$ within the comprehensive operating space
$\Omega$ marks a significant advancement in MetaBBO methodologies. Unlike traditional approaches that treat each optimization algorithm and problem in isolation, $\Lambda$ leverages a holistic view, enhancing performance across a diverse set of tasks.

2) We instantiate the landscape analyser $\Lambda$ through a neural network model parameterized by $\theta$. Specifically, $\Lambda_{\theta}$ is implemented using a two-stage attention-based neural network. Our design enables flexible integration with most existing MetaBBO algorithms as an effective replacement for their traditional hand-crafted landscape analysis mechanisms. Given the non-differential nature of our objective in BBO, we adopt an Evolution Strategy [, ] to train $\Lambda_{\theta}$. This training is carried out within a multitask neuroevolution [] paradigm, aiming to optimize $\theta$ to maximize the expected performance across representative MetaBBO tasks.

3) We propose using the trained $\Lambda_{\theta}$ in two distinct ways. First, having been trained on a diverse range of tasks, $\Lambda_{\theta}$ is capable of being directly applied to new, unseen MetaBBO algorithms, achieving zero-shot generalization. Second, $\Lambda_{\theta}$ also supports fine-tuning in conjunction with the meta-training of new MetaBBO algorithms, allowing it to adapt its performance for specific optimization challenges.

Through extensive experiments, we validate the effectiveness of the proposed NeurELA framework, as illustrated on the right side of Figure 1. These results demonstrate that it is possible to pre-train an universal neural landscape analyser $\Lambda_{\theta}$ over a group of MetaBBO tasks and then generalize it to unseen ones. Our work pioneers the integration of automatic feature extraction in MetaBBO, transforming these methods into fully end-to-end systems. This advancement significantly streamlines optimization processes, enhancing efficiency and adaptability across related fields, and reducing the need for manual expertise.

## 2 Related Works

Meta-Black-Box Optimization.
Existing MetaBBO paradigms can be classified into following categories with respect to role they play at the low-level optimization process: 1) Hyper-parameters configurator: this line of works meta-trained neural network-based policy to dynamically configure the hyper-parameters of the backend BBO optimizer, e.g., the scale factor ($F$) and crossover rate ($Cr$) in Differential Evolution (DE) algorithm [, ]. Initial researches (e.g., RLPSO []) employed simple MLP structures to parameterize the meta-level policy, which was enhanced by the latest works such as GLEET [], which facilitated Transformer [] architecture to boost the sequential decision making along the low-level optimization. 2) Update rule selector: works in this line flexibly select desired update rules for the low-level optimization process. Several initial works learns to select one mutation operator from a pre-defined DE mutation operator pool to attain diverse optimization behaviours [, , ]. Then the research scope extends to the refinement of the operators pool for optimizing more complex problems [], and dynamically selecting the entire algorithm[]. 3) Complete optimizer generator: works in this line aims to meta-learning the neural network-based policy as a complete optimizer. Lange et al. [, ] proposed meta-training several attention blocks as the complete updating process of CMAES []  , which was followed by Song et al. [], proposing meta-training Transformer-based language model as an entire BBO method. Such end-to-end deep learning systems encountered interpretability issue. To this end, SYMBOL [] proposed generating symbolic update rules for the low-level optimization process, promoting the interpretable and automatic discovery of black-box method. GLHF [] carefully design a Transformer-based neural network that imitates the evolutionary operators in DE algorithm. As for the designs of optimization status extraction, a great majority of existing MetaBBO algorithms borrow the idea from well-known landscape analysis researches [, ]. To make those landscape features compatible with the MetaBBO paradigm, existing MetaBBO algorithms more or less facilitated feature selection and feature aggregation procedure. Take a recent work GLEET [] as an example, it aggregates the representative ELA features [] into a $9$-dimensional high-level profiling features, which helps reduce the training complexity and enhance the overall optimization performance. However, we have to note that such hand-crafted feature processing requires deep expertise in optimization.

Landscape Analysis.
Landscape Analysis originates from the emergence of Automated Algorithm Selection (AAS) [, ], which selects an optimal algorithm out of a collection of algorithms for any given optimization problem. Generally, Landscape analysis is formulated as a function: $\Lambda:(X,y)\rightarrow s$, that maps the sampled candidates $X$ and their corresponding objective values $y$ to a feature vector $s$, each dimension of which indicates some problem-specific properties of the target optimization problem. The feature vector $s$ can be subsequently fed into machine learning algorithms (e.g., Support Vector Machines [] and Random Forests []) to address some specific algorithmic tasks []. Over the years, a wide range of Landscape Analysis feature groups have been developed, and structurally summarized in [, , , ], of which the mostly utilized ones are: 1) Fitness Distance Correlation [] includes six features profiling the distances of the sampled candidates both in decision space and the objective space, as well as the correlation in-between the two spaces. 2) Dispersion [] features describe the degree of dispersity for the best quantiles of the sampled candidates. 3) Classical ELA [] proposed using six groups of low-level metrics such as local search, skewness of objective space and approximated curvature in (second) first-order as a comprehensive abstract of basic landscape properties (e.g., global structure, multimodality, separability). Generally, classic ELA requires
adequate samples and enormous computational efforts to conclude all of its fifty low-level feature values. 4) Information Content [] comprises five features about the smoothness, ruggedness and neutrality of an optimization problem. It first applies random walk sampling (e.g., Metropolis Random Walk []) for each of the sampled candidates, then calculates its features by statistic metrics. 5) Nearest Better Clustering [] is a collection of features that describe the distributional correlation of the nearest better neighbors and the nearest neighbors of the sampled candidates.

While these hand-crafted landscape analysis features provide a systematic understanding of optimization problems, they inevitably hold high computing complexity hence consume additional computational resources. Besides, the intricate correlations in-between these feature groups [] and their sensitivities to the random sampling strategy [] necessitate the expert-level feature selection and feature aggregation procedures. To address these issues, several early-stage studies have recently proposed learning neural networks (e.g., convolutional neural network [] and point cloud transformer []) as the feature mapping function. [] first proposed using Deep-CNN to extract landscape features from fitness map of moderate initial sample points. The neural network is trained to correctly predict algorithm configuration of modular CMA-ES algorithms. However, solely feeding the fitness map might introduce information loss. To address such issue, [] propose a novel point cloud transformer (PCT) based approach, which embeds sample points information and the fitness map together as the input. This work generally achieves superior landscape feature prediction accuracy. Another technical bottleneck of [] is the supervised training, which might leads the learned landscape representation overfit to a certain AAS/AAC task. To refine this issue, [] propose self-supervised learning through AutoEncoder, which adopted fitness map as the target to be reconstructed. More recently, [] propose DeepELA to adrress both the infomation loss and the supervised trainng issues. It introduces MHA and trains it by a contrastive learning loss InfoNCE to ensure aligned representation across different sampling strategies. In this paper, our NeurELA presents several novelty compared with the above learning-based ELA frameworks: a) NeurELA aims for MetaBBO tasks yet the previous works generally aim for classical AS or AC tasks. MetaBBO tasks are more akin to Dynamic AS or AC tasks (DAC, DAS), where the configuration decision have to be customized in each generation of optimization of a given problem. b) Unlike previous works, which provide static and invariant problem properties, NeurELA offers dynamic optimization status information. c) compared with [], our architecture designs do not rely on a pre-defined dimension upper bound, making it more generic and versatile for various MetaBBO tasks.

