1 Introduction
1.1 Contributions
1.2 Illustrative example
1.3 Related work
2 Background
2.1 Numerical abstract domains
2.1.1 Abstract operations on intervals
2.1.2 Abstract operations on zonotopes
2.2 kNN classifiers
2.3 Stability and robustness
2.4 Individual fairness
3 Abstract verification of kNN
3.1 Abstract distance
3.2 Abstract kNN classification
3.2.1 Step\(_1\): Computing and ordering abstract distances
3.2.2 Step\(_2\): Computing score bounds for labels
3.2.3 Step\(_3\): Refining lower bounds
3.2.4 Step\(_4\): Abstract classification
3.2.5 Remarks
3.3 Regression tasks
3.4 Instantiating to different abstractions and perturbations
4 Equivalence of data poisoning and input perturbation for the maximum norm
4.1 Intervals
4.2 Zonotopes
4.3 Arbitrary abstractions
5 Dealing with categorical features
6 Experimental evaluation
6.1 Setup
Dataset | |D| training | |T| test | #feat | #feat with one-hot | #labels | kNN accuracy % | |||
---|---|---|---|---|---|---|---|---|---|
\(k=1\) | \(k=3\) | \(k=5\) | \(k=7\) | ||||||
Stability | |||||||||
Australian | 483 | 207 | 14 | 39 | 2 | 77.8 | 80.2 | 82.6 | 82.6 |
BreastCancer | 479 | 204 | 10 | 10 | 2 | 92.6 | 94.6 | 93.6 | 93.6 |
Diabetes | 556 | 230 | 8 | 8 | 2 | 70.9 | 72.2 | 70.0 | 71.3 |
Fourclass | 604 | 258 | 2 | 2 | 2 | 100 | 100 | 100 | 100 |
Letter | 15,000 | 5000 | 16 | 16 | 26 | 95.7 | 94.6 | 94.2 | 94.3 |
Pendigits | 7494 | 3498 | 16 | 16 | 10 | 97.7 | 97.8 | 97.5 | 97.5 |
Satimage | 4435 | 2000 | 36 | 46 | 6 | 88.8 | 90.3 | 89.5 | 90.1 |
Fairness | |||||||||
Compas | 4222 | 1056 | 10 | 370 | 2 | 58.4 | 59.1 | 60.2 | 61.1 |
German | 800 | 200 | 20 | 56 | 2 | 73.0 | 71.5 | 74.5 | 77.0 |
\(\epsilon \) | Australian | BreastCancer | Diabetes | Fourclass | Letter | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | \(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | \(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | \(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | \(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | |
0.001 | 98.5 | 99.5 | 99.0 | 99.0 | 100 | 98.5 | 99.5 | 100 | 93.4 | 94.3 | 93.4 | 95.2 | 100 | 99.6 | 100 | 100 | 96.9 | 94.4 | 93.4 | 93.5 |
0.005 | 96.6 | 96.1 | 93.2 | 95.1 | 97.5 | 98.5 | 98.5 | 97.5 | 73.4 | 69.1 | 63.9 | 65.2 | 99.6 | 99.6 | 99.2 | 99.2 | 90.5 | 86.7 | 83.6 | 81.2 |
0.01 | 92.2 | 93.2 | 91.3 | 94.2 | 93.6 | 95.5 | 97.0 | 96.5 | 45.2 | 39.1 | 36.9 | 34.3 | 99.2 | 98.8 | 96.1 | 95.7 | 75.3 | 67.2 | 60.8 | 56.1 |
0.02 | 88.8 | 88.4 | 86.9 | 89.8 | 85.7 | 86.2 | 86.7 | 88.7 | 14.3 | 12.6 | 10.8 | 9.1 | 87.6 | 86.0 | 81.7 | 81.0 | 40.2 | 32.6 | 27.9 | 25.0 |
