# Robust Classification via Support Vector Machines

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Background and Problem Definition

#### 2.1. Problem Definition

#### 2.2. Loss Function

- (i)
- Hinge loss:${L}_{H}\left(u\right):=max\{0,u\}$;
- (ii)
- Truncated hinge loss ($a\ge 1$):${L}_{TH}\left(u\right):=min\{max\{0,u\},a\}$;
- (iii)
- Pinball loss ($a\le 0$):${L}_{P}\left(u\right):=max\{au,u\}$;
- (iv)
- Pinball loss with ‘ϵ zone’ ($\u03f5\ge 0$ and $a,b\le 0$):${L}_{PEZ}\left(u\right):=max\{0,u-\u03f5,au+b\}$;
- (v)
- Truncated pinball loss ($a\le 0$ and $b\ge 0$):${L}_{TP}\left(u\right):=max\{u,min\{au,b\}\}$.

#### 2.3. Fisher Consistency

**Theorem**

**1.**

#### 2.4. Related Work

## 3. Robust SVM

#### 3.1. Single Perturbation SVM

**Assumption**

**1.**

#### 3.2. Extreme Empirical Loss SVM

#### 3.3. Recommendations Related to the Use of the Two New Formulations

## 4. Numerical Experiments

- (i)
- (ii)
- (iii)

#### 4.1. Synthetic Data

#### 4.1.1. Synthetic Non-Contaminated Data

- SP-SVM: $\alpha \in {\mathcal{A}}_{SP}=\{0.50,0.51,\cdots ,0.60\}$;
- EEL-SVM: $\alpha \in {\mathcal{A}}_{EEL}=\{0,0.01,0.02\}$;
- $pin$-SVM: $\tau \in {\mathcal{T}}_{pin}=\{0.1,0.2,\cdots ,1\}$;
- $\overline{pin}$-SVM: $(\tau ,s)\in {\mathcal{T}}_{\overline{pin}}\times \mathcal{S}$, where ${\mathcal{T}}_{\overline{pin}}=\{0.25,0.5,0.75\}$ and $\mathcal{S}:=\{0.25,0.5,0.75,1\}$.

#### 4.1.2. Synthetic Contaminated Data

#### 4.2. Real Data Analysis

`awgn`with different signal noise ratios (SNR); perturbations were separately introduced 10 times for each dataset before training; and the average classification accuracy was reported so that the sampling error was alleviated.

- SP-SVM: $\alpha \in {\mathcal{A}}_{SP}=\{0.50,0.51,\cdots ,0.56,0.58,0.60\}$;
- EEL-SVM: $\alpha \in {\mathcal{A}}_{EEL}=\{0,0.05,0.10,\cdots ,0.3\}$;
- $pin$-SVM: $\tau \in {\mathcal{T}}_{pin}=\{0.1,0.2,\cdots ,0.8,1\}$;
- $\overline{pin}$-SVM$:s\in \{0.01,0.1,0.3,0.5,0.7,1\}$ and $\tau =0.5$;
- Ramp-KSVCR: $\u03f5\in \{0.1,0.2,0.3\},t\in \{1,3,5\},s=-1$.

#### 4.3. Interpretable Classifiers

- US mortgage lending data that are downloaded from the Home Mortgage Disclosure Act (HMDA) website3; specifically, we collected the 2020 data for two states, namely, Maine (ME) and Vermont (VT), with a focus on subordinated lien mortgages;

## 5. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Notation

$\mathbf{x}$ | vector of features |

y | label |

$\mathbf{w}$, b | coefficients of the hyper-plane |

$\varphi (\xb7)$ | kernel function |

C | penalty hyper-parameter for misclassifications |

$\tau $ | additional hyper-parameter for the $pin$-SVM and the $\overline{pin}$-SVM |

s | additional hyper-parameter for the $\overline{pin}$-SVM and the Ramp-KSVCR |

r | percentage of noisy points in the synthetic data |

$\alpha $ | additional hyper-parameter for the SP-SVM and the EEL-SVM |

## Appendix A

#### Appendix A.1. Proof of Theorem 1

#### Appendix A.2. Explicit Solution for (8)

#### Appendix A.3. Explicit Solution for (11)

## Notes

1 | See https://archive.ics.uci.edu/ml/index.php (accessed on 5 January 2021). |

2 | See https://www.csie.ntu.edu.tw/~cjlin/libsvmtools/datasets/ (accessed on 5 January 2021). |