## 3 Neural Exploratory Landscape Analysis

### 3.1 Framework

We introduce a multi-task operating space, denoted as $\Omega$, which serves as the foundational basis for our landscape analyser, $\Lambda$. The space $\Omega$ is mathematically defined as:

$$ $\Omega=\left\{\mathbb{T}_{k}=\left(\mathbb{A}_{k},\mathbb{D}_{k}\right)\mid k= 1,2,\ldots,K\right\}$ (1) $$

where each $\mathbb{T}_{k}$ represents a task configuration consisting of a MetaBBO algorithm $\mathbb{A}_{k}$ and a corresponding set of optimization problems $\mathbb{D}_{k}$. This structured collection of tasks is meticulously designed to optimize our proposed landscape analyser $\Lambda_{\theta}$ over a variety of algorithms and problems.

In accordance with the MetaBBO protocol, each algorithm $\mathbb{A}_{k}$ is composed of three fundamental elements: the meta-level neural policy $\Pi_{\phi}$, the low-level optimizer $\Gamma$, and the initial landscape feature analyser, denoted as $\Lambda_{0}$.
As illustrated in Figure 1, the meta-training of $\mathbb{A}_{k}$ adheres to a bi-level optimization paradigm. Given a problem instance $f$ sampled from $\mathbb{D}_{k}$, at each optimization step $t$, the low-level BBO method $\Gamma$ interacts with $f$ to attain a raw observation $\{X^{t}_{i},y^{t}_{i}\}_{i=1}^{m}$ of the current population of candidates solutions, denoted as $o^{t}$, where $m$ is the population size in population-based search methods [, , ]. Subsequently, the observation $o^{t}$ is further processed by the original landscape analyser, $\Lambda_{0}$, to extract optimization status features $s^{t}=\Lambda_{0}\left(o^{t}\right)$. Based on these features, the meta-level neural policy $\Pi_{\phi}$ dictates a design choice $c^{t}$ from the configuration space $\mathbb{C}$ conditioned on the current optimization status: $c^{t}=\Pi_{\phi}(s^{t})$. Then, the low-level BBO method $\Gamma$ loads $c^{t}$ to optimize $f$ for next time step $t+1$. The meta-performance $r^{t}$ is calculated by evaluating the performance gain achieved by implementing $c_{t}$. The ultimate goal of the meta-training phase is to maximize the overall meta-objective $F_{\text{metaBBO}}$, defined as:

$$ $\underset{\phi}{\max}F_{\text{metaBBO}}=\mathbb{E}_{f\sim\mathbb{D}_{k},\Pi_{ \phi}}\left[\sum_{t=0}^{T}r^{t}\right],$ (2) $$

where the expectation is over all problem instances from $\mathbb{D}_{k}$ and spans the total length $T$ of the low-level optimization horizon.

In our NeurELA, we parameterize a neural network-based landscape analyser $\Lambda_{\theta}$ (where $\theta$ denote the learnable parameters), which substitutes the original landscape analyser $\Lambda_{0}$, to produce the optimization status features as $s^{t}=\Lambda_{\theta}(o^{t})$. Our aim is to develop a universal $\Lambda_{\theta}$ for diverse MetaBBO algorithms that perform comparably to, or better than, the original hand-crafted $\Lambda_{0}$ in the corresponding MetaBBO tasks. To objectively measure the performance of $\Lambda_{\theta}$, we define an evaluation metric $\Upsilon(\Lambda_{\theta}|\mathbb{T}_{k})$ as the relative performance of $\Lambda_{\theta}$ against $\Lambda_{0}$ in the context of MetaBBO task $\mathbb{T}_{k}=\left(\mathbb{A}_{k},\mathbb{D}_{k}\right)$. Concretely, we first meta-train two versions of the MetaBBO algorithms on $\mathbb{D}_{k}$: $\mathbb{A}_{k}(\Pi_{\phi},\Gamma,\Lambda_{0})$ and $\mathbb{A}_{k}^{\prime}(\Pi_{\phi},\Gamma,\Lambda_{\theta})$, and then evaluate $\Upsilon(\Lambda_{\theta}|\mathbb{T}_{k})$ as:

$$ $\Upsilon\left(\Lambda_{\theta}\mid\mathbb{T}_{k}\right)=\frac{1}{P\times Q} \sum_{p=1}^{P}\sum_{q=1}^{Q}Z_{p,q},\text{ where }Z_{p,q}=-\frac{f_{p,q}^{*}- \mu_{p}}{\sigma_{p}}$ (3) $$

where $P=\left|\mathbb{D}_{\text{test}}\right|$ is the number of test problems (the dataset $\mathbb{D}_{k}$ is divided into two distinct subsets: $\mathbb{D}_{\text{train}}$ for training and $\mathbb{D}_{\text{test}}$ for testing), $Q=50$ represents the number of independent runs to obtain average performance, $f_{p,q}^{*}$ denotes the objective value achieved by testing $\mathbb{A}_{k}^{\prime}$ on the $p$-th problem instance and $q$-th run, and $\mu_{p}$ and $\sigma_{p}$ are the mean and standard deviation of the objective values obtained by testing $\mathbb{A}_{k}$ on the $p$-th problem over $50$ independent runs. Note that we follow the Z-score normalization design in [] to evaluate the performance gain $Z_{p,q}$.

The meta-objective of our NeurELA framework is defined as maximizing the expected relative performance of the neural network-based landscape analyser, $\Lambda_{\theta}$, across all MetaBBO tasks defined within the multi-task space $\Omega$. Mathematically, this is expressed as:

$$ $\underset{\theta}{\max}F_{\text{metaELA }}=\mathbb{E}_{\mathbb{T}_{k}\sim \Omega}\left[\Upsilon\left(\Lambda_{\theta}\mid\mathbb{T}_{k}\right)\right]= \frac{1}{K}\sum_{k=1}^{K}\Upsilon\left(\Lambda_{\theta}\mid\mathbb{T}_{k}\right)$ (4) $$

Figure: Figure 2: Left: The architecture of the basic attention block ($Attn$). Right: The computation graph of the two-stage attention mechanism (Ts-Attn).
Refer to caption: extracted/5881667/pics/neuralnetwork.png

### 3.2 Architecture of Λ θ subscript Λ 𝜃 \Lambda_{\theta} roman_Λ start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT

Our architecture is designed for compatibility with various MetaBBO algorithms. We introduce a Population Information Embedding (PIE) module that preprocesses observations $o^{t}=$ $\left\{X_{i}^{t},y_{i}^{t}\right\}_{i=1}^{m}$, allowing $\Lambda_{\theta}$ to adapt to different search ranges and objective scales. Additionally, a two-stage attention-based encoder (Ts-Attn) enhances $\Lambda_{\theta}$ ’s information extraction capabilities and scalability.

