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Automatic Unsupervised Ensemble Outlier Model Selection--Extended Version

2026 · arxiv_cs

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Automatic Unsupervised Ensemble Outlier Model Selection–Extended Version

Hong-Phuc Phan * 1 Tuan-Anh Vu * 2 Tung Kieu * 3 Son Ha Xuan 4 Bin Yang 5 Christian S. Jensen 3

arXiv:2605.16567v1 [cs.LG] 15 May 2026

Abstract

1. Introduction Outlier detection plays a critical role in applications like fraud detection, network security, medical diagnosis, and system monitoring (Chandola et al., 2009; Ruff et al., 2021). However, in many real-world scenarios where ground-truth labels are unavailable, and outlier detection must be performed in a fully unsupervised manner (Zimek et al., 2013). Further, contamination rates are often unknown, and data distributions vary widely across tasks (Han et al., 2022; Zhang et al., 2026). These factors make it difficult to not only detect anomalies but also evaluate and compare detection models.

Unsupervised outlier detection is attractive because it eliminates the need for labeled data. Moreover, forming multi-model ensembles can improve detection robustness. However, composing an ensemble without labeled data is challenging. Naively composed ensembles can suffer from ensemble saturation, where redundant or unreliable detection models degrade performance and incur unnecessary computation. We propose MetaEns, an automatic unsupervised framework for selecting ensembles of outlier detection models. Using labeled meta-datasets, MetaEns learns a model that predicts marginal ensemble gains, estimating the expected improvement from adding a candidate model to a partially constructed ensemble. At test time, this learned signal is combined with a submodular-inspired proxy objective that enforces diminishing returns through diversityaware discounting and family-level risk regularization, thereby enabling greedy sequential selection with adaptive early stopping. As a result, MetaEns constructs compact, high-quality ensembles without access to ground-truth labels. Experiments on 39 real-world datasets show that MetaEns consistently outperforms state-of-theart unsupervised selectors and ensemble baselines, achieving higher average precision while using fewer models.

Existing studies have proposed diverse unsupervised outlier detectors based on density estimation (Breunig et al., 2000), isolation mechanisms (Liu et al., 2008), reconstruction errors (Zong et al., 2018), and deep representations (Ruff et al., 2018). Despite their success in specific settings, no single detector performs reliably across diverse datasets. This observation has motivated the use of ensemble methods, which aim to improve robustness by aggregating multiple detectors (Campos et al., 2021; Kieu et al., 2019). In supervised learning, ensembles can be trained and validated using labeled data. In contrast, constructing effective ensembles for unsupervised outlier detection remains an open challenge (Zimek et al., 2013). A key challenge is to select models when no labels are available (Marques et al., 2020). Without ground-truth feedback, it is unclear which detection models are reliable on a given dataset or whether adding a new detector will improve or degrade ensemble performance (Rayana et al., 2016; Aggarwal & Sathe, 2015). As a result, many unsupervised ensemble methods rely on fixed aggregation strategies, such as averaging scores from all available detectors or selecting a fixed number of top-ranked detection models (Zhao et al., 2019a). Methods employing these strategies suffer from ensemble saturation: beyond a small ensemble size, adding more detectors yields diminishing or even negative returns due to redundancy, conflicting rankings, or systematically poor models. Moreover, fixed-size ensembles are inherently inflexible and cannot adapt to dataset-specific complexity.

* Equal contribution 1 Department of Software Engineering, FPT University, Vietnam 2 Department of Information Technology, Can Tho University of Technology, Vietnam 3 Department of Computer Science, Aalborg University, Denmark 4 School of Business, RMIT University, Vietnam 5 School of Data Science and Engineering, East China Normal University, China. Correspondence to: Tung Kieu <[email protected]>, Son Ha Xuan <[email protected]>.

Recent unsupervised model selection methods attempt to address these challenges by leveraging meta-learning across labeled auxiliary datasets (Hospedales et al., 2022). No-

Automatic Unsupervised Ensemble Outlier Model Selection–Extended Version

tably, frameworks such as MetaOD (Zhao et al., 2021) and ELECT (Zhao et al., 2022) learn to recommend a single detector for a new unlabeled task based on task similarity or historical performance patterns. While effective, these methods are limited to singleton selection and do not address the more general problem of adaptive ensemble construction, where multiple complementary detectors may be required to capture diverse outlier patterns (Cheng et al., 2020).

enable adaptive ensemble construction with early stopping. • We provide experimental results across 39 datasets, supported by ablation studies, showing that compact, adaptively sized ensembles can outperform larger fixed ensembles in fully unsupervised settings.

2. Preliminaries

We address this gap by formulating unsupervised ensemble outlier model selection as a sequential decision problem. Our key insight is that although the true marginal benefit of adding a detector to an ensemble is unobservable at test time, its structure can be learned offline from labeled metadatasets. Building on this idea, we propose MetaEns, a framework that learns to predict the marginal ensemble gain of candidate detectors conditioned on the current ensemble state. At inference time, MetaEns greedily constructs an ensemble by maximizing a submodular-inspired proxy objective (Nemhauser et al., 1978) that integrates the predicted gain with explicit mechanisms for diversity control and risk mitigation.

Definition 2.1 (Dataset). A dataset X is a finite collection of data instances (or data points) X = {x1 , x2 , . . . , xN }, where each instance xi ∈ Rd is a d-vector. We denote |X| = N as the cardinality of the dataset. Definition 2.2 (Unsupervised Outlier Detection). Given a dataset X ∈ RN ×d , an unsupervised outlier detection model, or detector, learns a scoring function f : X → RN that assigns an outlier score oi = f (xi ) to each instance xi ∈ X, where larger values indicate a higher likelihood of being an anomaly. A decision function can be derived by thresholding the scores at a user-defined level τ , yielding outlier labels ŷi = I(oi > τ ). The resulting outlier set is defined as follows.

Specifically, MetaEns introduces two principles that are crucial for unsupervised ensemble construction. First, we enforce diminishing returns through similarity-based discounting that penalizes candidates that introduce redundancy with already selected detectors (Kulesza & Taskar, 2012). Second, we incorporate family-risk regularization, which discourages the selection of multiple detectors from algorithmic families with a history of poor or unstable performance. Together, these components yield a proxy objective that favors compact, diverse ensembles and naturally supports adaptive early stopping when no further improvement is expected.

O = {xi ∈ X | ŷi = 1} Definition 2.3 (Unsupervised Outlier Model Selection). Let Ω = {f1 , f2 , . . . , fK } be the set of K candidate outlier detection models. Each fi ∈ Ω can be seen as a (detector, configuration) tuple, where the configuration denotes a set of hyperparameters of the detector. Let X = {x1 , x2 , . . . , xN } be an unlabeled dataset. Each outlier detection model fi acts as an outlier scoring function fi : RN ×d 7→ RN , which assigns an outlier score to each instance in X. The task of unsupervised outlier model selection is then to choose the model f ∗ as follows. f ∗ = arg min Γ(fi , X)

We evaluate MetaEns on a benchmark of 39 real-world anomaly detection datasets (Han et al., 2022) using a large pool of 297 candidate detectors spanning multiple algorithmic families. Extensive experiments show that MetaEns is able to consistently outperform strong unsupervised baselines and recent meta-learning approaches. Notably, MetaEns achieves higher detection accuracy while selecting fewer models than fixed-size ensembles, and it exhibits strong resilience by recovering performance even when an initially selected detector performs poorly.

fi ∈Ω

Here, Γ(·) is an unsupervised evaluation criterion that estimates the quality of model fi based solely on the distributional properties of its output scores without using labeled outlier or normal instances. Problem Definition: Unsupervised Ensemble Outlier Model Selection. Let Ω = {f1 , f2 , . . . , fK } be a set of K candidate outlier detection models. Each model fi ∈ Ω can be seen as a (detector, configuration) tuple, where the configuration denotes a specific set of hyperparameters of the detector. Let X = {x1 , x2 , . . . , xN } be an unlabeled dataset. Each outlier detection model fi ∈ Ω acts as an outlier scoring function fi : RN ×d 7→ RN that assigns an outlier score to each instance in X. A

In summary, our contributions are threefold: • We formulate the problem of unsupervised ensemble outlier model selection and cast it as a sequential decision process without access to labels. • We propose MetaEns, a meta-learning framework that predicts marginal ensemble gains and combines these with a diversity- and risk-aware proxy objective to 2

Automatic Unsupervised Ensemble Outlier Model Selection–Extended Version

tion terminates once no candidate is expected to provide positive utility.

candidate ensemble P ⊆ Ω is evaluated as a whole, not as an independent sum of member-model criteria. Specifically, we first aggregate member scores into a single ensemble score vector, oP =

Fig. 1 summarizes the interaction between offline metatraining and online selection.

1 X f (X). |P |

3.2. Offline Meta-training Algorithmic families. We partition the candidate pool Ω into a fixed set of algorithmic families F, where each family groups detectors sharing the same underlying algorithmic paradigm. For instance, all LOF variants (differing only in k or distance metric) belong to the same family, as do all IForest variants (differing in number of estimators or subsampling ratio). The 297-model pool spans 8 base families: IForest, LOF, kNN, HBOS, OCSVM, LODA, ABOD, and COF. Family assignments are fixed from metadata and do not change across datasets or training steps. We use a single family-risk percentile, the 10th percentile of oracle marginal gains, for all families so that a family is penalized only when its worst-case historical contributions are systematically negative. This percentile is fixed globally using only metatraining trajectories and is never tuned on the target dataset. It is a conservative lower-tail statistic: lower percentiles may miss families with occasional catastrophic failures, whereas higher percentiles can over-penalize average families that still provide useful orthogonal diversity. We evaluate the sensitivity to this family definition in App. A.1 and the sensitivity to the risk percentile in App. A.2.

f ∈P

The task of unsupervised ensemble outlier model selection is to choose a set of models P ⊆ Ω as follows. P ∗ = arg max ψ(P ; X) P ⊆Ω

s.t.

|P | ≤ η.

Here, the criterion ψ(P ; X) estimates the quality of the aggregated ensemble score vector oP without using any ground-truth labels, and η is a budget on the ensemble size.

Definition 2.4 (Meta-datasets). We assume access to a collection of labeled meta-datasets M = {(M1 , y1 ), (M2 , y2 ), . . . , (ML , yL )} where each Mi ∈ RNMi ×dMi is a dataset and yi ∈ {0, 1}NMi provides ground-truth anomaly labels. These meta-datasets are used exclusively for evaluating and selecting candidate outlier detection models. At test time, we are given an unlabeled dataset X = {x1 , x2 , . . . , xN } that satisfies X ∼ PX and PX ̸= PMi , ∀i, i.e., the test distribution differs from the distributions underlying the meta-datasets.

Let M = {(Mℓ , yℓ )}L ℓ=1 be the labeled meta-datasets, and let Ω = {f1 , . . . , fK } denote the candidate detector pool (detector, configuration). For a dataset (M, y) ∈ M and an ensemble (model set) P ⊆ Ω, we define the ensemble score vector as the mean of the member scores:

3. Methodology 3.1. Framework Overview MetaEns encompasses two phases: offline meta-training and online model selection.

oP =

1 X f (M), |P |

(1)

f ∈P

Offline meta-training. Given a collection of labeled metadatasets, we simulate sequential ensemble construction and compute the true marginal gain of adding a candidate detector to a partial ensemble using ground-truth labels. These oracle gains provide supervision for learning a meta-model that predicts the expected marginal gain conditioned on the current ensemble context.

where f (M) ∈ R|M| is the outlier-score vector produced by f on M. We use the mean as the default aggregation rule because it preserves continuous score information from all selected detectors while reducing sensitivity to any single poorly calibrated model. In App. A.3, we compare this choice against median, max, and min aggregation and find that mean aggregation provides the strongest and most stable performance.

Online model selection. At test time, MetaEns is applied to a new unlabeled dataset. The procedure starts by selecting a high-quality primary detector that serves as an anchor. MetaEns then expands the ensemble by greedily adding detectors with high predicted utility. Since the predicted gain is a learned one and may be noisy, we introduce a submodular-inspired proxy objective that (i) discounts redundant candidates to induce diminishing returns and (ii) penalizes risky algorithm families based on meta-training history. This proxy enables adaptive early stopping: selec-

Oracle marginal gain. We define the true marginal gain of adding fi ∈ Ω \ P to P as the improvement in Average Precision: G(fi | P ) = AP(oP ∪{fi } , y) − AP(oP , y),

(2)

where AP(·, ·) is computed from the ensemble score vector and the ground-truth label vector. We use AP as it is 3

Automatic Unsupervised Ensemble Outlier Model Selection–Extended Version (A) Offline Meta-Training Ground Truth

Train Classifier

Target

Simulate Sequential Path (Offline) Meta-datasets Init: Sample Primary (diverse)

Greedy Add

Calculate Gain

Training Tuples

Train Classifier

Target

Repeat Family Risk

Model Pool

(B) Online Sequential Selection

Transfer Transfer

Candidate Pool

Candidate model Target Dataset

Transfer

Current Ensemble

Primary

Family Risk (from Marginal Gains)

Classifier

State

Context-Aware State Final Ensemble

ELECT Selector

Classifier

YES (Add

Predicted Submodular Gain Utility

to current ensemble

NO (Adaptive Stop)

)

Figure 1. Overview of MetaEns. (A) Offline Meta-Training: Oracle-greedy rollouts on labeled meta-datasets M generate state–gain pairs for partial ensembles. These pairs supervise a two-part gain predictor consisting of classifier fcls , which estimates whether a candidate will improve the ensemble, and regressor freg , which estimates the positive gain magnitude. Family-risk priors πF are computed from lower-tail oracle gains to identify detector families with historically harmful additions. (B) Online Selection: On an unlabeled target dataset, MetaEns starts from a primary detector and greedily expands the ensemble by maximizing a proxy utility that combines predicted gain, redundancy discounting γ, and family-risk penalty. Selection stops adaptively when no remaining candidate has positive utility, yielding a compact label-free ensemble.

threshold-independent and robust under severe class imbalance, which is common in outlier detection.

candidate and the most recent selection, (ii) compatibility between the candidate and the current ensemble, and (iii) the selection stage via |P |. We deliberately restrict MetaEns to score-level summaries rather than raw inputs, learned embeddings, or detector internals. This score-only design makes the representation independent of the original data dimensionality and avoids assumptions about whether a detector is classical, neural, image-based, or text-based. As a result, the same meta-feature map can be applied across heterogeneous detector pools and modality-transfer settings, which we evaluate empirically in Section 4.

Generating training trajectories. To obtain informative supervision, we construct meta-training trajectories using an oracle greedy policy. For each meta-dataset, we initialize the ensemble with a primary model f1∗ chosen to maximize AP(f (M), y). We then iteratively add models that maximize the true gain in Eq. 2. This strategy exposes the meta-model to high-quality partial ensembles, avoiding training dominated by arbitrary or low-signal states.

We compute a base feature extractor on normalized score vectors that includes correlation measures (e.g., Spearman correlation, cosine similarity), distributional statistics (e.g., entropy, kurtosis), and overlap-based agreement (e.g., Jaccard similarity over top-ranked instances; see App. A.5 for ∗ full definitions). Concretely, we compute: (i) ϕfi ,fi−1 ∈ dbase d base ∗ R from (ofi , ofi−1 ), (ii) ϕfi ,P ∈ R from (ofi , oP ), dbase ∗ ∗ and (iii) ϕfi−1 ∈ R from (o , o ). The final state ,P fi−1 P is:

We empirically verify that alternative strategies such as ε-greedy exploration or reduced meta-dataset sizes consistently underperform oracle greedy rollouts, confirming that aligning training with test-time behavior is both sufficient and sample-efficient (see App. A.4). At each step i, for every candidate f ∈ Ω \ P , we com∗ pute a state representation ϕ(f, fi−1 , P ) and its oracle gain G(f | P ), obtaining supervised pairs {(ϕ(·), G(·))} across multiple ensemble sizes and datasets. These pairs are used to learn a predictor of marginal gains.