0.03 | 84.0 | 83.0 | 85.9 | 85.5 | 78.9 | 83.3 | 84.8 | 86.2 | 4.7 | 3.0 | 1.7 | 1.3 | 70.1 | 68.6 | 67.0 | 62.4 | 15.4 | 12.0 | 9.84 | 8.60 |
0.05 | 79.7 | 80.1 | 83.5 | 82.6 | 66.6 | 68.6 | 75.4 | 75.0 | 0.8 | 0.8 | 0 | 0 | 34.1 | 34.5 | 29.8 | 28.6 | 1.2 | 1.1 | 1.1 | 1.0 |
0.07 | 78.2 | 78.2 | 77.7 | 78.2 | 34.8 | 45.1 | 51.4 | 58.8 | 0.4 | 0 | 0 | 0 | 15.1 | 15.5 | 14.3 | 13.5 | – | – | – | – |
0.10 | 69.0 | 64.2 | 65.7 | 66.6 | 5.3 | 5.8 | 6.3 | 19.1 | 0 | 0 | 0 | 0 | 5.4 | 5.0 | 5.0 | 4.2 | – | – | – | – |
\(\epsilon \) | Pendigits | Satimage | \(\epsilon \) | Compas | German | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | \(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | \(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | \(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | |||||
0.001 | 99.5 | 99.1 | 99.2 | 99.1 | 93.6 | 91.8 | 92.0 | 91.7 | 0 | 56.4 | 62.0 | 68.0 | 70.5 | 85.5 | 83.5 | 82.5 | 85.0 | |||
0.005 | 96.5 | 96.1 | 95.8 | 95.4 | 68.3 | 65.5 | 64.2 | 63.3 | 0.001 | 46.5 | 52.3 | 57.7 | 60.0 | 84.5 | 83.5 | 81.5 | 84.5 | |||
0.01 | 92.1 | 91.8 | 91.1 | 90.7 | 44.4 | 43.5 | 43.7 | 43.9 | 0.002 | 40.5 | 45.9 | 49.3 | 51.7 | 84.0 | 82.5 | 81.0 | 84.0 | |||
0.02 | 79.0 | 78.4 | 77.9 | 77.5 | 20.8 | 19.9 | 20.0 | 20.4 | 0.005 | 26.3 | 31.8 | 33.7 | 36.3 | 83.5 | 79.5 | 80.5 | 82.0 | |||
0.03 | 62.6 | 63.6 | 63.2 | 63.0 | 12.1 | 12.1 | 11.8 | 12.2 | 0.01 | 16.9 | 20.8 | 22.7 | 26.0 | 82.0 | 76.0 | 78.0 | 80.0 | |||
0.05 | 28.3 | 29.5 | 29.2 | 28.6 | 8.4 | 8.4 | 8.4 | 8.4 | 0.02 | 11.0 | 13.7 | 14.5 | 16.9 | 78.0 | 74.0 | 74.0 | 73.0 | |||
0.07 | 8.1 | 8.9 | 9.1 | 9.1 | 6.3 | 6.4 | 6.2 | 6.4 | 0.03 | 9.1 | 10.6 | 11.5 | 13.6 | 73.5 | 69.5 | 70.0 | 68.0 | |||
0.10 | 0.2 | 0.1 | 0.06 | 0.03 | 2.9 | 3.0 | 2.9 | 2.9 | 0.05 | 5.8 | 7.0 | 7.4 | 8.9 | 67.5 | 62.5 | 59.5 | 58.0 |
\(\epsilon \) | Australian | BreastCancer | Diabetes | Fourclass | Letter | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | \(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | \(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | \(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | \(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | |
0.001 | 100 | 99.5 | 99.5 | 99.5 | 100 | 99.0 | 99.5 | 100 | 95.2 | 96.5 | 96.5 | 96.0 | 100 | 99.6 | 100 | 100 | 96.9 | 94.4 | 93.4 | 93.5 |
0.005 | 98.5 | 97.5 | 94.6 | 95.6 | 98.0 | 98.5 | 99.0 | 99.5 | 85.2 | 82.1 | 83.4 | 83.9 | 99.6 | 100 | 100 | 99.6 | 92.3 | 89.7 | 88.2 | 87.7 |
