On Group Sufficiency Under Label Bias
Authors: Haoran Zhang
Olawale Elijah Salaudeen
Marzyeh Ghassemi
Venue: NeurIPS 2025 poster ()
Keywords: algorithmic fairness
label noise
label bias
group sufficiency
Primary area: social_and_economic_aspects_of_machine_learning
TL;DR: We characterize the problem of achieving group sufficiency under label bias, and introduce a regularizer that restores fairness without sacrificing accuracy.

Abstract: Real-world classification datasets often contain label bias, where observed labels differ systematically from the true labels at different rates for different demographic groups. Machine learning models trained on such datasets may then exhibit disparities in predictive performance across these groups. In this work, we characterize the problem of learning fair classification models with respect to the underlying ground truth labels when given only label biased data. We focus on the particular fairness definition of group sufficiency, i.e. equal calibration of risk scores across protected groups. We theoretically show that enforcing fairness with respect to label biased data necessarily results in group miscalibration with respect to the true labels. We then propose a regularizer which minimizes an upper bound on the sufficiency gap by penalizing a conditional mutual information term. Across experiments on eight tabular, image, and text datasets with both synthetic and real label noise, we find that our method reduces the sufficiency gap by up to 7.2% with no significant decrease in overall accuracy.

On Group Sufficiency Under Label Bias
Haoran Zhang Olawale Salaudeen Marzyeh Ghassemi Massachusetts Institute of Technology {haoranz, olawale, mggassemm/mmit.co edu
Abstract
Real-world classification datasets often contain label bias, where observed labels differ systematically from the true labels at different rates for different demographic groups. Machine learning models trained on such datasets may then exhibit dispar- ities in predictive performance across these groups. In this work, we characterize the problem of learning fair classification models with respect to the underlying ground truth labels when given only label biased data. We focus on the particular fairness definition of group sufficiency, i.e. equal calibration of risk scores across protected groups. We theoretically show that enforcing fairness with respect to label biased data necessarily results in group miscalibration with respect the true labels. We then propose a regularizer which minimizes an upper bound on the sufficiency gap by penalizing conditional mutual information term. Across experiments on eight tabular, image, and text datasets with both synthetic and real label noise, we find that our method reduces the sufficiency gap by up to 7.2% with no significant decrease in overall accuracy.
Introduction
Machine learning systems are increasingly being deployed in safety critical settings, from parole recommendations to medical triage and financial lending 2 B.. However, prior work has shown that such systems often exhibit biases the form performance discrepancies between demographic There exists range of work which propose algorithmic approaches to correct in-processing
groups 4
5 6
these disparities, spanning from pre-processing 7 processing [54 However, these methods rarely consider the source of these biases. Prior work has shown that naively enforcing these fairness definitions can have unintended side effects, leading to worse performance for all [17 18 [9 20] and exacerbating other types of biases (2 22 [6]
[10] 12 [3], to post-
In this work, we take a different approach and tackle fair classification from data-centric perspec- tive (23), focusing on datasets that contain significant differences in label eπrors between protected groups, also known as label bias (24. For example, in the medical diagnosis setting. women with heart disease are underdiagnosed at higher rates than men with heart disease, and are instead more likely be misdiagnosed with mental health conditions 255. Similarly, Black patients are less likely to be prescribed opioid painkillers than White patients in emergency departments, even after adjusting for level of pain (26). Finally, when screening resumes for job applications, applicants with African American names are much less likely to receive an interview than those with White names 2
When downstream models are trained on these label-biased datasets (i.e. to diagnose, prescribe, or hire), the model may learn to predict noisy labels and propagate disparities in the historical data (28 29), leading the observed performance disparities. When the training dataset contains label bias, our goal is to learn fair and performant model with respect to the true unobserved labels.
Learning under label noise is well-studied problem 30 31 32 33 34], where methods aim to learn the best model with respect to the true labels, without any consideration for per-group performance (i.e. without considering label bias). Though there has been limited work looking at fairness under
39th Conference on Neural Information Processing Systems (NeurIPS 2025).

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(a)
(b)
Underfying
Subgroup Callbration
Men
Women
Healthy
O
Sick
Fraciion Positive
Marginal
T
705
0.5
Miediagnoses
T
0
Documentation Errors
0.5
0.5/
Calbbraton Error Cr (f)
+
Sufficiency Gap suf
Predicted Probability
Observed Nolay Labels
Figure 1: Our Problem Setup. (a) Observed labels are corrupted from unobserved true labels at different rates for each gender, which may be caused by misdiagnoses [25] or noisy automated processes used to extract labels [42). (b) Training a model on these noisy labels and evaluating calibration
with respect to labels results in different calibration curves for each group.
Specifically, each group's 's calibration curve differs from the marginal curve. We propose a method to achieve group sufficiency (zero sufficiency gap for all groups) under these conditions.
label bias
35 36
37
these works all focus on the fairness definitions of demographic parity and
equal odds 115
Here, we study the problem of learning classification models under label bias which satisfy a different fairness property group sufficiency, which corresponds to equal calibration of the risk score across groups 21338] A group sufficient score can then be post-processed to achieve perfect calibration for all groups via group-agnosti post-processing (21). Importantly, group sufficiency is incompatible with demographic parity and equal odds on the risk score under different group base rates, so not possible to achieve all fairness definitions simultaneously [22]. Here, focus on group sufficiency, as prior work have shown that it is requirement to prevent discrimination or maximize utility in many contexts 39 40 41 For example, Loi and Heitz (40) argue, from moral philosophy perspective, that risk scores being miscalibrated for particular demographic group may constitute discrimination against that group.
start by theoretically characterizing how class-attribute conditioned label noise distorts per-group calibration and derive necessary and sufficient conditions for group insufficiency under label bias (Section 4.. This demonstrates that naive models trained on noisy labeled data are necessarily unfair with respect to the true underlying labels, and motivates the need for specialized methods. Building on this theory, we propose a regularizer term, CMI-REG, based on penalizing conditional mutual information (Section 5). CMI-REG We evaluate CM with 7 base losses on 8 datasets spanning tabular, text, and image domains, finding that effectively reduce the sufficiency gap while maintaining, and often improving, overall performance over the base losses. To the best of our knowledge, our work is the first to study this problem setting and propose method achieve this fairness definition.
be added to any existing off-the-shelf noise-robust loss.
Contributions. Our work makes the following contribution
1. We theoretically establish necessary and sufficient conditions for which label bias causes lack of group sufficiency under class-attribute conditioned noise.
2. We propose method to achieve sufficiency under label bias, assuming atts-bet conditioned noise. We theoretically prove that our method minimizes an upper bound on the sufficiency gap. Our method a regularizer term that can be flexibly added to any existing loss-based noisy label learning method.
3. We empirically evaluate our method on 8 datasets with both synthetic and real-world noise, showing that it achieves sufficiency with respect to the true labels without significant decrease in accuracy over the baselines.
'Code: http:///ttttttooooorrrrrtttttttt

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2 Related Work
2.1 Learning Under Label Noise
There has been long line of work on the problem of learning classification models under label noise. Under uniform label noise, symmetric loss functions such as mean absolute error (MAE) are theoretically noise robust [45] This then motivated line of work which adapts MAE have better optimization properties and empirical performance, including improved MAE [44), symmetric cross entropy 45). generalized cross entropy (GCE) 30), and generalized Jensen-Shanner divergence [46).
Under class conditioned noise (where the probability of observing noisy label depends only on the true label), one direction focuses estimating the class transition matrix. When the class transition matrix is known, a simple weighting approach is sufficient 48). Methods to estimate the transition matrix have been proposed, given small dataset with clean labels 44 (50,, anchor points for each
class 52), or by examining the transition matrix an ERM model trained on noisy labels [53] 54]
Liu and Guo [3 proposes peer loss which, in the binary case, only requires knowledge of the sum of the off-diagonal transition matrix elements.
Under instance dependent label noise, there are theoretical guarantees in general. Methods that have done well empirically in this area include those based on training dynamics, e.g. area under the margin [55]. DataMap (56). first-time k-epoch learning [57], early regularization (58). as well as methods based on training multiple simultancous models such as Co-teaching [59 and DivideMix (32]
However, given that all methods in this category are agnostic of the group attribute and do not explicitly enforce fairness condition, they do not explicitly satisfy any faimess properties, and thus do not succeed in our problem setup as we will later show empirically.
2.2 Sufficiency as a Fairness Definition
Sufficiency of the risk score is commonly used group faimess definition in the classification setting, which corresponds to equal calibration curves for all groups It is a slightly weaker definition than calibration by group (i.e. all groups being perfectly calibrated) (2]] Prior work has shown that under different per-group prevalences of the label, it is impossible to simultaneously satisfy sufficiency and probabilistic equal odds 22 60). Liu al. [8] show that ERM models inherently prefer sufficiency, and that intervening to achieve probabilistic equal odds necessarily worsens this condition. Empirically, this has been observed in images [9 61 and tabular data 622 Raghavan (39 discuss the implications of this impossibility, making a case that the choice of fairesss metric is highly context dependent. Similarly, Loi and Heitz [40] discuss, from philosophical view, when group calibration is a required condition to avoid discrimination.
In order to achieve sufficiency, prior methods have been proposed based on mutual information 63 and multi-objective optimization (64 65). One related concept is multicalibration [66]. which seeks achieve approximate calibration for all computationally identifiable subgroups. However, multicalibration has been found to be far less sample efficient than standard post-hoc calibration methods [67] Both multicalibration (when applied provided attributes) and standard post-hoc calibration methods require access to the group attribute during inference, which our method will not assume. Regardless, no prior work has studied the setting of group sufficiency under label noise, which is the focus of this work.
2.3 Fairness Under Label Bias
Prior works on fairness under label bias have only focused on demographic parity and equal odds. Wick et al. [68] use simulated data to show that enforcing statistical parity on label-biased data may already achieve the same fairness definition on clean data, with increased accuracy. Wang et al. (35] study the problem of equal odds under class-attribute conditioned label noise, and propose two solutions based on an alternate constraint using a surrogate loss, and a group-weighted peer loss (1 Zhang et al. 37 learn fair representations under class-conditional noise for the same definitions. Wu et al. (36) study fairness under instance-dependen noise but targets counterfactual fairness [69], which has been shown to have similar drawbacks as equal odds [70]. None of these works consider group sufficiency, nor do they provide guarantees for calibration error under noise.
3