∗ ∗ ∗ ϕ(fi , fi−1 , P ) = (ϕfi ,fi−1 , ϕfi ,P , ϕfi−1 ,P , |P |).

State representation. The state representation ∗ ϕ(fi , fi−1 , P ) encodes three factors essential to sequential ensemble construction: (i) redundancy between the

(3)

Marginal gain modeling. Positive marginal gains become increasingly sparse as the ensemble grows: once a strong partial ensemble has been formed, most remaining 4

Automatic Unsupervised Ensemble Outlier Model Selection–Extended Version

Proxy marginal utility. Given a current ensemble P (with ∗ the last selected model being fi−1 ), the marginal utility of adding candidate fi is: ∆U (fi | P ) = γ(fi , P ) · Ĝ(fi | P ) − λfam πF (fi ) , (5)

candidates are redundant or harmful and therefore have G(fi | P ) ≤ 0. The resulting gain distribution is zeroinflated, with many non-positive targets and a small tail of genuinely useful additions. A single regressor trained on this distribution can minimize squared error by predicting small positive values for many candidates, which is undesirable because it can still trigger the selection of redundant models. We therefore use a two-part hurdle-style model that separates whether a candidate improves the ensemble from how much it improves the ensemble when improvement occurs: Ĝ(fi | P ) = fcls (fi | P ) · freg (fi | P ),

where γ(fi , P ) ∈ (0, 1] discounts redundant candidates, and λfam πF (fi ) ≥ 0 penalizes families with historically negative tail behavior. The hyperparameters β and λfam control the strength of redundancy discounting and familyrisk regularization.

(4)

Redundancy discount.

where fcls (fi | P ) = P(G(fi | P ) > 0) predicts the probability of improvement and freg (fi | P ) = E[G(fi | P ) | G(fi | P ) > 0] predicts the gain magnitude conditioned on being positive. This decomposition lets the classifier act as a gate for sparse positive gains, while the regressor focuses only on the scale of useful additions.

γ(fi , P ) =

We define the discount factor as 1 , 1 + β · simmax (fi , P )

(6)

where simmax (fi , P ) is the maximum similarity between fi and any model in P :  max sim(fi , fj ) if |P | > 0, simmax (fi , P ) = fj ∈P (7) 0 if P = ∅,

For each training pair (ϕ, G), the classification target is ycls = I(G > 0). The regression target is yreg = max(0, G), optimized only on samples with G > 0. This design is validated by the ablation in Table 2: replacing the two-part architecture with a single gain predictor reduces AP from 0.4308 to 0.4133 and worsens the average rank from 59.3 to 87.

where sim(fi , fj ) is the Jaccard similarity between the sets of the top-ktop ranked instances of fi and fj . We use the maximum (rather than average) similarity to prevent selecting near-duplicates of any existing ensemble member, which is particularly harmful without validation labels.

Why ExtraTrees? We instantiate both fcls and freg with ExtraTrees (Geurts et al., 2006). A single decision tree has high variance and can overfit the moderate-size metatraining corpus, especially because our state representation contains dense, nonlinear score-statistics derived from pairwise detector behavior. ExtraTrees reduces this variance through an ensemble of randomized trees; compared with Random Forests, its more randomized split thresholds provide stronger regularization, and compared with boosting models such as XGBoost or LightGBM, its bagging-style variance reduction better matches our goal of transferring to unseen anomaly-detection datasets rather than fitting residual errors on the meta-training corpus. Empirically, we compare six meta-model families in App. A.6; ExtraTrees achieves the best AP (0.4308 ± 0.0064) and average rank (59.3 ± 6.96), outperforming alternatives such as XGBoost, Random Forest, and multilayer perceptrons while retaining competitive training and inference costs.

The discount γ(·) induces diminishing returns in the following sense: since simmax (fi , P ∪ {f }) ≥ simmax (fi , P ) for any added model f , the redundancy penalty is nondecreasing with ensemble growth, dampening the marginal utility of redundant candidates. We therefore treat the objective as submodular-inspired rather than formally submodular: a strict guarantee would require the learned black-box gain predictor Ĝ(fi | P ) itself to satisfy diminishing returns for all detector subsets, which is difficult to enforce without constraining the predictor and potentially degrading marginal-gain accuracy. Our proxy instead preserves the practical benefits needed here—greedy efficiency, redundancy control, and natural early stopping—while allowing flexible learned interactions among heterogeneous detectors. Family-risk regularization. To reduce downside risk, we introduce a family-level prior that penalizes candidates from algorithmic families with negative lower-tail historical gains. Let F : Ω → F map each candidate to an algorithm family. For a family F ∈ F, we define the family risk as the 10th percentile of oracle gains observed during meta-training:

3.3. Online Model Selection At test time, the true gain G(·) is unavailable. We therefore select ensembles using a proxy objective that combines the predicted gain Ĝ with explicit redundancy and risk control. Our goal is to induce diminishing returns behavior—a hallmark of submodular maximization—without requiring the learned predictor to satisfy submodularity.

RiskF (M) = Q0.10 ({G(f | P ) | f ∈ MF , |P | ≥ 1}) , (8) and convert it into a non-negative penalty: πF = max (0, −RiskF (M)) 5

(9)

Automatic Unsupervised Ensemble Outlier Model Selection–Extended Version

Here, MF collects candidates in family F across all metadatasets and all ensemble states encountered during metatraining. Intuitively, if a family occasionally produces strongly negative marginal gains, it receives a higher penalty. This is to avoid harmful additions in the absence of labels. For previously unseen families (MF = ∅), we set RiskF (M) = 0 and thus πF = 0, treating them as neutral rather than risky. This avoids injecting a manual penalty against novel detector families before observing any negative evidence, and keeps the framework extensible to zeroshot use of future detectors. In practice, all models in our main pool belong to known families, so this rule mainly affects plug-and-play extensions.

for IForest, and kernel types for OCSVM. The full specification of the algorithms and hyperparameter grids is detailed in App. A.10. Baselines. We compare MetaEns against 19 unsupervised baselines and a supervised greedy oracle upper bound, spanning standard detectors, deep learning methods, and metalearning or ensemble-based selectors. For ensemble baselines that depend on an ensemble size parameter k, we report the best-performing k (selected to maximize average AP), giving these fixed-size baselines an oracle advantage. The baselines include standard methods such as random selection (Singleton), classical detectors (e.g., LOF); naı̈ve and random ensemble methods (e.g., IForest, Random Ensemble, Mega Ensemble, RandNet (Chen et al., 2017)); deep learning approaches (e.g., RDA (Zhou & Paffenroth, 2017), DAGMM (Zong et al., 2018), DeepSVDD (Ruff et al., 2018), and recent deep detectors ROBOD (Ding et al., 2022), LUNAR (Goodge et al., 2022), DTE-C (Livernoche et al., 2024), and TCCM (Li et al., 2025) evaluated under the same unsupervised protocol); and meta-learning or ensemble-based selectors, including MetaOD (Zhao et al., 2020), LSCP (Zhao et al., 2019a), and the state-of-the-art ELECT (Zhao et al., 2022). We further consider ensemble variants built on ELECT (top-k aggregation and random expansion) to assess whether simple rank-based or hybrid strategies suffice. Finally, we report a supervised greedy oracle, which iteratively selects models using ground-truth labels as an upper bound. Detailed descriptions are provided in App. A.11.

3.4. Framework Algorithm The detailed framework algorithm is provided in App. A.7, where we present the offline meta-training and online model selection procedure in pseudocode form. 3.5. Complexity Analysis We conduct a computational complexity analysis to clarify how the proposed framework scales with the size of the candidate model pool and dataset size. The detailed computational complexity analysis is given in App. A.8.

4. Experiments 4.1. Experimental Settings Datasets. We use the 39-dataset benchmark introduced in ELECT (Zhao et al., 2022), which consists of independent, real-world tabular datasets drawn from the Outlier Detection Data Sets (ODDS)1 and UCI Machine Learning Repository (Campos et al., 2016) repositories. The datasets vary substantially, with the number of samples n ranging from 129 to 49,097 (median n = 1456) and the dimensionality d ranging from 4 to 400 (median d = 21). Contamination rates range from 0.03% to 35% (median: 2.3%). The collection covers three data-type categories: numeric-only (18 datasets), categorical-only (2 datasets), and mixed-type (19 datasets). Details of the datasets are provided in App. A.9.

Hyperparameter & Implementation Details. No method uses target labels for tuning. Baselines use fixed defaults (PyOD/Scikit-learn) or the best fixed ensemble size by benchmark AP, while MetaEns tunes only on labeled meta-training data via leave-one-dataset-out validation. Detailed hyperparameter settings are provided in App. A.2, and implementation details are included in App. A.12. Metrics. We evaluate all methods using five metrics: Average Precision (AP), Average Rank (AR), ROC-AUC, Precision@π, and Max F1-score. Details of metrics are provided in App. A.13.

Candidate Model Pool. We construct a large candidate pool of 297 unsupervised outlier detection models spanning 8 widely-used algorithmic families: Isolation Forest (IForest), Local Outlier Factor (LOF), k-Nearest Neighbors (kNN), Histogram-based Outlier Score (HBOS), OneClass SVM (OCSVM), LODA, ABOD, and COF. This diverse pool is generated by systematically varying hyperparameters within each family. For instance, we vary k among {5, 10, 15, . . . , 100} for kNN and LOF, contamination factors 1

4.2. Main Results Table 1 compares MetaEns with unsupervised baselines on 39 benchmark datasets. The per-dataset results are in App. A.14. Overall, MetaEns achieves the best performance across metrics. Notably, it consistently outperforms the strongest meta-learning baseline ELECT, despite both methods sharing the same primary detector, demonstrating the effectiveness of our sequential partner selection strategy.

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Automatic Unsupervised Ensemble Outlier Model Selection–Extended Version Table 1. Main performance comparison on the 39 benchmark datasets. We report Average Precision (AP), Average Rank over datasets (lower is better), ROC-AUC, Precision@π where π is the number of true anomalies, Max-F1 over all thresholds, and the average selected ensemble size. Best results are in bold; second-best are underlined. Method abbreviations: Singleton is random single-model selection, Mega Ensemble averages all 297 detectors, and ELECT Top-k aggregates the top-k models selected by ELECT.

Rank ↓

ROC-AUC ↑

Prec@π ↑

Max-F1 ↑

Ens Size

Theoretical Upper Bound Greedy Oracle 0.6877

1.0

0.8968

0.6504

0.6906

Single Model Baselines Singleton 0.3495 ± 0.0255 LOF 0.3513 ± 0.0038 Global Best 0.3787 ± 0.0074

147.9 ± 22.0578 120.1 ± 2.2336 122.7 ± 9.0314

0.7081 ± 0.0248 0.7439 ± 0.0063 0.7583 ± 0.0049

0.3304 ± 0.0283 0.3297 ± 0.0037 0.3593 ± 0.0093

0.4075 ± 0.0190 0.4169 ± 0.0051 0.4266 ± 0.0075

1 1 1

Naı̈ve & Random Ensembles IForest 0.3858 ± 0.0016 RandNet 0.3460 ± 0.0018 Mega Ensemble 0.3970 ± 0.0000 Random Ensemble 0.3759 ± 0.0175

117.0 ± 5.8119 170.6 ± 2.9136 100.0 ± 0.0000 124.3 ± 13.6874

0.7699 ± 0.0018 0.6865 ± 0.0029 0.7737 ± 0.0000 0.7477 ± 0.0106

0.3619 ± 0.0033 0.3189 ± 0.0034 0.3782 ± 0.0000 0.3608 ± 0.0172

0.4337 ± 0.0022 0.3927 ± 0.0027 0.4443 ± 0.0000 0.4290 ± 0.0140

200 20 297 3

Deep Learning Baselines RDA 0.2742 ± 0.0065 DAGMM 0.2958 ± 0.0124 DeepSVDD 0.2073 ± 0.0115 ROBOD 0.3135 ± 0.0026 LUNAR 0.3024 ± 0.0045 DTE-C 0.3144 ± 0.0012 TCCM 0.2929 ± 0.0144

211.9 ± 11.4741 221.6 ± 6.8508 247.5 ± 6.8516 208.0 ± 2.1602 172.0 ± 18.6815 199.7 ± 9.2376 188.3 ± 28.2902

0.7063 ± 0.0064 0.6676 ± 0.0134 0.5905 ± 0.0164 0.6665 ± 0.0001 0.6566 ± 0.0062 0.7469 ± 0.0087 0.6745 ± 0.0126

0.2721 ± 0.0090 0.3013 ± 0.0135 0.2116 ± 0.0147 0.2973 ± 0.0024 0.3026 ± 0.0098 0.3083 ± 0.0033 0.2834 ± 0.0226

0.3515 ± 0.0068 0.3681 ± 0.0129 0.2968 ± 0.0073 0.3621 ± 0.0002 0.3745 ± 0.0074 0.3971 ± 0.0016 0.3690 ± 0.0111

1 1 1 1 1 1 1

Meta-Learning & Ensemble Methods LSCP 0.3484 ± 0.0173 MetaOD 0.3989 ± 0.0024 ELECT+Random 0.3981 ± 0.0060 ELECT (Top-1) 0.4069 ± 0.0063 ELECT (Top-10) 0.4117 ± 0.0050

124.3 ± 11.1161 101.0 ± 7.6594 102.2 ± 9.2232 85.8 ± 7.8712 83.2 ± 6.8118

0.7560 ± 0.0096 0.7547 ± 0.0014 0.7719 ± 0.0068 0.7734 ± 0.0046 0.7785 ± 0.0038

0.3441 ± 0.0208 0.3746 ± 0.0032 0.3778 ± 0.0103 0.3861 ± 0.0059 0.3856 ± 0.0055

0.4251 ± 0.0123 0.4392 ± 0.0022 0.4449 ± 0.0067 0.4519 ± 0.0051 0.4546 ± 0.0043

1 1 10 1 10

59.3 ± 6.9610

0.7867 ± 0.0045

0.4042 ± 0.0063

0.4681 ± 0.0069

2.2

Method

MetaEns (Ours)

AP ↑

0.4308 ± 0.0064

Naı̈ve aggregation strategies perform substantially worse. Methods that average many detectors (Mega Ensemble) or randomly form ensembles (Random Ensemble) fail to filter weak or redundant models, highlighting the necessity of informed selection. Similarly, single detectors and static meta-learners such as Global Best fall behind adaptive model selection methods.

We also compare with ensemble variants built on ELECT. Simply aggregating top-ranked models provides only marginal improvements over the single-model selector, whereas MetaEns’s context-aware expansion yields clear gains. Across additional evaluation metrics, MetaEns remains among the top performers while using compact, adaptive ensembles, demonstrating both effectiveness and efficiency.

Deep learning baselines show limited effectiveness on these fully unsupervised tabular anomaly detection tasks, consistent with large-scale tabular studies (Grinsztajn et al., 2022; Shwartz-Ziv & Armon, 2022) and ADBench (Han et al., 2022). This should not be read as a general limitation of deep anomaly detection; rather, it reflects the difficulty of tuning neural objectives without labels on heterogeneous tabular datasets with small sample sizes, mixed feature types, and severe class imbalance. Recent methods such as ROBOD, LUNAR, DTE-C, and TCCM improve over older deep baselines but still trail MetaEns, suggesting that cross-family model selection is more effective here than committing to a single neural architecture family.