0.01 | 94.6 | 94.6 | 92.7 | 94.6 | 96.0 | 96.5 | 98.5 | 97.5 | 72.1 | 73.0 | 73.0 | 72.1 | 99.6 | 99.6 | 99.6 | 99.2 | 84.9 | 82.2 | 81.0 | 79.6 |
0.02 | 93.7 | 90.8 | 91.3 | 92.2 | 91.6 | 92.6 | 97.0 | 94.6 | 60.4 | 56.5 | 64.3 | 65.6 | 94.5 | 96.5 | 97.6 | 98.0 | 66.7 | 64.1 | 63.9 | 61.7 |
0.03 | 91.3 | 87.9 | 88.4 | 92.2 | 89.7 | 91.1 | 92.6 | 94.6 | 50.4 | 54.7 | 57.3 | 52.6 | 90.3 | 89.9 | 91.0 | 89.9 | 54.0 | 52.6 | 52.8 | 51.7 |
0.05 | 85.0 | 85.0 | 89.8 | 87.9 | 88.2 | 92.1 | 92.1 | 94.6 | 23.9 | 28.7 | 28.7 | 29.5 | 74.0 | 79.4 | 80.6 | 84.5 | 37.7 | 28.3 | 26.8 | 25.9 |
0.07 | 81.6 | 81.6 | 85.9 | 86.9 | 78.9 | 84.8 | 87.7 | 89.7 | 10.4 | 7.3 | 20.4 | 10.4 | 67.4 | 58.5 | 59.6 | 58.1 | – | – | – | – |
0.10 | 84.0 | 85.5 | 85.5 | 86.9 | 73.5 | 83.8 | 78.4 | 79.9 | 3.4 | 0.4 | 19.5 | 6.9 | 37.9 | 37.6 | 46.9 | 51.9 | – | – | – | – |
\(\epsilon \) | Pendigits | Satimage | \(\epsilon \) | Compas | German | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | \(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | \(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | \(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | |||||
0.001 | 99.6 | 99.2 | 99.3 | 99.2 | 95.5 | 93.8 | 94.7 | 95.5 | 0 | 56.4 | 62.0 | 68.0 | 70.5 | 85.5 | 83.5 | 82.5 | 85.0 | |||
0.005 | 98.4 | 98.0 | 98.3 | 97.9 | 84.8 | 84.7 | 84.5 | 85.2 | 0.001 | 47.9 | 53.6 | 58.7 | 61.9 | 85.0 | 83.5 | 81.5 | 84.5 | |||
0.01 | 96.4 | 96.8 | 97.1 | 96.8 | 75.4 | 77.6 | 78.9 | 81.1 | 0.002 | 44.0 | 49.5 | 55.3 | 56.8 | 85.0 | 82.5 | 81.5 | 84.0 | |||
0.02 | 92.7 | 93.9 | 93.7 | 93.4 | 74.9 | 75.9 | 79.5 | 80.2 | 0.005 | 33.2 | 38.5 | 43.7 | 47.1 | 84.0 | 81.0 | 81.0 | 83.5 | |||
0.03 | 88.4 | 91.5 | 91.4 | 91.9 | 66.7 | 68.3 | 70.8 | 73.3 | 0.01 | 25.1 | 30.2 | 31.9 | 37.1 | 82.5 | 76.5 | 78.5 | 82.0 | |||
0.05 | 77.2 | 83.4 | 84.1 | 83.7 | 55.6 | 57.7 | 57.5 | 60.1 | 0.02 | 17.6 | 21.5 | 24.5 | 27.1 | 80.0 | 76.5 | 76.0 | 78.5 | |||
0.07 | 61.8 | 67.7 | 68.4 | 68.4 | 49.7 | 41.0 | 42.7 | 51.5 | 0.03 | 13.5 | 17.2 | 19.3 | 21.3 | 75.0 | 75.0 | 73.0 | 74.0 | |||
0.10 | 38.4 | 41.2 | 49.0 | 52.5 | 41.8 | 35.0 | 35.7 | 35.9 | 0.05 | 9.1 | 12.9 | 16.1 | 16.8 | 70.5 | 69.5 | 63.5 | 69.0 |
\(\epsilon \) | Australian | BreastCancer | Diabetes | Fourclass | Letter | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | \(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | \(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | \(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | \(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | |
0.001 | 100 | 100 | 100 | 100 | 99.5 | 100 | 99.0 | 99.5 | 95.2 | 96.0 | 98.7 | 98.7 | 100 | 100 | 100 | 100 | 97.9 | 96.3 | 95.2 | 95.6 |
0.005 | 97.1 | 98.5 | 97.5 | 99.0 | 96.5 | 99.5 | 98.0 | 98.5 | 78.7 | 80.4 | 80.0 | 77.3 | 99.6 | 99.6 | 99.2 | 98.8 | 91.7 | 88.6 | 85.4 | 83.0 |
0.01 | 95.1 | 96.6 | 96.1 | 97.1 | 96.5 | 99.0 | 96.5 | 97.0 | 62.6 | 59.5 | 56.5 | 54.3 | 99.2 | 99.6 | 98.4 | 97.6 | 82.6 | 75.1 | 69.4 | 65.2 |
0.02 | 92.7 | 91.3 | 91.7 | 94.6 | 92.1 | 93.1 | 91.6 | 93.1 | 31.3 | 25.6 | 23.9 | 23.0 | 93.4 | 91.4 | 86.8 | 85.2 | 54.6 | 45.3 | 40.2 | 36.9 |
0.03 | 91.3 | 88.8 | 89.3 | 92.2 | 86.7 | 88.7 | 87.7 | 88.2 | 13.0 | 9.1 | 7.8 | 6.1 | 78.2 | 75.5 | 75.5 | 72.8 | 31.0 | 24.8 | 21.2 | 19.2 |
0.05 | 85.9 | 84.0 | 85.5 | 85.9 | 75.0 | 77.9 | 81.3 | 83.8 | 3.0 | 1.7 | 0.8 | 0.8 | 44.9 | 42.6 | 40.3 | 37.6 | 7.7 | 6.6 | 5.7 | 5.2 |
0.07 | 84.5 | 82.1 | 85.5 | 83.5 | 65.2 | 65.6 | 67.6 | 74.5 | 1.3 | 1.3 | 0 | 0 | 24.0 | 23.6 | 22.8 | 19.3 | – | – | – | – |
0.10 | 82.1 | 80.1 | 85.0 | 83.0 | 37.7 | 38.7 | 42.6 | 52.9 | 0.8 | 0.8 | 0 | 0 | 9.3 | 8.5 | 8.1 | 8.1 | – | – | – | – |
\(\epsilon \) | Pendigits | Satimage | \(\epsilon \) | Compas | German | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | \(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | \(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | \(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | |||||
0.001 | 99.6 | 99.4 | 99.4 | 99.6 | 93.8 | 93.8 | 93.7 | 94.3 | 0 | 58.2 | 62.6 | 69.1 | 71.0 | 85.0 | 83.0 | 83.5 | 82.5 | |||
0.005 | 98.3 | 98.0 | 98.1 | 97.8 | 73.7 | 72.8 | 72.4 | 72.3 | 0.001 | 48.2 | 54.5 | 61.1 | 62.4 | 85.0 | 83.0 | 83.0 | 82.0 | |||
0.01 | 96.6 | 96.1 | 95.7 | 95.0 | 56.0 | 54.9 | 54.0 | 54.2 | 0.002 | 45.5 | 50.9 | 56.7 | 57.6 | 85.0 | 82.5 | 82.5 | 82.0 | |||
0.02 | 91.4 | 90.5 | 89.5 | 89.0 | 31.4 | 31.0 | 31.8 | 32.7 | 0.005 | 35.2 | 41.4 | 45.4 | 46.0 | 85.0 | 81.5 | 82.5 | 81.5 | |||
0.03 | 82.0 | 81.9 | 81.0 | 79.9 | 19.3 | 18.5 | 18.3 | 18.6 | 0.01 | 25.6 | 30.7 | 32.8 | 36.4 | 83.0 | 78.0 | 81.0 | 81.5 | |||
0.05 | 58.9 | 60.6 | 59.7 | 58.9 | 9.6 | 9.6 | 9.6 | 9.8 | 0.02 | 17.8 | 20.8 | 23.8 | 27.4 | 80.0 | 75.5 | 78.0 | 78.5 | |||
0.07 | 36.3 | 38.4 | 37.5 | 36.8 | 8.2 | 8.1 | 7.9 | 7.8 | 0.03 | 13.5 | 16.6 | 19.5 | 21.7 | 76.0 | 72.5 | 75.0 | 73.5 | |||
0.10 | 13.5 | 13.4 | 13.2 | 13.2 | 4.7 | 4.6 | 4.6 | 4.5 | 0.05 | 10.2 | 12.7 | 14.3 | 16.4 | 72.0 | 68.5 | 69.0 | 69.5 |