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Finally, Mhasawade et al. [24] propose various candidate causal graphs to model the label bias scenario and derive cases when sufficiency is satisfied without further intervention, but do not propose any algorithmic solutions in the remaining cases. Under their framework, we study the Y + Y; and Y; J. A setting, under which they provide no guarantees. In contrast, we provide a theoretical characterization of this setting and propose method to achieve group sufficiency under label bias.
3 Preliminaries
Let X X' denote input feature, Y E 21, K} the true (clean) label, and Y (1,2...., K} the observed (noisy) label. Additionally, let G E represent a group attribute. We have a dataset D {a Gis yt B0) with (1 9b yı, ūi) p. where the true labels ys are unobserved. and we only have access to the noisy labels Ø:-
In practice, we want classifiers whose predicted probabilities accurately reflect true real-world outcomes, both overall and consistently across different demographic or group attributes. To quantify how well a model achieves these goals, we introduce the standard definition of group sufficiency, which quantifies whether predictions provide equal information about outcomes, regardless of group membership.
Definition 3.1 (Group Sufficiency). A multiclass classifier : - A is group sufficient with respect to label Y and group it satisfies:
Y T G f(X).
This condition implies that the prediction f(X) captures information about Y that renders the group attribute G irrelevant given the model's output.
Definition 3.2 (Group Sufficiency Gap). The per-group sufficiency gap, as well as the overall sufficiency gap for a model f, are defined as (38]:
Sufg(f)
Σ K
E
E
BE
f(X)al E[1y= f(X)x, - 9/1l1
Suf(f) EEf
Similarly, let Suf denote the sufficiency gap with respect to Y. The group sufficiency gap quantifies the extent to which model's predictions provide unequal information about the true label Y across different groups G; measures deviations from the ideal group sufficiency condition Y G ((X). This gap is zero when f(X) is group sufficient-Lemma. A.1
To formally analyze the effect of label noise and group membership on calibration and sufficiency. we introduce explicit assumptions about the Sata-generating process. These assumptions clarify the relationship between noisy and true labels, conditional independence, and the structure of the noise across different groups.
Assumption 3.1 (Class-Attribun Conditioned Noise). The noisy label Ý is conditionally independent of the input given the true label and group attribute G:
Y X (Y,G).
Definition 3.3 Stroup-Condition Transition Matrix). For each group g e G. define row-stochastic noise transition matrix T E [0, [0,1]xx K where
TV = P(Ý
Assumption 3.2 (Invertible Group-Conditionall Transition Matrix). T is invertible for all groups (g E 9).
These assumptions are standard in the literature on learning under label noise 48 71 Importantly, the set of singular matrices has Lebesgue measure zero within the space of all row-stochastic matrices (72), and strictly diagonally dominant matrices-whie naturally arise when noise rates are sufficiently small-are are always invertible (73].
Definition 3.4 (Label Bias). Let T E [0 where Tú, P( j Y =
Egeg P(G g). If 3g such that T # T. then say that the observed labels Y are label biased relative to Y.

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Main Result and Problem Statement
First, we characterize the necessary and sufficient conditions for which we would expect non-zero sufficiency gap.
Theorem 4.1 (Closed-Fooe Form Decomposition of the Sufficiency Gap). Let 8 = f(X) A K-1 be the model score, Tg (6) = p(Ý = [(X) = s, G = the group-conditional noisy label posterior, and क (s)) P(V = [(X) = 8) the marginal counterpart. Then under Assumptions 3.1 and 3.2 for any p 1,
1/2
Sff =
- K
(TT) (TT) + A((
(1)
posterior shift
noise matrix mismatch
where ( #f(()) and 4T = -
Corollary 4.1.1 (Necessary Condition for Sufficiency Gap). Under Assumptions 3.1 and 3.2 suppose that f satisfies group sufficiency with respect to noisy labels (i.e. Y I G f(X)). Then, if 3g g such that Suf, (f) > 0, it must be that T + T.
Thus, when f is fair with respect to the noisy labels (which ERM models trained noisy data have a tendency to be, as the sufficiency gap is upper bounded by its excess risk [38]), label bias is necessary for sufficient gap. The contrapositive of this corollary also shows that under uniform label noise which is group agnostic, a group sufficient classifier with respect to noisy labels (implying Suf(f) = 0) will also be group sufficient to the true labels. This implies that if we know apriori that the noise rates do not differ across groups (i.e. there is no label bias), enforcing group sufficiency with respect to the noisy labels is valid mitigation strategy. Thus, for the rest of this work, we will deal with the case of non-group-agnostic noise.
Remark 4.2 (Sufficient Condition for Sufficiency Gap). Theorem 4.1 implies that under label bias (i.e. T + 7) and a non-trivial classifier (f(X) + 0), we have that Suf, ( > 0, so long as the 0) and ((((X)) is not in the null space
model is group-sufficient for the noisy labels (A+(((X)) of ATo. Specifically, even though T and T are different, the marginal क does not have sufficient mass in the directions that they are different, then the sufficiency gap is unaffected since the inverses of I project स to the same output and cancel out, resulting in Suf, (f) = O.
We define and provide similar bounds for the per-group calibration error in Appendix A Further, characterize the behavior Suf, 9 under perturbations to the transition matrix 79, which may happen when 9 is estimated from observational data. All proofs can be found in Appendix A
Problem Statement. Our goal is to learn a risk score fo : X' + K- satisfying Definition 3.1 Specifically, given loss function (((x, v), RR. We would additionally like to achieve:
min E (x,Y))xxxx [((XX)), fo)l #E0
s.l. YLG fo(X)
5
Methodology
To guide the design of our method, we derive an upper bound on the group sufficiency gap in terms of the conditional mutual information between the label and group, given the model's prediction. This result highlights that reducing the dependence between Y and G conditioned on f(X) sufficient to control the sufficiency gap:
Proposition 5.1 (Conditional Mutual Information Bound on Sufficiency Gap).
22 G S(X)) K V P(G g)G=g)
V2
Suf S
Suf(f) < K VI(Y;G f(X)).
This bound motivates our core objective: to minimize the sufficiency gap by minimizing the con- ditional mutual information I(Y; f(X)). However, since the true labels Y are unobserved in
5

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practice. directly estimating this quantity is non-trivial. In the following section, we describe how we approximate this objective using the noisy labels Y. Formal proofs are provided in Appendix A
propose differentiable regularizer that approximates and minimizes the conditional mutual information upper bound using the observed noisy labels.
Under Assumptions 3.1 and 3.2
min
Cfre Zt. Us) + XR(fo).
where : + V. A > O, and Epred is pointwise loss function, potentially selected for robustness label noise, e.g., Peer Loss [31 or GCE (30]
CMI Regularizer
We can decompose I(Y; G f(X)) into the difference of two entropy terms:
GYY; (V;GIS(X)) S(X)) H(Y (x))) - H( [x(X), G) 1E log.logoPP PYY fo(X))
log. fo(X), G)]
For a mini-batch B of size B, each entropy term can be estimated using a Monte Carlo estimator:
A (o(X),
1
Σ
log P(Y = = fo()) = G 9:))
B
LEB
1
Σ
log py (. fe(X) = fe(x:)) G gi)).
B
where (- (- fa(X) fa(x;), G 9.) [P =
f fo( G g.., P(Y K
[X( = Jo(x,), - 9:)], and (-)g, denotes taking the 74-th element of the K dimeensiional vector.
As we do not observe ys. we can apply the backward correction applied elementwise:
[48] to this estimator, where log is
YY [o((),
Σ (-T
log PY (-1 = fe(x.), 9:)), in
(4)
B
iBB
and similarly
1
AY fo(x))
Σ
(-T log Posp(( ly fe( = fo(x.)))
(5)
B
SE
Note that the population entropy H(Y S(X)) does not depend G. However, only observe Y. estimator must invert the noise process that generated Y. Under Assumption 3.1 the unbiased backward-correecion of requires the group-specific matrix T-11 for the i-th sample. Using the marginal matrix - would bias the estimator when there is label bias.
Heads for Entropy Estimation To estimate Py given only noisy data, parameterize two functions h&s : AK-1 + K and he K-1 x G + A each by single linear layer, which we then learn by using a standard cross entropy loss with the forward correction (48):
h = arg min
log (f((())))
(6)
B
YEB
h arg min
log
YEB
Thus, we have that
R(fo) = ((): = (-T log (o(F)())))) -(-T log (fo(ss)
(i))
B
9.
iEB
(8)
single linear layer is each used to parameterize h& and hgs as similar models are frequently learned for post-hoc calibration (74). In practice, hs and hy operate the logits of f instead of the output

