FairSplit: Decomposing the Embedding Space for Fair Classification
Authors: Andrii Shkabrii, Anna Beer, Ylli Sadikaj, Claudia Plant
Organizations: Faculty of Computer Science, University of Vienna, Vienna, Austria · Doctoral School Computer Science, University of Vienna, Vienna, Austria · Webster Vienna Private University, Vienna, Austria · Research Network Data Science, University of Vienna, Vienna, Austria
Fair classification faces a twofold challenge: maintaining high classification accuracy while limiting reliance on sensitive information. Our method FairSplit achieves this by learning a decomposition of the embedding space into a Fair Space used for prediction and a Sensitive Space that isolates sensitive information. During training, the model must simultaneously learn the classification task and separate fair from sensitive information. The Sensitive Space gives sensitive attributes a destination, so the Fair Space can be used for prediction without relying on them. This decomposition is guided via a learnable mask and the Hilbert-Schmidt Independence Criterion (HSIC) as a measure of statistical dependence. We provide theoretical guarantees showing that the demographic parity gap is upper-bounded by both (i) the learned dimensionality of the Fair Space and (ii) the HSIC between the Fair Space and sensitive attributes. FairSplit effectively handles multiclass tasks and non-binary sensitive attributes, overcoming key limitations of existing fair classification methods. Experiments on established fairness benchmarks show that FairSplit achieves competitive accuracy-fairness trade-offs. Code is available at https://github.com/shkabrii/fairsplit.
Figures & tables
Figure 1: Embedding spaces on CelebA-S , where the task is to predict whether a person is smiling and gender is the sensitive attribute, i.e., the protected characteristic on which predictions should not depend, visualized with t-SNE and colored by the target (left) and the sensitive attribute (right). Each side shows three panels: the Vanilla embedding, trained by empirical risk minimization (ERM) without fairness constraints, and the Fair Space and Sensitive Space of FairSplit . In the Vanilla (ERM) embedding, smiling / not smiling correlates with female / male . FairSplit instead classifies from the Fair Space, which is guided towards independence from the sensitive attribute.
Method
Multiclass
G>2
DRAlign [ 37 ]
✗
✗
DiffMCDP [ 28 ]
✗
✗
PRemover [ 31 ]
✗
✗
MMD-B-Fair [ 11 ]
✗
✗
FairMixup [ 8 ]
♠
✗
AdvDebias [ 55 ]
♠
♠
Table 1: Method capabilities. Support for multiclass targets and non-binary sensitive attributes ( G>2 ). ✓ supported, ✗ binary only, ♠ indicates that it should be possible, but has not been tested in the respective papers.
Figure 2: Overview of FairSplit . An encoder maps each input to an embedding z , and a learned binary mask b splits it into a Fair Space zf=b⊙z and a Sensitive Space zs=(1−b)⊙z . Only the Fair Space is passed to the classifier. During training, HSIC regularization reduces the dependence between the Fair Space and the sensitive attribute and increases it in the Sensitive Space, which gives sensitive information a destination outside the classifier’s input. The mask is updated in closed form by comparing, per dimension, how strongly the embedding varies within classes and within sensitive groups, so that the size of the Fair Space is learned end-to-end rather than fixed in advance.