4.3. Ablation Study To understand the contribution of each component in MetaEns, we perform ablations by selectively disabling key mechanisms. Table 2 shows the ablation study. The full model performs best overall. Removing β shows a performance decrease, as β controls intra-family similarity while λfam controls family-level risk. Removing family-risk regularization (λfam = 0) causes the largest performance drop, showing that family-aware risk control is the most critical component for robust ensemble

Automatic Unsupervised Ensemble Outlier Model Selection–Extended Version Table 2. Ablation study on the 39 benchmark datasets. ∆AP measures the degradation relative to the full model. Setting β = 0 removes the redundancy/diversity discount, setting λfam = 0 removes family-risk regularization, and the single-part gain model replaces the two-part classifier–regressor architecture with one direct gain predictor. Variant

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ization strategies and analyze its behavior when the starting model performs poorly. We pair MetaEns with heterogeneous primary selectors, including the meta-learning method ELECT, density-based method LOF, tree-based IForest, and a random singleton baseline. For each configuration, we compare the primary selector’s AP with the final AP achieved by MetaEns. Detailed scatter plots and per-selector analyses are shown in Fig. 2. Across all selectors, MetaEns consistently improves over the starting model, demonstrating that its partner selection mechanism is selector-agnostic and not tied to a particular initialization strategy. Improvements are especially pronounced in challenging cases where the primary model underperforms. Rather than propagating initial errors, MetaEns effectively recovers performance by selecting complementary detectors from diverse algorithmic families. Even when initialized with weak or biased selectors, the method constructs strong ensembles, indicating that gains stem from diversity-aware partner selection rather than reliance on the primary model quality.

construction. Replacing the two-part meta-model with a single gain predictor leads to a consistent but moderate degradation, supporting our design choice of separating improvement probability and magnitude. A detailed sensitivity analysis of all proxy objective hyperparameters (β, λfam , τ1 , τ2 ) across search grids and 3 random seeds is provided in App. A.2. Additional family-granularity and single-family analyses are in App. A.1 and App. A.15. 4.4. Statistical Significance

4.7. Failure Mode Analysis

To validate the performance improvements of MetaEns, we conduct paired one-sided Wilcoxon signed-rank tests comparing our method against all baselines across the 39 datasets. MetaEns achieves statistically significant AP improvements over all baselines, including recent deep detectors. The rank-based view is important because the benchmark is heterogeneous: average AP can be dominated by easier datasets with high anomaly prevalence, whereas rank summarizes whether a method is consistently competitive across datasets with different contamination rates and feature types. Full p-values and win rates are provided in App. A.16.

While MetaEns consistently outperforms baselines on average, per-dataset results (App. A.14) reveal systematic underperformance on a small number of datasets. Notably, MetaEns trails strong single detectors on lowdimensional datasets such as Glass, HeartDisease, Cardiotocography, and Vertebral (5–13 features), where classical detectors with strong inductive biases (e.g., LOF, IForest) are already near-optimal, leaving little margin for ensemble improvement. This is a known limitation of ensemble methods in general (Zimek et al., 2013). Figure 2 further identifies failure modes via the Rescue Zone (primary AP < 0.4): when all candidate detectors in the pool perform poorly, no complementary partner exists and MetaEns cannot recover. In such cases, adaptive early stopping prevents further degradation. Distribution shift between test and meta-training datasets is partially mitigated by our score-based, distribution-agnostic state representation (Table A.9): the meta-model learns patterns of interdetector behavior from normalized score vectors rather than raw input features, enabling transfer across heterogeneous regimes.

4.5. Modality Transfer to Image and Text Anomaly Detection Because MetaEns uses only normalized score vectors, it transfers beyond tabular data. On 20 ADBench image/text extracted-feature datasets (15 MVTec-AD, 5 text), MetaEns improves over the strongest baseline overall and on images (+0.0257 AP), with significant wins across baselines; see App. A.17. This supports the score-only design: the metamodel does not depend on raw feature dimensionality, input modality, or detector internals, and can reuse the same state representation for image, text, and tabular anomaly scores.

4.8. Model Diversity Analysis To illustrate the difference between MetaEns and baseline selection strategies, we project the 297-model candidate pool into a two-dimensional space using t-SNE (van der Maaten & Hinton, 2008) based on prediction-correlation distance. This embedding reveals that models naturally

4.6. Robustness to Initialization A central question is whether MetaEns’s gains depend on a specific primary selector or on strong initial models. To investigate this, we evaluate MetaEns under diverse initial8

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Figure 2. Robustness analysis across four different primary selectors: ELECT, LOF, IForest, and Random Selection. Each panel compares the primary model’s performance (x-axis) against the final MetaEns ensemble (y-axis). Points above the diagonal indicate improvement. The shaded red “Rescue Zone” highlights where the primary model fails (AP < 0.4). MetaEns consistently rescues performance in these failure modes across all primary selectors, demonstrating that its diverse partner selection logic is robust and selector-agnostic.

cluster by algorithmic family, as detectors within the same family tend to produce similar outlier rankings.

filtering, showing that the default ExtraTrees configuration achieves the best balance between predictive performance and computational efficiency.

Figure 3 shows the resulting visualization on four representative datasets (Speech, WBC, Waveform, and Shuttle), overlaid with the models selected in our experiments. Background points correspond to all 297 detectors, colored by family, and exhibit clear family-level clustering. In all cases, ELECT-10 selects models concentrated within a single family cluster (e.g., HBOS for Speech and IForest for WBC), indicating limited diversity. In contrast, MetaEns selects models distributed across multiple distinct family clusters. For example, on Speech, selections span four families (HBOS, IForest, OCSVM, ABOD), while on WBC, selections cover three (IForest, OCSVM, HBOS).

4.11. Effect of Expanded Model Pools To evaluate MetaEns’s scalability and robustness to model pool composition, we conduct additional experiments with an expanded pool of 310 models that incorporates neural networks. A detailed analysis of the expanded model pool is provided in App. A.19. 4.12. Effect of Ensemble Size To understand how ensemble size affects performance, we analyze the relationship between the number of models and detection quality. A detailed analysis of the effect of ensemble size is provided in App. A.20.

This visualization supports the observation that MetaEns does not rely solely on the initial model chosen by ELECT. Although both methods share the same primary detector, MetaEns’s family-risk regularization encourages selection of complementary models from different algorithm families. The dispersion of selected detectors suggests that the framework captures diverse decision patterns, which is particularly important in unsupervised settings where validation labels are unavailable. Quantitative diversity measurements and additional discussion are provided in App. A.18.

5. Related Work 5.1. Unsupervised Outlier Detection Unsupervised outlier detection has been studied across diverse methodological paradigms. Statistical methods such as HBOS (Aryal et al., 2021) identify anomalies in low-density regions under feature independence assumptions. Density- and distribution-based methods, including LOF (Breunig et al., 2000), COPOD (Li et al., 2020), and ECOD (Li et al., 2023), quantify deviations using local density or tail probabilities of empirical distributions. Treebased methods such as IForest (Liu et al., 2008) exploit random partitioning to isolate anomalies, while distancebased methods including kNN (Chehreghani, 2016) and ODIN (Hautamäki et al., 2004) detect outliers via spatial isolation. One-class classifiers, such as OCSVM (Schölkopf et al., 1999) and SVDD (Tax & Duin, 2004), learn decision boundaries enclosing normal data, whereas subspace-based methods like PCA (Chapel & Friguet, 2014) rely on reconstruction errors. More recently, deep learning–based meth-

4.9. Pool Size Analysis A detailed analysis of the effect of candidate pool size is provided in App. A.15. The study shows how MetaEns’s performance evolves as the number of available detectors increases, demonstrating that the method scales well and remains stable while benefiting from greater model diversity. 4.10. Effects of Meta-model Architectures We study the choice of meta-model architectures in App. A.6. The comparison evaluates several alternative learning families for marginal gain prediction and candidate 9

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ods learn expressive representations of normality. These include autoencoders (AEs) (Goodge et al., 2020), variational autoencoders (VAEs) (Xu et al., 2018), and generative adversarial networks (GANs) (Lim et al., 2018). Extensions such as Deep SVDD (Ruff et al., 2018), DAGMM (Zong et al., 2018), and RDA (Zhou & Paffenroth, 2017) integrate representation learning with one-class objectives or density estimation to improve robustness. Recent methods further explore graph neural representations (LUNAR) (Goodge et al., 2022), diffusion-based anomaly scoring (DTE-C) (Livernoche et al., 2024), and flow-matching formulations (TCCM) (Li et al., 2025). Transformer-based models (Kim et al., 2024) have also been employed to capture complex dependencies. Despite their effectiveness, these methods are often sensitive to architectural choices and hyperparameters, motivating ensemble-based strategies to improve generalization.

upfront, limiting adaptivity in unsupervised settings. Unlike these approaches, MetaEns focuses on label-free model selection over heterogeneous detector families rather than score reweighting within a fixed architecture. 5.3. Model Selection Model selection is well established in supervised learning via information-theoretic criteria such as AIC and BIC (Schwarz, 1978), cross-validation (Stone, 1974), structural risk minimization (Vapnik, 1998), as well as modern hyperparameter optimization techniques including Bayesian optimization (Snoek et al., 2012) and automated machine learning frameworks (Feurer et al., 2015). In contrast, unsupervised outlier model selection is considerably more challenging due to the lack of labels, extreme class imbalance, and heterogeneous anomaly patterns. Early approaches rely on internal validation heuristics derived from score distributions or stability under perturbations (Goix, 2016; Marques et al., 2020), which often correlate weakly with true detection performance. Recent studies explore metalearning approaches (Vilalta & Drissi, 2002; Hospedales et al., 2022) that leverage labeled meta-datasets to generalize model selection across tasks. Representative methods include MetaOD (Zhao et al., 2020) and ELECT (Zhao et al., 2022), which focus primarily on selecting individual detectors. Ensemble-based model selection remains underexplored: LSCP (Zhao et al., 2019a) performs instancewise detector selection but assumes local data consistency and requires constructing the full ensemble in advance. In contrast, MetaEns performs sequential ensemble model selection without assuming local consistency and supports adaptive early stopping, enabling compact, high-performing ensembles in fully unsupervised settings.

5.2. Ensembles for Outlier Detection Ensemble methods aim to improve robustness and stability by combining multiple detectors (Aggarwal & Sathe, 2015). Common aggregation strategies include score averaging, ranking, and voting; however, naively combining all detectors often incurs high computational cost, performance degradation due to weak models, and limited adaptivity. Several ensemble designs address these issues implicitly. Feature-bagging ensembles promote diversity through random subspaces (Noto et al., 2010; Lazarevic & Kumar, 2005), while tree-based ensembles such as IForest (Liu et al., 2008) embed diversity via randomized construction. Sequential ensembles refine performance by reweighting instances across rounds, as exemplified by XGBOD (Zhao & Hryniewicki, 2018). Stacking-based approaches combine heterogeneous detectors, including autoencoder (Chen et al., 2017) and GAN ensembles (Han et al., 2021), to preserve complementary behaviors. ROBOD (Ding et al., 2022) specifically addresses hyperparameter sensitivity in deep outlier detection by constructing a scalable homogeneous hyper-ensemble over neural detectors. Yet, most existing methods rely on fixed aggregation schemes, a single architecture family, or require constructing the full ensemble

6. Conclusion We address the problem of unsupervised ensemble outlier model selection, where ensembles must be constructed without access to labels. We propose MetaEns, a meta-learned framework that predicts marginal ensemble gains and com10

Automatic Unsupervised Ensemble Outlier Model Selection–Extended Version

bines these with a submodular-inspired proxy objective to guide adaptive ensemble construction with early stopping. Extensive experiments on 39 real-world datasets show that MetaEns can consistently outperform strong unsupervised baselines while selecting substantially smaller ensembles, highlighting the importance of family-level diversification and risk control in unsupervised settings.

Aryal, S., Baniya, A. A., Razzak, I., and Santosh, K. SPAD+: an improved probabilistic anomaly detector based on onedimensional histograms. In Proceedings of the International Joint Conference on Neural Networks (IJCNN), pp. 1–7, 2021. Breiman, L. Random forests. Mach. Learn., 45(1):5–32, 2001.

Future work includes exploring richer meta-representations to improve gain prediction under distribution shift, extending the framework to streaming or non-stationary data, and developing theoretical or uncertainty-aware variants of the proxy objective to better understand unsupervised ensemble selection.

Breunig, M. M., Kriegel, H., Ng, R. T., and Sander, J. LOF: identifying density-based local outliers. In Proceedings of the ACM SIGMOD International Conference on Management of Data (SIGMOD), pp. 93–104, 2000. Campos, D., Kieu, T., Guo, C., Huang, F., Zheng, K., Yang, B., and Jensen, C. S. Unsupervised time series outlier detection with diversity-driven convolutional ensembles. Proc. VLDB Endow., 15(3):611–623, 2021.

Broader Impact Unsupervised outlier detection is used widely in high-impact domains such as fraud detection, cybersecurity, healthcare monitoring, and scientific data analysis. By enabling adaptive and data-driven ensemble construction without requiring labeled data, MetaEns has the potential to improve the robustness and reliability of outlier detection systems deployed in practice. In particular, the ability to construct compact ensembles can reduce computational cost and energy consumption, which is beneficial for large-scale or resource-constrained settings.

Campos, G. O., Zimek, A., Sander, J., Campello, R. J. G. B., Micenková, B., Schubert, E., Assent, I., and Houle, M. E. On the evaluation of unsupervised outlier detection: Measures, datasets, and an empirical study. Data Min. Knowl. Discov., 30:891–927, 2016. Chandola, V., Banerjee, A., and Kumar, V. Anomaly detection: A survey. ACM Comput. Surv., 41(3):15:1–15:58, 2009. Chapel, L. and Friguet, C. Anomaly detection with score functions based on the reconstruction error of the kernel PCA. In Proceedings of the European Conference on Machine Learning and Knowledge Discovery in Databases (ECML PKDD), pp. 227–241, 2014.

Limitations This work has several limitations. First, MetaEns relies on labeled meta-datasets to learn transferable patterns of marginal ensemble gain. While our experiments suggest strong generalization across diverse datasets, performance may degrade if test tasks differ substantially from the metatraining distribution; empirically, underperformance is most pronounced on low-dimensional datasets (d ≤ 13) where single detectors are already near-optimal (see Sec. 4.7). The score-based, distribution-agnostic state representation partially mitigates this risk, as confirmed by statistically significant improvements across heterogeneous benchmarks spanning contamination 0.20%–45.78% and dimensionality 5–1,555. Second, the family-risk regularization depends on a predefined mapping of detectors to algorithmic families, which may be coarse and require domain knowledge to define for new detectors. Finally, we focus on batch settings and do not address streaming or non-stationary data, which we leave for future work.

Chehreghani, M. H. K-nearest neighbor search and outlier detection via minimax distances. In Proceedings of the SIAM International Conference on Data Mining (SDM), pp. 405–413, 2016. Chen, J., Sathe, S., Aggarwal, C. C., and Turaga, D. S. Outlier detection with autoencoder ensembles. In Proceedings of the SIAM International Conference on Data Mining (SDM), pp. 90–98, 2017. Chen, T. and Guestrin, C. Xgboost: A scalable tree boosting system. In Proceedings of the ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (SIGKDD), pp. 785–794, 2016. Cheng, L., Wang, Y., Liu, X., and Li, B. Outlier detection ensemble with embedded feature selection. In Proceedings of the AAAI Conference on Artificial Intelligence (AAAI), pp. 3503–3512, 2020.

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Automatic Unsupervised Ensemble Outlier Model Selection–Extended Version Table A.1. Ablation on family definition granularity. ∆AP measures performance degradation relative to the full model.

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72.0 65.0 59.3

−0.0313 −0.0112 —

−0.0313), demonstrating that MetaEns does not require a perfectly optimal taxonomy. However, broad groupings can cause collateral penalties: for instance, a strong kNN candidate may be suppressed because a LOF model underperformed, despite their differing sensitivity to local versus global anomaly patterns. The fine-grained definition ensures that a family’s historical failure penalizes only its own hyperparameter variants, without unfairly suppressing distinct but conceptually related algorithms.

Zhao, Y., Rossi, R. A., and Akoglu, L. Automatic unsupervised outlier model selection. In Proceedings of the Advances in Neural Information Processing Systems (NeurIPS), pp. 4489–4502, 2021. Zhao, Y., Zhang, S., and Akoglu, L. Toward unsupervised outlier model selection. In Proceedings of the IEEE International Conference on Data Mining (ICDM), pp. 773–782, 2022.