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Algorithm 1: Single Step Update of CMI-REG
Input:
Batch ((x, Bi: }t)]
Per-group transition matrices {1' E [0, 1]] JoEG, A 0, 0.2 Learning rates a Qg > 0:
Current classifier and auxiliary heads: fo, hy
(2, 14);;
Output: Updated fe, heg hw [o(x:) Vi E ;
// logits
for j = to C' do
// update heads, freeze 0
aA
V
log (T (z))))
// Eq. (6)
B
BEB
hA
log ( 9i))
// Eq.
KEB
5 Epred B
Σ prea Ri, W:);
1EB
6 Compute Rn using Equation 8
Lpred AR
8 A 6- 0 Ox VoL:
return fo, hag Bap
probabilities. Given a batch B, we first update ha and hi using gradient descent on Equations (6) and (7) for C iterations, and then update fo using Equation (3) once. As the backward corrected entropy estimators (Equations and can produce entropies outside of [o, log K], we clip estimated entropies to this range. The full algorithm can be found in Algorithm [ When the two heads
are perfectly learned, R(fe) is an unbiased estimator of the conditional mutual information (see
Lemma A.5)
Directly Regularizing Sufficiency (SUF-REG) In parallel, consider direct penalty the sufficiency gap by plugging in estimates of से and fg into Theorem 4. Concretely, and f are estimated by two linear heads and trained in an alternating manner similar CMI-REG. We refer to this regularizer term as SUF-REG, and it can be combined with any frred similar to CMI-REG. We emphasize that SUF-REG is also novel contribution that has not been explored in prior work. This variant avoids
estimation but can be higher-variance. An algorithm outline for SUF-REG can be found in Appendix
6 Experiments
Datasets and Noise Types We evaluate our method using the 8 datasets shown in Table D which span tabular, image, and text modalities, and a range number of classes. For datasets with synthetic noise, experiment with two synthetic noise types: Group Uniform (where uniform noise is added only for one group), and Group Asymmetric (where class-specific noise is only added for one group). Additional details of how the per-group transition matrices constructed can be found in Appendix Datasets with real noise have label noise derived from multiple annotators (in the case of civilcomments and cifar10ns) from human annotators of noisy web data (for clothingim). Further details of dataset construction can be found Appendix B.I
B.2
Training We train MLP, ImageNet- ResNet-18 (75). and BERT-base [76] models for tabular, image, and text modalities respectively, optimized using the Adam optimizer [77]. All datasets are divided into 60%/20%%/20% training/validatioonnest splits. For synthetic noise, to mitigate the confounding effect of different marginal noise rates depending on group proportions, we balance the groups by subsampling the majority group during both training and testing. Hyperparameters were selected using a random hyperparameter search 78 with runs. For the full hyperparameter grid for each method, see Appendix B.3 Each hyperparameter setting was repeated three times with different random seeds, which affects the dataset split, model initialization, and random noise (in the case of synthetic noise).
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Table 1: Datasets used this paper. All attributes contain two groups. n. number of samples, K;
number of classes. Data processing details can be found in Appendix B.I
Dataset
Modality
Task
Attribute
K
Noise Type
adult 80
Tabular
Income 2 50k
Gender
32,561
Synthetic
1sac
Tabular
Student passes the har
Race
18,337
Synthetic
crime 82 83
Tabular
Binned rate of violent crime
Primary ethnicity
1,994
Synthetic
income 84
Tabular
Binned income
Race
1,445,699
Synthetic
gradea 85
Tabular
Student passes exam
Gender
856
Real
civilcomennts 86
Text
Comment is toxic
Contains identity
448,000
Real
clothingim 88
Image
Type of clothing
Contains face
1,072,409
14
Real
cifar10na [88 89
Image
Image classification
Image is grayscale
60,000
10
Real
To select the optimal hyperparameters, we experiment with three model selection strategies: (1) Noisy Brier: We select the model with the lowest overall Brier score on the noisy validation set. The
Brier score is chosen the selection criteria as it accounts for both performance and calibration. (2) Theory: We select the model with the smallest value from Theorem 4.1 on the noisy validation set, which requires knowledge of the transition matrices but not any clean labels, (3) Clean Brier: We select the model with the lowest overall Brier score on the clean validation set, which requires a validation set with clean labels. For our method (and baselines) which require transition matrices, we experiment with two settings: (1) Known: we use the true T9, (2) Estimated: we estimate T using Confident Learning [54]
Baselines
We consider the following off-the-shelf loss functions for ppred: Cross Entropy (CE),
Backward
48 Forward 48). Generalized Cross Entropy (GCE) [30, a-weighted peer loss
(Peer)
[31
Early Learning Regularization (ELR) (58), and Determinant Mutual Information
(DMI) 779
We compare our methods (CMI-REG and SUF-REG) against each of the baseline
losses. We also compare against the group-weighted peer loss for achieving equal odds from Wang et al. 335 ("GroupPeer"). Note that this method is only applicable when K = 2. We additionally compare against applying multicalibration ("MC") post-processing using Algorithm from Hébert- Johnson et al. [66]] (specifically. using the implementation from Hansen et al. [67]). using the single attribute to define C, and defining the calibration set to be a 40% subset of the original training set. This proc-scccssing done the scores of an ERM model trained on labels produced by per-group stochastic flipping according to the row-normalized (Tg)-1. Finally, as oracle, we train an ERM model on the clean training data.
Metrics All metrics shown evaluated the cleanly labeled test To evaluate overall performance, we use the one-vs-rest AUROC macro-averaged across classes. To evaluate fairness, compute the sufficiency gap Suf(f) in Definition 3.2 with p 1, by using binning estimator with 10 equally sized bins.
7 Results
We present results for the four tabular datasets with synthetic group asymmetric noise, clean Brier selection, and known transition matrices in the main paper. Additional experimental results for group uniform noise, real-world noise, other selection criteria, estimated transition matrices, and additional synthetic noise levels can be found in Appendix D
On Performance and Fairness In Table 2, we show the overall performance and sufficiency gap of SUF-REG and CMI-REG when added to seven different base losses. We find that in the large majority of cases, the addition of either regularizer results in lower sufficiency gap, with CMI-REG exhibiting marginally better fairness. However, the confidence intervals are generally large as calibration error estimation has high variance [90]. Regardless, the model which achieves the lowest sufficiency gap consistently regularized by CMI-REG. The base losses which function best with these regularization terms are GCE and Peer loss. Interestingly, in several cases, adding the regularizer term actually improves overall performance. Similar results for group uniform noise and real-world noise can be found in Appendix D When the estimated transition matrix is used instead (Appendix [D.5), we find that regularizers still generally improve fairness, though the performance improvements become limited.
8