CelebA-A ( Δπ=40.0 )
Adult ( Δπ=19.8 )
Δ DP ↓
ABCC ↓
MCDP ↓
Δ EOpp ↓
Δ EOdd ↓
AP ↑
Δ DP ↓
ABCC ↓
MCDP ↓
Δ EOpp ↓
Δ EOdd ↓
AP ↑
ERM
52.8 ± 0.8
35.9 ± 0.8
53.4 ± 0.8
36.8 ± 1.5
75.0 ± 2.1
84.5 ± 0.4
19.4 ± 0.6
18.7 ± 0.2
36.6 ± 0.6
9.3 ± 2.0
18.2 ± 2.3
76.7 ± 0.5
AdvDebias
45.9 ± 3.4
31.2 ± 1.6
46.8 ± 3.8
26.0 ± 4.4
57.1 ± 7.6
84.9 ± 0.7
6.1 ± 5.7
7.7 ± 5.0
16.7 ± 11.1
21.8 ± 9.5
25.3 ± 8.9
72.8 ± 2.6
FairMixup
37.2 ± 1.1
19.5 ± 0.9
39.5 ± 1.2
19.5 ± 2.4
39.8 ± 2.5
83.1 ± 0.1
1.1 ± 0.6
4.6 ± 0.5
17.2 ± 2.4
32.6 ± 2.8
36.0 ± 3.3
72.9 ± 0.4
DRAlign
31.7 ± 4.9
15.4 ± 2.9
33.6 ± 4.3
12.4 ± 4.8
29.9 ± 9.1
80.3 ± 1.2
0.7 ± 0.5
6.1 ± 0.3
19.8 ± 0.9
34.2 ± 1.0
37.8 ± 1.0
73.6 ± 0.4
HSIC (Y,S)
23.9 ± 1.4
13.6 ± 0.5
27.0 ± 1.7
3.3 ± 1.5
11.6 ± 3.0
81.3 ± 0.5
1.9 ± 0.5
4.1 ± 0.3
11.2 ± 1.1
27.8 ± 1.7
31.1 ± 2.0
73.0 ± 0.4
Table 2: Performance on binary settings. CelebA-A and Adult are binary classification tasks with gender as the sensitive attribute, and we measure utility by average precision (AP). For each method, we consider all configurations that reach at least 95% of ERM’s validation AP and report the one with the lowest Δ DP, without further optimizing AP. Bold and underline mark the best and second-best results. Results are averaged over 5 seeds. Δπ is the base-rate gap between groups ( Section 4.1 ).
UTKFace-AR ( Δπ=25.9 )
COMPAS ( Δπ=26.6 )
Δ DP ↓
ABCC ↓
MCDP ↓
Δ EOpp ↓
Δ EOdd ↓
F1 ↑
Δ DP ↓
ABCC ↓
MCDP ↓
Δ EOpp ↓
Δ EOdd ↓
AP ↑
ERM
27.9 ± 1.4
27.1 ± 2.6
36.2 ± 2.4
29.1 ± 5.2
35.9 ± 3.0
73.8 ± 0.7
35.2 ± 3.3
17.9 ± 1.8
58.8 ± 4.5
24.9 ± 4.7
55.4 ± 8.7
69.1 ± 2.4
AdvDebias
28.7 ± 3.0
21.1 ± 1.9
32.9 ± 2.5
32.7 ± 7.8
38.5 ± 7.9
74.7 ± 0.8
26.4 ± 7.7
13.9 ± 3.1
47.8 ± 10.9
29.5 ± 5.7
44.7 ± 12.4
68.4 ± 2.1
HSIC (Y,S)
11.9 ± 1.3
12.0 ± 0.8
20.4 ± 2.4
18.1 ± 2.7
24.1 ± 3.5
70.3 ± 1.1
20.5 ± 3.1
6.7 ± 0.7
35.8 ± 4.6
28.3 ± 9.2
36.1 ± 9.8
67.9 ± 2.3
FairSplit
12.6 ± 1.1
9.1 ± 1.0
18.7 ± 1.2
14.9 ± 4.6
22.1 ± 4.9
70.4 ± 1.5
16.8 ± 3.6
1.2 ± 0.4
32.9 ± 6.2
23.8 ± 6.7
31.5 ± 7.6
66.8 ± 2.4
Table 3: Performance on multiclass and intersectional settings. UTKFace-AR has a 3-class age target and 3 racial groups, with F1 as utility. COMPAS has a binary target and an intersectional gender × age attribute, with AP as utility. Methods restricted to binary targets or sensitive attributes are omitted ( Table 1 ). As in Table 2 , all configurations achieve at least 95% of ERM’s validation utility, and bold and underline mark the best and second-best results. Results are averaged over 5 seeds. With more than two groups, Δπ is the largest base-rate gap over pairs of groups and classes.