A.2. Hyperparameter Settings Baseline tuning protocol. To preserve the fully unsupervised evaluation setting, target-dataset labels are never used to tune any baseline. Single-model baselines are evaluated with fixed default configurations from standard implementations (e.g., PyOD and Scikit-learn) across all datasets. For baselines with an ensemble-size parameter k, we report the best fixed k according to average AP over the benchmark, giving these baselines an oracle advantage over normal unsupervised deployment. The supervised greedy oracle is reported only as an upper bound and is not a deployable unsupervised method.

Zhou, C. and Paffenroth, R. C. Anomaly detection with robust deep autoencoders. In Proceedings of the ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (SIGKDD), pp. 665–674, 2017. Zimek, A., Campello, R. J. G. B., and Sander, J. Ensembles for unsupervised outlier detection: challenges and research questions a position paper. SIGKDD Explor., 15 (1):11–22, 2013. Zong, B., Song, Q., Min, M. R., Cheng, W., Lumezanu, C., Cho, D., and Chen, H. Deep autoencoding gaussian mixture model for unsupervised anomaly detection. In Proceedings of the International Conference on Learning Representations (ICLR), 2018.

MetaEns hyperparameters. We configure the following hyperparameters, which remain fixed across all 39 test datasets to ensure fair comparison. For primary model selection via ELECT (Zhao et al., 2022), we adopt the default configuration. For marginal gain prediction, we train a two-part meta-model using ExtraTrees: (i) a classifier component for predicting improvement probability (yp = I(g > 0)) with balanced class weights, and (ii) a regression component for predicting improvement magnitude (ym = max(0, g)) optimized for MAE loss. For sequential selection, we set acceptance thresholds τ1 and τ2 for first-partner selection and iterative expansion, respectively. For diversity modulation, we configure discount strength β, family-risk weight λfam , and Jaccard similarity top-k threshold ktop . The ensemble size budget is η with adaptive early stopping. All hyperparameters are tuned via leave-one-out cross-validation on meta-training data. Table A.2 lists the search grids for all tunable proxy objective hyperparameters.

A. Appendix A.1. Ablation on Family Definition To assess the sensitivity of MetaEns to the granularity of the family definition, we compare three variants: (1) no familyrisk penalty (λfam = 0), (2) a coarse-grained definition mapping the 8 base algorithms into 3 broad super-families, and (3) our default fine-grained definition using all 8 base families independently. The coarse-grained grouping maps algorithms as follows: Density/Proximity (LOF, kNN, COF), Isolation/Tree (IForest), and Linear/Probabilistic (HBOS, OCSVM, LODA, ABOD).

Sensitivity analysis. Table A.2 summarizes the search grids used. ables A.3, A.5, A.4, and A.6 report sensitivity across 39 datasets with 3 seeds. β is effectively insensi-

Fine-grained family definitions yield the strongest performance. Even the coarse-grained variant substantially outperforms the no-penalty baseline (∆AP = −0.0112 vs. 14

Automatic Unsupervised Ensemble Outlier Model Selection–Extended Version Table A.2. Hyperparameter search grids. Hyperparameter

Role

Search Grid

τ1 , τ 2 β λfam

Acceptance thresholds Diversity discount strength Family-risk weight

{−0.01, 0, 0.001/0.005, 0.01, 0.05, 0.1} {0, 0.5, 1, 2, 3, 5, 10} {0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.5}

Table A.5. Sensitivity to λfam (family-risk weight).

tive across the entire search range (AP variance <0.0005). λfam must be at least 0.2 to activate family-risk regularization; once active, all larger values yield stable performance, consistent with Table 2. Setting τ too high causes premature early stopping; the default small positive threshold (0.001/0.005) does not require fine-tuning. For the familyrisk percentile, the 10th percentile is a fixed global lower-tail choice rather than a target-dataset-tuned parameter. It penalizes families with occasional catastrophic negative gains while preserving families whose average behavior may still contribute orthogonal diversity.

λfam

AP (Mean ± Std) ↑ ROC-AUC ↑ Rank ↓

0 0.3995 ± 0.0041 0.2 (def.) 0.4308 ± 0.0064 0.4 0.4231 ± 0.0065 0.6 0.4231 ± 0.0065 0.8 0.4231 ± 0.0065 1.0 0.4231 ± 0.0065 1.5 0.4231 ± 0.0065

0.7111 0.7867 0.7802 0.7802 0.7802 0.7802 0.7802

72.0 59.3 68.3 68.3 68.3 68.3 68.3

Table A.6. Sensitivity to τ1 , τ2 (acceptance thresholds). Table A.3. Sensitivity to β (diversity discount strength). τ1 , τ 2 β

AP (Mean ± Std) ↑ ROC-AUC ↑ Rank ↓

0 0.4303 ± 0.0057 0.5 0.4304 ± 0.0057 1 0.4304 ± 0.0057 2 0.4308 ± 0.0064 3 (def.) 0.4308 ± 0.0064 5 0.4306 ± 0.0058 10 0.4308 ± 0.0058

0.7843 0.7842 0.7842 0.7867 0.7867 0.7841 0.7841

−0.01, −0.01 0.4257 ± 0.0060 0, 0 0.4257 ± 0.0060 0.001, 0.005 (def.) 0.4308 ± 0.0064 0.01, 0.01 0.4041 ± 0.0052 0.05, 0.05 0.4133 ± 0.0000 0.1, 0.1 0.4133 ± 0.0000

60.5 60.2 60.2 59.3 59.3 59.8 59.8

AP (Mean ± Std) ↑ Median Rank ↓

0% (no risk penalty) 0.4269 ± 0.0066 5% 0.4272 ± 0.0066 10% (def.) 0.4308 ± 0.0064 25% 0.3368 ± 0.0048 50% (median) 0.3142 ± 0.0087

0.7836 0.7836 0.7867 0.7461 0.7792 0.7792

68.7 68.7 59.3 86.8 79.1 79.1

outlier ensembles: it preserves the continuous ranking signal contributed by all selected detectors while reducing dependence on any single base model. In contrast, max aggregation is highly sensitive to isolated false positives from a poorly calibrated detector, while min aggregation requires near-unanimous agreement and is therefore sensitive to false negatives. Median aggregation is more robust to extreme scores, but can discard fine-grained score variation from competent detectors.

Table A.4. Sensitivity to the family-risk percentile τrisk used in Eq. 8. The default 10% threshold matches the main result reported in Table 1. τrisk

AP (Mean ± Std) ↑ ROC-AUC ↑ Rank ↓

67.7 67.7 59.3 143.3 170.0

To verify this design choice, we conduct a focused aggregation ablation across three random seeds on the 39 benchmark datasets, changing only the operator used to combine selected detector scores. Table A.7 shows that mean aggregation achieves the best AP. It significantly outperforms max and min aggregation under a one-sided Wilcoxon test at α = 0.05. The gap between mean and median is positive but not statistically significant, indicating that the median is a reasonable robust alternative but not a better default in our setting.

Table A.4 shows that MetaEns is robust in the lower-tail regime from 0% to 10%, indicating that performance does not depend on a precisely fine-tuned percentile. In contrast, aggressive thresholds such as 25% and 50% substantially degrade AP and rank because they over-penalize candidate families and suppress useful ensemble diversity. The 10th percentile, therefore, provides a conservative trade-off between mitigating downside risk and preserving cross-family complementarity.

These results support mean aggregation as the most stable default for MetaEns. This focused ablation isolates the score-combining operator and is therefore complementary to the main benchmark results in Table 1, where MetaEns achieves AP 0.4308 ± 0.0064 and average rank 59.3 ± 6.96.

A.3. Ablation on Ensemble Score Aggregation Equation 1 defines the ensemble score as the mean of member detector scores. We use the mean because it is the standard and most stable aggregation rule in unsupervised 15

Automatic Unsupervised Ensemble Outlier Model Selection–Extended Version Table A.7. Ablation on ensemble score aggregation. p-values are from one-sided Wilcoxon tests against mean aggregation; ∗ indicates significance at α = 0.05. Combiner Mean (default) Median Max Min

AP (Mean ± Std) ↑

∆AP

p-value

0.4308±0.0064 0.4182±0.0028 0.3778±0.0251 0.3736±0.0057

— −0.0122 −0.0526 −0.0567

— 0.377 0.049∗ 0.022∗

Max normalized independently for each detector and dataset. Table A.9 provides the complete specification of all feature dimensions. Feature Aggregation for Ensemble Context. For ensembles with |P | > 1, set-level context features are computed via mean pooling to ensure permutation invariance: ∗ ϕfi−1 ,P =

A.4. Ablation on Training Trajectory Strategies ϕfi ,P =

We evaluate two alternative offline training designs against the default oracle greedy rollout: (A) ε-greedy extension, where the greedy argmax is replaced by random sampling with probability ε; and (B) reduced meta-dataset coverage, using 25/50/75% of labeled meta-datasets. Results are averaged over 3 seeds on the 39-dataset benchmark.

Mean AP ↑

Avg. Rank ↓

Greedy (ours, ε=0) ε-greedy (ε=0.1) ε-greedy (ε=0.2)

0.4308±.0064 0.4156±.0069 0.4162±.0036

59.3±7.0 70.7±10.8 77.7±1.2

25% meta-datasets 50% meta-datasets 75% meta-datasets 100% (ours)

0.3358±.0079 0.3743±.0053 0.4122±.0013 0.4308±.0064

109.0±0.0 96.0±3.5 92.0±14.0 59.3±7.0

∗ ϕfi−1 ,f

∗ f ∈P \{fi−1 }

1 X ϕf,fi |P | f ∈P

Here, the first equation measures the variance between the last selected model and the current ensemble. The second equation measures the variance between the candidate model and the current ensemble. When |P | = 1 (i.e., only ∗ the primary detector has been selected), ϕfi−1 ,P is zeropadded, while ϕfi ,P is computed directly from the primary detector. This design maintains a fixed 61-dimensional representation across all selection stages and allows the meta-model to distinguish early-stage expansion from later iterative selection.

Table A.8. Ablation on training trajectory strategies.

Strategy

1 |P | − 1

A.6. Effects of Meta-Model Architectures The effectiveness of MetaEns depends on the underlying models used to predict marginal gains (freg ) and filter candidates (fcls ). In this section, we justify our default choice of ExtraTrees by comparing it with five alternative learning families: Random Forest (Breiman, 2001), XGBoost (Chen & Guestrin, 2016), LightGBM (Ke et al., 2017), Multi-Layer Perceptrons (Goodfellow et al., 2016), and Linear Models (James et al., 2013). All methods are evaluated using the same meta-features and training protocol.

Adding exploration (ε > 0) degrades both AP and rank, as random steps create a train/test distribution mismatch—the deployed policy is purely greedy, so training should match this structure. Meta-dataset coverage improves monotonically, confirming that diversity across meta-tasks is critical for transferable gain prediction.

To ensure a fair comparison, we employ strong, standardized configurations across baselines. Tree ensembles (Random Forest, ExtraTrees, XGBoost, LightGBM) use 500 estimators for classification tasks and 800 for regression tasks. Gradient boosting models (XGBoost, LightGBM) adopt a subsample ratio of 0.8 to improve generalization. The Multi-Layer Perceptrons model consists of two hidden layers with sizes (100, 50), trained for 500 epochs. Linear Models use Logistic Regression and Ridge Regression with default regularization.

A.5. State Representation Features State Representation. The state representation ∗ ϕ(fi , fi−1 , P ) is a fixed 61-dimensional feature vector that encodes the interaction between a candidate model ∗ fi , the most recently selected detector fi−1 , and the current ensemble P . It consists of three 20-dimensional ∗ feature blocks and one scalar feature, namely: (i) ϕfi−1 ,fi , capturing pairwise interactions between the last selected ∗ model and the candidate; (ii) ϕfi−1 ,P , summarizing interactions between the last selected model and the existing ensemble; (iii) ϕfi ,P , summarizing interactions between the ensemble and the candidate; and (iv) |P |, the current ensemble cardinality.

Table A.10 summarizes the results. Train (s) denotes the average wall-clock time to train the meta-model on the meta-training set (leave-one-out protocol), while Infer (s) measures the time required to select models for a new dataset. ExtraTrees consistently achieves the best predictive performance, with the highest AP (0.4308) and best average rank (59.3), while maintaining competitive train-

Each 20-dimensional block is composed of 15 base pairwise features and 5 context features, computed from normalized outlier score vectors of ∈ [0, 1]N , where scores are Min– 16

Automatic Unsupervised Ensemble Outlier Model Selection–Extended Version Table A.9. Feature specification for pairwise model comparison. All features are computed from normalized outlier score vectors op , oq ∈ [0, 1]N . Feature

Type

Mathematical Definition

Base Pairwise Features (15 dimensions) pearson tail pearson jaccard rel kurtosis tail pos disagreement centrality cand std pseudo ap pseudo roc tail entropy score dist l2 cosine dist tail divergence same family same as ensemble count

Correlation Correlation Set similarity Moment ratio Distributional Distributional Distributional Pseudo-label Pseudo-label Distributional Distance Distance Distance Categorical Set overlap

cov(o ,o )

Pearson correlation: ρ(op , oq ) = σ pσ q p q Pearson correlation on U = top-kp ∪ top-kq where k = ⌈0.1N ⌉ Jaccard index: J = |top-kp ∩ top-kq |/|top-kp ∪ top-kq | Relative kurtosis: kurt(oq )/ max(kurt(op ), 10−3 ) Mean positive difference: Ei∈U [max(0, oq,i − op,i )] 1 PK Correlation with pool mean: ρ(oq , ō) where ō = K j=1 oj Standard deviation: σ(oq ) Average Precision treating top-kp as positive class, oq as predictions ROC AUC treating top-kp as positive P class, oq as predictions Shannon entropy: H(oq ) = − 10 b=1 pb log pb where pb is 10-bin histogram L2 distance of sorted vectors: ∥sort(op ) − sort(oq )∥2 o ·o Cosine distance: 1 − ∥o ∥p ∥oq ∥ p 2 q 2 Mean absolute difference: Ei∈U [|oq,i − op,i |] Binary indicator: I[F (p) = F (q)] where F maps models to families Count of partners s ∈ S with J(q, s) > 0.5; equals 0 when S = ∅

Primary-Context Features (5 dimensions) prim std prim entropy prim centrality prim kurtosis prim skewness

Distributional Distributional Distributional Moment Moment

Standard deviation of primary: σ(op ) Shannon entropy of primary score distribution: H(op ) Centrality of primary: ρ(op , ō) Excess kurtosis of primary: kurt(op ) Skewness of primary: skew(op )

Set-Level Scalar Features (1 dimension) |P |

Cardinality

Number of partners: |S| ∈ {0, 1, 2, . . .}

Table A.10. Comparison of meta-model architectures for MetaEns selection quality. Method

AP (Mean ± Std) ↑

Rank ↓

ROC-AUC ↑

Train (s) ↓

Infer (s) ↓

ExtraTrees (Ours) Linear Models XGBoost Random Forest LightGBM Multi-Layer Perceptrons

0.4308 ± 0.0064 0.4056 ± 0.0015 0.4110 ± 0.0055 0.4043 ± 0.0062 0.3899 ± 0.0058 0.3711 ± 0.0115

59.3 ± 6.9610 105.2 ± 2.5000 108.0 ± 8.2000 77.0 ± 7.1000 99.0 ± 8.5000 110.0 ± 12.5000

0.7867 ± 0.0045 0.7737 ± 0.0010 0.7664 ± 0.0040 0.7617 ± 0.0050 0.7600 ± 0.0042 0.7607 ± 0.0090

4.1518 6.9150 1.7408 15.5561 1.9412 35.6667

1.6839 1.0327 1.4910 1.5200 1.3405 10.7667

A.7. Framework Algorithm

ing (4.15s) and inference (1.68s). XGBoost provides faster training but trails in AP (0.4110) and rank (108.0), suggesting overfitting under the relatively small meta-training regime. Linear Models offer the fastest inference (1.03s) but cannot capture the non-linear relationships required for accurate selection, resulting in substantially worse ranking performance. Random Forest is competitive in rank but is slower to train and less accurate than ExtraTrees; this supports the benefit of fully randomized splits as additional regularization. Multi-Layer Perceptrons perform poorly both in accuracy (lowest AP 0.3711) and computational efficiency, indicating that neural architectures are less suitable for this structured meta-learning task. Overall, the comparison shows that ExtraTrees provides the best accuracy–efficiency tradeoff for our two-part marginal-gain predictor.