PIE. PIE normalizes observation $o^{t}$ using two min-max normalization operations: first on the candidate solutions $\{X^{t}_{i}\}_{i=1}^{m}$ against the search range, and second on the objective values $\{y^{t}_{i}\}_{i=1}^{m}$ using the extremum values at time step $t$. This ensures unified representation and generalization by scaling all values to $[0,1]$. For a $d$-dimensional optimization problem, the normalized observations $o^{t}$ are then reorganized into per-dimensional tuples $\{\{(X_{i,j}^{t},y_{i}^{t})\}_{i=1}^{m}\}_{j=1}^{d}$, with a shape of $d\times m\times 2$. These tuples undergo a linear transformation via a mapping matrix $W_{\mathrm{emb}}\in\mathbb{R}^{2\times h}$, producing the population information encoding $E^{t}$, which has the shape $d\times m\times h$. Here, $h$ represents the hidden dimension used in the subsequent two-stage attention module.

Ts-Attn. The computation graph of the Ts-Attn is illustrated on the right side of Figure 2, with a basic component being the attention block ($Attn$) of hidden dimension $h$. As illustrated in the left side of Figure 2, the $Attn$ block follows the design of the original Transformer [], except that the layer normalization [] is used instead of batch normalization []. Ts-Attn receives $E^{t}$ and then advances the information sharing at both cross-solution and cross-dimension levels.
1) Cross-solution attention: We utilize an Attn block to enable same dimensions shared by different candidate solutions within the population to exchange information. 2) Cross-dimension attention: A second Attn block is implemented to further promote the sharing of information across different dimensions within each candidate. To achieve this, the output from the cross-solution phase must be transposed into the shape $m\times d\times h$ and augmented with cosine/sine positional encodings to maintain the dimensional order within a candidate. After these two phases of information processing, the resulting output $s^{t}$ provides comprehensive optimization status features, suitable for different MetaBBO algorithms. This highly parallelizable, attention-based architecture significantly boosts the scalability of NeurELA as the number of candidates $m$ or dimensions $d$ increases. Additional technical details about the architecture of $\Lambda_{\theta}$ can be found in Appendix A.1.

### 3.3 Training Method

Figure: Algorithm 1 Pseudo code of training NeurELA

Neuroevolution of $\Lambda_{\theta}$. The meta-objective of our NeurELA, as defined in Eq. (4), is non-differentiable, since the evaluation of the relative performance $\Upsilon\left(\Lambda_{\theta}\mid\mathbb{T}_{k}\right)$ in Eq. (3) is not directly derived from the parameter ${\theta}$ of $\Lambda_{\theta}$. We in turn employ an Evolution Strategy, denoted as $ES$, to neuroevolve $\Lambda_{\theta}$ towards maximizing its expected relative performance for multiple MetaBBO tasks in $\Omega$. The pseudo code of the training is presented in Algorithm 1. The $ES$ first initializes a group of $N$
landscape analysers, then iteratively searches for ${\theta^{*}}$ through $maxGen$ generations of optimization (lines $4$ to $16$). Note that the meta-training of the $K$ MetaBBO tasks, which involves training their neural-policies over the training problem set, is nested within the optimization loop (lines $5$ to $12$) and is extremely time-consuming. We hence introduce Ray [], an open-source framework for parallel processing in machine learning applications, to distribute the meta-training pipelines to multiple CPUs. This parallelization significantly enhances the overall efficiency of training NeurELA.

Zero-shot Generalization. The pre-trained $\Lambda_{\theta^{*}}$ demonstrates robust performance boosts across $K$ representative MetaBBO tasks, enabling seamless integration into unseen MetaBBO tasks. Specifically, $\Lambda_{\theta^{*}}$ can replace the original $\Lambda_{0}$ in an unseen MetaBBO algorithm, $\mathbb{A}_{\text{unseen }}$. This enables $\mathbb{A}_{\text{unseen }}$ to concentrate on refining its meta-level neural policy $\Pi_{\phi}$ during training on the designated problem set, utilizing the capabilities of $\Lambda_{\theta^{*}}$ without requiring any adaptation.

Fine-tuning Adaptation. If $\Lambda_{\theta^{*}}$ does not perform well in zero-shot generalization, we provide a flexible alternative: fine-tuning it for unseen MetaBBO tasks. In this case, $\Lambda_{\theta^{*}}$ is integrated into $\mathbb{A}_{\text{unseen }}$ and co-trained with its $\Pi_{\phi}$ during its meta-training process. This method adjusts $\Lambda_{\theta^{*}}$ to better align with the specific characteristics of $\mathbb{T}_{\text{unseen }}$.

## 4 Experimental Results

In this section, we discuss the following research questions: RQ1: Can NeurELA generalize the pre-trained landscape analyser $\Lambda_{\theta^{*}}$ to unseen MetaBBO tasks? RQ2: If the zero-shot performance is not as expected, is it possible to efficiently fine-tune NeurELA for the underperforming MetaBBO task? RQ3: What does NeurELA have learned? RQ4: How efficient is the pre-trained landscape analyser in computing landscape features? RQ5: How to choose the Evolutionary Strategy for training our NeurELA? RQ6: What is the relationship between the model complexity and the generalization performance? Below, we first introduce the experimental settings and then address $\text{RQ1}\sim\text{RQ6}$.

Training Setup. The operating space $\Omega$ comprises $K=3$ MetaBBO tasks. In specific, we first select three representative MetaBBO algorithms LDE [], RLEPSO []and RL-DAS [] from existing MetaBBO literature. Then we set the associated problem set $\mathbb{D}_{k}$ for these MetaBBO algorithms as the BBOB testsuites in COCO [], which includes a variety of optimization problems with diverse landscape properties. We have to note that the selected algorithms cover diverse MetaBBO scenarios such as auto-configuration for control parameters and auto-selection for evolutionary operators, hence ensuring the generalization of our NeurELA. We follow the original settings of these MetaBBO algorithms during their meta-training, as suugested in their papers. For the settings about the neural network, we set the hidden dimension $h=16$ for the neural network modules in the landscape analyser $\Lambda_{\theta}$. With a single-head attention in $Attn$, $\Lambda_{\theta}$ possess a total number of $3296$ learnable parameters. We adopt Fast CMA-ES [] as $ES$ for its searching efficiency and robust optimization performance. During the training, we employ a population of $N=10$ $\Lambda_{\theta}$s within a generation and optimize the population for $maxGen=50$ generations. For each generation, we parallel the $N\times K=30$ meta-training pipelines across $30$ independent CPU cores using Ray []. For each low-level optimization process, we set the maximum functiona evaluations as 20000. The experiments are run on a computation node of a Slurm CPU cluster, with two Intel Sapphire Rapids 6458Q CPUs and $256$ GB memories. Due to the limitation of the space, we present more technical details such as the train-test split $\{\mathbb{D}_{\text{train}},\mathbb{D}_{\text{test}}\}$, the control parameters of Fast CMAES and the use of open-sourced software in Appendix A.2 $\sim$ A.5.