Backbone
Method
AP ↑
Δ DP ↓
ABCC ↓
Δ EOpp ↓
Δ EOdd ↓
df/m
ViT-B/16
ERM
88.9 ± 0.2
23.0 ± 0.5
17.2 ± 0.1
22.3 ± 1.1
33.5 ± 1.1
768 / 768
(fine-tuned)
FairSplit
86.4 ± 0.3
9.5 ± 1.6
5.8 ± 1.4
8.1 ± 2.6
9.5 ± 1.9
552 / 768
CLIP ViT-L/14
ERM
96.5 ± 0.1
21.4 ± 0.9
20.0 ± 0.6
4.9 ± 0.5
23.6 ± 2.4
256 / 256
(frozen)
FairSplit
93.1 ± 0.4
9.1 ± 1.3
6.4 ± 0.7
0.8 ± 0.3
4.8 ± 1.6
174 / 256
CLIP ResNet-50
ERM
96.6 ± 0.1
23.4 ± 0.4
21.0 ± 0.3
6.1 ± 0.5
29.4 ± 1.5
256 / 256
(frozen)
FairSplit
93.6 ± 0.4
12.2 ± 2.4
7.9 ± 1.5
0.9 ± 0.5
8.5 ± 5.0
168 / 256
Table 4: FairSplit on large pretrained models (5 seeds, gender as the sensitive attribute). ERM uses the same architecture as FairSplit . ViT-B/16 is fine-tuned end-to-end on FairFace to predict whether a person is at least 30 years old. For the frozen backbones, the Fair Space is learned for predicting high cheekbones on CelebA. A new linear head is then trained on this fixed Fair Space to predict a different attribute, age. df/m is the size of the Fair Space relative to the full embedding.
Figure 3: Trade-off between performance and fairness (higher performance is better) across four datasets. Each row is one dataset (CelebA-A, Adult, UTKFace-AR, COMPAS) with its own y-axis (AP for binary classification, F1 for the multiclass UTKFace-AR), and each column is one fairness metric. ERM ( ★ ★ ) is a single point. The ideal model (✚) lies in the top-left corner of every panel. FairSplit matches or outperforms competing methods across most settings and achieves the best ABCC on every dataset in Tables 2 and 3 .
Figure 8
Appendix figures & tables20 assets
Supplementary material from the paper’s appendix.
Appendix
Symbol
Description
Data and Labels
x,xi
Input features
y,yi
Class label ∈{1,…,K}
y,yi
One-hot encoding of class label
s,si
Sensitive attribute ∈{1,…,G}
s,si
One-hot encoding of sensitive attribute
Appendix
Table 6: Summary of notation used in this paper.
Name
Modality
Target
Samples
K
G
Target Dist. (%)
Sens. Dist. (%)
Adult
Tabular
Income
45,222
2
2
75.2 / 24.8
32.5 / 67.5
COMPAS
Tabular
2-year recidivism
5,278
2
4
53.0 / 47.0
58.8 / 21.7 / 14.6 / 5.0
CelebA-A
Image
Attractive
202,599
2
2
48.7 / 51.3
58.3 / 41.7
UTKFace-AR
Image
Age (3-class)
23,705
3
3
32.2 / 44.1 / 23.7
42.5 / 19.1 / 38.4
FairFace
Image
Age ≥30
97,698
2
2
54.2 / 45.8
53.0 / 47.0
Appendix
Table 7: Dataset statistics. K and G denote the number of target classes and sensitive groups, respectively. Target Dist. shows the class distribution, and Sens. Dist. shows the sensitive group distribution. FairFace is used in Section 4.3 , where each seed samples 60,000 training and 10,000 test images. Groups are ordered Female/Male (Adult, CelebA-A), White/Black/Other (UTKFace-AR), Male/Female (FairFace), and Male-Older/Male-Young/Female-Older/Female-Young (COMPAS).