Offline Meta-training Procedure. Algorithm 1 summarizes the offline meta-training phase of MetaEns. Given labeled meta-datasets M = {(Mℓ , yℓ )} and a detector pool Ω, we simulate oracle greedy ensemble construction by iteratively expanding an ensemble P ⊆ Ω. Starting from an oracle-selected primary detector f1∗ , at each step we evaluate every candidate fi ∈ Ω \ P using its state repre∗ sentation ϕ(fi , fi−1 , P ) and compute the true marginal gain G(fi | P ) in Average Precision. These state–gain pairs are used to train a two-part meta-model that estimates the probability and magnitude of marginal improvement, yielding a predictor Ĝ(fi | P ) that is later used to guide label-free online ensemble selection. Online Model Selection Procedure. Algorithm 2 describes the online ensemble selection procedure of MetaEns for a new unlabeled dataset. Starting from a primary detector f1∗ selected by an unsupervised criterion, we iteratively con17

Automatic Unsupervised Ensemble Outlier Model Selection–Extended Version

Algorithm 1 MetaEns Offline Meta-training: Oracle Rollouts for Marginal-Gain Supervision

Algorithm 2 MetaEns: Online Selection via SubmodularInspired Proxy Maximization

Require: Labeled meta-datasets M = {(Mℓ , yℓ )}L ℓ=1 ; detector pool Ω; budget η; rollout length T ≤ η; state function ∗ ϕ(fi , fi−1 ,P) Ensure: Trained meta-models (fcls , freg ) 1: Initialize Dcls ← ∅, Dreg ← ∅ 2: for all (M, y) ∈ M do 3: (Oracle primary) 4: f1∗ ← arg maxf ∈Ω AP(f (M), y) 5: P ← {f1∗ } 6: flast ← f1∗ 7: for t = 2 to T do 8: for all fi ∈ Ω \ P do 9: ϕi ← ϕ(fi , flast , P ) 10: G(fi | P ) ← AP(oP ∪{fi } , y) − AP(oP , y) 11: ycls ← I(G(fi | P ) > 0) 12: Dcls ← Dcls ∪ {(ϕi , ycls )} 13: if ycls = 1 then 14: yreg ← G(fi | P ) 15: Dreg ← Dreg ∪ {(ϕi , yreg )} 16: end if 17: end for 18: f ∗ ← arg maxfi ∈Ω\P G(fi | P ) 19: if G(f ∗ | P ) ≤ 0 then 20: break 21: end if 22: P ← P ∪ {f ∗ }; flast ← f ∗ 23: if |P | ≥ η then 24: break 25: end if 26: end for 27: end for 28: Train classifier fcls on Dcls (balanced weights) 29: Train regressor freg on Dreg (positives only) 30: return (fcls , freg )

Require: dataset X, candidate pool Ω, meta-model M (·) producing Ĝ, state representation ϕ(·), family mapping F, thresholds τ1 , τ2 , budget η ≥ 1, diversity β, risk λfam , ktop Ensure: Ensemble P 1: Primary model selection using ELECT as S(·) 2: f1∗ ← S(X, Ω) 3: P ← {f1∗ } 4: Ωpool ← Ω \ {f1∗ } 5: Stage 1: Select first expansion model 6: for all fi ∈ Ωpool do 7: Ĝ(fi | P ) ← M.predict(ϕ(fi , f1∗ , P )) 8: end for 9: f2∗ ← arg maxfi ∈Ωpool Ĝ(fi | P ) 10: if Ĝ(f2∗ | P ) < τ1 then 11: return P 12: end if 13: P ← P ∪ {f2∗ } 14: Stage 2: Iterative expansion 15: while |P | < η do 16: ubest ← 0 17: for all fi ∈ Ωpool \ P do ∗ 18: Ĝ(fi | P ) ← M.predict ϕ(fi , f|P |−1 , P ) 19: if Ĝ(fi | P ) < τ2 then 20: continue 21: end if 22: Compute ∆U (fi | P ) 23: if ∆U (fi | P ) > ubest then 24: f ∗ ← fi 25: ubest ← ∆U (fi | P ) 26: end if 27: end for 28: if ubest ≤ 0 then 29: break 30: end if 31: P ← P ∪ {f ∗ } 32: end while 33: return P

struct an ensemble P by adding candidates fi ∈ Ω \ P that maximize the proxy marginal utility ∆U (fi | P ). The proxy utility combines the predicted marginal gain Ĝ(fi | P ) from the meta-model with redundancy discounting and familyrisk regularization, thereby enforcing diminishing returns as the ensemble grows. The selection process terminates automatically when maxfi ∆U (fi | P ) ≤ 0 or when the budget |P | = η is reached, yielding compact, dataset-adaptive ensembles without access to labels.

based features require O(|Ω|2 ·N ) operations, while Jaccard similarity over top-ktop instances requires O(|Ω| · N ) time for extracting top-ktop indices and O(|Ω|2 · ktop ) time for set intersections, becoming independent of N after ranking. The interactive selection phase has complexity O(η|Ω| · Cinf ) and is independent of N . For tree-based meta-models, Cinf = O(ntrees · dϕ ), where ntrees is the number of trees and dϕ is the feature dimensionality.

A.8. Complexity Analysis We conduct a computational complexity analysis to clarify how the proposed framework scales with the size of the candidate model pool and dataset size.

A.9. Datasets We conduct experiments on the 39-dataset benchmark introduced in ELECT (Zhao et al., 2022), which comprises independent, real-world tabular anomaly detection tasks sourced from the Outlier Detection Data Sets (ODDS)2 and the UCI Machine Learning Repository (Campos et al.,

MetaEns operates in O(|Ω|2 · N + η|Ω| · Cinf ) time, where |Ω| is the size of the candidate model pool, N is the number of samples in the target dataset, η is the ensemble budget, and Cinf is the meta-model inference cost. The O(|Ω|2 · N ) term corresponds to one-time feature pre-computation of similarity statistics for the target dataset: correlation-

https://giftpathao.com

Automatic Unsupervised Ensemble Outlier Model Selection–Extended Version

2016). Detailed characteristics of all datasets used in our experiments are summarized in Table A.11.

MetaOD (Zhao et al., 2020): A model selection as a cold-start recommendation problem using matrix factorization on historical meta-features; (16) LSCP (Zhao et al., 2019a): An ensemble framework that selects competent base detectors for local regions of test instances; (17) ELECT (Zhao et al., 2022): The current state-of-the-art model selector, which identifies the single best model based on performance-driven task similarity. (18) ELECT+k: An ensemble baseline that aggregates the top-k models ranked by ELECT. This tests whether simple rank-based selection is sufficient compared to our context-aware approach. (19) ELECT+r: A hybrid baseline that initializes with the high-quality primary model selected by ELECT but expands the ensemble with random partners.

A.10. Candidate Model Pool We construct a diverse candidate pool of M = 297 unsupervised outlier detection models by systematically varying hyperparameters within 8 widely-used algorithmic families. For each dataset, all 297 models are pre-fitted and their anomaly scores cached for efficient meta-learning experiments. Table A.12 summarizes the configuration. A.11. Baselines We compare the proposed MetaEns framework against 19 unsupervised baselines categorized into four groups. For ensemble baselines dependent on a size parameter k, we report the best k performance—the fixed ensemble size that yields the highest average AP across the benchmark—to ensure a strong competitive baseline.

Theoretical Upper Bound: (20) Greedy Oracle: A fully supervised upper bound that utilizes ground-truth labels to iteratively select the candidate maximizing marginal AP at each step. This baseline quantifies the maximum potential performance achievable by a greedy sequential selection strategy.

Single Baselines: (1) Singleton: Selects a single model uniformly at random from the candidate pool; (2) LOF (Breunig et al., 2000); (3) Global Best: A static meta-learner selecting the single model with the highest average performance across all training datasets;

A.12. Implementation Details We implement the proposed framework using PyTorch 1.13 (Paszke et al., 2019) for neural components and Scikitlearn 1.2 (Pedregosa et al., 2011) and PyOD 1.0 (Zhao et al., 2019b) for base outlier detectors in Python 3.9. All experiments were executed on a high-performance computing cluster equipped with dual NVIDIA RTX 3090 GPUs (24GB VRAM), AMD EPYC 7742 64-Core Processors, and 512GB RAM. The source code is available at https://anonymous.4open.science/r/MetaEns. For preprocessing, numeric features are scaled using RobustScaler (median removal and scaling according to the interquartile range) to preserve the integrity of global outliers. Categorical features are transformed via target encoding with leave-one-out regularization (α = 10) to prevent data leakage. We add binary missingness indicators and impute missing values with median/mode. Since raw anomaly scores from different algorithms operate on vastly different scales, we strictly apply Min-Max Normalization per model per dataset to all model outputs before aggregation. To ensure reproducibility, we fix random seeds to 42 for all stochastic components.

Naı̈ve & Random Ensemble Baselines: (4) IForest (Liu et al., 2008): A tree-based ensemble that isolates anomalies using random partitions; (5) RandNet (Chen et al., 2017): An ensemble of autoencoders with randomized connectivity to mitigate overfitting and enhance diversity; (6) Random Ensemble: An ensemble of k randomly selected models; (7) Mega Ensemble: A naı̈ve ensemble that averages outlier scores from all 297 models in the pool. Deep Learning Baselines: (8) RDA (Zhou & Paffenroth, 2017): A robust deep autoencoder that decomposes input into low-dimensional manifold and sparse noise components; (9) DAGMM (Zong et al., 2018): An end-to-end framework jointly optimizing a compression network and a Gaussian Mixture Model for density estimation; (10) DeepSVDD (Ruff et al., 2018): Maps data into a minimum volume hypersphere to extract common factors of variation; (11) ROBOD (Ding et al., 2022): A deep hyper-ensemble that aggregates multiple MLP detectors trained under varied hyperparameters and unsupervised contamination assumptions; (12) LUNAR (Goodge et al., 2022): A graph-based contrastive representation learner for tabular anomaly detection; (13) DTE-C (Livernoche et al., 2024): A diffusionbased generative model for tabular anomaly detection with a transductive scoring protocol aligned to our benchmark; (14) TCCM (Li et al., 2025): A flow-matching generative model for tabular anomaly detection, evaluated with the same protocol as DTE-C. Meta-Learning

and

Ensemble

Baselines:

A.13. Evaluation Metrics Let {(xi , yi )}N i=1 denote an evaluation dataset, where yi ∈ {0, 1} indicates whether instance xi is anomalous (yi = 1) or normal (yi = 0). Let o = (o1 , . . . , oN ) denote the outlier score vector produced by a method, where P larger values inN dicate higher anomaly likelihood. Let π = i=1 yi denote the number of anomalies in the dataset. All ranking-based metrics are computed by sorting instances in descending

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Automatic Unsupervised Ensemble Outlier Model Selection–Extended Version Table A.11. Summary of benchmark datasets. Dataset

#Samples

Dimensionality

Contamination (%)

49,534 7,129 450 2,114 214 270 1,966 5,393 9,868 768 1,013 4,207 340 223 367 198 3,443 4,819 7,200 452 683 214 351 1,600 148 11,183 7,603 3,062 5,216 6,870 768 6,435 5,803 3,686 3,772 240 1,456 378 129

27 21 259 21 7 13 1,555 10 16 8 9 57 9 9 30 33 21 5 6 274 9 9 33 32 18 6 100 166 64 16 8 36 36 400 6 6 12 30 13

3.04 7.49 45.78 22.04 4.21 44.44 18.72 9.46 0.20 34.90 1.28 39.91 9.12 4.48 2.72 23.74 2.90 5.33 7.42 14.60 34.99 4.21 35.90 6.25 4.05 2.32 9.21 3.17 2.88 2.27 34.90 31.64 1.22 1.65 2.47 12.50 3.43 5.56 7.75

ALOI Annthyroid Arrhythmia Cardiotocography Glass HeartDisease InternetAds PageBlocks PenDigits Pima Shuttle SpamBase Stamps WBC WDBC WPBC Waveform Wilt annthyroid arrhythmia breastw glass ionosphere letter lympho mammography mnist musk optdigits pendigits pima satellite satimage-2 speech thyroid vertebral vowels wbc wine

order of oi .

ROC-AUC. The ROC-AUC measures the probability that an anomalous instance receives a higher outlier score than a normal instance:

Average Precision (AP). Average Precision summarizes the area under the precision–recall curve: AP =

N 1X

P I y(k) = 1 P (k),

(10)

i:yi =1

j:yj =0 I(oi > oj )

π(N − π)

(13) where o+ and o− denote outlier scores of anomalous and normal instances, respectively.

where y(k) is the label of the instance ranked at position k, and P (k) is the precision at cutoff k: k 1X I y(j) = 1 k j=1

−

AUC = P(o > o ) =

k=1

P (k) =

Type Numeric Mixed Mixed Numeric Numeric Mixed Mixed Numeric Numeric Numeric Numeric Numeric Numeric Numeric Categorical Numeric Numeric Numeric Mixed Mixed Numeric Numeric Numeric Numeric Categorical Numeric Numeric Numeric Numeric Numeric Numeric Numeric Numeric Numeric Mixed Numeric Numeric Categorical Numeric

Precision@π. Precision@π evaluates the accuracy of the top-ranked predictions at the exact number of ground-truth anomalies and is defined as

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Average Rank (AR). Let rfi ,Xj denote the rank of method fi on dataset Xj according to Average Precision, where rank 1 indicates the best-performing method. The Average Rank of method fi is defined as follows. 1 X ARfi = rfi ,Xj , (12) |D|

Precision@π =

π 1X I y(k) = 1 , π

(14)

k=1

PN where π = i=1 yi is the number of anomalous instances in the dataset and y(k) is the label of the instance ranked at position k. This metric reflects practical inspection scenarios where only the top-ranked instances are examined.

Xj ∈D

where D denotes the set of benchmark datasets. Lower values indicate better overall performance. 20

Automatic Unsupervised Ensemble Outlier Model Selection–Extended Version Table A.12. Candidate model pool specification (M = 297 total models). Family

Count

Hyperparameter Grid

kNN (Chehreghani, 2016)

method ∈ {largest, mean, median}; k ∈ {1, 5, 10, 15, 20, 25, 50, 60, 70, 80, 90, 100}

LOF (Breunig et al., 2000)

metric ∈ {euclidean, manhattan, minkowski}; k ∈ {1, 5, 10, 15, 20, 25, 50, 60, 70, 80, 90, 100}

IForest (Liu et al., 2008)

nestimators ∈ {10, 20, 30, 40, 50, 75, 100, 150, 200}; max samples ∈ {0.1, 0.2, . . . , 0.9}

HBOS (Goldstein & Dengel, 2012)

nbins ∈ {5, 10, 20, 30, 40, 50, 75, 100}; tolerance ∈ {0.1, 0.2, 0.3, 0.4, 0.5}

OCSVM (Schölkopf et al., 1999)

kernel ∈ {linear, poly, rbf, sigmoid}; ν ∈ {0.1, 0.2, . . . , 0.9}

LODA (Pevný, 2016)

nbins ∈ {5, 10, 15, 20, 25, 30}; ncuts ∈ {10, 20, 30, 40, 50, 75, 100, 150, 200}

ABOD (Kriegel et al., 2008)

nneighbors ∈ {3, 5, 10, 15, 20, 25, 50}

COF (Tang et al., 2002)

nneighbors ∈ {3, 5, 10, 15, 20, 25, 50}

Total

297

Max F1-score. For a decision threshold τ applied to outlier scores, predicted labels are defined as follows.

