CBE: A Complexity-adaptive Boundary-aware Ensemble Method for Imbalanced Data

Xun Wei

cs.LGOct 29, 2026

CBE: A Complexity-adaptive Boundary-aware Ensemble Method for Imbalanced Data

Authors: Xun Wei

Organizations: School of Industrial Software, Jiangxi University of Science and Technology, Nanchang 330044, China

Abstract

In many real-world classification tasks, datasets exhibit class imbalance, which can degrade the generalization performance of conventional learning algorithms. Data complexity has recently been recognized as a more objective indicator of learning difficulty than the imbalance ratio alone, because class overlap and complex decision boundaries often impair classifiers more severely than imbalance itself. Samples near the decision boundary are inherently difficult to classify, which has motivated many researchers to focus on such instances to improve classifier performance. However, most under-sampling approaches fail to differentiate their sampling of boundary samples across datasets of varying complexity. To overcome this limitation, we propose a complexity-adaptive boundary-aware ensemble (CBE) framework that links the sampling weight distribution to data complexity and dynamically adjusts the boundary preference of under-sampling accordingly. We also introduce an overall complexity measure that combines metrics at multiple levels and use it to divide imbalanced datasets into three categories. Extensive experiments on 24 benchmark datasets from UCI and KEEL show that the proposed method outperforms many existing methods in terms of F-measure and G-mean on Easy and Median datasets, without degrading performance on hard cases.

Figures & tables

Category	$N1$	$N3$	$L2$	OCM
Easy	Low	Low	Low	Low
Median	Low/Median	Low/Median	High/Median	Median
Hard	High	High	High	High

TABLE I : Dataset Categorization via Complexity Measures

Fig. 1 : Synthetic imbalanced datasets illustrating the proposed categorization based on complexity measures. The four datasets range from Easy to Hard, each containing 900 negative samples and 100 positive samples. In particular, we show two kinds of Median datasets: one with low N1 and N3 and high L2, and the other with median N1, N3, and L2.

Dataset	$N1$	$N3$	$L2$	OCM	Category
Clear Blob	0.1	0.1	0.06	0.09	Easy
Moon	0.31	0.23	0.38	0.28	Median
Spiral	0.2	0.11	1.0	0.32	Median
Overlap Blob	0.82	0.65	1.0	0.77	Hard

TABLE II : Complexity Measures on Four Synthetic Datasets.

Panel of Figure 4 — Fig. 2 : (a) shows the sampling probabilities of majority instances, encoded by the intensity of the blue color. CDWS applies an adaptive sampling intensity to boundary majority instances. On datasets with low complexity, it assigns closer examples higher weights to preserve critical boundary information, while it weakens excessive boundary preference on hard datasets. (b) shows the procedure of the CBE framework. It first calculates the overall complexity of the dataset; then performs weighted under-sampling of the majority examples based on this complexity to create a balanced training subset; finally, it trains base classifiers on multiple subsets and obtains the ensemble model.

Dataset	Size	Features	Imbalance Ratio	N1	N3	L2	OCM
mfeat-fac	2000	216	9.00	0.050	0.020	0.000	0.025
mfeat-pix	2000	240	9.00	0.045	0.025	0.000	0.026
mfeat-kar	2000	64	9.00	0.070	0.050	0.010	0.048
newthyroid2	215	5	5.14	0.171	0.029	0.029	0.072
vowel0	988	13	9.98	0.133	0.000	0.167	0.073
vehicle	845	18	3.25	0.176	0.060	0.035	0.090

TABLE III : Details of the datasets, including basic information and data complexity measures. The 24 datasets are divided into three groups by OCM (Easy, Median, Hard) and listed in ascending order of OCM.

	Predicted Positive Class	Predicted Negative Class
Actual Positive Class	TP(true positive)	FN(false negative)
Actual Negative Class	FP(false positive)	TN(true negative)

TABLE IV : Confusion Matrix

	CART	Ada	RF	UAda	URF	BB	BRF	RUSB	EE	CBE
mfeat-fac	0.064	0.540	0.245	0.036	0.064	0.169	0.106	0.016	0.105	0.169
mfeat-pix	0.012	0.198	0.091	0.067	0.045	0.134	0.076	0.017	0.077	0.139
mfeat-kar	0.045	0.596	0.288	0.104	0.06	0.139	0.102	0.028	0.109	0.138
newthyroid2	0.000	0.003	0.034	0.001	0.033	0.081	0.051	0.004	0.019	0.026
vowel0	0.002	0.066	0.058	0.010	0.035	0.089	0.058	0.007	0.038	0.050
vehicle	0.001	0.094	0.052	0.059	0.042	0.099	0.068	0.010	0.064	0.071

TABLE V : Training Times (in Seconds). The last row shows the average training time of each method over the 24 datasets.