Baselines. During the following comparison experiments, we consider two baselines for comparison: 1) Original, which denotes the original hand-crafted landscape analyser functions ($\Lambda_{0}$) in the MetaBBO algorithms. 2) ELA, which denotes the traditional ELA feature groups summarized in Section 2 and defined in [], which we implement by the open-sourced Pflacco package [].

### 4.1 Zero-shot Performance (RQ1)

Figure: Figure 3: Zero-shot performance of NeurELA on unseen MetaBBO algorithms and problem sets.
Refer to caption: x1.png

Recall that an MetaBBO task is composed of an MetaBBO algorithm and an associated optimization problem set (i.e., $\mathbb{T}_{k}=\left(\mathbb{A}_{k},\mathbb{D}_{k}\right)$). We hence generalize the pre-trained landscape analyser $\Lambda_{\theta^{*}}$ towards three zero-shot cases: 1) five unseen MetaBBO algorithms DEDDQN [], RLPSO [], GLEET [], MelBA [] and GLHF []; 2) two unseen problem set BBOB-Noisy [] and Protein-Docking benchmark []; 3) both the unseen MetaBBO algorithms and problem sets. Those tested MetaBBO tasks cover a wide range of meta-learned black-box optimizers. Besides, the problem sets include challenging realistic application, e.g., Protein-Docking. We evaluate the performance of the ELA and our NeurELA by calculating the relative optimization performance (Eq. (3)) against the Original baseline in the given MetaBBO tasks. Since the performance of Original baseline would be 0 0 within this setting, we report the exponential of the calculated relative performances in Figure 3, where the bars denote the average relative performance, the error bars denote the $95\%$ bootstrapped confidence interval, for $10$ runs of meta-training. For BBOB-Noisy set, we set the maximum function evaluations as 20000, while for Protein-Docking set, we set it to 1000. The three zero-shot cases are split by the dashed lines.

The results in Figure 3 show that: 1) Over all of the three zero-shot cases, NeurELA generally achieves superior performance against original feature extraction designs (Original) in these MetaBBO algorithms, as well as the traditional landscape analysis features (ELA). 2) The results of integrating ELA into the evaluated MetaBBO tasks are quite noisy, which might be due to the intricate correlations in-between the feature groups. Such strong correlations might challenge the information processing ability of the neural policies $\Pi_{\phi}$ in some MetaBBO algorithms. 3) We observe that our NeurELA significantly boosts the performance of Original baseline on unseen MetaBBO algorithms & problem sets case, which not only validates the effectiveness of our design, but also indicates that the specific feature extraction designs (Origin) in existing MetaBBO algorithms might overfits to the problem set it is meta-trained on. 4) Our NeurELA consistently outperforms the traditional ELA features on the BBOB-Noisy testsuites and Protein-Docking benchmark, which comprises a wide range of optimization problems with either different noise models and levels (e.g., Gaussian noise and Cauchy noise []) or intricate realistic problem features. It indicates that our NeurELA shows robust generalization ability to unseen MetaBBO tasks, which involve both unseen MetaBBO algorithm and unseen realistic optimization problems.

Figure: Figure 4: Performance gain curves of fine-tuning NeurELA for a specific MetaBBO task. The three tasks are selected to illustrate scenarios where the pre-trained NeurELA outperforms, matches, or underperforms the original baseline.
Refer to caption: x2.png

### 4.2 Fine-tuning Performance (RQ2)

We select three representative MetaBBO task from the zero-shot performance analysis (Figure 3) to validate fine-tuning adaption efficiency of our NeurELA. The selected MetaBBO tasks includes: 1) LDE on BBOB problem set, where NeurELA surpasses the Original baseline in LDE. 2) RLPSO on Protein-Docking problem set, where NeurELA equally performs against the Original baseline in RLPSO. 3) GLEET on BBOB problem set, where NeurELA does not achieves the expected zero-shot performance. Concretely, we integrate the pre-trained $\Lambda_{\theta^{*}}$ into the corresponding MetaBBO algorithm, then fine-tune it in conjunction with the meta-training over the given problem set. We depict the performance gain curves during the meta-training in Figure 4. We additionally plot the performance gain curves of the meta-training during the zero-shot generalization for comparison. The results showcase the flexible adaption potential of our NeurELA: 1) fine-tuning NeurELA on specific MetaBBO tasks where our NeurELA outperforms, equally performs or underperforms against the Original baseline would consistently introduce significant performance boosts. 2) the fine-tuning strategy shows remarkable efficiency in adapting NeurELA for new MetaBBO algorithm dynamics and optimization problem structures, which only requires less than $20\%$ meta-training epochs to achieve the similar zero-shot performance.

### 4.3 What has NeurELA learned? (RQ3)

Figure: Figure 5: Low-level optimization features obtained by traditional ELA and our NeurELA, through PCA dimension reduction.
Refer to caption: x3.png

In this section, we presents the advantage of the landscape features learned by NeurELA against traditional ELA features under the MetaBBO settings. Concretely, we take LDE [] as a showcase. In LDE, the meta-level policy (a LSTM-based RL agent) outputs the mutation strength factor $F^{t}\in(0,1)$ for the low-level DE per optimization step $t$. $F^{t}$ indicates the exploration-exploitation tradeoff during the optimization (larger value indicates exploration, vice versa). We first record the per step populations during the optimization process of LDE ($10$ runs on 10-D Rastrigin in CoCo-BBOB testsuites). We then simply classify the recorded populations as exploration population ($F^{t}>0.5$) and exploitation population ($F^{t}\leq 0.5$). We now illustrate the low-rank visualization of the ELA feature and our NeurELA feature for the recorded populations in Figure 5. The data points denoted the 2-dimensional features obtained by first feeding the populations into ELA/NeurELA and then using PCA [] transformation on the output landscape features. Red points and blue points represent the features for the exploration and exploitation population respectively.
The results validate that NeurELA is more sensitive than traditional ELA features on the dynamics of low-level optimization. Since traditional ELA is mainly used for describing an optimization problem, it might falls short in providing informative low-level optimization status for MetaBBO methods. This explains the consistent superior performance of our NeurELA (see Figure 3). Compared with the traditional ELA features, our NeurELA can be meta-trained on MetaBBO tasks to provide sensitive and accurate low-level optimization status for the meta-level policy. As shown in Figure 5, the decision bound between the exploration population and the exploitation population is clear, which is helpful for the meta-level policy to balance the exploration-exploitation tradeoff within the low-level optimization.

### 4.4 In-depth Study (RQ4 ∼ similar-to \sim ∼ RQ6)

Computational Efficiency Comparison (RQ3). We compare the computational efficiency of our NeurELA with Original and ELA as either the amount of candidate samples $m$ or the dimension of the optimization problem
$d$ scales. We report the average wall-time required by each baseline and our NeurELA to produce the landscape features (for $1000$ runs). The results show that: 1) Our NeurELA achieves comparative computational efficiency with the specific hand-crafted designs in existing MetaBBO algorithms, while introducing significant performance boosts; 2) When the sample size $m$ increases, our NeurELA becomes more time-efficient than the original feature extraction designs in existing MetaBBO algorithms. This is primarily owning to the two-stage attention architecture in our NeurELA, which can be efficiently computed on GPUs in parallel. 3) The traditional ELA features encounters the curse of dimensionality, which makes itself impractical for MetaBBO algorithms to solve high-dimensional optimization problems.