Average Precision (AP)

ŷi (τ ) = I(oi ≥ τ )

0.44

(15)

Precision and recall at threshold τ are given as follows. P I(ŷi (τ ) = 1 ∧ yi = 1) Prec(τ ) = i P i I(ŷi (τ ) = 1) P (16) I(ŷi (τ ) = 1 ∧ yi = 1) Rec(τ ) = i P i I(yi = 1) 2 · Prec(τ ) · Rec(τ ) Prec(τ ) + Rec(τ )

0.42 0.398

0.40 0.38

0.390

0.407

0.401

0.392

0.380

Individual Seeds Mean AP ±1 Std Dev

0.36 0

The F1-score at threshold τ is defined as follows. F1(τ ) =

0.431

100

150

200

Candidate Pool Size

250

300

Figure A.1. Impact of candidate pool size on MetaEns performance. Results are averaged over 10 random seeds for each pool size. The shaded area represents ±1 standard deviation.

(17)

The Max F1-score is defined as follows. MaxF1 = max F1(τ ) τ

297 detectors. For each pool size, we randomly sample the specified number of models and apply the full MetaEns selection procedure. To account for variability introduced by sub-pool sampling, experiments are repeated with 10 random seeds and results are averaged.

(18)

A.14. Detailed Experimental Results Table A.13 reports the detailed performance of MetaEns and all baselines on each of the 39 benchmark datasets. For each dataset, we report Average Precision (AP) scores computed under the same evaluation protocol described in Section 4. This table complements the aggregated results in the main paper by providing a per-dataset view of method behavior, enabling fine-grained comparison and reproducibility analysis.

Figure A.1 shows the relationship between pool size and final ensemble performance measured by AP. The blue curve denotes the average AP across the 10 runs, and the shaded region indicates ±1 standard deviation. Performance improves steadily as the pool size increases, indicating that MetaEns effectively leverages a richer set of candidate models. Gains begin to plateau around 250 models, suggesting diminishing returns beyond this point. This trend indicates that the full pool of 297 models is sufficient to capture the diversity needed for robust anomaly detection across the benchmark datasets.

A.15. Pool Size Analysis We study the performance and robustness of MetaEns by varying the size of the candidate model pool from 10 to 21

Automatic Unsupervised Ensemble Outlier Model Selection–Extended Version Table A.13. Detailed Performance Comparison Across 39 Benchmark Datasets. For each method, we report Average Precision (AP) and rank in parentheses (lower rank is better, 1–19, among compared methods). Best results in bold. Per-dataset values are from a single representative seed (seed=42); the 3-seed averaged AP for MetaEns is 0.4308 (Table 1). Abbreviations: RS = Random Selection, RE = Random Ensemble (k=3), ELECT = ELECT (Top-1), ELECT-10 = ELECT (Top-10). Dataset

IForest

LOF

RDA

DAGMM

DeepSVDD

RandNet

ROBOD

LUNAR

DTE-C

TCCM

LSCP

MetaOD

ELECT-1

ELECT-10 MetaEns

0.039 (11) 0.104 (12) 0.675 (14) 0.412 (8) 0.134 (11) 0.598 (1) 0.331 (15) 0.367 (16) 0.006 (13) 0.466 (11) 0.129 (7) 0.476 (9) 0.249 (11) 0.556 (12) 0.647 (11) 0.233 (7) 0.078 (8) 0.048 (13) 0.195 (13) 0.424 (14) 0.895 (9) 0.076 (16) 0.636 (17) 0.140 (7) 0.782 (10) 0.184 (9) 0.246 (16) 0.548 (13) 0.038 (12) 0.195 (10) 0.477 (11) 0.604 (9) 0.709 (11) 0.027 (3) 0.464 (9) 0.095 (12) 0.178 (9) 0.612 (6) 0.246 (6)

0.034 (16) 0.113 (10) 0.765 (2) 0.437 (6) 0.153 (8) 0.541 (7) 0.527 (5) 0.465 (11) 0.005 (15) 0.516 (2) 0.069 (16) 0.480 (7) 0.307 (8) 0.882 (3) 0.647 (10) 0.231 (9) 0.061 (12) 0.045 (17) 0.314 (7) 0.479 (4) 0.969 (4) 0.093 (10) 0.809 (6) 0.087 (15) 0.944 (3) 0.221 (6) 0.265 (13) 1.000 (1) 0.051 (6) 0.279 (3) 0.500 (6) 0.660 (4) 0.926 (4) 0.018 (18) 0.557 (7) 0.096 (11) 0.138 (13) 0.608 (7) 0.213 (12)

0.074 (6) 0.129 (8) 0.755 (6) 0.302 (16) 0.092 (19) 0.574 (2) 0.366 (14) 0.531 (5) 0.019 (5) 0.514 (3) 0.355 (4) 0.355 (18) 0.333 (4) 0.875 (5) 0.691 (7) 0.232 (8) 0.131 (4) 0.053 (8) 0.204 (12) 0.464 (7) 0.392 (18) 0.083 (13) 0.799 (9) 0.244 (4) 0.857 (6) 0.121 (12) 0.379 (5) 0.090 (17) 0.021 (18) 0.044 (16) 0.493 (7) 0.397 (18) 0.142 (16) 0.020 (11) 0.335 (13) 0.088 (17) 0.385 (4) 0.650 (3) 0.290 (2)

0.032 (19) 0.139 (6) 0.751 (7) 0.473 (3) 0.197 (5) 0.525 (11) 0.455 (10) 0.409 (13) 0.006 (14) 0.436 (16) 0.090 (11) 0.533 (4) 0.333 (5) 0.827 (9) 0.716 (4) 0.239 (5) 0.055 (16) 0.041 (19) 0.366 (5) 0.502 (1) 0.967 (5) 0.116 (7) 0.758 (13) 0.088 (12) 0.976 (2) 0.221 (7) 0.262 (14) 1.000 (2) 0.043 (10) 0.255 (7) 0.465 (12) 0.660 (5) 0.915 (7) 0.027 (5) 0.380 (12) 0.092 (15) 0.106 (15) 0.582 (11) 0.234 (8)

0.036 (14) 0.135 (7) 0.748 (9) 0.408 (9) 0.120 (14) 0.571 (3) 0.482 (9) 0.385 (15) 0.007 (11) 0.500 (6) 0.117 (8) 0.483 (6) 0.333 (3) 0.864 (8) 0.697 (5) 0.231 (10) 0.066 (9) 0.043 (18) 0.285 (8) 0.445 (11) 0.979 (2) 0.085 (12) 0.803 (8) 0.130 (8) 0.877 (5) 0.247 (3) 0.278 (12) 1.000 (3) 0.054 (5) 0.260 (6) 0.503 (5) 0.680 (2) 0.944 (2) 0.021 (10) 0.497 (8) 0.087 (18) 0.233 (7) 0.597 (8) 0.255 (5)

0.035 (15) 0.129 (9) 0.739 (12) 0.398 (10) 0.136 (10) 0.566 (4) 0.405 (11) 0.390 (14) 0.007 (10) 0.488 (9) 0.114 (9) 0.439 (10) 0.256 (10) 0.604 (11) 0.684 (8) 0.230 (11) 0.081 (7) 0.055 (6) 0.247 (10) 0.431 (13) 0.941 (8) 0.080 (14) 0.745 (14) 0.220 (5) 0.822 (8) 0.198 (8) 0.290 (9) 0.882 (10) 0.050 (7) 0.214 (9) 0.521 (1) 0.613 (8) 0.754 (10) 0.028 (2) 0.434 (10) 0.090 (16) 0.195 (8) 0.616 (5) 0.274 (3)

0.039 (10) 0.083 (17) 0.669 (15) 0.264 (18) 0.192 (6) 0.458 (16) 0.296 (16) 0.436 (12) 0.020 (4) 0.451 (13) 0.040 (17) 0.399 (14) 0.196 (13) 0.274 (18) 0.092 (19) 0.243 (4) 0.029 (19) 0.283 (1) 0.161 (15) 0.313 (15) 0.883 (10) 0.195 (2) 0.949 (1) 0.311 (2) 0.151 (18) 0.171 (11) 0.393 (3) 0.450 (15) 0.023 (16) 0.035 (18) 0.405 (15) 0.508 (15) 0.100 (17) 0.019 (13) 0.309 (14) 0.116 (5) 0.458 (3) 0.241 (18) 0.093 (17)

0.038 (12) 0.090 (15) 0.637 (16) 0.362 (13) 0.121 (12) 0.441 (18) 0.277 (18) 0.262 (17) 0.014 (7) 0.409 (19) 0.175 (5) 0.349 (19) 0.113 (19) 0.537 (13) 0.551 (13) 0.206 (19) 0.033 (18) 0.089 (3) 0.128 (17) 0.257 (19) 0.782 (12) 0.052 (18) 0.599 (18) 0.094 (10) 0.519 (14) 0.100 (15) 0.225 (17) 0.856 (11) 0.018 (19) 0.083 (13) 0.431 (13) 0.574 (12) 0.394 (12) 0.023 (9) 0.146 (16) 0.144 (4) 0.067 (17) 0.415 (16) 0.112 (15)

0.049 (7) 0.109 (11) 0.501 (19) 0.308 (15) 0.107 (17) 0.546 (6) 0.202 (19) 0.227 (18) 0.068 (1) 0.451 (12) 0.388 (3) 0.367 (17) 0.164 (16) 0.368 (17) 0.316 (16) 0.266 (2) 0.056 (15) 0.046 (14) 0.120 (18) 0.309 (16) 0.320 (19) 0.041 (19) 0.565 (19) 0.081 (19) 0.056 (19) 0.070 (19) 0.207 (18) 0.043 (18) 0.032 (13) 0.036 (17) 0.398 (16) 0.479 (16) 0.063 (18) 0.027 (4) 0.049 (19) 0.103 (7) 0.057 (19) 0.447 (14) 0.081 (18)

0.040 (9) 0.066 (19) 0.747 (10) 0.531 (1) 0.117 (16) 0.536 (9) 0.525 (6) 0.551 (4) 0.002 (19) 0.410 (18) 0.022 (19) 0.404 (13) 0.123 (18) 0.394 (16) 0.630 (12) 0.229 (12) 0.058 (13) 0.054 (7) 0.135 (16) 0.460 (8) 0.686 (15) 0.205 (1) 0.735 (16) 0.100 (9) 0.897 (4) 0.073 (18) 0.392 (4) 0.998 (8) 0.070 (3) 0.182 (11) 0.368 (18) 0.527 (14) 0.961 (1) 0.019 (14) 0.176 (15) 0.156 (3) 0.140 (12) 0.596 (9) 0.220 (9)

0.042 (8) 0.068 (18) 0.737 (13) 0.520 (2) 0.118 (15) 0.452 (17) 0.503 (8) 0.483 (7) 0.002 (18) 0.422 (17) 0.025 (18) 0.421 (12) 0.146 (17) 0.501 (14) 0.501 (15) 0.207 (18) 0.066 (10) 0.052 (9) 0.114 (19) 0.450 (10) 0.728 (14) 0.154 (5) 0.742 (15) 0.085 (16) 0.808 (9) 0.097 (16) 0.333 (8) 0.440 (16) 0.043 (11) 0.228 (8) 0.382 (17) 0.587 (10) 0.889 (9) 0.019 (12) 0.118 (18) 0.174 (2) 0.063 (18) 0.417 (15) 0.108 (16)

0.142 (4) 0.093 (14) 0.750 (8) 0.222 (19) 0.172 (7) 0.514 (13) 0.371 (13) 0.192 (19) 0.038 (2) 0.527 (1) 0.160 (6) 0.377 (16) 0.174 (15) 0.750 (10) 0.512 (14) 0.227 (14) 0.141 (3) 0.083 (4) 0.173 (14) 0.442 (12) 0.748 (13) 0.171 (4) 0.914 (3) 0.383 (1) 0.640 (12) 0.105 (13) 0.350 (7) 0.025 (19) 0.021 (17) 0.033 (19) 0.504 (4) 0.332 (19) 0.033 (19) 0.024 (8) 0.120 (17) 0.099 (8) 0.615 (1) 0.378 (17) 0.239 (7)

0.033 (18) 0.102 (13) 0.607 (17) 0.270 (17) 0.144 (9) 0.482 (14) 0.295 (17) 0.568 (2) 0.013 (8) 0.438 (14) 0.512 (1) 0.392 (15) 0.213 (12) 0.111 (19) 0.197 (18) 0.275 (1) 0.035 (17) 0.147 (2) 0.655 (1) 0.291 (17) 0.786 (11) 0.172 (3) 0.916 (2) 0.269 (3) 0.323 (16) 0.176 (10) 0.412 (2) 0.517 (14) 0.029 (14) 0.048 (15) 0.431 (14) 0.580 (11) 0.142 (15) 0.018 (17) 0.729 (2) 0.115 (6) 0.362 (6) 0.203 (19) 0.256 (4)

0.038 (13) 0.171 (4) 0.591 (18) 0.315 (14) 0.120 (13) 0.420 (19) 0.373 (12) 0.575 (1) 0.009 (9) 0.437 (15) 0.112 (10) 0.431 (11) 0.189 (14) 0.400 (15) 0.275 (17) 0.216 (17) 0.063 (11) 0.049 (10) 0.243 (11) 0.267 (18) 0.569 (17) 0.078 (15) 0.809 (7) 0.088 (14) 0.403 (15) 0.102 (14) 0.442 (1) 0.688 (12) 0.028 (15) 0.173 (12) 0.349 (19) 0.540 (13) 0.380 (14) 0.038 (1) 0.565 (6) 0.193 (1) 0.089 (16) 0.472 (13) 0.123 (14)

0.077 (5) 0.083 (16) 0.746 (11) 0.392 (11) 0.213 (2) 0.480 (15) 0.576 (2) 0.564 (3) 0.015 (6) 0.474 (10) 0.084 (13) 0.494 (5) 0.334 (2) 0.895 (1) 0.800 (1) 0.265 (3) 0.168 (2) 0.079 (5) 0.257 (9) 0.459 (9) 0.588 (16) 0.125 (6) 0.790 (11) 0.209 (6) 0.173 (17) 0.082 (17) 0.371 (6) 0.994 (9) 0.072 (2) 0.066 (14) 0.490 (8) 0.415 (17) 0.393 (13) 0.019 (15) 0.410 (11) 0.000 (19) 0.369 (5) 0.542 (12) 0.000 (19)

0.033 (17) 0.155 (5) 0.768 (1) 0.442 (5) 0.253 (1) 0.523 (12) 0.525 (7) 0.467 (9) 0.005 (17) 0.499 (7) 0.071 (15) 0.479 (8) 0.345 (1) 0.877 (4) 0.678 (9) 0.227 (15) 0.056 (14) 0.045 (16) 0.336 (6) 0.482 (2) 0.979 (1) 0.116 (8) 0.910 (4) 0.091 (11) 1.000 (1) 0.238 (4) 0.246 (15) 1.000 (4) 0.055 (4) 0.266 (5) 0.505 (3) 0.658 (6) 0.921 (6) 0.018 (16) 0.612 (4) 0.098 (10) 0.127 (14) 0.591 (10) 0.211 (13)

0.164 (2) 0.197 (3) 0.761 (4) 0.429 (7) 0.209 (4) 0.538 (8) 0.534 (4) 0.480 (8) 0.007 (12) 0.509 (4) 0.090 (12) 0.559 (2) 0.309 (7) 0.874 (6) 0.760 (3) 0.229 (13) 0.115 (6) 0.048 (12) 0.452 (3) 0.474 (6) 0.955 (6) 0.092 (11) 0.793 (10) 0.085 (17) 0.720 (11) 0.269 (2) 0.282 (11) 1.000 (5) 0.050 (8) 0.289 (2) 0.480 (10) 0.658 (7) 0.927 (3) 0.026 (6) 0.651 (3) 0.094 (13) 0.155 (11) 0.639 (4) 0.215 (10)