Metric
Name
Binary
Multiclass / Multi-group
AP ↑
Average Precision
∑r(Rr−Rr−1)Pr
–
F1 ↑
F1 Score
2PR/(P+R)
K1∑k2PkRk/(Pk+Rk)
Δ DP ↓
Demographic Parity
∣ΔP(y^=2)∣
maxk,g,g′∣ΔPg,g′(y^=k)∣
Δ EOpp ↓
Equal Opportunity
∣ΔTPR∣
maxk,g,g′∣ΔTPRg,g′(k)∣
Δ EOdd ↓
Equalized Odds
∣ΔTPR∣+∣ΔFPR∣
maxk,g,g′(∣ΔTPRg,g′(k)∣+∣ΔFPRg,g′(k)∣)
ABCC ↓
Area Between CDF Curves
∫01∣ΔF(t)∣dt
maxk,g,g′∫01∣ΔFg,g′(k)(t)∣dt
Appendix
Table 8: Performance and fairness metrics. y^∈{1,…,K} is the predicted class, y the true class, and s∈{1,…,G} the sensitive attribute, with class 2 as the positive class for binary targets. P and R are precision and recall, and Pr,Rr their values at the r -th threshold. With Pg(⋅)=P(⋅∣s=g) , TPRg(k)=Pg(y^=k∣y=k) , FPRg(k)=Pg(y^=k∣y=k) , and Fg(k) the CDF of the predicted probability of class k in group g , Δ(⋅)g,g′ denotes the difference between groups g and g′ . Binary metrics drop the indices.
Hyperparameter
Benchmarks
ViT-B/16
Frozen backbones
Optimizer
Adam
Adam
Adam
Learning rate
10−3
10−4
3×10−4
StepLR step size ( γ=0.1 )
50
100
200
Training steps
150
300
600
Warm-up steps (fairness methods)
50
50
50
Batch size
1024 (tabular), 128 (image)
128
1024
Appendix
Table 9: Training hyperparameters shared by all methods. The mask moving-average rate applies to FairSplit only.
Method
Parameter
Adult
CelebA-A
UTKFace-AR
COMPAS
AdvDebias
λ
2
0.2
0.35
1
FairMixup
λ
0.8
1
—
—
DRAlign
λ
0.5
0.2
—
—
β
0.01
0.01
—
—
HSIC (Y,S)
λ
70
50
300
200
σ
1
5
5
1
Appendix
Table 10: Selected best hyperparameters per baseline method and dataset. Among the configurations whose validation AP (validation F1 for UTKFace-AR) reaches at least 95% of ERM’s, we select the one with the lowest mean validation Δ DP (see Section 4 ). Cells marked “—” indicate the method is not applicable to the dataset’s setting.
Parameter
Adult
CelebA-A
UTKFace-AR
COMPAS
ViT-B/16
Frozen
λf
0.025
0.01
0.1
0.01
0.01
0.01
λs
0.05
0.025
0.05
0.2
0.025
0.05
ρf
0.4
0.5
1.3
0.3
1.5
1.0
ρs
0.005
0.5
0.1
0.005
2.0
0.5
σ
2
5
5
2
5
5
Appendix
Table 11: Selected FairSplit hyperparameters per dataset, with the same selection rule as in Table 10 . σ is fixed at the modality default (tabular vs. image). For the frozen backbones, the hyperparameters are selected on CLIP ViT-L/14 and reused unchanged on the other two backbones.
Figure 5: Bound of Theorem 3.2 and the demographic parity gap of the logits (top), and the demographic parity gap of the predicted classes (bottom), as the HSIC regularization strength ρf varies. Shaded regions are the mean ± std over 5 seeds, and the dashed line marks the reported ρf .
UTKFace-AR
CelebA-A
Encoder
m
F1 ↑
Δ DP ↓
Red.
df
AP ↑
Δ DP ↓
Red.
df
ResNet-18
512
73.8 → 70.4
27.9 → 12.6
55%
233
84.5 → 80.4
52.8 → 18.3
65%
334
ResNet-34
512
73.8 → 69.1
28.9 → 12.8
56%
243
84.5 → 78.1
53.2 → 18.4
65%
297
ResNet-50
2048
74.8 → 72.2
27.9 → 14.9
47%
1003
85.9 → 80.6
52.4 → 25.5
51%
1230
EfficientNetV2-S
1280
73.2 → 73.8
29.1 → 15.9
45%
1161
84.9 → 82.6
50.6 → 31.2
38%
777
MobileNetV4-S
960
70.4 → 67.5
28.5 → 13.9
51%
584
84.0 → 74.7
51.6 → 19.3
63%
419
Appendix
Table 12: Encoder robustness on UTKFace-AR (3-class age, 3 racial groups) and CelebA-A. FairSplit uses the same hyperparameters for every encoder, and only the encoder is changed, with no per-architecture retuning. Since the hyperparameters are not tuned per encoder, the 95% utility gate of the main tables is not applied here. Utility and Δ DP columns show ERM → FairSplit , and Red. is the relative reduction of Δ DP. Mean over 5 seeds.