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Table 2: Clean test-set AUROC and sufficiency gap (Suf) of our regularizers when added to off-the- shelf loss functions, for tabular datasets with synthetic group asymmetric noise and known transition matrices. Models are selected using the clean Brier score. The best method within each base loss is bolded, and the best model overall is also underlined.
incone
adult
lsac
crine
AUROCT
Sul
AUROCT
Suf
AUROCTT T
Suf 1
AUROCTT
Suf
CE
(0.000)
0.000
0,899 (0.000)
0.029 (0.003)
0.581 (0.049)
0.108 (0.003)
0.741 (0.025)
0,075 (0.01 ()
SUT-RGG
0.232 (0.000)
0.035 (0.007)
0.894 (0.000
0.029 (0.004)
0.712 (0022)
0.089 (0.007)
0.804 (0,021)
0.057 (8.015)
CE + CMI-REG
0.736 (0.00)
0.033 (0.001
0,903 (0.005)
0.016 (0.004)
0.708 (0.077)
0.072 (0.026)
0.748 (0.030)
0.065 (0.007)
Forward
0.801 (0.0022
0.038 (0.0022
0.857 (0.0011
0.024 (0.004)
0.4%6 (0.066
0.097 (0.001)
0.788 (0.011)
0.056 (0.010)
Forsard SuE-REG
0.812 (0.000)
0.059 (0.003)
0.898 (0.0000
0.032 (0.001)
0.690 (0.011)
0.091 (0.016)
0.789 (0.006)
0,053 (0.006)
Forward CMI-RRG
0.802 (0.0013
0.019 (0.001)
0.887 (0.001)
0.023 (0.066
0.664 (0.247)
0.075 (0.052)
0.797 (0,000)
0.052 (0.016)
Backward
0.794 (0.002)
0.042 (0.002)
0.857 (0.001)
0.021 (0.003)
0.473 (0.042)
0.100 (0.003)
0.711 (0.007)
(0.00)
Backward SUP-REG
0,800 (0.0022)
0J032 (0.003)
0,888 (0.0011
0.025 (0.004)
0.631 (0.028)
0.097 (0.017)
0.721 (0.023)
0.4773 (0.002)
Backward CMI-REG
0.797 (0.003)
0.036 (0.005)
0,858 (0.001)
0.018 (0.007)
0.452 (0.032)
0.091 (0.009)
0.739 (0.065)
0.066 (0.005)
DMI
(0.10(
0.077 (0.01))
(0.001)
(0.00%)
0,476 (0.184)
0.088 (0.000)
0.406 (0.033)
0.087 (0.033)
DMI SUP-REG
0.591 (0.111)
0.045 (0.01)
0.877 (0.003)
0.028 (0.006)
0,478 (0.167)
0.090 (0.000)
0.395 (0.033)
0.088 (0.033)
DMI CMI- REG
0..90(0.119)
0.018 (0.017)
0.555 (0.0022
0.020 (0.006)
0.478 (0.170)
0.090 (0.001)
0.484 (0.042)
0.107 (0.023)
Fi
0.731 (0:001)
0.040 (0.005)
0.898 (0.001)
0.030 (0.)
0 364 (0. 266)
0.061 (0.050)
0.662 (0.091)
0.092 (0.012)
SUF-REG
0,754
0.025 (0.005)
(0.005
0.022 (0.002)
(0.054)
(0.001)
(0.051)
0.054
ELR CMI-REG
0.738 (0.003)
0.033 (0.002)
0,600 (0.010)
0.019 (0.001)
0.319 (0.040)
0.089 (0.014)
0.664 (0.0770)
0,088 (0.061)
GCE
0.730
0.041 (0.004)
0.894 (0.002)
0.022 (0.001)
0.462 (0.028)
0.099 (0.002)
0.739 (0.025)
0.077 (0.014)
GCE SUP-RRG
0.748 (0:001)
0.041 (0.001)
0,884 (0.0055
0.027 (0.000)
0.590 (0.02)
0.0997 (0.0044)
0.771 (0.047)
0.061 (0.008)
GCE CMI-REG
0,735 (0.0000
0.036 (0.002)
0.894 (0.008)
0.022 (0.002)
0.257 (0.039)
0.050 (0.018)
0.771 (0.014)
0.062 (0.000)
Poer
0.230(0.022)
0.030 (0.001)
0.899 (0.000)
0.029 (0.004)
(0.063)
0.110 (0.003)
0.681 (0.015)
0.069 (0.003)
Peer SUF-REG
0.732 (0.002)
0.035 (0.005)
0,890 (0.0000
0.033 (0.003)
0.590 (0.032)
0.098 (0.001)
0.734 (0.032)
0.077 (0.000)
Poet CMI- REG
0.686 (0.002)
0.054 (0.003)
0 (0.0033
0.028 (0.002)
0.551 (0.058)
0.096 (0.006)
0.630 (0.055)
0.070 (0.021)
MCC
0,561 (0.019)
0.078 (0.010)
0.787 (0.0044
0.017 (0.001)
0.500 (0.000)
0.107 (0.007)
0 0.500 (0.000)
0.104 (0.012)
GroupReer
(0.001)
0.035 (0.004)
(0.010)
0.095 (0.023)
Oricle
0.854 (0.001)
0.003 (0.000) 0.919 (0.0007
0.013 (0.001)
0.838 (0,011)
0.018 (0.009)
0,825 (0.01 3) 0.046 (0:010)
We find that naively adapting a post-hoc calibration method (multicalibration) results in generally worse performance and limited faimnsss gains, especially 1sac and crime. We hypothesize this stems from the limited effectiveness of the noise-aware aabel-flippig procedure used to construct training labels, and MC's sample-inefficie nature [67] combined with the smaller, class-imbalanced datasets.
Pareto Fronts In Figure 2 we show the Pareto front of AUROC vs. sufficiency gap for two datasets. We find that CMI-REG and SUF-REG find that these two regularizers consistently achieve the best AUROC-sufficiency trade-off when comparing across all base losses (Appendix D.3).
the base losses in almost all cases. We also
On Model Selection In Appendix D.4, we examine the impact of varying the model selection criteria. As expected, selecting using clean Brier gives the best performance and lowest sufficiency gap. Selecting using the theoretical expression (Theorem (.1)) on the noisy validation set gives similarly low sufficiency gaps, but compromises significantly on performance. Selecting using the noisy Brier gives mediocre performance with slightly worse sufficiency. Across all selection criteria, selected models with regularization terms have lower sufficiency gaps than the base model in almost all cases.
8 Discussion
On Performance and Fairness Adding either CMI-REG or SUF-REG to seven noise-robust base losses rarely harmed AUROC and, in surprising number of cases, improved AUROC, by up to 25.4% in one instance. We attribute this to the fact that both regularizers bias optimization toward decision boundaries that are locally calibrated with respect to the true labels across groups, and we speculate such decision boundaries are also more likely to coincide with high performing solutions with respect to the clean data.
Comparing CMI-REG and SUF-REG Even though CMI-REG minimizes an upper bound while SUF-REG minimizes the direct sufficiency gap, we observe that CMI-REG performs better than SUF-REG. Across eight combinations of datasets and noise types, CMI-REG achieves the lowest sufficiency gap seven times, and SUF-REG once. We attribute this to optimization and finite sample effects, First, SUF-REG must backpropagate through inverted T inside of p-norm, which may be

---

Clean Sul
CE
Forward
Backward
DMI
OCE
Peer
0.075
0.050
0.6730.700.72
0.7
0.75
0.65
0.70
Clean Test AUROC
(a) crime
Sal
CF
Forward
Backward
DMI
FLR
6CF
Peer
0.10
0.05
Clean Test AUROC
Suf Suf-Reg
Ayre CMi-eeg
(b) lsac
Figure 2: Pareto fronts of AUROC versus Suf of CMI-REG, SUF-REG, and baseline loss for tabular datasets with synthetic group asymmetric noise and known transition matrices, A curve towards the bottom right of the plot desirable achieves the best fairess- accuraac trade-off. We observe that CMI-REG and SUF-REG ptreto-dominaee the base loss for most losses. Pareto curves for remaining datasets can be found in Appendix D.3
more numerically challenging than simply backpropagating through inverted T multiplied by the oog-likeihhod, leading to abnormal gradient behavior. Second, CMI-REG consists of two convex cross-entropy terms, which may lead smoother optimization landscape than the piecewise-ling linear absolute error terms (when = 1) inside SUF-REG.
Conclusion
In this work, we studied the problem of achieving group sufficiency under label bias. We have shown theoretically that under certain conditions, label bias is necessary and sufficient to induce sufficiency gap. To address this, we introduced CMI-REG and SUF-REC two drop-in regularizers that provably upper bound the sufficiency gap, and in practice, consistently reduce the sufficiency gap across eight datasets.
Limitations Our analysis rests on the attribbt conditioned noise model (Assumption 3.1 When noise is instead deeenddn [36] 9I 92], the regularizers SUF-REG and CMI-REC will become biased. Theoretically studying the effect of instance-dependens label bias is an area of future work. Next, our method assumes knowledge (or estimation) of the transition matrices T and this estimation problem is impossible in general without further information assumptions [93 52 These assumptions may be violated in real-world datasets, and applying our method using an imperfect empirically observed. Finally, measure
transition matrix will result in worse performance fairness with respect to a single protected attribute at a time. Studying label bias between intersections of (potentially many) attributes is another area of future work.
Acknowledgments and Disclosure of Funding
This research was supported in part by an Optum Labs Research Award, a National Science Foundation (NSF) 22-586 Faculty Early Career Development Award (#2339381), a Gordon & Betty Moore Foundation award, a Google Research Scholar award, and the A12050 Program at Schmidt Sciences.
10