0.158 (3) 0.218 (2) 0.764 (3) 0.444 (4) 0.212 (3) 0.532 (10) 0.536 (3) 0.466 (10) 0.005 (16) 0.507 (5) 0.079 (14) 0.557 (3) 0.327 (6) 0.885 (2) 0.694 (6) 0.225 (16) 0.118 (5) 0.046 (15) 0.395 (4) 0.480 (3) 0.955 (7) 0.104 (9) 0.813 (5) 0.088 (13) 0.852 (7) 0.235 (5) 0.282 (10) 1.000 (6) 0.048 (9) 0.272 (4) 0.487 (9) 0.671 (3) 0.924 (5) 0.026 (7) 0.604 (5) 0.093 (14) 0.165 (10) 0.668 (1) 0.215 (11)

Average Std Dev

0.342 (10.56) 0.398 (8.31) 0.330 (9.49) 0.392 (9.03) 0.397 (8.00) 0.369 (9.08) 0.276 (12.18) 0.275 (14.18) 0.208 (14.62) 0.347 (11.00) 0.314 (12.82) 0.302 (10.67) 0.314 (10.67) 0.293 (12.08) 0.348 (9.08) 0.408 (7.95) 0.413 (7.08) 0.414 (7.00) 0.435 (6.23) 0.246 0.308 0.248 0.302 0.301 0.268 0.223 0.227 0.171 0.285 0.255 0.242 0.231 0.209 0.267 0.313 0.294 0.297 0.285

Table A.14. Single-family pool analysis. Restricted pools remove cross-family diversity. Candidate Pool

AP ↑

Rank ↓

IForest-only (81) 0.3552 ± 0.0015 N/A LOF-only (36) 0.3369 ± 0.0028 N/A Full 8-family (297) 0.4308 ± 0.0064 59.3 ± 6.9

0.164 (1) 0.223 (1) 0.757 (5) 0.384 (12) 0.102 (18) 0.562 (5) 0.579 (1) 0.494 (6) 0.021 (3) 0.491 (8) 0.396 (2) 0.594 (1) 0.269 (9) 0.865 (7) 0.768 (2) 0.235 (6) 0.190 (1) 0.048 (11) 0.531 (2) 0.475 (5) 0.972 (3) 0.058 (17) 0.759 (12) 0.082 (18) 0.556 (13) 0.284 (1) 0.160 (19) 1.000 (7) 0.114 (1) 0.359 (1) 0.516 (2) 0.698 (1) 0.895 (8) 0.017 (19) 0.746 (1) 0.099 (9) 0.491 (2) 0.663 (2) 0.366 (1)

Table A.15. Statistical significance testing: Paired Wilcoxon signed-rank test comparing MetaEns against baselines across 39 datasets (one-sided test, α = 0.05).

Ens. Size 2.0 2.0 2.2

To separate algorithm diversity from hyperparameter selection, we further run a single-family pool analysis. In this setting, MetaEns may still select and combine multiple hyperparameter variants, but all candidates come from the same algorithmic family. Table A.14 shows that performance falls substantially for IForest-only and LOF-only pools, while the average ensemble size remains close to the full setting. Thus, the loss is mainly due to missing crossfamily candidates rather than an inability to form ensembles, indicating that algorithm diversity is the primary driver and hyperparameter selection provides complementary gains.

Ours

Baseline

p-value

∆AP

Win%

MetaEns MetaEns MetaEns MetaEns MetaEns MetaEns MetaEns MetaEns MetaEns MetaEns MetaEns MetaEns MetaEns MetaEns MetaEns MetaEns MetaEns MetaEns

Random Selection IForest LOF Global Best Mega Ensemble Random Ensemble RDA DAGMM DeepSVDD ROBOD LUNAR DTE-C TCCM RandNet LSCP MetaOD ELECT-1 ELECT-10

0.0000 0.0151 0.0022 0.0050 0.0096 0.0002 0.0001 0.0000 0.0000 0.0000 0.0004 0.0010 0.0000 0.0013 0.0047 0.0390 0.0314 0.0386

+0.0939 +0.0378 +0.1052 +0.0438 +0.0384 +0.0662 +0.1599 +0.1605 +0.2272 +0.1173 +0.1284 +0.1164 +0.1379 +0.0884 +0.0876 +0.0275 +0.0222 +0.0213

79.5% 64.1% 69.2% 71.8% 66.7% 74.4% 76.9% 84.6% 84.6% 79.5% 71.8% 71.8% 76.9% 71.8% 66.7% 61.5% 66.7% 61.5%

A.17. Modality Transfer Experiments We further evaluate whether MetaEns can be used as a plug-and-play selector outside tabular anomaly detection. Since the state representation depends only on detector-score vectors, no architectural change is needed for image or text settings. We use ADBench extracted features: CV by ResNet18 (512-dimensional ResNet-18 features) or all 15 MVTec-AD image categories (bottle, cable, capsule, carpet, grid, hazelnut, leather, metal nut, pill, screw, tile, toothbrush, transistor, wood, and zipper) and NLP by BERT (768dimensional BERT features) for 5 text datasets. The compared methods are MetaEns, random singleton selec-

A.16. Statistical Significance Analysis As discussed in Sec. 4.4, we use paired Wilcoxon signedrank tests to compare MetaEns against each baseline across the 39 benchmark datasets. The test is non-parametric and is therefore suitable for paired comparisons across heterogeneous datasets without assuming normally distributed AP differences. We use a one-sided alternative hypothesis that MetaEns achieves higher AP than the baseline and set α = 0.05. Table A.15 reports the corresponding p-values, mean AP differences, and dataset-level win rates. 22

Automatic Unsupervised Ensemble Outlier Model Selection–Extended Version Table A.16. Modality-transfer results on 20 non-tabular datasets. Setting

#Data

Overall Image (MVTec-AD) Text

20 15 5

MetaEns AP ↑

Best Base. AP ↑

Table A.18. Quantitative diversity of selected ensembles on the 39 tabular datasets. Distinct families count unique base algorithms; pairwise Jaccard is the mean top-10% overlap between selected detectors.

Gap

0.4914 ± 0.0007 LUNAR: 0.4717 ± 0.0045 +0.0197 0.6230 ± 0.0009 ELECT-1: 0.5973 ± 0.0035 +0.0257 0.0966 LUNAR: 0.0977 ± 0.0021 −0.0012

Method ELECT (Top-10) Random Ensemble Mega Ensemble MetaEns (Ours)

Table A.17. Dataset-level win/loss and one-sided Wilcoxon signedrank tests for modality-transfer experiments over 20 non-tabular datasets. Baseline ELECT-1 ELECT-10 LUNAR ROBOD DTE-C TCCM Mega Ensemble Random Singleton

Win/Loss

p-value

14 / 6 17 / 3 15 / 5 18 / 2 18 / 2 18 / 2 18 / 2 19 / 1

0.0148 0.0047 0.0266 1.61 × 10−4 6.7 × 10−5 1.05 × 10−4 5.1 × 10−4 6.7 × 10−5

Distinct Families ↑

Pairwise Jaccard ↓

2.1 1.0 8.0 2.2

0.68 — 0.44 0.36

Expanded Pool Composition. The expanded pool adds 13 neural network variants spanning 5 deep learning families: AutoEncoder (Chen et al., 2017) (4 architectural variants), Variational AutoEncoder (Xu et al., 2018) (3 variants), SO GAAL (Liu et al., 2020) (2 configurations), MO GAAL (Liu et al., 2020) (2 configurations), and DeepSVDD (Ruff et al., 2018) (2 variants). These models represent diverse deep learning paradigms, including reconstruction-based methods, adversarial approaches, and deep one-class classification. Table A.19 provides the complete specification of the 310-model expanded pool.

tion, mega ensemble, ELECT-1, ELECT-10, ROBOD, LUNAR, DTE-C, and TCCM, all under the same unsupervised evaluation protocol. Table A.16 reports the headline AP results. Across all 20 datasets, MetaEns reaches 0.4914±0.0007 AP, outperforming the strongest baseline, LUNAR (0.4717 ± 0.0045). On MVTec-AD image data, MetaEns obtains 0.6230 ± 0.0009 AP, improving over ELECT-1 (0.5973±0.0035). Table A.17 further reports win/loss statistics and one-sided Wilcoxon tests, confirming significant gains against all compared baselines over the 20 non-tabular datasets.

Overall Performance Comparison. Table A.20 presents the scalability analysis results comparing MetaEns performance on the original (297 models) versus expanded (310 models) pools. MetaEns demonstrates robust performance across both pool configurations, maintaining competitive efficacy with only a modest decrease in Average Precision (0.0186) and an increase in median rank (13.0). Detailed Dataset-by-Dataset Results. Table A.21 presents the complete performance comparison of all methods on the expanded 310-model pool. The table shows Average Precision (AP) values and model ranks (1–310, where lower ranks indicate better performance) for each method across all 39 benchmark datasets, with best results for each dataset highlighted in bold.

A.18. Model Diversity Visualization As a qualitative complement to Table A.18, we project the 297-model candidate pool into a two-dimensional space using t-SNE (van der Maaten & Hinton, 2008) based on prediction-correlation distance. The embedding shows that detectors naturally cluster by algorithmic family, since variants within the same family tend to produce similar outlier rankings.

Results Analysis. The combined results demonstrate several key findings:

Figure 3 visualizes four representative datasets (Speech, WBC, Waveform, and Shuttle). ELECT-10 selections often concentrate within a single family cluster, whereas MetaEns selects models spanning multiple clusters. This supports the quantitative finding that MetaEns constructs compact but complementary ensembles rather than simply adding many redundant high-ranked detectors.

• Scalability: MetaEns successfully handles the expanded 310-model pool with minimal performance impact, validating its applicability to larger model collections.

A.19. Effect of Expanded Model Pools

• Robustness: Performance remains stable across different pool compositions, indicating robust meta-learning that generalizes across model types.

To evaluate MetaEns’s scalability and robustness to model pool composition, we conduct additional experiments with an expanded pool of 310 models that incorporates neural network variants alongside the original 297 classical detectors.

• Discriminative Selection: Analysis reveals that despite several neural network models achieving competitive individual performance, MetaEns predominantly selects classical models, demonstrating intelligent cross-dataset pattern recognition. 23

Automatic Unsupervised Ensemble Outlier Model Selection–Extended Version Table A.19. Expanded candidate model pool specification with neural network variants (M = 310 total models). Family

Count

Hyperparameter Grid

kNN (Chehreghani, 2016)

method ∈ {largest, mean, median}; k ∈ {1, 5, 10, 15, 20, 25, 50, 60, 70, 80, 90, 100}

LOF (Breunig et al., 2000)

metric ∈ {euclidean, manhattan, minkowski}; k ∈ {1, 5, 10, 15, 20, 25, 50, 60, 70, 80, 90, 100}

IForest (Liu et al., 2008)

nestimators ∈ {10, 20, 30, 40, 50, 75, 100, 150, 200}; max samples ∈ {0.1, 0.2, . . . , 0.9}

HBOS (Goldstein & Dengel, 2012)

nbins ∈ {5, 10, 20, 30, 40, 50, 75, 100}; tolerance ∈ {0.1, 0.2, 0.3, 0.4, 0.5}

OCSVM (Schölkopf et al., 1999)

kernel ∈ {linear, poly, rbf, sigmoid}; ν ∈ {0.1, 0.2, . . . , 0.9}

LODA (Pevný, 2016)

nbins ∈ {5, 10, 15, 20, 25, 30}; ncuts ∈ {10, 20, 30, 40, 50, 75, 100, 150, 200}

ABOD (Kriegel et al., 2008)

nneighbors ∈ {3, 5, 10, 15, 20, 25, 50}

COF (Tang et al., 2002)

nneighbors ∈ {3, 5, 10, 15, 20, 25, 50}

Neural Network Extensions Autoencoder (Chen et al., 2017)

Architecture variants: [128,64,64,128], [64,32,16,16,32,64]

Variational Autoencoder (Xu et al., 2018)

Encoder-decoder pairs: ([64,32],[32,64]), ([128,64],[64,128]), ([32,16],[16,32])

SO GAAL (Liu et al., 2020)

Single-objective GAN: ∈ {0.0005, 0.001}

MO GAAL (Liu et al., 2020)

Multi-objective GAN: stop epochs=20; k ∈ {5, 20}

DeepSVDD (Ruff et al., 2018)

Deep SVDD: pre-training ∈ {True, False}

Classical Total Neural Network Total Grand Total

297 13 310

Traditional anomaly detection methods Deep learning-based detection methods Complete expanded model pool

• Competitive Performance: MetaEns achieves the best performance on 10 out of 39 datasets (25.6%) on the expanded pool, demonstrating consistent competitive behavior across diverse anomaly detection tasks.

[64,32,32,64], [32,16,16,32],

stop epochs=20;

adaptive behavior contrasts sharply with fixed-size baselines: ELECT Top-10 always uses 10 models, while Random Ensemble uses a fixed best k = 3. The ability to adjust ensemble size based on dataset characteristics is a crucial advantage of our approach.

The stable performance across pool configurations validates MetaEns’s design principle of leveraging comprehensive cross-dataset information for intelligent model selection, supporting its applicability as new model architectures emerge in the anomaly detection landscape.

The performance comparison in Table 1 reveals important insights about ensemble construction strategies. ELECT Top10, despite using 10 models, achieves only 0.4117 AP—a marginal 0.0048 improvement over ELECT (Top-1) with 0.4069 AP. This negligible gain from expanding to 10 models suggests that simple rank-based aggregation without diversity consideration leads to redundant model selection. In contrast, MetaEns achieves 0.4308 AP with an average of only 2.2 models, demonstrating that carefully selected small ensembles can substantially outperform large ensembles of top-ranked models.

A.20. Ensemble Size Impact Analysis To understand how ensemble size affects performance, we analyze the relationship between the number of models and detection quality. Figure A.2 presents the Average Precision (AP) and ROC AUC as a function of ensemble size for several representative methods.