m=32
m=64
m=128
m=256
m=512
Δ DP ↓
ERM
19.8 ± 0.8
19.4 ± 0.6
18.5 ± 0.5
18.2 ± 0.3
17.5 ± 0.2
FairSplit
3.3 ± 1.2
1.0 ± 0.9
3.2 ± 2.6
1.3 ± 0.5
2.5 ± 2.4
AP ↑
ERM
76.8 ± 0.5
76.7 ± 0.5
77.0 ± 0.5
77.3 ± 0.4
77.5 ± 0.4
FairSplit
71.5 ± 0.8
72.9 ± 0.9
71.6 ± 1.2
70.8 ± 1.2
74.3 ± 1.0
ABCC ↓
ERM
18.8 ± 0.3
18.7 ± 0.2
18.5 ± 0.2
18.6 ± 0.2
18.5 ± 0.1
FairSplit
0.9 ± 0.2
2.4 ± 0.7
1.5 ± 0.3
0.8 ± 0.2
4.3 ± 0.5
Appendix
Table 13: Embedding-size ablation on Adult (5 seeds).
Backbone
Task
Method
AP ↑
Δ DP ↓
ABCC ↓
Δ EOpp ↓
Δ EOdd ↓
df/m
CLIP ViT-L/14
A
ERM
93.7 ± 0.0
27.8 ± 0.4
26.0 ± 0.2
15.8 ± 0.4
18.3 ± 1.0
256 / 256
FairSplit
89.3 ± 0.4
13.1 ± 1.2
9.0 ± 0.6
1.2 ± 0.7
12.1 ± 1.0
174 / 256
B
ERM
96.5 ± 0.1
21.4 ± 0.9
20.0 ± 0.6
4.9 ± 0.5
23.6 ± 2.4
256 / 256
FairSplit
93.1 ± 0.4
9.1 ± 1.3
6.4 ± 0.7
0.8 ± 0.3
4.8 ± 1.6
174 / 256
CLIP ResNet-50
A
ERM
94.5 ± 0.0
27.1 ± 0.2
26.3 ± 0.1
14.0 ± 0.5
15.7 ± 0.5
256 / 256
FairSplit
90.8 ± 0.7
15.4 ± 1.3
10.5 ± 0.9
1.5 ± 1.2
10.9 ± 0.6
168 / 256
Appendix
Table 14: Transfer of the Fair Space from Task A ( High_Cheekbones ) to Task B ( Young ) on three frozen backbones (5 seeds). Task B uses a fresh linear head on the frozen representation.
Linear probe
MLP probe
Backbone
Task
Δπ
z
zf
zs
zf
df/m
AP ↑
Δ DP ↓
CLIP ViT-L/14
Smiling
14
98.1 → 99.0
50.4 ± 0.3
99.0 ± 0.0
95.9 ± 1.1
167 / 256
97.9 → 97.0
15.2 → 8.5
Young
24
98.5 → 99.0
61.0 ± 6.9
99.0 ± 0.0
89.0 ± 1.9
91 / 256
97.6 → 95.5
23.2 → 4.9
High_Cheekbones
25
98.4 → 99.0
54.7 ± 2.1
99.0 ± 0.0
97.2 ± 0.9
174 / 256
93.7 → 89.3
27.8 → 13.1
Wavy_Hair
31
98.8 → 99.0
64.4 ± 4.5
99.0 ± 0.0
94.8 ± 1.1
115 / 256
85.6 → 83.3
35.4 → 26.6
ResNet-18
Attractive
40
85.7 → 94.8
88.2 ± 2.4
94.8 ± 0.2
94.5 ± 0.3
334 / 512
84.5 → 80.4
52.8 → 18.3
Appendix
Table 15: Balanced accuracy (%, 5 seeds) of probes predicting the sensitive attribute from the full embedding z , the Fair Space zf , and the Sensitive Space zs of the same trained model, where random guessing scores 50% . CLIP ViT-L/14 is frozen (Appendix G.3 ), and ResNet-18 is the CelebA-A model of Table 2 . For z , AP, and Δ DP, arrows show ERM → FairSplit with the same architecture. Lower probe accuracy means that s is harder to predict.