3 | See https://ffiec.cfpb.gov (accessed on 15 December 2021). |

4 | See https://www.kaggle.com/datasets/buntyshah/auto-insurance-claims-data (accessed on 5 January 2021). |

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**Figure 1.**Classification boundaries for five SVM classifiers if DU is induced by (

**a**) normal distribution, (

**b**) Student’s t distribution with 5 degrees of freedom and (

**c**) Student’s t distribution with 1 degree of freedom.

**Figure 2.**ME loan amount deviations from the population mean based on full data and sub-populations with low/high minority and low/high income percentages.

**Figure 3.**VT loan amount deviations from the population mean based on full data and sub-populations with low/high minority and low/high income percentages.

**Table 1.**Distance (13) between various SVM classifiers and Bayes classifier for non-contaminated synthetic data. Lowest distance along each row in bold.

C-SVM | pin-SVM | $\overline{\mathit{pin}}$-SVM | SP-SVM | EEL-SVM | |
---|---|---|---|---|---|

$N=50$ | 0.5185 | 0.3531 | 0.4956 | 0.3968 | 0.5222 |

$N=100$ | 0.3763 | 0.1132 | 0.1809 | 0.1014 | 0.3477 |

$N=200$ | 0.0185 | 0.0337 | 0.0349 | 0.2166 | 0.0397 |

**Table 2.**Distance (13) between various SVM classifiers and Bayes classifier for contaminated synthetic data. Lowest distance along each row in bold.

Normal distribution | ||||||

$r=0.05$ | $r=0.10$ | |||||

$N=50$ | $N=100$ | $N=200$ | $N=50$ | $N=100$ | $N=200$ | |

C-SVM | 1.3443 | 0.8397 | 0.6623 | 1.9649 | 1.1455 | 0.8675 |

$pin$-SVM | 0.0537 | 0.0704 | 0.1880 | 0.5350 | 0.2955 | 0.2996 |

$\overline{pin}$-SVM | 0.6378 | 0.3073 | 0.3984 | 1.2398 | 0.7334 | 0.4611 |

SP-SVM | 0.3574 | 0.2305 | 0.2994 | 1.0827 | 0.5938 | 0.2813 |

EEL-SVM | 1.3432 | 0.8431 | 0.6788 | 1.8921 | 1.1682 | 0.8785 |

Student’s t distribution (5 degrees of freedom) | ||||||

$r=0.05$ | $r=0.10$ | |||||

$N=50$ | $N=100$ | $N=200$ | $N=50$ | $N=100$ | $N=200$ | |

C-SVM | 1.5685 | 1.1520 | 0.6795 | 2.4753 | 1.4983 | 0.9863 |

$pin$-SVM | 0.8546 | 0.2966 | 0.1710 | 2.1801 | 0.6861 | 0.3322 |

$\overline{pin}$-SVM | 1.2229 | 0.5929 | 0.3410 | 1.6040 | 0.9491 | 0.6492 |

SP-SVM | 0.7485 | 0.7405 | 0.3754 | 1.3408 | 0.8801 | 0.4539 |

EEL-SVM | 1.5901 | 1.1560 | 0.6895 | 2.4864 | 1.5077 | 1.0025 |

Student’s t distribution (1 degree of freedom) | ||||||

$r=0.05$ | $r=0.10$ | |||||

$N=50$ | $N=100$ | $N=200$ | $N=50$ | $N=100$ | $N=200$ | |

C-SVM | 1.7930 | 1.8189 | 2.0358 | 3.1206 | 2.6941 | 3.2549 |

$pin$-SVM | 0.8546 | 1.1983 | 1.6466 | 2.1801 | 2.0853 | 2.1472 |

$\overline{pin}$-SVM | 1.2002 | 1.4458 | 1.3479 | 2.5982 | 1.9612 | 2.4647 |

SP-SVM | 0.4277 | 1.2384 | 1.2675 | 2.1628 | 1.8261 | 1.8463 |

EEL-SVM | 1.7669 | 1.8558 | 2.0280 | 3.0685 | 2.6788 | 3.2757 |

**Table 3.**Computational time ratios of SVM classifiers compared to C-SVM classifier for contaminated synthetic data. Lowest computational time ratio along each row in bold.