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A Proof of Lemmas and Theorems
A.1 Definitions
Definition A.1 (Calibration Error). The marginal calibration emoor of multiclass classifier f : X -
A K-1 with respect to and p-norm (p 2 1) is defined as 990 94
1/2
CE())
Σ K
B(J(()}
B[ [(X)x)PP)
When K = 2 and 1, this is equivalent to the definition of Expected Calibration Error (ECE).
Definition A.2 (Per-Group Calibration Error). Let denote a particular group:
1/p
CF2(f)= Σ K
BIIJ(X)
-Bly== f(X),, G= G g|
CE(f) and (f) are defined similarly with respect to Y
A.2 (f) and Group Sufficiency
Lemma A.1. When K = 2, f is group sufficient O Vg E g: (uf =0
Proof. (=). Write by S: P(Y = 1 S) = S, P(Y = 0 5) =1 S.
fXX) (0. 1]. Because 2, the prediction is completely characterised
Fix any group g E G and any score value E [0, 1] with Pr(S s, G g) 0. Group sufficiency gives the conditional independence
P(Y = y S = a, G G= =) P(Y = 5
E 10,
Hence
8, = g] P(Y = 115= =) = P(Y = 1| E[1y=1111 S= s). Substituting this equality into the definition of the sufficiency gap (Definition 3.2 yields Suf, (f) 0.
SS 1
(=) for K = 2. Assume K = 2 and that (ff = 0 for every group g. Let s be any score value taken by f(X) and let E G be fixed. Write (s(s) = P(Y = 1 f(X) = s, = 9) and p(s) = P(Y = 1 f(X) = s). Because Suf, 0, the absolute difference in Definition 3.2 is
zero, so Po (s) = p(s) almost surely. For binary labels, P(Y = 0 = P(Y = 1 .), hence
P(Y 01 f(X) G - g) = (p)(s) = 1 p(s) P(Y 0 ((x) x),
so the conditional distributions of Y given f(X) P(Y y, G = s(X) 8) P(Y f(x) 8) P(G S(X) 8), i.e.
identical for every group. Therefore
Y LG f(X), establishing group sufficiency.
A.3 Bound on CE
Theorem A.2 (Lower Bound on CE2(f)). Let A = ((T)T)-1 and sg(X) =
K (1k=) B[yyE f(X);, G = g). Then, under Assumption 3.1
K Ells [ sl
1/pl
CE 2 (E,(f) -
K
k=1
17

---

For non-zero noise rate (e.g. (o..).Ta.o) * (0,0), 0)), this gives that Cf, > 0, where fg is the Bayes optimal classifier with respect to noisy labels. Thus, learning the best model solely on the noisy labels will give us an uncalibrated model with respect to the true labels.
Proof. Fix group 9 G and abbreviate all expectations and probabilities conditionally G (we omit the conditioning in the notation for clarity). For every E {l...., _K} define
(X)
E = {(X)x)
Tx(X) = E 1y {(
Under Assumptions
3.1 and 3.2
Bayes' rule gives
K
(x) =
Σ
A9 #x(X),
A =- =
((T)T))
t=1
Hence
f(X)x - rx( f(X)x (X) +
Σ (1k= - (x(X).
=xs((X)
By definition of (X) we have the decomposition
f(X): (X) (X) + 5
an (= J(X)x #x)
Next, introduce Lhe over the product space (X, k) (conditionally on G g):
(x)())))) 1/p
||0/|p
B
With this notation
1 K K
||allp (f2
||
Σ k=1
Bl 1s (()))))) 1/P
||a + s = CB,(f).
The L space (p 1) satisfies the Minkowski inequality:
||a 1 2 |||||p - ||s"
Substituting the three norms identified above yields
K Σ k=1
1/2
CE2 2 (E2 - ||00 = (E)
B [((((()))))
which is exactly the claimed lower bound.
A.4 Proof of Theorem 4.1
Closed-Form Decomposition of the Sufficiency Gap. Let = f(X) E A be the model score, 7g
P(Ý fXx s,G. G 9) the group-conditions noisy label posterior, and
7(s) = P(Ý - ((X) 3) the marginal counterpart. Let T E [0. K where Ti,; p(Ý
y - = Egeg TV, P(G g). Then under Assumptions 3.1 and 3.2 for any p 2
1/2
8
Sff
BB
+ ((17))T-11- (TT)- #((())
posterior
noise matrix mismatch
Proof. Let f(X) A K and set
(8) P(Y = ss G = =g, T(s) = P(Y = 18))
1
* (s) P( = - s, G g), #(s) = P( ()))
18

---

From Definition 3.1 we have
Suf = K [s)) =(s)]B | - gl-
(9)
Assumption 3.1 implies
mg(s) (T9) [T(s))
and #(s) T (s)
(10)
and Assumption 3.2 gives
Tg (s)
((T9)))-17((5) and T(s) (TT)-
(11)
Subtracting the two identities in (11)
(s) (s)) (())))))(((())))))))))
-(s)) (1(9))))--1
(12)
Substituting 12 into 19
yields
9l 1/P (13)
Suf
E
Equation
13
is exactly the closed-form decomposition stated Theorem 4.1
A.5 Errors in Estimation of T
Finally, estimating T from finite samples can be difficult. Proposition A.4 bounds the true sufficiency gap error with respect to the empirical sufficiency gap, mediated by estimating T, T from finite samples.
Lemma A.3 (Concentration of transition matrix estimates). Let T99 and I be the empirical frequency- count estimates of the true transition matrices 79 and T from ng and n i.i.d. samples, respectively. Assume each class appears at least ng/ (resp. n/ K) times. Then for any 5 E (0, 1), with probability at least 1 6,
T - T 12 S ng
T - T Sn
where
2K2 log(2K/0)
2K log(2K/0)
17
ng
Proof. Fix i [K] and let m, ng/ K be the number of samples with Y By a vector-valued Hoeffding bound,
P(II TV. 2 S 2cxp
m,e²
S 2exp
ngg
2K
Taking union bound over E [K] gives
P max I!T - T.llk 2 28) < 2K exp
ngg²
2K
Setting 6
2K log(2K/5)
ng ensures the RHS at most
Then using 1 - T3 S [ 1
/ K max; |/
T ll.. we have
2K log log(2K/5)
2K2 log(2K/5)
T 78 S VK.
ng
ng
The same argument applies to 1|T 1l12. completing the proof.
19

---

Proposition A.4. Suppose Assumptions 3.1 and 3.2 hold. Let T and T be estimates of T and T, respectively, such that with probability at least
| 1 S ng
and I|7 - 2 S
Assume the estimates are well-conditioned, i.c. aglly and an 1, where ag [I((T9)T)- and -I(TT)-- Define
ag
a²
C Then for some constant Md depending only on (from the sufficiency gap definition), with probability least 1 - 6,
and C 1 om)
ag
( - () - (M,CaMg + Cn)+ where
(x))+ (0,x)
Proof. Let {1[T9 1/ S fg. TIT f11 S 7). By Lemma A.3 P(E) 2 6, and we work on this event.
Let A ((T9)))- and B A (TT)-1, and define their empirical analogues A
((f9)T)-1 and 1 As (T -1 By the Neumann-series lemma, for any matrix A and perturba-
tion A such that ||-A- |22|| A 2 < 1, we have
1|
[I -) - 1- A 1221
Applying this to 799 and T, and using the definitions 0g = ||.49 ||2 and a = =a==[(T] )|(TTT)-1 ) we obtain
|| A ag79g 0g
a²11 Cn.
Cy and |
Now define the vector-valued sufficiency for group and similarly, define the empirical estimate
A A + #BB
2, o Aa + # B9.
By triangle inequality and sub-multiplicativity of norms,
_ 2011p S | 1 A llp || B-
Note that | llpp 2 and |||p since both are probability distributions. Moreover, by triangle inequality on the matrices,
|B3 12 | A + - < | - +|( -
Combining the bounds above, we obtain
22 zdlpp I 2C + (Cy + Cn)) = (2 + 1)C + Cn = MyCo9 + Cn,
for M, = (or more generally any Mp 2 2).
We now translate this into a bound on the sufficiency gap. By Minkowski's inequality, intiouullliiieuuuliiy....
1/p
1
SSf - Suf, S
Rl - -2n1-2y 2y1b²
S (M,Ci + C44)+
This bound holds on event E, which occurs with probability at least - 5. This completes the proof.
A.6 Proof of Corollary 4.1.1
Here, we prove the contrapositive of the corollary: Under Assumptions 3.1 and 3.2 if Y 1L LGI f(X) and 2 T for all g G. then Suf for all g E G.
Proof. Applying Theorem T T implies that ATs = 0, and y G | S(X) implies (s) i(s) for all and g, and xx = 0. Thus, Suf (f) 0.
D
20