Adult
UTKFace-AR
COMPAS
Method
Linear
MLP
Linear
MLP
Linear
MLP
ERM
71.7 ± 0.8
79.1 ± 0.9
69.4 ± 0.8
68.2 ± 0.9
45.1 ± 0.9
48.5 ± 0.6
AdvDebias
80.0 ± 2.5
80.1 ± 1.9
70.0 ± 1.2
66.4 ± 1.8
48.7 ± 0.5
47.8 ± 0.7
FairMixup
75.9 ± 0.6
80.6 ± 0.7
—
—
—
—
DRAlign
72.1 ± 2.4
79.7 ± 0.9
—
—
—
—
HSIC (Y,S)
72.0 ± 1.6
78.6 ± 2.5
64.8 ± 1.8
62.2 ± 1.8
44.1 ± 0.7
47.5 ± 0.6
Appendix
Table 16: Balanced accuracy (%, 5 seeds) of probes predicting the sensitive attribute from the input of each method’s classifier, the Fair Space zf for FairSplit and the full embedding z for the baselines. Random guessing scores 50% on Adult, 33% on UTKFace-AR, and 25% on COMPAS. The probed models are those of Tables 2 and 3 . Lower probe accuracy means that s is harder to predict, and the lowest value per column is in bold. A dash (—) marks baselines that support only binary targets and binary sensitive attributes, so they cannot be applied to UTKFace-AR (three age classes, three racial groups) or COMPAS (four groups).
Method
Head
AP/F1 ↑
Δ DP ↓
ABCC ↓
MCDP ↓
Δ EOpp ↓
Δ EOdd ↓
df
CelebA-A
ERM
Linear
84.5 ± 0.4
52.8 ± 0.8
35.9 ± 0.8
53.4 ± 0.8
36.8 ± 1.5
75.0 ± 2.1
512
FairSplit
Linear
80.4 ± 1.1
18.3 ± 1.1
9.4 ± 1.3
19.6 ± 1.8
2.6 ± 0.3
5.2 ± 1.7
334
ERM
Non-linear
85.3 ± 0.4
53.7 ± 1.9
37.5 ± 1.2
54.0 ± 2.0
37.3 ± 3.0
75.6 ± 4.9
512
FairSplit
Non-linear
81.7 ± 0.5
23.1 ± 3.6
13.5 ± 1.5
24.3 ± 3.3
4.4 ± 2.2
11.0 ± 5.0
369
UTKFace-AR
Appendix
Table 17: FairSplit with linear and non-linear classification heads on zf (5 seeds). The utility metric is AP on CelebA-A and F1 on UTKFace-AR. For each dataset and head, bold marks the better of ERM and FairSplit .
Δ DP ↓
ABCC ↓
MCDP ↓
Δ EOpp ↓
Δ EOdd ↓
AP ↑
ERM
54.0 ± 0.4
34.2 ± 0.8
54.4 ± 0.5
42.4 ± 1.8
79.3 ± 1.6
84.1 ± 0.3
AdvDebias
52.3 ± 2.7
35.1 ± 1.1
53.6 ± 1.9
35.4 ± 8.8
73.3 ± 7.0
85.4 ± 0.3
HSIC (Y,S)
19.4 ± 1.9
10.2 ± 1.1
23.1 ± 2.8
2.3 ± 1.2
7.2 ± 4.0
80.7 ± 0.8
FairSplit
18.2 ± 1.6
9.7 ± 1.0
22.8 ± 2.8
1.8 ± 1.3
7.1 ± 3.5
80.7 ± 0.7
Appendix
Table 18: CelebA-A with an imposed 95/5 group imbalance in the training data (5 seeds). Each method keeps its hyperparameters from Tables 10 and 11 . Best in bold , second-best underlined .