Evolution Strategy Selection (RQ4). Within this paper, we employ a neuroevolution process to optimize the parameters of the landscape analyzer $\Lambda_{\theta}$, which presents a large-scale optimization challenge as $\Lambda_{\theta}$ comprises $3296$ learnable parameters. To identify an effective $ES$, we implement several candidate evolution strategy variants: Fast CMAES [], SEP-CMAES [], R1ES, RMES [], and the original CMAES [], and report the overall zero-shot performance (unseen algorithms & unseen problem sets) achieved by training our NeurELA with these candidate strategies for $10$ runs in Figure 7. The results reveal that Fast CMAES dominates the other candidates, and is hence selected as the neural network optimizer $ES$ for our NeurELA.

Figure: Table 1: The average wall time (in seconds) for computing features.
Refer to caption: x4.png

Model Complexity (RQ5).
We discuss the relationship between the model complexity and the zero-shot performance (unseen MetaBBO algorithm & problem sets) of our NeurELA. Concretely, We pre-train NeurELA under $6$ different model complexities, with various hidden dimensions, i.e., $h=(16,64)$, and the number of the Ts-Attn module, i.e., $l=(1,3,5)$. We additionally pre-train three MLP baselines, which substitute the Ts-Attn module in NeurELA with a linear feed-forward layer, which holds a shape of $h\times h,h=(16,64,128)$. We report both the zero-shot performance ($y$-axis) and the computational efficiency ($x$-axis, presented as the consumed wall time for computing the landscape features) in Figure 7, where the dashed lines denotes the performance and wall-time of the Original baseline. $\#para$ denotes the number of the learnable parameters. The results show that: 1) a significant performance gap is observed between the MLP baselines and our Ts-Attn module ($h=16,l=1$). It validates the effectiveness of our Ts-Attn design, which enhance the feature extraction of our NeurELA by encouraging the information sharing at both the cross-solution and cross-dimension levels; 2) As the model complexity increases, the performance of the Ts-Attn module drops rapidly. It reveals that the increased number of learnable parameters challenges the optimization ability of the backend $ES$. Given the limited computational resources, it is difficult to identify the optimal parameters ${\theta^{*}}$.

Limitations. As a pioneer work on using deep neural network for landscape analysis in MetaBBO, our NeurELA show possibility of learning an universal landscape analyser with minimal expertise requirement and appealing optimization performance. However, a major limitation is that the training efficiency of NeurELA is not very satisfactory. As discussed in the Model Complexity part, the zero-shot performance drops rapidly when the model complexity increases. This is due to the relatively limited large-scale optimization ability of existing evolution strategies. Unfortunately, the meta-objective of NeurELA (Eq. (4)) is not differentiable hence cannot benefit from back propagation methods. We denote this limitation as an important future work to address.

## 5 Conclusion

In this paper, we introduce NeurELA to complement the last mile in making the MetaBBO paradigms entirely end-to-end: the automatic extraction of optimization status features. To this end, we propose a landscape analyser parameterized by a two-stage attention-based neural network. The analyser enables flexible replacement for the original hand-crafted landscape analysis mechanisms in existing MetaBBO algorithms. We employ a neuroevolution paradigm to optimize it by maximizing performance over a multi-task MetaBBO operating space. The experimental results showcase that our NeurELA serves two key functions: 1) it can be seamlessly integrated into unseen MetaBBO algorithms for impressive zero-shot performance, and 2) it can also be fine-tuned together with new MetaBBO algorithms to adapt its performance for specific optimization challenges. While it possesses certain limitation (e.g., low training efficiency), our work represents a significant step toward realizing a fully automatic MetaBBO algorithm—a paradigm that learns to optimize without the need for manual expertise.

## Appendix A Technical Details

### A.1 Technical Details of Λ θ subscript Λ 𝜃 \Lambda_{\theta} roman_Λ start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT

Within this paper, the aim of the landscape analysis is to profile the dynamic optimization status of the current optimization process. That is, given a $d$-dimensional target optimization problem $f$, at any time step $t$, the optimization process maintains a population of $m$ candidate solutions $\{X_{i}^{t}\in\mathbb{R}^{d}\}_{i=1}^{m}$, and their corresponding objective values $\{y_{i}^{t}=f(X_{i}^{t})\}_{i=1}^{m}$. We consider an end-to-end neural network structure that receives the candidates population and their corresponding objective values as input, and then outputs $h$-dimensional dynamic optimization status $s_{i}^{t}$ for each $X_{i}^{t}$. This optimization status feature aggregates the information of the optimization problem and the current candidate population, hence can be used for dynamic landscape analysis in MetaBBO algorithms. We have to note that the core challenges in designing such a neural landscape analyser locate at: 1) generalizability: it should be able to handle optimization problems with different searching ranges and objective value scales; 2) scalability: it should be capable of computing the dynamic optimization status efficiently as the amount of the sampled candidates or the dimensions of the problem scales. We address the above two challenges by designing a two-stage attention based neural network structure as the landscape analyser ($\Lambda_{\theta}$) in NeurELA. We now introduce the architecture of the $\Lambda_{\theta}$ and establish its overall computation graph step by step. For the convenience of writing, we omit superfix for time step $t$.

Pre-processing Module. To make NeurELA generalizable across different problems with various searching ranges and objective value scales, we apply min-max normalization over the searching space and the objective value space. Concretely, for a specific $d$-dimensional optimization problem $f$ (suppose a minimization problem), we acquire its searching range $\{[lb^{j},ub^{j}]\}_{j=1}^{d}$, where $lb^{j}$ and $ub^{j}$ represent the lower bound and the upper bound at $j$-th dimension. Then we normalize each $X_{i}$ in the candidate population by $X_{i}^{j}=\frac{X_{i}^{j}-lb^{j}}{ub^{j}-lb^{j}}$, where $X_{i}^{j}$ denotes the $j$-th dimension of $X_{i}$. After the min-max normalization over the searching space, we min-max normalize the objective values within this time step, $y_{i}=\frac{y_{i}-y_{min}}{y_{max}-y_{min}}$, where $y_{min}$ and $y_{max}$ denotes the lowest and highest achieved objective values in this time step. We have to note that by normalizing the $X_{i}$ and $y_{i}$ within the range of $[0,1]$, we attain universal representation for different optimization problems, ensuring the generalizability of the subsequent neural network modules. The normalized $\{X_{i}\}_{i=1}^{m}$ and $\{y_{i}\}_{i=1}^{m}$ are then re-organized as a collection of meta data $\{\{(X_{i}^{j},y_{i})\}_{i=1}^{m}\}_{j=1}^{d}$ with the shape of $d\times m\times 2$. We then embed the meta data with a linear mapping $W_{emb}\in\mathbb{R}^{2\times h}$ as the final input encoding $s$, of which the shape is $d\times m\times h$. $h$ denotes the hidden dimension of the subsequent two-stage attention module.