---

A.7 Proof of Proposition 5.1
Proof. For clarity we give the proof for p = 1; the extension to p > 1 follows by replacing absolute values with |-| and using the generalized Pinsker inequality [95].
Fix group 9 and let Z f(X) A k be the (predicted) score vector. For every score value 2, define the conditional label distributions
Priz=: and Py|z==,G=9-
Because Ex= k= ly=k = 1, the form of the per-group sufficiency gap (Definition 3.2 with p 1) can be written as
1
2 B2c=o/TV(Py K
(f)
K E2|c=1|1 Pri Priz,==lli]]
where Q)) } P 1 denotes total variation.
For any two discrete distributions, TV(P,Q) S ViDKI.PPIQO. Applying this point-wise and Jensen's 's inequality,
v2 K
Sff s
V ol Priz)]
(A)
By the chain rule for KL divergences,
I(Y:G] Z) Ez P(G= Z) D || Priz)] (B)
g¹
Because P(G - 9 () < for every Z, I(Y; Z) 2 P(G=9) g) [ (ri(,, || Pr|z)]
Plugging this equation into (A) gives
V2 I(Y:GI 2)
Suf <
K P)
which is the first inequality.
Let Pg = P(G = g). and ag = Ex [ (r (yiiz, || Pr|z)]
We have, for each group,
V2
Suf s
v VT
K
Hence
V2
Suf(f) Σ Pg (f)
< Σ
Py Poyag.
(SI)
K
g
Write ug == VPg and vg = vp ag Then Ug Py ag and
Σ Po ag = Σ Ug.
By the Cauchy-Schwarz Schwarz inequality.
E <
Σ
Σ
Σ Pg
Pg ag)
I(Y 12).
g
Taking square roots yields
Σ Po & VI(YG Z).
(S2)
Substituting (S2) into (S1) gives
V2
Sui(f) < K V /I(;GIJ((X))
21

---

A.8 Unbiasedness and Convergence of the CMI Estimator
Lemma A.5 (Unbiasednesss of R(fo)) Under the assumption that Vx, E B (o(fo(z))) P(Y
fo(r:) and (f(((())) = =(Y fo( 9:), we have that
E[R(fo)] = I(Y; f(X))
Proof. Let (- 8;) == |- logpy py (1 s.)..... py s:)]. and hy (: 8t) similarly. Let [l denote taking the ÿi-th element of this K K-dimensional vector.
To clarify the problem, start with the definition of entropy:
H(Y ((()) = ExExxxllllopp (Y ((X)),,
which we can estimate given a minibatch {( 9t, Us) by applying the (unbiased) Monte Carlo estimator, defining s; = fxx) for convenience:
B
B by (: 8:)lp.
H(Y f(X)) = B
Σ
log py (yi s:) = B
Since we do not observe yis we propose instead to use the backward-coorection estimator shown in Equation
B
(Y ((x))
1
Σ
T by si))s..
B
1=1
We want to show that this estimator is unbiased, i.e. Ex.CYr (f [(())) NY f(x)). First, observe that
( (TT--h((-))
[s)))
Σ P( c
Vi: G gi) [T -T- |s)]
K
T9:
[T hy (. [8))]
Wiie
(1)
c=1
y. Tg. T hr(: 85)
g:
hy (. (si) y1
log py (3/s ss).
which is no longer a function of g:. Similarly,
Eviro [Ta' hy ss)] log (Y s2).
Then,
E Exaxy (A'(Y [f(x)) = EX,G E (X [(())]]
= Ex.,Y ( [(X)I]]
(Assumption 3.1
--.c.yl-logp (Y (XX)
(Substituting (1)
1Ex.yl-logyy (Y ((()
H((Y I(()).
Thus, A' is an unbiased estimator of H(Y IS(X)).
22

---

Lemma A.6 sample convergence of the CMI estimator). Let In be the CMI estimator defined in Equation estimated with IID samples (I, 9s. Ut) Assume that Vz, B (f,()(z;)) P(Y So (x:) and hys (fo(z:) P(Y fo(s)),g)) Further assume that there exists E (0, 1/2) such that for all (s, (.
Pris(y 2 6,
PC ()) 2 PY,ais(V.g 26 c.
Then, under Assumptions
3.1 and 3.2 we have that, with probability 1- 8,
log(6/0) O(n -1/2).
1I, I ((X)) < 332llog(1e))
Further, combining with Proposition 5.1 we have that
Suf 5f) / 0(n
Proof. Let S f(X) and suppose the conditional probability heads oracle, i.e., Pyis Pyls. PGIS - PG|s. and py, G = PY,GIS (approximation error = 0). Define the bounded random variables
Øy PY SS)
&G
pG 8) ΦYG - poy, G S).
By the e-lower bound, each d. E [0, log( (e]() almost surely. With E the population expectation and E, the empirical mean over an i.d. sample of size n, the plug-in estimator and the population CMI can be written
I, = E, + E, [da] En(oval,
I(Y:G) S) B[ør] + Bjoal E[eral.
because H(Y 5) E[ HGG (S) B[éa] and H(, G S) E[dyal-
Therefore
in - TY: GY;G|S) [ s - [øy]]] + (E., [)(o]| I - [dy
Each term is a deviation of the empirical mean of bounded random variable. By Hoeffding's inequality, for any > 0 and any bounded Z E [0, BJ.
2nd²
(I E||Z| 2 t) S 2 exp
B
Applying this with B = log(1/ 1/8) and a union bound over the three variables (dy, d0; dY we obtain that with probability least 1 5,
1
1
[n 1(Y; G|Y:)|)) SS s log
V
2n
log
B Experimental Details
B.1 Data Processing
income We use the ACS income dataset from Ding al. 184. using data from all 50 states and Puerto Rico for 2018. Gender used as the protected attribute, with women being the group with noise added. We create 3-class classification problem from the dataset by defining class to be less than $30,000, class 2 to be between $30,000 and $50,000, and class to be greater than $50,000. This dataset is governed by the terms of use provided by the US Census Bureau.
adult We use the Adult dataset from the UCI Machine I Learning repository 80 Gender is used as the protected attribute, with women being the group with noise added. This dataset licensed under Creative Commons Attribution 4.0 International (CC BY 4.0) license.
23

---

crime We use the Communities and Crime dataset from the UCI Machine Learning repository [8]] Following Denis al. [96]. we create a 5-class classification problem by dividing the continuous crime rate into 5 equally sized bins. Following Denis et al. [961, we define the binary attribute to be whether community majority Black, with the non-Black group being noised. This dataset licensed under Creative Commons Attribution 4.0 International (CC BY 4.0) license.
1sac We use the law school admissions dataset from Wightman 81 where the goal is to predict whether someone passes the bar given their academic history. We use race as the protected attribute, and add noise to White applicants.
grades We use the dataset from Lenders and Calders 85| where the goal is to predict whether high school student will pass an exam. We use the biased human labels from the ranking strategy as observed labels, and the eight categorical features from the Kaggle version of the dataset as features.
cifarions We combine the CIFAR-1ON dataset from Wei et al. [88. with the CIFAR- dataset from Wang et al. [89] Specifically, we use the nomaan-dervee noisy labels from CIFAR-1O1 and augment the images to create an attribute by randomly turning 5% of images belonging to the second class to be grayscale, and the remaining classes to be 95% grayscale. The attribute is then whether the image is grayscale or not. Note that CIFAR-ION is released under the Creative Commons CuibbttonnNonnCommrrii 4.0 license.
We use the CivilComments dataset from the WILDS library 97 where the goal is to classify toxicity given string of text. We choose toxicity as the binary target, and the presence of any identity as the protected attribute. Note that each sample in CivilComments has been labeled by multiple annotators. For the noisy labels, we observe labels with Ber(p), where is the proportion labelers who said that the sample is severely toxic. Note that severe toxicity is different label than the toxicity true label.
civil commentss 86
The objective to predict the type of clothing from an image scraped from the web. The noisy label is derived from keyword scraping, while the true label is derived from human labeling. Note that only the validation and test sets contain clean labels. To create an attribute for this dataset, run MTCNN [98] from the facenet-pytor the image. We found that 64.8% of images in the dataset contain a face. The dataset is released for academic use under an unknown license.
clothingim 87
repo detect the presence of face in
B.2
Synthetic Noise
For both synthetic noise types, we add noise to the labels for only one of the groups.
B.2.1 Group Uniform
Given noise rate 7) E (0, 1), We noise one group with the uniform noise transition matrix:
"
n
K
K 7
11
1 - 1)
K
K
=
"1
1 1-
K-1
K-1
B.2.2 Group Asymmetric
Given noise rate v E (0, 1), for K = 2, we noise one group with the following transition matrix:
0 n T =
24