Δπ
AP ↑
Δ DP ↓
AP cost ↓
ERM
FairSplit
ERM
FairSplit
5
97.5 ± 0.0
96.6 ± 0.3
4.0 ± 0.2
0.7 ± 0.5
0.9
15
97.5 ± 0.0
96.4 ± 0.2
14.4 ± 0.5
6.6 ± 0.7
1.1
25
97.6 ± 0.1
95.3 ± 0.3
24.7 ± 0.3
12.6 ± 1.1
2.3
35
97.8 ± 0.1
93.9 ± 0.3
35.4 ± 0.3
17.4 ± 1.6
3.9
45
98.0 ± 0.1
91.3 ± 0.4
45.7 ± 0.3
20.3 ± 1.9
6.7
Appendix
Table 19: Varying only the base-rate gap Δπ (frozen CLIP ViT-L/14, Smiling , 5 seeds).
Figure 6: Hyperparameter ablation on CelebA-A (5 seeds). We vary λf,λs,ρf,ρs (left to right) while fixing the other three at their defaults (dashed line). Rows: AP, parity gaps ( Δ DP, ABCC, MCDP), error-rate gaps ( Δ EOpp, Δ EOdd), and the learned Fair Space dimensionality df . Fairness improvements are driven primarily by ρf , and the method is robust to λf . Increasing λs and ρs shrinks df .
Variant
AP/F1 ↑
Δ DP ↓
ABCC ↓
MCDP ↓
Δ EOpp ↓
Δ EOdd ↓
df
CelebA-A
ERM
84.5 ± 0.4
52.8 ± 0.8
35.9 ± 0.8
53.4 ± 0.8
36.8 ± 1.5
75.0 ± 2.1
512
No Sensitive Space
80.7 ± 0.7
26.6 ± 1.4
16.1 ± 1.0
26.8 ± 1.0
5.5 ± 0.9
18.4 ± 2.8
512
Unregularized ( λs=ρs=0 )
80.1 ± 0.8
22.2 ± 2.4
12.4 ± 1.7
22.8 ± 1.6
1.9 ± 0.9
9.1 ± 2.4
391
Unregularized +λs
79.9 ± 1.2
20.1 ± 2.7
10.7 ± 1.8
21.6 ± 2.0
2.4 ± 1.3
8.1 ± 2.5
366
Unregularized +ρs
80.9 ± 0.6
21.4 ± 1.8
11.2 ± 1.0
22.4 ± 1.2
1.4 ± 2.2
6.8 ± 2.8
371
Appendix
Table 20: Component ablation on CelebA-A and UTKFace-AR (5 seeds). The utility metric is AP on CelebA-A and F1 on UTKFace-AR. The best value per metric within each dataset is in bold . † One of the five seeds collapses to a constant predictor.
AP ↑
Δ DP ↓
ABCC ↓
ERM
84.5 ± 0.4
52.8 ± 0.8
35.9 ± 0.8
Kernel family ( σ=5 )
Gaussian RBF
80.4 ± 1.1
18.3 ± 1.1
0 9.4 ± 1.3
Matérn 3/2
80.5 ± 1.5
18.6 ± 2.9
10.0 ± 2.1
Laplacian
78.1 ± 2.5
35.4 ± 6.6
16.4 ± 4.2
Gaussian RBF bandwidth
Appendix
Table 21: HSIC kernel family and Gaussian RBF bandwidth on CelebA-A (5 seeds). Only the kernel or σ changes, and ERM is reported as a reference without fairness regularization.
Method
Adult (s)
vs. ERM
CelebA-A (s)
vs. ERM
ERM
11.4
1.00 ×
27.7
1.00 ×
AdvDebias
10.6
0.93 ×
28.6
1.03 ×
FairMixup
12.4
1.09 ×
35.8
1.29 ×
DRAlign
13.8
1.21 ×
48.6
1.76 ×
HSIC (Y,S)
11.9
1.05 ×
28.4
1.03 ×
DiffMCDP
16.3
1.43 ×
28.8
1.04 ×
Appendix
Table 22: Wall-clock training time per run (in seconds), averaged over 5 seeds on a single NVIDIA A100. “vs. ERM” is the multiplicative cost relative to the ERM baseline.