Normal distribution | ||||||

$r=0.05$ | $r=0.10$ | |||||

$N=50$ | $N=100$ | $N=200$ | $N=50$ | $N=100$ | $N=200$ | |

$pin$-SVM | 4.2695 | 15.5138 | 33.9715 | 4.4855 | 19.2942 | 39.8558 |

$\overline{pin}$-SVM | 2.5515 | 30.1658 | 22.9012 | 3.2739 | 36.6373 | 32.7994 |

SP-SVM | 2.8101 | 7.6634 | 15.3068 | 2.9372 | 9.2248 | 16.7844 |

EEL-SVM | 2.0301 | 6.6263 | 14.0950 | 2.1713 | 8.6631 | 16.1476 |

Student’s t distribution (5 degrees of freedom) | ||||||

$r=0.05$ | $r=0.10$ | |||||

$N=50$ | $N=100$ | $N=200$ | $N=50$ | $N=100$ | $N=200$ | |

$pin$-SVM | 3.1421 | 12.6893 | 41.3835 | 4.4825 | 17.0174 | 44.2939 |

$\overline{pin}$-SVM | 2.4289 | 42.1007 | 26.7222 | 4.0141 | 61.5340 | 27.6636 |

SP-SVM | 2.2494 | 6.1618 | 19.6716 | 2.9885 | 7.7810 | 17.8432 |

EEL-SVM | 2.3466 | 5.6473 | 17.2680 | 3.1776 | 7.7076 | 17.6688 |

$r=0.05$ | $r=0.10$ | |||||

$N=50$ | $N=100$ | $N=200$ | $N=50$ | $N=100$ | $N=200$ | |

$pin$-SVM | 3.2366 | 11.4355 | 34.5926 | 4.4800 | 15.0441 | 41.0832 |

$\overline{pin}$-SVM | 3.4272 | 25.4353 | 42.4561 | 5.6555 | 45.9922 | 62.0633 |

SP-SVM | 2.4380 | 5.9118 | 15.5805 | 3.0326 | 7.3598 | 17.8881 |

EEL-SVM | 2.1472 | 5.1893 | 14.5320 | 3.0242 | 7.1204 | 16.3805 |

Data | Number of | Training | Testing | |
---|---|---|---|---|

Features | Sample Size | Sample Size | ||

(I) | Fourclass | 2 | 580 | 282 |

(II) | Diabetes | 8 | 520 | 248 |

(III) | Breast cancer | 10 | 460 | 223 |

(IV) | Australian | 14 | 470 | 220 |

(V) | Statlog | 13 | 180 | 90 |

(VI) | Customer | 7 | 300 | 140 |

(VII) | Trial | 17 | 520 | 252 |

(VIII) | Banknote | 4 | 920 | 452 |

(IX) | A3a | 123 | 3185 | 29,376 |

(X) | Mushroom | 112 | 2000 | 6124 |

**Table 5.**Classification accuracy (in %) of all SVM classifiers across all datasets. Highest accuracies along each row in bold. Each row signifies the original data (reported as “NA”, i.e., no contamination) or their contaminated variants (with SNR values 1, 5, 10).

Data | SNR | C- | pin- | $\overline{\mathit{pin}}$- | SP- | EEL- | Ramp- |
---|---|---|---|---|---|---|---|