---

For K > 2, we use the pairfipp noise type from Zhu et al. 199
?
0
0
P
0
0
1-
17
1]
1
For both noise types, 7/ 0.75 for all datasets, except for 1sac where 1/ 0.4. We choose these noise rates as they produce ERM models with better-than-chance performance, while having significant sufficiency gaps.
B.3 Hyperparameter Grids
For all datasets and methods, we use a random hyperparameter search with 20 iterations.
We the following hyperparameter grids which common across all methods:
Batch size: Rand((,,11) for tabular datasets, 2Rnn((4,6) for text and image datasets
Number of steps: Rand(750, 2000) for tabular datasets, Rond(2000, 4000) for text and image datasets.
Learning rate: 10
For the tabular datasets, we use an MLP varying the following architecture hyperparameters:
Depth: Rand(2, 5)
Width: Rand(32, 256)
Dropout: Unif(O, 0.5)
We use the following hyperparameter grids for specific base losses:
GCE Unif (0, 1)
ELR A: Unif( 1.0, 5.0)
ELR 8: Unif(0..5, 0.9)
Peer loss a: Unif(0,
We the following hyperparameter grids for the MC baseline:
A: (0-33)
Number of iterations: Rand(50, 200)
We use the following hyperparameter grids for SUF-REG and CMI- REG:
Learning rate for he and hy: 10Unis(-3.-2)
C: Rand(1, 8)
(00mmf(-005.1)
Warm up steps: Rand(20, 100)
C SUF-REG Method
In order to directly minimize Theorem 4. we parameterize R(s) by model (oo(()))) and (s) by model heg (fx(s))) learning v and o with learning 0, similar to CMI-REG. The algorithm is found in Algorithm
(0x)),))) g), with both being linear layers operating on the logits. We then alternate
25

---

Algorithm 2: Sufficiency- Learning under Class-Attribute Label Noise
Input: Batch B = [(x, gi. 3t)]
Per-group transition matrices {T E [0, 1]] JoEs, A 02 1; Learning rates a, ag > 0:
Current classifier and auxiliary heads: fe, h
Epred(2i 24. Ik));
Output: Updated fo, has he 2, 6 fxx) Vi E B:
// logits
for j = 1 to C' do
1/ update heads, freeze 0
an.
V d
(log has (z)))
0
V
(log hig (2i, 9i))
B
KEB
fprdd 1
prea (ii, ;;);;
B VEB
RB= = Σ
1
[((TT:) (h (0TT))--ho(a0
K
IEB
C = Cpred + ARB
8 4 0 a VeL:
return fo, has hai
D Additional Experimental Results
D.1 Varying Synthetic Noise Type
In Tables and we present results for group-asymmetric and group-uniform synthetic noise respectively at the default noise levels. Across eight combinations of dataset and noise types, we find that our regularizer achieves the lowest sufficiency gap in all cases (with CMI-REG being the best in seven of eight cases).
Table 3: Clean test-set AUROC and sufficiency gap (Suf) of our regularizers when added to off-the-shel loss functions, for tabular datasets with synthetic group-uniform noise and known transition matrices. Models are selected using the clean Brier score. The best method within each base loss is bolded, and the best model overall is also underlined.
Sncome
adult
1sac
crine
AUROCT
Suf
AUROCT
Suf
AUROCT
Suf
AUROCT 1
Suf
0.762 (0.00)
0.046 (0.000)
0.634 (0.014)
0.112 (0.005)
0.587 (0.049)
0.108 (0.009)
0.613 (0,019)
0.089 (0.301)
CE+ SLF-REG
0.670
0.025 (0.003)
0.755 (0.050)
0.036 (0.010)
0.563 (0.008)
0.114 (0.006)
0.641 (0.013)
0.096 (0.006)
CE + CMI-REG
0,795 (0.007)
0.058 (0.004)
0.698 (0.1
0.073 (0.018)
0.596 (0.005)
0.101 (0.008)
0.625 (0.020)
0.085 (0.010)
Forward
0.761 (0.003)
0.043 (0.001)
0.539 (0.082)
0.100 (0.013)
0.457 (0.093)
0.095 (0.004)
0.602 (0.040)
0.081 (0.018)
Furward SUP-REG
0.677 (0.001)
0.024 (0.00))
0.848 (0.006
0.039 (0.015)
0.500 (0.0K))
0.107 (0.007)
0.639 (0.084)
0.086 (0.002)
Farwad CMI-REG
0.744 (0.02))
0,011 (0.005)
0.846 (0.009)
0.029 (0.010)
0.496 (0.066
0.107 (0.007)
0.618 (0.010)
0.093 (0.0044
Backward
0,764 (0.000)
0.042 (0.002)
0,540 (0.081)
0.100 (0.013)
0.476 (0.
0.096 (0.011)
0.605 (0.0440
0.079 (0.008)
Backward SUP-RER
0.766 (0.001)
0.043
0.696 (0.013)
0.045 (0.002)
0.459 (0.097)
0.092 (0.010)
0.609 (0.003)
0.074 (0.011)
Backward CMI-REG
0,768 (0.005)
0.035 (0.001)
0.878 (0.023)
0.028 (0.008)
(0.113)
0.093 (0.013)
(0.049)
0.082 (0.010)
DMI
0.677 (0055)
0.09 (0.03)
0.366 (14)
0.098 (0.010)
"0474 (0.183)
0,088 (0.002)
0.419 (0.007)
0.102 (0.003)
DMI SUP-REG
0.617 (0.032)
0.038 (0.014)
0.707 (0.040)
106 (0.003)
0.477 (0.175)
0.088 (0.000)
0.644 (0.038)
0.097 (0.004)
DMI CMI-RES REG
0.619 (0.033)
0.029 (0.010)
0.757 (0.012)
0.099 (0.008)
0.478 (0.178)
0.088 (0.000)
0,477 (0.057)
0.102 (0.017)
FLR
0.767 (0.001)
0.048
0.555
0.099 (0.017)
(0.265)
0.061 (0.051)
0.610 (0.085)
0.090 (0.005)
ELR+ SUP-REG
0.764 (0.005)
0.045 (0.002)
0.685 (57)
0.079 (0.027)
0.340 (0.007)
0.096 (0.004)
0.616 (0.027)
0.092 (0.016)
ELR + CMI-REG
0.784 (0.021)
0.031 (0.014)
0.806 (0.067)
0.060 (0.019)
0.230 (00026)
0.841 (0.003)
0.599 (0.042)
0.096 (0)2005
GCE
1.768 (0.D0T)
0.047 (0.004)
0.540 (0.085)
100 (0.013)
0.459 (0.043)
0.09% (0.003)
0.604 (0.039)
0.085 (0.014)
SUP-RRG
0.722 (0.056)
0.019
(0.010)
0.061 (0.01)
582
0.105 (0.005)
0.637
0.081 (0.015)
CMI-REG
0.347 (0:007)
0.011 (0.001)
0.848 (0.044))
0,043 (0.020)
0.704 (0.00)
0.086 (0.005)
0.609 (0.015)
0.072 (0.001)
Peer
0.767 (0.000)
0.045 (0.004)
0.110 (0.005)
0.092 (0.004)
0055% (0.066)
0. (0.008)
0.616 (0.001)
(0.023)
Peet SUF- REG
0.745 (0.039)
0.024 (0.011)
0.509 (0.008)
0.092 (0.004)
0.640 (0. (0.175)
0 (0.025)
0.670 (0.025)
0.061 (0.008)
Peer CMI-RES
0,742 (0.013)
0.019 (0.002)
0.756 (0.021)
0.098 (0.003)
0.590 (0.052)
0.114 (0.011)
0.606 (0.031)
0.102 (0.005)
MC
0.805 (0.002)
0.000 (0.002)
0.861 (0.015)
0.015 (0.011)
0.500 (0.000)
0.107 (0.007)
0.508 (0.013)
0.101 (0.008)
GroupPeer
0.572 (0.062)
0.076 (0.00%)
0.635 (0.056)
0.100 (0.006)
Oracle
0.854 (0.001)
0.003 (0.000)
0.919 (0.002)
0.013 (0.001)
0.838 (0.011)
0.018 (0.009)
0.855 (0.013)
0.046 (0.010)
26