Two-stage Attention Block. We construct a two-stage attention block (Ts-Attn) to aggregate optimization status information across candidate solutions and across each dimension of the decision variables. The overall computation graph of the Ts-Attn is illustrated in the right of Figure 2, of which a basic component is the attention block ($Attn$). As illustrated in the left of Figure 2, the $Attn$ block mainly follows designs of the original Transformer [], except that the layer normalization [] is used instead of batch normalization []. Given a group of $L$ input encoding vectors $X_{in}\in\mathbb{R}^{L\times h}$, Eq. (5) details the computation of the $Attn$ block.

$$ $\displaystyle g$ $\displaystyle=\text{LN}(X_{in}+\text{MHSA}(X_{in}))$ (5) $\displaystyle v$ $\displaystyle=\text{FF}^{(2)}(\text{ReLU}(\text{FF}^{(1)}(g)))$ $\displaystyle o$ $\displaystyle=\text{LN}(g+v)$ $$

where MHSA, LN and FF denote the multi-head self-attention [] (with the hidden dimension of $h$), layer normalization [] and linear feed forward layer respectively. The output $o$ holds identical shape with the input $X_{in}$. In our Ts-Attn block, we employ an $Attn$ block $Attn_{inter}$ for the first cross-solution information sharing stage, and the other $Attn$ block $Attn_{intra}$ for the second cross-dimension information sharing stage (illustrated in the right of Figure 2). The Ts-Attn receives the input encoding $s$ of the current candidate population, and then advances the information sharing in both cross-solution and cross-dimension level. The computation is detailed in Eq. (6).

$$ $\displaystyle H$ $\displaystyle=\text{Attn}_{inter}(S)$ (6) $\displaystyle H$ $\displaystyle=\text{Transpose}(H,d\times m\times h\rightarrow m\times d\times h )+\text{PE}$ $\displaystyle H_{out}$ $\displaystyle=\text{Attn}_{intra}(H)$ $\displaystyle F_{indiv}$ $\displaystyle=\text{MeanPooling}(H_{out},m\times d\times h\rightarrow m\times h)$ $\displaystyle F_{pop}$ $\displaystyle=\text{MeanPooling}(F_{indiv}),m\times h\rightarrow h)$ $$

At the first stage, we let the input encoding $s$ (attained from the pre-processing module) pass through $Attn_{inter}$. Since we group the encodings of the same dimension of all candidates in $s$, the $Attn_{inter}$ promotes the optimization information sharing across candidates in current population. From the first stage, we obtain a group of hidden features $H$ with the shape of $d\times m\times h$. At the second stage, we first transpose $H$ into the shape of $m\times d\times h$ to regroup all dimensions of a candidate together. We then add $cos$/$sin$ positional encoding (PE) over the transposed $H$ to inform the order of different dimensions in a candidate. We then let $H$ pass through $Attn_{intra}$ to advance the information sharing among the different dimensions within the same candidate. The output of $Attn_{intra}$ holds the shape of $m\times d\times h$. At last, we apply MeanPooling on $H_{out}$ to get the landscape feature for each candidate $F_{indiv}$ in the population, and apply a second MeanPooling on $F_{indiv}$ to get the landscape feature for the whole candidate population $F_{pop}$. We have to note that we calculate both $F_{indiv}$ and $F_{pop}$ to make our NeurELA compatible with diverse MetaBBO algorithms, which either require the landscape feature of the whole population (e.g., []) or require a separate landscape feature for each candidate (e.g., []). The highly parallelizable attention-based neural-network architecture ensure the scalability of our method as the amount of the sampled candidates or the dimensions of the problem increases.

Now we summarize the end-to-end workflow of the neural landscape analyser ($\Lambda_{\theta}$) in our NeurELA. At any time-step $t$ within the optimization process, the pre-processing module transforms the information of the candidate population (i.e., $\{X_{i}^{t}\}_{i=1}^{m}$ and $\{y_{i}^{t}\}_{i=1}^{m}$) into the input encoding $s$. Then the Ts-Attn module transforms $s$ into the dynamic landscape features $F_{inidiv}^{t}$ and $F_{pop}^{t}$.

### A.2 Train-test split of BBOB testsuites

BBOB contains 24 synthetic problems owning various landscape properties [] (e.g. multi-modality, global structure, separability and etc.). Table 2 list all of the 24 problems according to their category. Due to the diversity of properties, how to split these problems into train-test set becomes a key issue to ensure the training performance and its generalization ability. Our fundamental principle to split is to maximize the inclusion of representative landscape properties as possible. Specifically we visualize these 24 problems under 2D setting, and then select 12 representative problems into train set. We also provide contour map of problems in train set in Figure 8 and test set in Figure 9. Moreover, to avoid possible issue [] coming from fixed optima which is often located in $[0,...,0]$ in current benchmark problems (this might facilitate model to overfit to this fixed point), we thus add random offset $O$ into each problems, that is to convert $y=f(x)$ into $y=f(x-O)$. This operation is inserted into both train set $\mathbb{D}_{\text{train}}$ and test set $\mathbb{D}_{\text{test}}$.

### A.3 Train-test split of BBOB-Noisy and Protein-Docking testsuites

We summarize some key characteristic of this two testsuits as follows.

- •
BBOB-Noisy: this testsuits contains 30 noisy problems from COCO []. They are obtained by further inserting noise with different models and levels into problems in BBOB testsuits. BBOB-Noisy is characterized by its noisy nature and often used to examine robustness of certain optimizers.
- •
Protein-Docking: this testsuits contains 280 instances of different protein-protein complexes []. These problems are characterized by rugged objective landscapes and are computationally expensive to evaluate.

We follow train-test split for these two testsuites defined in MetaBox []. Under easy mode in MetaBox, 75$\%$ of instances are allocated into training and the remaining 25$\%$ are used in testing. Training or further fine-tuning on these two testsuites in our experiments are executed in the train set, and then validate performance of corresponding MetaBBO algorithms in the test set $\mathbb{D}_{\text{test}}$.

**Table 2: Overview of the BBOB testsuits.**
|  | No. | Functions |
| --- | --- | --- |
| Separable functions | 1 | Sphere Function |
|  | 2 | Ellipsoidal Function |
|  | 3 | Rastrigin Function |
|  | 4 | Buche-Rastrigin Function |
|  | 5 | Linear Slope |
| Functions<br>with low or moderate<br>conditioning | 6 | Attractive Sector Function |
|  | 7 | Step Ellipsoidal Function |
|  | 8 | Rosenbrock Function, original |
|  | 9 | Rosenbrock Function, rotated |
| Functions with<br>high conditioning<br>and unimodal | 10 | Ellipsoidal Function |
|  | 11 | Discus Function |
|  | 12 | Bent Cigar Function |
|  | 13 | Sharp Ridge Function |
|  | 14 | Different Powers Function |
| Multi-modal<br>functions<br>with adequate<br>global structure | 15 | Rastrigin Function (non-separable counterpart of F3) |
|  | 16 | Weierstrass Function |
|  | 17 | Schaffers F7 Function |
|  | 18 | Schaffers F7 Function, moderately ill-conditioned |
|  | 19 | Composite Griewank-Rosenbrock Function F8F2 |
| Multi-modal<br>functions<br>with weak<br>global structure | 20 | Schwefel Function |
|  | 21 | Gallagher’s Gaussian 101-me Peaks Function |
|  | 22 | Gallagher’s Gaussian 21-hi Peaks Function |
|  | 23 | Katsuura Function |
|  | 24 | Lunacek bi-Rastrigin Function |
| Default search range: [-5, 5]^D |  |  |

Figure: (a) Function 1
Refer to caption: x6.png

Figure: (a) Function 3
Refer to caption: x18.png

### A.4 License of Used Open-sourced Assets

Our codebase can be accessed at [https://github.com/GMC-DRL/Neur-ELA](https://github.com/GMC-DRL/Neur-ELA). In Table 3 we listed several open-sourced assets used in our work and their corresponding licenses.