SVM | SVM | SVM | SVM | SVM | KSVCR | ||

(I) | NA | 99.29% | 99.29% | 99.65% | 99.65% | 99.65% | 92.20% |

10 | 99.65% | 99.65% | 99.65% | 99.61% | 99.75% | 91.95% | |

5 | 99.65% | 99.65% | 99.65% | 99.54% | 99.72% | 91.67% | |

1 | 99.54% | 99.61% | 99.61% | 99.50% | 99.65% | 91.84% | |

(II) | NA | 77.02% | 79.84% | 79.84% | 80.24% | 78.63% | 80.24% |

10 | 76.98% | 76.49% | 78.10% | 79.64% | 77.26% | 77.10% | |

5 | 76.69% | 76.57% | 77.54% | 78.02% | 76.45% | 79.48% | |

1 | 76.49% | 77.70% | 76.90% | 77.66% | 74.96% | 79.44% | |

(III) | NA | 93.72% | 93.27% | 94.17% | 94.62% | 93.72% | 95.96% |

10 | 93.90% | 94.75% | 93.86% | 93.32% | 94.44% | 95.29% | |

5 | 93.86% | 94.57% | 94.17% | 94.13% | 94.08% | 94.80% | |

1 | 93.81% | 93.86% | 93.86% | 93.86% | 94.04% | 95.11% | |

(IV) | NA | 88.64% | 88.18% | 89.55% | 88.18% | 89.09% | 89.55% |

10 | 85.82% | 85.23% | 85.77% | 85.45% | 85.32% | 83.82% | |

5 | 80.68% | 80.50% | 82.41% | 80.59% | 78.45% | 77.73% | |

1 | 76.59% | 75.86% | 77.86% | 76.14% | 76.23% | 75.77% | |

(V) | NA | 82.22% | 82.22% | 82.22% | 83.33% | 78.89% | 80.00% |

10 | 80.22% | 80.56% | 77.56% | 80.22% | 82.44% | 81.00% | |

5 | 80.00% | 79.33% | 79.89% | 79.33% | 81.33% | 77.00% | |

1 | 78.67% | 78.22% | 80.00% | 78.44% | 76.89% | 80.22% | |

(VI) | NA | 92.14% | 91.43% | 90.71% | 92.14% | 92.86% | 91.89% |

10 | 92.86% | 92.50% | 92.50% | 92.71% | 93.21% | 89.93% | |

5 | 92.93% | 92.86% | 91.43% | 93.07% | 93.07% | 91.36% | |

1 | 92.57% | 92.93% | 91.57% | 92.57% | 92.86% | 90.79% | |

(VII) | NA | 100.00% | 100.00% | 100.00% | 100.00% | 100.00% | 100.00% |

10 | 99.56% | 99.80% | 99.33% | 99.76% | 99.60% | 99.48% | |

5 | 94.72% | 94.60% | 94.44% | 94.84% | 94.52% | 93.97% | |

1 | 88.13% | 88.13% | 85.99% | 88.29% | 88.21% | 86.98% | |

(VIII) | NA | 100.00% | 100.00% | 100.00% | 100.00% | 100.00% | 100.00% |

10 | 99.73% | 99.71% | 99.80% | 99.76% | 99.89% | 99.85% | |

5 | 99.38% | 99.18% | 99.05% | 99.25% | 99.36% | 99.05% | |

1 | 97.94% | 97.92% | 97.61% | 97.94% | 97.83% | 96.75% | |

(IX) | NA | 82.81% | 83.20% | 83.36% | 83.67% | 81.71% | 84.04% |

10 | 80.50% | 81.05% | 80.57% | 81.12% | 81.15% | 81.13% | |

5 | 78.57% | 78.56% | 77.94% | 78.35% | 78.26% | 78.00% | |

1 | 76.34% | 76.74% | 75.94% | 76.01% | 76.02% | 76.25% | |

(X) | NA | 99.87% | 99.87% | 99.87% | 99.87% | 99.87% | 99.87% |

10 | 98.37% | 98.84% | 99.38% | 98.31% | 98.29% | 99.03% | |

5 | 93.02% | 93.79% | 94.72% | 93.05% | 92.92% | 93.77% | |

1 | 85.36% | 85.63% | 85.82% | 85.46% | 85.47% | 85.55% | |

Average rank | 3.25 | 3.18 | 3.18 | 2.93 | 3.05 | 3.53 |

C- | pin- | $\overline{\mathit{pin}}$- | SP- | EEL- | Ramp- | |
---|---|---|---|---|---|---|