Condition
AP ↑
Δ DP ↓
ABCC ↓
ERM
84.5 ± 0.4
52.8 ± 0.8
35.9 ± 0.8
FairSplit (clean)
80.4 ± 1.1
18.3 ± 1.1
0 9.4 ± 1.3
Noisy 10%
82.8 ± 0.7
28.3 ± 2.6
16.2 ± 1.5
Noisy 30%
85.5 ± 0.2
53.1 ± 0.9
31.5 ± 0.9
Noisy 50%
85.6 ± 0.3
54.0 ± 0.4
32.2 ± 0.6
Missing 10%
81.0 ± 0.9
21.8 ± 1.9
11.6 ± 1.2
Appendix
Table 23: Robustness on CelebA-A under noisy and missing sensitive attributes (5 seeds). “Noisy p ” flips each sensitive label with probability p . “Missing p ” drops the sensitive label for a fraction p of samples (these samples contribute only to the cross-entropy loss). “Shuffled” randomly permutes all sensitive labels. ERM and clean FairSplit are reported as references.
Ensuring fair and equitable treatment across diverse groups, particularly in multi-class classification tasks, poses a significant challenge due to the persistent biases inherent in machine learning models. Most existing bias mitigation techniques are tailored to binary settings, and the presence of multi-dimensional outputs and complex fairness mechanisms makes their extension to multi-class scenarios neither straightforward nor effective. In this paper, we investigate two fundamental, unresolved challenges in fair classification: (i) characterizing the optimal accuracy-fairness frontier in multi-class settings, and (ii) designing practical algorithms that attain this optimum in different training phases. To tackle these challenges, we first specify an analytically tractable probabilistic formulation of the optimal classifier under fairness constraints. Building upon this, we propose two attribute-blind algorithms to enforce fairness requirements in practice: an in-processing approach for fairness intervention during training via the reduction approach, and a post-processing approach for fine-tuning output probabilities with plug-in estimation. Theoretical analysis reveals that both methods converge to the optimal accuracy-fairness Pareto frontier. Experiments conducted on multiple datasets demonstrate the superior performance of our methods in balancing accuracy and fairness.
Li Zhang, Yuyuan Li, XiaoHua Feng +3
College of Computer Science and Technology, Zhejiang University · School of Communication Engineering, Hangzhou Dianzi University
Ensuring fairness is essential as machine learning increasingly informs consequential decisions. However, many fairness-aware methods focus on the outputs of individual predictors, without directly controlling sensitive information retained in the underlying representations. We propose Deep Fair Learning (DFL), which combines distance covariance regularization with predictive loss to jointly learn representations and downstream predictors, promoting fairness at both levels while preserving task-relevant information. Its marginal and class-conditional formulations target independence and separation, respectively. Under suitable regularity conditions, we establish non-asymptotic joint excess-risk rates and convergence of the learned representation up to natural invariances. We further derive fairness-inheritance bounds linking representation-level dependence to downstream disparities over suitable predictor classes, extending fairness guarantees beyond the jointly trained predictor. Experiments on tabular, text, and image benchmarks show that DFL achieves lower fairness gaps than competing methods in many evaluated settings while maintaining competitive predictive accuracy, with fairness gains largely preserved after downstream retraining.
Enze Shi, Yiqun Xiao, Linglong Kong +1
Department of Mathematical and Statistical Science, University of Alberta.
We revisit the foundations of fairness and its interplay with utility and efficiency in settings where the training data contain richer labels, such as individual types, rankings, or risk estimates, rather than just binary outcomes. In this context, we propose algorithms that achieve stronger notions of evidence-based fairness than are possible in standard supervised learning. Our methods support classification and ranking techniques that preserve accurate subpopulation classification rates, as suggested by the underlying data distributions, across a broad class of classification rules and downstream applications. Furthermore, our predictors enable loss minimization, whether aimed at maximizing utility or in the service of fair treatment. Complementing our algorithmic contributions, we present impossibility results demonstrating that simultaneously achieving accurate classification rates and optimal loss minimization is, in some cases, computationally infeasible. Unlike prior impossibility results, our notions are not inherently in conflict and are simultaneously satisfied by the Bayes-optimal predictor. Furthermore, we show that each notion can be satisfied individually via efficient learning. Our separation thus stems from the computational hardness of learning a sufficiently good approximation of the Bayes-optimal predictor. These computational impossibilities present a choice between two natural and attainable notions of accuracy that could both be motivated by fairness.