**Table 3: Used open-sourced tools and their licenses.**
| Used scenario | Asset | License |
| --- | --- | --- |
| Top-level optimizer | [PyPop7](https://pypi.org/project/pypop7/) [] | GPL-3.0 license |
| MetaBBO algorithms implementation<br>Low-level train-test workflow | [MetaBox](https://github.com/GMC-DRL/MetaBox) [] | BSD-3-Clause license |
| Parallel processing | [Ray](https://github.com/ray-project/ray) [] | Apache-2.0 license |
| ELA feature calculation | [pflacco](https://pypi.org/project/pflacco/) [] | MIT license |

### A.5 Control-parameters of E ⁢ S 𝐸 𝑆 ES italic_E italic_S

Fast CMAES
We grid-search three key hyper-parameters in Fast CMAES, including the mean value $\mu$ and sigma value $\sigma$ of the initial Gaussian distribution used for sampling, learning rate of evolution path update $c$. We list the grid search options in Table 4 and choose the best setting according to training performance on BBOB. Besides, for other control-parameters of the Fast CMAES, we follow the default settings listed in its original paper [].

**Table 4: Grid-search of control-parameters of Fast CMAES.**
| Control-parameters | Grid options | Selected setting |
| --- | --- | --- |
| Initial mean value $\mu$ | [$\mathbf{\mathrm{0}}^{D}$, $\mathbf{\mathcal{R}}^{D}$] | $\mathbf{\mathcal{R}}^{D}$ |
| Initial sigma value $\sigma$ | [$0.1$, $0.3$] | $0.3$ |
| Learning rate of evolution path update $c$ | [$2.0/(D+5.0)$, $6.0/(D+5.0)$] | $2.0/(D+5.0)$ |

Other candidate evolution strategy variants
We follows the default settings as implementations in PyPop7 [] for other candidate top-level optimizers. We made a comparision study among SEP-CMAES [], R1ES, RMES [], original CMAES [] and Fast CMAES under their default settings and finally select Fast CMAES as the default top-level optimizer of this work.

## Peer Reviews

confidence: 4
contribution: 3
initial_rating: 6
presentation: 2
questions: 1. I suppose that the computational cost of the proposed approach is larger than the existing MetaBBO because it requires the training of a landscape feature extractor as the outer loop for MetaBBO. What is the exact computational time/cost for the proposed method compared to existing MetaBBO methods?
1. If the baseline methods can use the same computational budgets, how is the performance gain of the proposed approach? For instance, extra budgets may be used to optimize hyperparameters or to select traditional ELA features in the baseline MetaBBO.
1. Training the landscape feature extractor in the current setting seems challenging because it should optimize the high-dimensional parameters by the evolution strategy. Is there any empirical evidence for the convergence of ES as the outer loop optimizer?
1. What kind of BBO or MetaBBO algorithms can be used with the proposed NeurELA? The authors might assume the population-based BBO algorithms or evolutionary algorithms. Is it possible to combine the NeurELA with other kinds of BBO methods?
1. What is the exact number of dimensions of the NeurELA features in the experiment? What is the impact of the dimensionality of the EeurELA features on the performance?
1. I could not find the definition of the performance metric in the experimental evaluation, e.g., in Figure 3. Could you provide a detailed definition of the performance metric in the experimental results?
soundness: 3
strengths: - The motivation for optimizing the feature extractor in MetaBBO is reasonable.
- The technical novelty of this paper is to present automating approach for extracting landscape features in MetaBBO.
- The effectiveness of the proposed NeurELA is experimentally demonstrated for several MetaBBO and BBO problems.
summary: This paper proposes an automatic construction method of landscape features for meta black-box optimization (MetaBBO). The proposed approach, termed neural exploratory landscape analysis (NeurELA), trains the attention-based neural network extracting landscape features to improve the MetaBBO algorithms. While existing MetaBBO methods rely on human-crafted landscape features, the proposed NeurELA can automate the design process for landscape features in MetaBBO. The experimental result demonstrates that the proposed NeurELA outperforms existing MetaBBO methods on unseen MetaBBO tasks. In addition, the authors show that fine-tuning process can lead to further performance improvement.
weaknesses: - The proposed formulation seems to be a tri-level optimization problem of training landscape feature extractor, training meta-level policy, and optimizing a target objective function. Therefore, using the proposed NeurELA increases the whole computational cost compared to existing MetaBBO methods.
- Training the landscape feature extractor is performed in a neuroevolution manner. It seems hard to scale for a large neural network as the feature extractor. In addition, it is not clear that the current setting, i.e., optimizing 3,296 parameters for 500 evaluations by the evolution strategy, is sufficient for convergence.

---

confidence: 3
contribution: 3
initial_rating: 8
presentation: 4
questions: See weaknesses..
soundness: 4
strengths: - The new end-to-end pipeline is (imho) novel and interesting
- NeurELA consistently outperforms the baselines
- Many ablations are performed and thus we better understand the effectiveness of the approach and its parts
- The sharing of the code is appreciated
summary: The paper introduces Neural Exploratory Landscape Analysis (NeurELA), a novel framework designed to improve Meta-Black-Box Optimization (MetaBBO) by dynamically profiling optimization landscape features through a two-stage, attention-based neural network. Unlike traditional approaches that rely on human-crafted features, NeurELA learns these features automatically in an end-to-end manner. This is aimed at overcoming the limitations of existing Exploratory Landscape Analysis (ELA) methods, such as computational overhead and reliance on expert knowledge. The authors propose a novel neural-network based landscape analyzer, and propose a two-stage attention mechanism: this architecture facilitates robust feature extraction, capable of generalizing to various MetaBBO algorithms without manual adjustments. The novel framework is tested in three MetaBBO tasks against several baselines and ablations.
weaknesses: **Weaknesses before author's rebuttal. I believe that most of my comments have been (at least partly) by the new version/reply by the authors.**

- The presentation of the paper needs quite some work. Many typos are present and a few paragraphs are hard to read. Overall, the authors need to spend a bit more time in improving the presentation.
- I believe that a more detailed description of the MetaBBO tasks would greatly help the reader understand and appreciate the performance of the proposed framework.
- In Section 4.2 and Fig. 4 it is not clear to what "one epoch" corresponds to. For example, I do not understand how we can compare "ZeroShot" with "Finetune" in the same scale. If I understand correctly, the "ZeroShot" variant just uses the learned $\Lambda_{\theta}$ to select at each generation of the low-level optimizer the appropriate feature vector and configuration. On the contrary, the "Finetune" variant should run the whole meta-learning pipeline. Thus, how can we compare "epochs" of one to the other? We need more explanation here to appreciate the results.
- The computational overhead of NeurELA framework is quite big. Also compared to the original variant ($\Lambda_0$) the gains are not big even when a big number of samples ($m$) is used.

