Pairwise Difference Learning (pdll)

A scikit-learn compatible meta-learner that boosts tabular classifiers by learning from pairs of points instead of single points.

Pairwise Difference Learning (PDL) turns a hard multiclass problem into an easy binary one. Instead of asking "what class is this point?", PDL asks "are these two points the same class?" — a reformulation that squeezes more signal out of the same training data and consistently improves the accuracy of tree-based models such as Random Forest and Gradient Boosting.

This library provides a drop-in, scikit-learn compatible PairwiseDifferenceClassifier (PDC), as introduced in Belaid et al., Pairwise Difference Learning for Classification, Discovery Science 2024.

✨ Highlights

• 🔌 Drop-in — wraps any scikit-learn classifier; keeps .fit, .predict, .predict_proba, .score.
• 📈 Better accuracy — benchmarked on 99 OpenML datasets against 7 base learners (macro-F1); wins on compatible learners.
• 🧠 Principled — probabilistic reformulation of multiclass classification into a similarity task.
• 🪶 Lightweight — pure Python, only numpy, pandas, and scikit-learn as dependencies.

📖 Contents

Installation · Usage · How does it work? · Evaluation · Reference

Installation

To install the package, run the following command:

pip install -U pdll

Usage

from pdll import PairwiseDifferenceClassifier

from sklearn.ensemble import RandomForestClassifier
from sklearn.datasets import make_blobs

# Generate random data with 2 features, 10 points, and 3 classes
X, y = make_blobs(n_samples=10, n_features=2, centers=3, random_state=0)

pdc = PairwiseDifferenceClassifier(estimator=RandomForestClassifier())
pdc.fit(X, y)
print('score:', pdc.score(X, y))

y_pred = pdc.predict(X)
proba_pred = pdc.predict_proba(X)

Please consult examples/ directory for more examples.

How does it work?

The PDL algorithm works by transforming the multiclass classification problem into a binary classification problem. The algorithm works as follows:

Example 1: Graphical abstract

Example 2: PDC trained on the Iris dataset

Clic to show

We provide a minimalist classification example using the Iris dataset. The dataset is balanced, so the prior probabilities of each of the 3 classes are equal: p(Setosa) = p(Versicolour) = p(Virginica) = 1/3

Three Anchor Points

• Flower 1: y1 = Setosa
• Flower 2: y2 = Versicolour
• Flower 3: y3 = Virginica

One Query Point

• Flower Q: yq (unknown target)

Pairwise Predictions The model predicts the likelihood that both points have a similar class:

• g_sym(Flower Q, Flower 1) = 0.6
• g_sym(Flower Q, Flower 2) = 0.3
• g_sym(Flower Q, Flower 3) = 0.0

Given the above data, the first step is to update the priors.

Posterior using Flower 1:

• p_post,1(Setosa) = 0.6
• p_post,1(Versicolour) = (1/3 * (1 - 0.6)) / (1 - 1/3) = 0.2
• p_post,1(Virginica) = (1/3 * (1 - 0.6)) / (1 - 1/3) = 0.2

Similarly, we calculate for anchors 2 and 3:

• p_post,2(Setosa) = 0.35

• p_post,2(Versicolour) = 0.30

• p_post,2(Virginica) = 0.35

• p_post,3(Setosa) = 0.5

• p_post,3(Versicolour) = 0.5

• p_post,3(Virginica) = 0.0

Averaging over the three predictions:

Finally, the predicted class is the most likely prediction:

ŷ_q = arg max_{y ∈ Y} p_post(y) = Setosa

Evaluation

To reproduce the experiment of the paper, please run run_benchmark.py with a base learner and a dataset number, between 0 and 99. Example:

python run_benchmark.py --model DecisionTreeClassifier --data 0

Scores will be stored in ./results/tmp/ directory.

Experiment

We use 99 datasets from the OpenML repository. We compare the performance of the PDC algorithm with 7 base learners. We use the macro F1 score as a metric. The search space is inspired from TPOT a state-of-the-art library in optimizing Sklearn pipelines

Description of the search space per estimator

Estimator	# parameters	# combinations
DecisionTree	4	350
RandomForest	7	1000
ExtraTree	6	648
HistGradientBoosting	6	486
Bagging	6	96
ExtraTrees	7	1000
GradientBoosting	5	900

Search space per estimator

Estimator	Parameter	Values
DecisionTreeClassifier	criterion	gini, entropy
	max depth	None, 1, 2, 4, 6, 8, 11
	min samples split	2, 4, 8, 16, 21
	min samples leaf	1, 2, 4, 10, 21
RandomForestClassifier	criterion	gini, entropy
	min samples split	2, 4, 8, 16, 21
	max features	sqrt, 0.05, 0.17, 0.29, 0.41, 0.52, 0.64, 0.76, 0.88, 1.0
	min samples leaf	1, 2, 4, 10, 21
	bootstrap	True, False
ExtraTreeClassifier	criterion	gini, entropy
	min samples split	2, 5, 10
	min samples leaf	1, 2, 4
	max features	sqrt, log2, None
	max leaf nodes	None, 2, 12, 56
	min impurity decrease	0.0, 0.1, 0.5
HistGradientBoostingClassifier	max iter	100, 10
	learning rate	0.1, 0.01, 1
	max leaf nodes	31, 3, 256
	min samples leaf	20, 4, 64
	l2 regularization	0, 0.01, 0.1
	max bins	255, 2, 64
BaggingClassifier	n estimators	10, 5, 100, 256
	max samples	1.0, 0.5
	max features	0.5, 0.9, 1.0
	bootstrap	True, False
	bootstrap features	False, True
ExtraTreesClassifier	criterion	gini, entropy
	max features	sqrt, 0.05, 0.17, 0.29, 0.41, 0.52, 0.64, 0.76, 0.88, 1.0
	min samples split	2, 4, 8, 16, 21
	min samples leaf	1, 2, 4, 10, 21
	bootstrap	False, True
GradientBoostingClassifier	learning rate	0.1, 0.01, 1
	min samples split	2, 4, 8, 16, 21
	min samples leaf	1, 2, 4, 10, 21
	subsample	1.0, 0.05, 0.37, 0.68
	max features	None, 0.15, 0.68