The Random Ensemble baseline with best k = 3 achieves 0.3759 AP, performing worse than both the single ELECT selector (0.4069 AP) and MetaEns (0.4308 AP). This indicates that expanding ensembles with randomly chosen

A key finding from our experiments is that MetaEns employs adaptive ensemble sizing, selecting an average of 2.2 models per dataset across the 39-dataset benchmark. This 24

Automatic Unsupervised Ensemble Outlier Model Selection–Extended Version Table A.20. MetaEns scalability analysis: Performance on original (297) vs. expanded (310) model pools. Results demonstrate method robustness across different pool compositions. Pool Configuration

Size

AP (Mean ± Std) ↑

Rank ↓

ROC-AUC ↑

Ens Size

Classical Models Only Classical + Neural Networks

297 310

0.4308 ± 0.0064 0.4122 ± 0.0068

59.3 ± 6.9610 72.3 ± 11.4018

0.7867 ± 0.0045 0.7823 ± 0.0044

2.2 2.3

Table A.21. Detailed Performance Comparison Across 39 Benchmark Datasets on Expanded Neural Network Pool (310 models). For each method, we report Average Precision (AP) and rank in parentheses (lower rank is better, 1–19, among compared methods). Best results in bold. Per-dataset values are from a single representative seed (seed=42). Abbreviations: RS = Random Selection, RE = Random Ensemble (k=3), ELECT-1 = ELECT (Top-1), ELECT-10 = ELECT (Top-10). Dataset

IForest

LOF

RDA

DAGMM

DeepSVDD

RandNet

ROBOD

LUNAR

DTE-C

TCCM

LSCP

MetaOD

ELECT-1

ELECT-10 MetaEns

0.039 (10) 0.104 (12) 0.675 (14) 0.412 (8) 0.134 (10) 0.598 (1) 0.339 (15) 0.325 (16) 0.006 (13) 0.466 (11) 0.129 (7) 0.476 (9) 0.267 (10) 0.483 (14) 0.647 (10) 0.233 (7) 0.078 (8) 0.048 (11) 0.195 (13) 0.424 (14) 0.895 (9) 0.076 (16) 0.636 (17) 0.140 (7) 0.782 (10) 0.184 (9) 0.246 (15) 0.506 (14) 0.037 (12) 0.195 (10) 0.474 (11) 0.571 (11) 0.709 (11) 0.026 (4) 0.464 (8) 0.095 (12) 0.169 (9) 0.612 (6) 0.246 (6)

0.034 (16) 0.113 (10) 0.765 (2) 0.437 (6) 0.153 (8) 0.541 (7) 0.527 (5) 0.465 (11) 0.005 (15) 0.516 (2) 0.069 (16) 0.480 (6) 0.307 (8) 0.882 (3) 0.647 (11) 0.231 (9) 0.061 (12) 0.045 (16) 0.314 (7) 0.479 (4) 0.969 (4) 0.093 (11) 0.809 (6) 0.087 (15) 0.944 (3) 0.221 (6) 0.265 (13) 1.000 (1) 0.051 (6) 0.279 (3) 0.500 (6) 0.660 (3) 0.926 (5) 0.018 (17) 0.557 (6) 0.096 (11) 0.138 (13) 0.608 (7) 0.213 (12)

0.074 (6) 0.129 (8) 0.755 (5) 0.302 (16) 0.092 (19) 0.574 (2) 0.366 (14) 0.531 (5) 0.019 (5) 0.514 (3) 0.355 (4) 0.355 (18) 0.333 (3) 0.875 (5) 0.691 (6) 0.232 (8) 0.131 (4) 0.053 (8) 0.204 (12) 0.464 (7) 0.392 (18) 0.083 (14) 0.799 (9) 0.244 (4) 0.857 (6) 0.121 (12) 0.379 (5) 0.090 (17) 0.021 (17) 0.044 (16) 0.493 (7) 0.397 (18) 0.142 (16) 0.020 (11) 0.335 (13) 0.088 (17) 0.385 (4) 0.650 (3) 0.290 (2)

0.032 (19) 0.139 (6) 0.751 (6) 0.473 (3) 0.197 (5) 0.525 (11) 0.455 (10) 0.409 (13) 0.006 (14) 0.436 (16) 0.090 (11) 0.533 (3) 0.333 (4) 0.827 (9) 0.716 (4) 0.239 (5) 0.055 (16) 0.041 (19) 0.366 (5) 0.502 (1) 0.967 (5) 0.116 (7) 0.758 (13) 0.088 (12) 0.976 (2) 0.221 (7) 0.262 (14) 1.000 (2) 0.043 (10) 0.255 (7) 0.465 (12) 0.660 (4) 0.915 (8) 0.027 (2) 0.380 (12) 0.092 (15) 0.106 (15) 0.582 (11) 0.234 (8)

0.036 (14) 0.136 (7) 0.746 (9) 0.410 (9) 0.119 (14) 0.567 (3) 0.476 (9) 0.393 (14) 0.007 (10) 0.497 (8) 0.119 (8) 0.479 (7) 0.332 (5) 0.858 (8) 0.686 (7) 0.230 (10) 0.067 (9) 0.044 (18) 0.290 (8) 0.444 (11) 0.979 (1) 0.085 (13) 0.806 (8) 0.134 (8) 0.877 (5) 0.252 (3) 0.274 (12) 1.000 (3) 0.053 (5) 0.258 (6) 0.501 (5) 0.682 (1) 0.944 (3) 0.021 (10) 0.486 (7) 0.087 (18) 0.228 (7) 0.597 (8) 0.253 (5)

0.035 (15) 0.129 (9) 0.737 (11) 0.400 (10) 0.131 (11) 0.566 (4) 0.390 (11) 0.368 (15) 0.007 (11) 0.488 (9) 0.114 (9) 0.439 (10) 0.265 (11) 0.605 (11) 0.684 (8) 0.229 (11) 0.080 (7) 0.055 (6) 0.247 (10) 0.431 (13) 0.940 (8) 0.095 (10) 0.752 (14) 0.225 (5) 0.827 (8) 0.198 (8) 0.295 (9) 0.818 (11) 0.051 (7) 0.211 (9) 0.520 (2) 0.609 (7) 0.729 (10) 0.026 (5) 0.434 (9) 0.090 (16) 0.192 (8) 0.616 (5) 0.272 (3)

0.039 (11) 0.083 (16) 0.669 (15) 0.264 (18) 0.192 (6) 0.458 (16) 0.296 (16) 0.436 (12) 0.020 (4) 0.451 (12) 0.040 (17) 0.399 (14) 0.196 (13) 0.274 (18) 0.092 (19) 0.243 (4) 0.029 (19) 0.283 (1) 0.161 (15) 0.313 (15) 0.883 (10) 0.195 (2) 0.949 (1) 0.311 (2) 0.151 (18) 0.171 (11) 0.393 (3) 0.450 (15) 0.023 (16) 0.035 (18) 0.405 (15) 0.508 (15) 0.100 (17) 0.019 (13) 0.309 (14) 0.116 (5) 0.458 (3) 0.241 (18) 0.093 (17)

0.038 (12) 0.090 (15) 0.637 (16) 0.362 (13) 0.121 (12) 0.441 (18) 0.277 (18) 0.262 (17) 0.014 (7) 0.409 (19) 0.175 (5) 0.349 (19) 0.113 (19) 0.537 (12) 0.551 (13) 0.206 (19) 0.033 (18) 0.089 (3) 0.128 (17) 0.257 (19) 0.782 (12) 0.052 (18) 0.599 (18) 0.094 (10) 0.519 (14) 0.100 (15) 0.225 (17) 0.856 (10) 0.018 (19) 0.083 (13) 0.431 (14) 0.574 (10) 0.394 (12) 0.023 (9) 0.146 (16) 0.144 (4) 0.067 (17) 0.415 (16) 0.112 (15)

0.049 (7) 0.109 (11) 0.501 (19) 0.308 (15) 0.107 (17) 0.546 (6) 0.202 (19) 0.227 (18) 0.068 (1) 0.451 (13) 0.388 (3) 0.367 (17) 0.164 (16) 0.368 (17) 0.316 (16) 0.266 (2) 0.056 (14) 0.046 (14) 0.120 (18) 0.309 (16) 0.320 (19) 0.041 (19) 0.565 (19) 0.081 (19) 0.056 (19) 0.070 (19) 0.207 (18) 0.043 (18) 0.032 (13) 0.036 (17) 0.398 (16) 0.479 (16) 0.063 (18) 0.027 (3) 0.049 (19) 0.103 (7) 0.057 (19) 0.447 (14) 0.081 (18)

0.040 (9) 0.066 (19) 0.747 (8) 0.531 (1) 0.117 (16) 0.536 (9) 0.525 (6) 0.551 (4) 0.002 (19) 0.410 (18) 0.022 (19) 0.404 (13) 0.123 (18) 0.394 (16) 0.630 (12) 0.229 (12) 0.058 (13) 0.054 (7) 0.135 (16) 0.460 (8) 0.686 (15) 0.205 (1) 0.735 (16) 0.100 (9) 0.897 (4) 0.073 (18) 0.392 (4) 0.998 (8) 0.070 (3) 0.182 (11) 0.368 (18) 0.527 (14) 0.961 (2) 0.019 (14) 0.176 (15) 0.156 (3) 0.140 (12) 0.596 (9) 0.220 (9)

0.042 (8) 0.068 (18) 0.737 (12) 0.520 (2) 0.118 (15) 0.452 (17) 0.503 (8) 0.483 (7) 0.002 (18) 0.422 (17) 0.025 (18) 0.421 (12) 0.146 (17) 0.501 (13) 0.501 (15) 0.207 (18) 0.066 (10) 0.052 (9) 0.114 (19) 0.450 (10) 0.728 (14) 0.154 (5) 0.742 (15) 0.085 (16) 0.808 (9) 0.097 (16) 0.333 (8) 0.440 (16) 0.043 (11) 0.228 (8) 0.382 (17) 0.587 (8) 0.889 (9) 0.019 (12) 0.118 (18) 0.174 (2) 0.063 (18) 0.417 (15) 0.108 (16)

0.142 (4) 0.093 (14) 0.750 (7) 0.222 (19) 0.172 (7) 0.514 (13) 0.371 (13) 0.192 (19) 0.038 (2) 0.527 (1) 0.160 (6) 0.377 (16) 0.174 (15) 0.750 (10) 0.512 (14) 0.227 (14) 0.141 (3) 0.083 (4) 0.173 (14) 0.442 (12) 0.748 (13) 0.171 (4) 0.914 (3) 0.383 (1) 0.640 (12) 0.105 (13) 0.350 (7) 0.025 (19) 0.021 (18) 0.033 (19) 0.504 (4) 0.332 (19) 0.033 (19) 0.024 (8) 0.120 (17) 0.099 (9) 0.615 (1) 0.378 (17) 0.239 (7)

0.033 (18) 0.102 (13) 0.607 (17) 0.270 (17) 0.144 (9) 0.482 (14) 0.295 (17) 0.568 (2) 0.013 (8) 0.438 (14) 0.512 (1) 0.392 (15) 0.213 (12) 0.111 (19) 0.197 (18) 0.275 (1) 0.035 (17) 0.147 (2) 0.655 (1) 0.291 (17) 0.786 (11) 0.172 (3) 0.916 (2) 0.269 (3) 0.323 (16) 0.176 (10) 0.412 (2) 0.517 (13) 0.029 (14) 0.048 (15) 0.431 (13) 0.580 (9) 0.142 (15) 0.018 (16) 0.729 (1) 0.115 (6) 0.362 (6) 0.203 (19) 0.256 (4)

0.038 (13) 0.171 (4) 0.591 (18) 0.315 (14) 0.120 (13) 0.420 (19) 0.373 (12) 0.575 (1) 0.009 (9) 0.437 (15) 0.112 (10) 0.431 (11) 0.189 (14) 0.400 (15) 0.275 (17) 0.216 (17) 0.063 (11) 0.049 (10) 0.243 (11) 0.267 (18) 0.569 (17) 0.078 (15) 0.809 (7) 0.088 (14) 0.403 (15) 0.102 (14) 0.442 (1) 0.688 (12) 0.028 (15) 0.173 (12) 0.349 (19) 0.540 (12) 0.380 (14) 0.038 (1) 0.565 (5) 0.193 (1) 0.089 (16) 0.472 (13) 0.123 (14)

0.077 (5) 0.083 (17) 0.746 (10) 0.392 (11) 0.213 (2) 0.480 (15) 0.576 (2) 0.564 (3) 0.015 (6) 0.474 (10) 0.084 (13) 0.494 (5) 0.334 (2) 0.895 (1) 0.800 (1) 0.265 (3) 0.168 (1) 0.079 (5) 0.257 (9) 0.459 (9) 0.588 (16) 0.125 (6) 0.790 (11) 0.209 (6) 0.173 (17) 0.082 (17) 0.371 (6) 0.994 (9) 0.072 (2) 0.066 (14) 0.490 (8) 0.415 (17) 0.393 (13) 0.019 (15) 0.410 (10) 0.000 (19) 0.369 (5) 0.542 (12) 0.000 (19)

0.033 (17) 0.155 (5) 0.768 (1) 0.442 (5) 0.253 (1) 0.523 (12) 0.525 (7) 0.467 (9) 0.005 (16) 0.499 (7) 0.071 (15) 0.479 (8) 0.345 (1) 0.877 (4) 0.678 (9) 0.227 (15) 0.056 (15) 0.045 (17) 0.336 (6) 0.482 (2) 0.979 (2) 0.116 (8) 0.910 (4) 0.091 (11) 1.000 (1) 0.238 (4) 0.246 (16) 1.000 (4) 0.055 (4) 0.266 (5) 0.505 (3) 0.658 (5) 0.921 (7) 0.018 (18) 0.612 (3) 0.098 (10) 0.127 (14) 0.591 (10) 0.211 (13)

0.164 (1) 0.197 (2) 0.761 (4) 0.429 (7) 0.209 (4) 0.538 (8) 0.534 (4) 0.480 (8) 0.007 (12) 0.509 (5) 0.090 (12) 0.559 (1) 0.309 (7) 0.874 (7) 0.760 (3) 0.229 (13) 0.115 (6) 0.048 (12) 0.452 (3) 0.474 (6) 0.955 (6) 0.092 (12) 0.793 (10) 0.085 (17) 0.720 (11) 0.269 (2) 0.282 (10) 1.000 (5) 0.050 (8) 0.289 (1) 0.480 (10) 0.658 (6) 0.927 (4) 0.026 (6) 0.651 (2) 0.094 (13) 0.155 (11) 0.639 (4) 0.215 (10)

0.158 (3) 0.218 (1) 0.764 (3) 0.444 (4) 0.212 (3) 0.532 (10) 0.536 (3) 0.466 (10) 0.005 (17) 0.507 (6) 0.079 (14) 0.557 (2) 0.327 (6) 0.885 (2) 0.694 (5) 0.225 (16) 0.118 (5) 0.046 (15) 0.395 (4) 0.480 (3) 0.955 (7) 0.104 (9) 0.813 (5) 0.088 (13) 0.852 (7) 0.235 (5) 0.282 (11) 1.000 (6) 0.048 (9) 0.272 (4) 0.487 (9) 0.671 (2) 0.924 (6) 0.026 (7) 0.604 (4) 0.093 (14) 0.165 (10) 0.668 (1) 0.215 (11)

Average Std Dev

0.337 (10.51) 0.398 (8.26) 0.330 (9.41) 0.392 (8.87) 0.396 (8.10) 0.367 (9.13) 0.276 (12.15) 0.275 (14.10) 0.208 (14.59) 0.347 (10.97) 0.314 (12.72) 0.302 (10.69) 0.314 (10.51) 0.293 (12.03) 0.348 (9.03) 0.408 (8.05) 0.413 (7.00) 0.414 (6.97) 0.422 (6.90) 0.243 0.308 0.248 0.302 0.300 0.264 0.223 0.227 0.171 0.285 0.255 0.242 0.231 0.209 0.267 0.313 0.294 0.297 0.283

Our adaptive stopping mechanism enables dataset-specific ensemble construction: simple datasets may benefit from small ensembles to avoid overfitting, while more complex datasets with diverse outlier patterns may require additional models to capture complementary perspectives. Fixed ensemble size baselines cannot adapt to this heterogeneity, leading to suboptimal performance across the benchmark.

0.70 (a) 0.65 0.60 0.55 0.50 0.45 0.40 0.35 1 2

0.90 (b) 0.85

ROC AUC

Average Precision (AP)

partners actively degrades performance by introducing lowquality models that add noise to the ensemble prediction. This finding validates our hypothesis that partner selection must be guided by both quality and diversity considerations.

0.164 (2) 0.197 (3) 0.726 (13) 0.384 (12) 0.102 (18) 0.562 (5) 0.579 (1) 0.494 (6) 0.021 (3) 0.512 (4) 0.436 (2) 0.530 (4) 0.269 (9) 0.875 (6) 0.768 (2) 0.235 (6) 0.166 (2) 0.048 (13) 0.568 (2) 0.475 (5) 0.977 (3) 0.058 (17) 0.771 (12) 0.082 (18) 0.556 (13) 0.284 (1) 0.160 (19) 1.000 (7) 0.114 (1) 0.285 (2) 0.522 (1) 0.535 (13) 0.965 (1) 0.017 (19) 0.406 (11) 0.099 (8) 0.491 (2) 0.665 (2) 0.364 (1)

Ensemble Size (k)

ELECT Top-k ELECT+Random

9 10

0.80 0.75 0.70

Greedy Oracle MetaEns

Ensemble Size (k)

9 10

Random

Figure A.2. Ensemble size impact on performance (AP) and robustness (ROC AUC). (Left) Average Precision vs ensemble size shows that MetaEns achieves rapid performance gains with small ensembles (k ≤ 3), then plateaus, confirming the effectiveness of the family-risk regularizer in adaptive ensemble stopping. (Right) ROC AUC demonstrates consistent robustness across different ensemble sizes. Greedy Oracle (with oracle access to test labels) provides an upper bound, while Random selection serves as the lower bound. ELECT performs comparably to single models (k = 1) across ensemble sizes due to its ranking-based selection strategy.

Automatic Unsupervised Ensemble Outlier Model Selection--Extended Version

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