---

confidence: 3
contribution: 1
initial_rating: 3
presentation: 3
questions: 1. What's the major difference between the proposed method and Deep-ELA? A more detailed explanation and corresponding experiments and/or ablation studies will help better support the paper's novelty and contribution.

2. Line 479: "This is primarily owning to the two-stage attention architecture in our NeurELA, which can be efficiently computed on GPUs in parallel." Was the training done on GPU? It was mentioned in line 356 that the training uses a Slurm CPU cluster. 

3. The wall time is too short to really tell a difference between algorithms since programming/system processing could have a larger effect if it's in milliseconds. Please consider using a more complicate benchmark or increasing the dimensions.
soundness: 3
strengths: 1. Quality: The paper validates the proposed method in detail by answering research questions, not only on the method's performance but also its adaptability, computational efficiency, and generalization capacity. 

2. Clarity: The paper is well-structured, with clear explanations of NeurELA’s architecture, training, and integration within MetaBBO tasks.
summary: The paper introduces Neural Exploratory Landscape Analysis (NeurELA), a framework for Meta-Black-Box Optimization (MetaBBO) that replaces traditional, human-crafted Exploratory Landscape Analysis (ELA) features with a fully neural network-based approach. NeurELA employs a two-stage, attention-based neural network trained via a neuroevolution to dynamically profile optimization landscapes, adapting its features in real time. The authors show that NeurELA enhances performance across various MetaBBO algorithms and generalizes effectively to unseen tasks for zero-shot generalization and fine-tuning.
weaknesses: 1. Originality: The proposed work is very similar to Seiler et al., 2024 (Deep-ELA), which also uses multi-head attention as the main component in the architecture. The only difference seems to be that Deep-ELA uses kNN embedding, while the proposed method uses a linear transformation to encode the population information, which is widely used in LLMs to generate embedding from tokens. 

2. Limited comparisons in experiments: The proposed work does not compare to any recent methods, e.g., Deep-ELA. 

3. Limited tasks: Although NeurELA is tested across a variety of MetaBBO algorithms and optimization problems, the experiments lack a detailed analysis of its performance in higher-dimensional optimization scenarios, where many MetaBBO algorithms struggle.

4. Interpretability and feature analysis: Although NeurELA shows promise in dynamically adapting landscape features, there is limited discussion on the interpretability of these features in relation to traditional ELA metrics.

---

confidence: 3
contribution: 3
initial_rating: 8
presentation: 3
questions: - What is the minimum number of training tasks needed for good zero-shot generalization?
- Could you provide more insight into what features the network learns compared to traditional ELA?

### Nitpicking
- L110: an universal neural landscape analyser... -> a universal
- L176: (Prager et al., 2021b) first proposed... -> wrong citation type, should use \citet instead of \citep
- L318: We hence introduce Ray Moritz et al. (2018), an open-source... -> wrong citation type + you don't introduce Ray, you "employ" Ray
- L350: functiona... -> funtion
- L367: Recall that an MetaBBO task is... -> that a MetaBBO
soundness: 3
strengths: - The approach is well-motivated and builds on established MetaBBO literature
- The experimental methodology is rigorous with proper baselines and ablations
- Results demonstrate clear improvements over traditional ELA features
- Comprehensive experiments across multiple MetaBBO algorithms
- Testing on both synthetic benchmarks and real-world problems (protein docking)
summary: The paper proposes Neural Exploratory Landscape Analysis (NeurELA), a framework to replace hand-crafted landscape analysis features in Meta Black-Box Optimization (MetaBBO) with a learned neural network approach.

The key contributions are:
- making the MetaBBO paradigm entirely end-to-end
- A two-stage attention-based neural network architecture for dynamic landscape feature extraction
- A multi-task training framework to learn from multiple MetaBBO algorithms simultaneously
- Demonstration of zero-shot generalization to unseen algorithms and problem sets
- Ability to fine-tune the pre-trained analyzer for specific tasks
weaknesses: - Could better analyze when/why zero-shot generalization fails
- The authors acknowledge training efficiency issues with larger models
- Lacks theoretical justification for why the two-stage attention architecture works well

Meta-review: The goal of the work is to learn a landscape featurizer for meta-blackbox optimization. The pipeline of the meta-BBO loop basically consists of:

1. A featurizer (e.g. attention-based architecture) which takes in the current trajectory of evaluations $(x,y)$ and outputs a feature.
2. This feature is then used to condition a separate blackbox optimizer (which itself needs to be neural-network based, e.g. an LSTM)
3. Since this bilevel problem is not differentiable through the featurizer, non-gradient based optimization methods (e.g. CMA-ES) must be used to train the featurizer.
4. The featurizer can be frozen and used directly over new test cases, or can also be fine-tuned if the new test situation has a large distribution shift from training (e.g. objective functions are different, or inner-loop algorithm is different)

Experiments are conducted over BBOB (deterministic and noisy) and protein design objectives, as well as when the inner-loop algorithm is changed.

Ablations show:
1. The Neural ELA features are cleanly separable for determining when the inner-loop algorithm should exploit or explore, whereas traditional ELA features are not so separable.
2. If the featurizer has too many learnable parameters, performance degrades since e.g. CMA-ES may suffer in high dimension, and that variants like FCMAES are the best for learning such parameters.

## Strengths
* After parsing the paper properly, the general logic makes sense. It's understandable why e.g. the featurizer needs to be trained with CMA-ES, and the whole setup is intuitive.
* It's interesting that these NeuralELA features can be transferrable (and worst-case just needs a bit of fine-tuning) to new algorithms as well, and not just new objective functions. This means that it's possible to construct universal features for representing the objective landscape, which can be sent to any inner-loop optimizer.

## Weaknesses
* I needed to re-read the paper multiple times to understand the nuances - The writing could be made much more clean. Currently the paper is too wordy and doesn't give enough "breathing space" between its text. Also, the notation can be reduced significantly, and I think the main method can be described much more simply in at most half a page.
* While the paper makes solid contributions, due to the way the paper is written at the moment, there aren't a lot of fundamental or profound conclusions. The paper could've been elevated, if e.g. training was done over a much larger set of algorithms and objectives, and that the featurizer's output became more universal and no longer required fine-tuning at all. 

Given the Reviewer scores, for now we can definitely accept the paper for poster presentation. I wouldn't move it to spotlight however, due to the weaknesses raised above.

Decision: Accept (Poster)