\(\epsilon \) | Australian | BreastCancer | Diabetes | Fourclass | Letter | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | \(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | \(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | \(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | \(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | |
0.001 | 100 | 100 | 100 | 100 | 99.5 | 100 | 99.0 | 99.5 | 96.5 | 96.0 | 98.7 | 98.7 | 100 | 100 | 100 | 100 | 97.9 | 96.3 | 95.2 | 95.7 |
0.005 | 98.0 | 99.0 | 98.5 | 99.0 | 97.5 | 99.5 | 99.0 | 99.0 | 85.2 | 87.8 | 88.2 | 87.8 | 99.6 | 99.6 | 100 | 100 | 94.3 | 92.1 | 91.0 | 90.1 |
0.01 | 97.1 | 97.1 | 96.6 | 98.0 | 96.5 | 98.5 | 99.0 | 98.0 | 79.1 | 81.3 | 77.3 | 79.1 | 99.2 | 99.6 | 99.6 | 99.2 | 88.0 | 86.0 | 84.2 | 82.8 |
0.02 | 95.1 | 92.2 | 95.1 | 96.6 | 94.1 | 95.1 | 94.6 | 96.5 | 66.5 | 66.9 | 65.2 | 64.7 | 90.7 | 94.5 | 94.5 | 96.5 | 68.3 | 66.6 | 65.0 | 63.3 |
0.03 | 93.7 | 89.3 | 93.7 | 94.6 | 92.6 | 97.0 | 95.1 | 94.6 | 50.0 | 52.6 | 55.2 | 57.3 | 81.4 | 83.3 | 84.5 | 86.0 | 51.4 | 51.3 | 50.5 | 49.1 |
0.05 | 89.3 | 87.4 | 89.8 | 88.8 | 89.2 | 93.1 | 92.6 | 95.5 | 25.6 | 30.0 | 31.7 | 38.7 | 56.2 | 64.7 | 69.7 | 68.9 | 28.7 | 27.8 | 27.1 | 25.8 |
0.07 | 85.0 | 83.0 | 87.9 | 85.5 | 77.4 | 86.7 | 91.6 | 95.1 | 10.8 | 8.2 | 11.3 | 10.0 | 36.0 | 37.2 | 44.1 | 49.2 | – | – | – | – |
0.10 | 82.6 | 80.1 | 85.5 | 84.5 | 66.6 | 75.9 | 75.9 | 78.9 | 4.3 | 1.7 | 14.3 | 2.1 | 19.7 | 22.0 | 31.0 | 32.1 | – | – | – | – |
\(\epsilon \) | Pendigits | Satimage | \(\epsilon \) | Compas | German | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | \(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | \(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | \(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | |||||
0.001 | 99.6 | 99.6 | 99.4 | 99.7 | 95.4 | 95.0 | 95.2 | 96.0 | 0 | 58.2 | 62.6 | 69.1 | 71.0 | 85.0 | 83.0 | 83.5 | 82.5 | |||
0.005 | 98.8 | 98.5 | 98.6 | 98.7 | 84.8 | 85.1 | 85.5 | 86.8 | 0.001 | 49.0 | 55.5 | 61.9 | 63.4 | 85.0 | 83.0 | 83.0 | 82.5 | |||
0.01 | 97.6 | 97.8 | 97.9 | 97.8 | 76.1 | 77.9 | 78.7 | 79.4 | 0.002 | 46.8 | 52.6 | 58.7 | 59.9 | 85.0 | 83.0 | 82.5 | 82.0 | |||
0.02 | 94.9 | 95.4 | 95.4 | 95.5 | 68.2 | 70.2 | 74.4 | 76.0 | 0.005 | 38.2 | 45.1 | 49.6 | 52.4 | 84.5 | 83.0 | 82.5 | 82.0 | |||
0.03 | 91.1 | 92.7 | 93.0 | 93.1 | 60.4 | 63.0 | 64.9 | 67.8 | 0.01 | 29.8 | 36.4 | 40.8 | 45.8 | 83.5 | 80.0 | 81.0 | 81.5 | |||
0.05 | 79.7 | 85.3 | 86.7 | 85.8 | 44.4 | 49.4 | 52.6 | 56.9 | 0.02 | 22.9 | 28.3 | 32.3 | 35.4 | 79.5 | 77.5 | 78.5 | 80.5 | |||