Fig. 3 : Correlation between data complexity measures and performance metrics.

Easy	mfeat-fac	mfeat-pix	mfeat-kar	newthyroid2	vowel0	vehicle	musk2	ecoli4	Average
CART	0.914 $\pm$ 0.032	0.854 $\pm$ 0.040	0.764 $\pm$ 0.056	0.849 $\pm$ 0.103	0.881 $\pm$ 0.081	0.846 $\pm$ 0.049	0.886 $\pm$ 0.019	0.700 $\pm$ 0.202	0.8370
Ada	0.963 $\pm$ 0.024	0.950 $\pm$ 0.021	0.843 $\pm$ 0.047	0.924 $\pm$ 0.085	0.984 $\pm$ 0.025	0.949 $\pm$ 0.021	0.907 $\pm$ 0.018	0.729 $\pm$ 0.252	0.9061
RF	0.955 $\pm$ 0.022	0.941 $\pm$ 0.022	0.843 $\pm$ 0.050	0.945 $\pm$ 0.063	0.949 $\pm$ 0.039	0.927 $\pm$ 0.029	0.916 $\pm$ 0.014	0.677 $\pm$ 0.288	0.8942
UAda	0.886 $\pm$ 0.077	0.944 $\pm$ 0.033	0.928 $\pm$ 0.069	0.843 $\pm$ 0.136	0.802 $\pm$ 0.101	0.901 $\pm$ 0.037	0.871 $\pm$ 0.021	0.519 $\pm$ 0.208	0.8367
URF	0.905 $\pm$ 0.045	0.926 $\pm$ 0.036	0.923 $\pm$ 0.035	0.927 $\pm$ 0.073	0.801 $\pm$ 0.065	0.875 $\pm$ 0.041	0.808 $\pm$ 0.030	0.603 $\pm$ 0.191	0.8459
BB	0.935 $\pm$ 0.033	0.954 $\pm$ 0.025	0.942 $\pm$ 0.023	0.957 $\pm$ 0.061	0.871 $\pm$ 0.049	0.900 $\pm$ 0.034	0.896 $\pm$ 0.017	0.742 $\pm$ 0.172	0.8996

TABLE VI : F-measure of Compared Methods with Standard Deviation on Easy, Median, and Hard Datasets.

F-measure	CART	Ada	RF	UAda	URF	BB	BRF	RUSB	EE	CBE
CART	—	0-1-7	0-1-7	2-2-4	2-1-5	0-1-7	0-1-7	0-4-4	0-2-6	0-0-8
Ada	7-1-0	—	4-2-2	6-1-1	6-1-1	4-2-2	4-2-2	6-2-0	3-4-1	1-2-5
RF	7-1-0	2-2-4	—	6-1-1	5-2-1	4-2-2	4-2-2	7-1-0	5-1-2	1-1-6
UAda	4-2-2	1-1-6	1-1-6	—	3-2-3	0-2-6	0-2-6	4-2-2	0-1-7	0-0-8
URF	5-1-2	1-1-6	1-2-5	3-2-3	—	0-0-8	0-0-8	3-4-1	0-1-7	0-0-8
BB	7-1-0	2-2-4	2-2-4	6-2-0	8-0-0	—	0-8-0	8-0-0	2-4-2	0-2-6

TABLE VII : Comparison of Pairwise t-tests on F-measure with Significance Level 0.05 on Easy, Median, and Hard Datasets.