SVM | SVM | SVM | SVM | SVM | KSVCR | |

C-SVM | 0.8349 | 0.7345 | 0.9688 | 0.3605 | 0.0506 | |

$pin$-SVM | 0.1651 | 0.5105 | 0.8881 | 0.1815 | 0.0268 | |

$\overline{pin}$-SVM | 0.2655 | 0.4895 | 0.8008 | 0.2668 | 0.0320 | |

SP-SVM | 0.0312 | 0.1119 | 0.1992 | 0.0714 | 0.0123 | |

EEL-SVM | 0.6395 | 0.8185 | 0.7332 | 0.9286 | 0.0721 | |

Ramp-KSVCR | 0.9494 | 0.9732 | 0.9680 | 0.9877 | 0.9279 |

**Table 7.**Summaries of HMDA datasets and their accuracy levels. Highest accuracy along each row in bold.

Dataset | Total | Testing | C-SVM | SP-SVM | EEL-SVM |
---|---|---|---|---|---|

Sample Size | Sample Size | ||||

ME | 4226 | 1396 | 70.77% | 70.77% | 70.13% |

VT | 1948 | 648 | 90.74% | 91.67% | 90.89% |

CI | 1000 | 330 | 83.33% | 83.33% | 83.93% |

**Table 8.**Kolmogorov–Smirnov distances in loan amount distributions of males versus females (MvsF), males versus joint (MvsJ) and females versus joint (FvsJ). Largest distances along each row corresponding to training and testing data in bold.

Training Data | Testing Data | ||||||
---|---|---|---|---|---|---|---|

MvsF | MvsJ | FvsJ | MvsF | MvsJ | FvsJ | ||

ME | $Y\in \{-1,1\}$ | 0.0853 | 0.0477 | 0.1330 | 0.0969 | 0.1248 | 0.2096 |

$Y=-1$ | 0.0540 | 0.0727 | 0.1215 | 0.1203 | 0.1122 | 0.2313 | |

$Y=1$ | 0.1091 | 0.0381 | 0.1472 | 0.0889 | 0.1266 | 0.1986 | |

VT | $Y\in \{-1,1\}$ | 0.0923 | 0.1009 | 0.1841 | 0.1355 | 0.1399 | 0.2000 |

$Y=-1$ | 0.0866 | 0.0449 | 0.1208 | 0.3095 | 0.1429 | 0.1667 | |

$Y=1$ | 0.1258 | 0.1579 | 0.2515 | 0.1235 | 0.1575 | 0.2108 |

**Table 9.**Probability of denied loan for the three classification methods. Closest value (per row) to ‘true’ probability in bold.

C-SVM | SP-SVM | EEL-SVM | True | |
---|---|---|---|---|

$Pr(Y=-1)$ | 2.3148% | 1.0802% | 5.5556% | 7.8704% |

$Pr(Y=-1|\mathrm{Female})$ | 3.9683% | 1.5873% | 6.3492% | 9.5238% |

$Pr(Y=-1|\mathrm{Male})$ | 7.8125% | 3.9063% | 10.1563% | 16.4062% |

$Pr(Y=-1|\mathrm{Joint})$ | 0.0000% | 0.0000% | 3.8071% | 4.5685% |

$Pr(Y=-1|\mathrm{Low}\phantom{\rule{4.pt}{0ex}}\mathrm{Income})$ | 2.9316% | 0.9772% | 5.5375% | 8.0495% |

$Pr(Y=-1|\mathrm{High}\phantom{\rule{4.pt}{0ex}}\mathrm{Income})$ | 1.7595% | 1.1730% | 5.5718% | 7.6923% |

$Pr(Y=-1|\mathrm{Low}\phantom{\rule{4.pt}{0ex}}\mathrm{Minority})$ | 1.8576% | 0.9290% | 5.5728% | 9.7720% |

$Pr(Y=-1|\mathrm{High}\phantom{\rule{4.pt}{0ex}}\mathrm{Minority})$ | 2.7692% | 1.2308% | 5.5385% | 6.1584% |

$CDD$ | −25.5944% | −25.2601% | −8.3263% | −11.3792% |

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## Share and Cite

**MDPI and ACS Style**

Asimit, A.V.; Kyriakou, I.; Santoni, S.; Scognamiglio, S.; Zhu, R.
Robust Classification via Support Vector Machines. *Risks* **2022**, *10*, 154.
https://doi.org/10.3390/risks10080154

**AMA Style**

Asimit AV, Kyriakou I, Santoni S, Scognamiglio S, Zhu R.
Robust Classification via Support Vector Machines. *Risks*. 2022; 10(8):154.
https://doi.org/10.3390/risks10080154

**Chicago/Turabian Style**

Asimit, Alexandru V., Ioannis Kyriakou, Simone Santoni, Salvatore Scognamiglio, and Rui Zhu.
2022. "Robust Classification via Support Vector Machines" *Risks* 10, no. 8: 154.
https://doi.org/10.3390/risks10080154