0.07 | 62.6 | 71.1 | 73.7 | 73.5 | 25.5 | 31.6 | 34.8 | 42.5 | 0.03 | 17.9 | 23.1 | 25.7 | 28.6 | 76.0 | 75.0 | 77.5 | 77.5 | |||
0.10 | 38.7 | 48.3 | 51.3 | 53.1 | 9.7 | 26.6 | 24.9 | 36.2 | 0.05 | 12.3 | 16.7 | 18.7 | 22.0 | 72.5 | 74.0 | 74.0 | 73.0 |
6.2 Results
Dataset | Intervals \({\mathcal {I}}\) | Zonotopes \({\mathcal {Z}}\) | ||
---|---|---|---|---|
\(\delta _1\) (s) | \(\delta _2\) (s) | \(\delta _1\) (s) | \(\delta _2\) (s) | |
Australian | 0.01 | 0.01 | 0.11 | 0.22 |
BreastCancer | 0.01 | 0.01 | 0.06 | 0.05 |
Diabetes | 0.11 | 0.07 | 0.55 | 0.58 |
Fourclass | 0.04 | 0.04 | 0.35 | 0.40 |
Letter | 5.44 | 4.97 | 21.89 | 22.64 |
Pendigits | 0.26 | 0.57 | 9.99 | 9.70 |
Satimage | 0.33 | 2.90 | 11.91 | 4.29 |
Compas | 15.30 | 18.48 | 140.82 | 239.99 |
German | 0.75 | 0.74 | 5.08 | 7.67 |
6.2.1 Robustness
\(\epsilon \) | Australian | BreastCancer | Diabetes | Fourclass | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | \(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | \(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | \(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | |
0.001 | 77.8 | 80.2 | 82.6 | 82.6 | 92.6 | 94.6 | 93.6 | 93.1 | 69.1 | 71.3 | 69.6 | 71.3 | 100 | 100 | 100 | 100 |
0.005 | 76.8 | 79.2 | 81.2 | 81.6 | 91.2 | 94.1 | 93.1 | 92.6 | 60.0 | 62.2 | 61.3 | 59.1 | 99.6 | 99.6 | 99.2 | 98.8 |
0.01 | 75.4 | 77.8 | 79.7 | 80.7 | 91.2 | 93.6 | 91.7 | 92.2 | 49.1 | 49.6 | 47.0 | 46.5 | 99.2 | 99.6 | 98.4 | 97.7 |
0.02 | 73.9 | 75.8 | 77.8 | 79.2 | 88.2 | 89.7 | 88.7 | 89.7 | 25.2 | 23.0 | 22.6 | 21.7 | 93.4 | 91.5 | 86.8 | 85.3 |
0.03 | 72.9 | 74.4 | 76.8 | 77.8 | 84.3 | 86.8 | 86.8 | 87.3 | 11.7 | 8.7 | 7.4 | 6.1 | 78.3 | 75.6 | 75.6 | 72.9 |
0.05 | 68.6 | 71.5 | 74.4 | 73.9 | 74.5 | 77.5 | 80.9 | 83.3 | 2.6 | 1.7 | 0.9 | 0.9 | 45.0 | 42.6 | 40.3 | 37.6 |
\(\epsilon \) | Letter | Pendigits | Satimage | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | \(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | \(k = 1\) | \(k = 3\) | \(k = 5\) | \(k = 7\) | |||||
0.001 | 94.7 | 93.6 | 92.9 | 93.0 | 97.6 | 97.5 | 97.4 | 97.4 | 86.1 | 87.4 | 87.0 | 87.5 | ||||
0.005 | 90.4 | 87.8 | 84.8 | 82.7 | 96.8 | 96.8 | 96.7 | 96.3 | 71.7 | 71.7 | 71.0 | 70.8 | ||||
0.01 | 82.2 | 75.0 | 69.4 | 65.1 | 95.6 | 95.3 | 94.9 | 94.1 | 55.4 | 54.6 | 53.8 | 53.8 | ||||
0.02 | 54.6 | 45.4 | 40.3 | 37.0 | 90.9 | 90.2 | 89.2 | 88.7 | 31.4 | 31.0 | 31.7 | 32.6 | ||||
0.03 | 31.0 | 24.9 | 21.3 | 19.2 | 81.7 | 81.7 | 80.8 | 79.8 | 19.3 | 18.6 | 18.3 | 18.6 | ||||
0.05 | 7.7 | 6.6 | 5.8 | 5.2 | 58.9 | 60.6 | 59.7 | 58.9 | 9.6 | 9.7 | 9.7 | 9.8 |