Easy	mfeat-fac	mfeat-pix	mfeat-kar	newthyroid2	vowel0	vehicle	musk2	ecoli4	Average
CART	0.946 $\pm$ 0.027	0.914 $\pm$ 0.032	0.858 $\pm$ 0.046	0.915 $\pm$ 0.071	0.937 $\pm$ 0.053	0.896 $\pm$ 0.039	0.931 $\pm$ 0.014	0.834 $\pm$ 0.178	0.9040
Ada	0.967 $\pm$ 0.021	0.951 $\pm$ 0.020	0.855 $\pm$ 0.042	0.945 $\pm$ 0.070	0.985 $\pm$ 0.024	0.968 $\pm$ 0.016	0.913 $\pm$ 0.017	0.779 $\pm$ 0.239	0.9204
RF	0.958 $\pm$ 0.021	0.944 $\pm$ 0.022	0.855 $\pm$ 0.044	0.951 $\pm$ 0.057	0.958 $\pm$ 0.035	0.954 $\pm$ 0.021	0.924 $\pm$ 0.013	0.706 $\pm$ 0.277	0.9061
UAda	0.973 $\pm$ 0.023	0.984 $\pm$ 0.011	0.975 $\pm$ 0.024	0.938 $\pm$ 0.055	0.966 $\pm$ 0.025	0.964 $\pm$ 0.014	0.958 $\pm$ 0.010	0.869 $\pm$ 0.160	0.9535
URF	0.977 $\pm$ 0.013	0.973 $\pm$ 0.013	0.966 $\pm$ 0.016	0.980 $\pm$ 0.036	0.972 $\pm$ 0.013	0.954 $\pm$ 0.016	0.939 $\pm$ 0.010	0.931 $\pm$ 0.071	0.9613
BB	0.976 $\pm$ 0.017	0.973 $\pm$ 0.016	0.955 $\pm$ 0.020	0.987 $\pm$ 0.033	0.979 $\pm$ 0.017	0.962 $\pm$ 0.014	0.956 $\pm$ 0.008	0.932 $\pm$ 0.091	0.9651

TABLE VIII : G-mean of Compared Methods with Standard Deviation on Easy, Median, and Hard Datasets.

G-mean	CART	Ada	RF	UAda	URF	BB	BRF	RUSB	EE	CBE
CART	—	0-1-7	0-1-7	0-0-8	0-0-8	0-0-8	0-0-8	0-0-8	0-0-8	0-0-8
Ada	7-1-0	—	5-2-1	1-3-4	2-0-6	1-1-6	1-1-6	2-1-5	1-1-6	0-1-7
RF	7-1-0	1-2-5	—	0-2-6	0-1-7	0-0-8	0-0-8	1-1-6	0-0-8	0-0-8
UAda	8-0-0	4-3-1	6-2-0	—	4-2-2	2-3-3	2-3-3	4-3-1	0-3-5	0-2-6
URF	8-0-0	6-0-2	7-1-0	2-2-4	—	1-3-4	1-3-4	5-3-0	0-2-6	0-2-6
BB	8-0-0	6-1-1	8-0-0	3-3-2	4-3-1	—	0-8-0	6-2-0	0-3-5	0-2-6

TABLE IX : Comparison of Pairwise t-tests on G-mean with Significance Level 0.05 on Easy, Median, and Hard Datasets.

Fig. 4 : Average ranks of all methods and critical difference diagram on three kinds of datasets for the Nemenyi test. Groups of methods that are not significantly different at 0.05 are connected.

Fig. 5 : F-measure and G-mean of different sampling strategies on the three categories of datasets.

Class-Imbalanced Learning

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From Imbalance to Evidence: Deep Nonparametric Testing for Complex Data under Severe Sample Imbalance

Guanhao Zhou, Xiufan Yu

Department of Applied and Computational Mathematics and Statistics University of Notre Dame

Two-Sample Testing Statistical Hypothesis Testing

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Comparing distributions under severe sample imbalance requires effectively using abundant majority observations while retaining sensitivity to differences involving the smaller sample. This challenge is particularly relevant for complex, non-Euclidean data, such as graphs, whose structural characteristics call for flexible testing procedures. Classifier two-sample tests (C2STs) provide such flexibility, but existing procedures are primarily designed for balanced or comparably sized samples. In this paper, we study C2STs in a highly imbalanced regime, and show that severe sample imbalance may substantially degrade testing performance. Different from conventional imbalanced learning approaches that treat sample imbalance merely as a nuisance for classifier training, we turn it into a source of repeated testing evidence. Building on this insight, we propose an Enhanced Imbalance-Aware Deep Classifier Two-Sample Test (EIAD-C2ST), a classifier-agnostic framework designed for sample imbalance that accommodates diverse data modalities by tailoring classifiers to data structure. For computationally expensive deep classifiers, we further develop a scalable representation-amortized implementation strategy. Theoretically, we establish rigorous guarantees on asymptotic size and characterize conditions for power consistency, allowing the imbalance ratio to diverge. We demonstrate the empirical gains using simulations and an application to blockchain forensics. The results reveal that the majority-sample surplus, when properly leveraged, can be organized into useful statistical evidence, rather than an obstacle, for distributional comparisons with complex data.

Abstract

Class-Imbalanced Learning

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