OpenML benchmark datasets

data_id	NumberOfClasses	NumberOfInstances	NumberOfFeatures	NumberOfSymbolicFeatures	NumberOfFeatures_post_processing	MajorityClassSize	MinorityClassSize
43	2	306	4	2	3	225	81
48	3	151	6	3	5	52	49
59	2	351	35	1	34	225	126
61	3	150	5	1	4	50	50
164	2	106	58	58	57	53	53
333	2	556	7	7	6	278	278
377	6	600	61	1	60	100	100
444	2	132	4	4	3	71	61
464	2	250	3	1	2	125	125
475	4	400	6	5	5	100	100
714	2	125	5	3	4	76	49
717	2	508	11	1	10	286	222
721	2	200	11	1	10	103	97
733	2	209	7	1	6	153	56
736	2	111	4	1	3	58	53
744	2	250	6	1	5	141	109
750	2	500	8	1	7	254	246
756	2	159	16	1	15	105	54
766	2	500	51	1	50	262	238
767	2	475	4	3	3	414	61
768	2	100	26	1	25	55	45
773	2	250	26	1	25	126	124
779	2	500	26	1	25	267	233
782	2	120	3	1	2	63	57
784	2	140	4	2	3	70	70
788	2	186	61	1	60	109	77
792	2	500	6	1	5	298	202
793	2	250	11	1	10	135	115
811	2	264	3	2	2	163	101
812	2	100	26	1	25	53	47
814	2	468	3	1	2	256	212
824	2	500	11	1	10	274	226
850	2	100	51	1	50	51	49
853	2	506	14	2	13	297	209
860	2	380	3	1	2	195	185
863	2	250	11	1	10	133	117
870	2	500	6	1	5	267	233
873	2	250	51	1	50	142	108
877	2	250	51	1	50	137	113
879	2	500	26	1	25	304	196
880	2	284	11	1	10	142	142
889	2	100	26	1	25	50	50
895	2	222	3	1	2	134	88
896	2	500	26	1	25	280	220
902	2	147	7	5	6	78	69
906	2	400	8	1	7	207	193
909	2	400	8	1	7	203	197
911	2	250	6	1	5	140	110
915	2	315	14	4	13	182	133
918	2	250	51	1	50	135	115
925	2	323	5	1	4	175	148
932	2	100	51	1	50	56	44
933	2	250	26	1	25	136	114
935	2	250	11	1	10	140	110
936	2	500	11	1	10	272	228
937	2	500	51	1	50	282	218
969	2	150	5	1	4	100	50
973	2	178	14	1	13	107	71
974	2	132	5	1	4	81	51
1005	2	214	10	1	9	138	76
1011	2	336	8	1	7	193	143
1012	2	194	29	27	28	125	69
1054	2	161	40	1	39	109	52
1063	2	522	22	1	21	415	107
1065	2	458	40	1	39	415	43
1073	2	274	9	1	8	140	134
1100	3	478	11	5	10	247	90
1115	3	151	7	5	6	52	49
1413	3	150	5	1	4	50	50
1467	2	540	21	1	20	494	46
1480	2	583	11	2	10	416	167
1488	2	195	23	1	22	147	48
1490	2	182	13	1	12	130	52
1499	3	210	8	1	7	70	70
1510	2	569	31	1	30	357	212
1511	2	440	9	2	8	298	142
1523	3	310	7	1	6	150	60
1554	5	500	13	5	12	192	43
1556	2	120	7	6	6	61	59
1600	2	267	45	1	44	212	55
4329	2	470	17	14	16	400	70
40663	5	399	33	21	32	96	44
40681	2	128	7	7	6	64	64
41568	3	150	5	1	4	50	50
41977	2	156	91	1	90	98	58
41978	2	156	81	1	80	94	62
42011	3	150	5	1	4	50	50
42021	3	150	5	1	4	50	50
42026	3	150	5	1	4	50	50
42051	3	150	5	1	4	50	50
42066	3	150	5	1	4	50	50
42071	3	150	5	1	4	50	50
42186	3	150	5	1	4	50	50
42700	3	150	5	1	4	50	50
43859	3	150	5	1	4	50	50
44149	2	296	14	1	18	159	137
44151	3	149	5	0	4	50	49
44344	3	150	5	1	4	50	50
45711	2	530	14	3	13	354	176

Score comparison

2D datasets Examples

Here we see the difference in the learned patterns between PDL and the base learner. In case PDL is compatible with the base learner (DecisionTree, RandomForest) then the scores improves. In case the base learner is not compatible with PDL (SVC, AdaBoost, ...) then the scores gets lower.

Reference

Please cite us if you use this library in your research:

@article{belaid2024pairwise,
  title={Pairwise Difference Learning for Classification},
  author={Belaid, Mohamed Karim and Rabus, Maximilian and H{\"u}llermeier, Eyke},
  journal={Discovery Science},
  year={2024}
}

The first commit correspond to the original implementation of the PDC algorithm

Acknowledgments: We would like to thank Tim Wibiral, Dorra ElMekki, Viktor Bengs, Muhammad Zeeshan Anwer, Muhammad Hossein Shaker, Alireza Javanmardi, Patrick Kolpaczki, and Maximilian Muschalik for their early comments on this work. We also acknowledge LRZ and IDIADA for computational resources.