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D.2 Real- World Noise
In Table we present results for datasets with real noise. We find that in the grades dataset, CMI- REG does show significant improvements in sufficiency gap without much decrease in performance. However, in the other datasets, the sufficiency gap is negligible when training with just cross- entropy noisy data. In addition, almost none of the other loss functions outperform simple cross-entropy with significance on overall AUROC. This indicates that real-world noise is highly non-clas--conditional and existing indicates that there the group attribute or noisy label have been selected when processing the datasets in way such that there is actually minimal label bias. Regardless, in these scenarios, we would hope that applying our method does not have a negative effect, i.c. that it leaves AUROC unchanged while maintaining negligbile sufficiency gap. We observe that this is the case for all three datasets.
methods fail as result. Further, it
Table 4: Clean test-set AUROC and sufficiency gap (Suf) for four datasets with real-world label noise. Models are selected using the clean Brier score. Within each base loss, the best value for every metric/dataset is bolded; the overall best non-Oracle value for each metric/dataset is additionally underlined.
grades
civilconmente
cifarione
clothingin
AUROC
Sui_
AUROC
Sul
AUROC 1
Sul
AUROC
Suf
0,681 (0.043)
0.929 (0.001)
0,004 (0.001)
0.007
CE + CMI-REG
0.683 (0.019)
0.047 (0.030)
0.929 (0.001)
0.005 (0.001)
0.984 (0.000)
0.007 (0.001)
0.952 (0.002)
0.007 (0.001)
Forward
0.689 (0.048)
0.055 (0.021)
0.906 (0.008)
0.010 (0.009)
0.988 (0.000)
0.003 (0.000)
0.943 (0.001)
0.009 (0.000)
Forward CMI-REG
0.674 (0.044)
0.054 (0.0 (0.014)
0.904 (0.010)
0.011 (0.008)
0.987 (0.001)
0.002 (0.000)
0.944 (0.003)
0,009 (0.002)
Backward
0.692 (0.032)
0.078 (0.0002)
0.833 (0.028)
0.013 (0.01
0.976 (0.000)
0.996 (0.000)
0,931 (0.002)
0.007 (0.000)
Backward= CMI-REG
0.688 (0.058)
0.920 (0.0002)
0.004
(0.001)
0.106 (0.000)
(0.001)
0.007 (0.000)
ELR
0.667 (0:035)
(0.0122)
0.928 (0.000)
0.004 (0.001)
0.978 (0.004)
0J003
0L929 (0.000))
0.008 (0.001)
FIR à CMI-REG
b.666 (0.040)
0.857 (0.009)
0.928 (0.000)
I.004 (0.000)
0.976 (0.000)
0J003 (0.001)
0.942 (0.000)
0.007 (0.000)
GCE
0.683 (0.026)
0.073 (0.017)
0.915 (0.001)
0.006 (0.000)
0.986 (0.001)
0.008 (0.002)
0.942 (0.001)
0.007 (0.000)
GCE+ CMI-REG
0,681 (0.053)
0,048 (0.009)
0.914 (0.002)
0.009 (0.005)
0.987 (0.000
0.004 (0.000)
0.940 (0.001)
0.007 (0.000)
Peer
0.681 (0.035)
0.067 (0.0220)
0.925 (0.004)
0.005 (0.003)
0.961
0.004 (0.000)
0 (0.000)
0.009 (0.000)
CMI-REG
(0:043)
(0.018)
0.924 (0.005)
(0.004)
0.962 (0.002)
0.003 (0.000)
0,894 (0.001)
0.009 (0.000)
Oracle
0.713 (0.075)
0.050 (0 (0.009)
0.936 (0.001)
0.004 (0.001)
D.3 Pareto Plots
To examine the trade-off between AUROC and sufficiency gap, we plot Pareto plots for these two metrics for each dataset with synthetic noise. We present two styles of Pareto plots. First, in Figure 3 we show each base loss in subplot, in order to examine whether our methods consistently improve the baseline for each individual loss. We note that this is fairly strict criteria, as there may be base losses that are incompatible with our loss. Regardless, we find that this occurs in most cases. Next, Figure 4. we aggregate all losses in single plot, order to show the best trade-off in the state-of-the-an before our work (i.e. base losses only) versus the two regularizers we contribute. We find that our regularizers advance the ofate-of-the-an by having either one -oortt-dominating the base loss in all cases.
27

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Clean Sul
CE
Forward
Backward
DMI
ELR
6CE
Peer
0.05
0.00
Clean Test AUROC
(a) income
CF
Forward
Barkward
DMI
RLR
GCF
Pear
0.03
Test
0.900
0.875
6.900
0.8250.850. 675
0.3
Clean Test AUROC
(b) adult
Forward
Backward
Pear
Test
0.075
0.050
0.7
0.8
0.7
0.8
0.6750.7000 725
0.6 0.7 AUROC
0.70
0.75
0.6
0.7
0.65
0.70
(c) crime
0.10
Clean
0.05
0.6
0.8
0.6
0.3
0.6
0.7
0.7
0.8
0.6
0.7
0.75
0.50
0.85
0.7
0.8
Clean Test AUROC
-
fyea + Suf-Req
-
fpod + CMi-Reg
(d) 1sac
Figure 3: Pareto fronts of AUROC versus Suf of CMI-REG, SUF-REG, and baseline loss for synthetic group asymmetric noise and known transition matrices. curve towards the bottom right of the plot is desirable as it achieves the best SUF-REG Pareto-dominate the base loss for most losses.
trade-off. We observe that CMI-REG and
28

---

0.06
0.030
fored
6pred
fored Suf-Reg
0.025
6prea
Suf-Reg
Apred CMI Reg
6pred
CMIlReg
Clean lest Suf
0.04
0.020
0.03
Clean
0.02
0.010
0.01
0.005
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.750
0.775
0.800 0.825 0.850 0.875 0.900
Clean Test AUROC
Clean Test AUROC
(a) income
(b) adult
fyed
Apeed Suf-Reg
0.060
foee + CMI-Rag
Clean Test Suf
0.055
Clean Test
0.030
fared
0.050
0.025
Apres Suf-Reg
fpres + CMI.Reg
0.045
0.020
0.040
0.015
0.035
0.675
0.700
0.725
D.750
0.775 0.800 0.825
0.80
0.81
0.82
0.83
0.84
0.85
Clean Test AUROC
Clean Test AUROC
(c) crime
(d) 1sac
Figure Pareto fronts of AUROC versus Suf of CMI-REG, SUF-REG, and baseline loss for synthetic group asymmetric noise and known transition matrices. A curve towards the bottom right of the plot desirable it achieves the best fairness-accuracy trade-off.
D.4 Impact of Model Selection
In T we compare the performance of different model selection criteria. Selection criteria are: (I) Clean Brier: We select the model with the lowest overall Brier score on the clean validation set, which requires a validation set with clean labels. (2) Default: We use default hyperparameter values reported in the papers for hyperparameters in fpred and use À
3 for SUF-REG and CMI-REG. (3) Theory: We select the model with the smallest value from Theorem 4. on the noisy validation set, which requires knowledge of the transition matrices but not any clean labels (4) Noisy Brier: We select the model with the lowest overall Brier score on the noisy validation set.
As expected, selecting using clean Brier gives the best performance and lowest sufficiency gap. Selecting using the theoretical expression (Theorem{4.) on the noisy validation set gives similarly low sufficiency gaps, but compromises significantly on performance. Selecting using the noisy Brier gives mediocre performance with slightly worse sufficiency. Across all selection criteria, selected models with regularization terms have lower sufficiency gaps than the base model in almost all cases.
D.5 Estimated Transition Matrix at Varying Noise Types
In Tables and 7 we compare our method versus the base losses when estimated transition matrices used at two other synthetic noise strengths for group-uniform noise (1 0.6 and イ 0.8 respectively), with clean Brier as the selection criteria. We find that with the estimated transition matrix, regularizers still generally improve fairness, though the performance improvements become limited relative to when the real transition matrix is used.
In Tables and 9
we show the performance and fairness of our method versus the base losses in the
hardest setting: when using the estimated transition matrix (via Confident Learning (54)). and when
using the noisy Brier score the selection metric. We find that even in this challenging setting where
29

---

Table 5: Comparison of AUROC and Suf under synthetic Group Uniform (GU) and Group Asym-
metric (GA) noise. Each entry is mean (std) across seven base losses.
AUROC
Suf
Clean Brier
Default
Noley Brier
Theory
Default
Nolay Bricr
Theory
Rise
0.744 (0:054)
0.642 (0.025)
0.630 (0.03)
0.645 (0.035)
(0.005)
0:047 (0.01)
0.047 (8:009)
0,029 (0:013)
GU
Base SUF- Reo
0.709 (0.058)
0.566 (0.019)
0.872 (0.061)
0.568 (0.092)
0,032 (0.012)
0.025 (0.006)
(0.00))
0.040 (0.013)
Bise CMI-Ruo
0.729 (0.000)
0.704 (0.064)
0.694 (8006)
0.597 (0.119)
0.065 (0.111)
0,024 (0.014)
0.019 (0:000)
0.036 (0.010)
incoem
Bise
0.738 (0.074)
0.677 (0.057)
[.681 (3.059)
0.643 (0.100
0.040 (0.007)
0.058 (0.020)
0.058 (B.012)
0,830 (0.00P)
GA Base StrF- ReG
0.738 (0.076)
0.689 (0.062)
0.711 (0.045)
0.558 (0.069)
0.009 (0.012)
(0025)
0.856 (0.011)
0.037 (0.028)
Bise CMI RBG
0.751 (0,074)
0.686 (0.061)
0,678 (00059)
0.605 (0.107)
0.835 (0.011)
0,054 (0.023)
0,058 (0.014)
0.035 (0.015)
Bise
0.529 (0,055)
0.619 (0.032)
1,400 (GLD48)
0.640 (0.129)
0,300 (0.110)
0.109 (0.013)
0,10F (0.008)
0.057 (0.035)
GU
Rase Sup-RRs
0.700 (0. 108)
1.666 103)
0.530 (0 126)
0.667 (0.162)
0.065 (0.028)
0.089 (0 026)
0.895 (0.000)
0.088 (0-028)
adult
Bise CMI RBG
0.836 (0.063)
0.610 (0.075)
0,684 (0.073)
0,706 (0.205)
0,053 (0.032)
0.095 (0.028)
0,091 (0.012))
0.067 (0,032)
Bise
0.893 (0.006
0.369 (0.052)
0.893 (0.005
6.876 (0.000)
0.026 (0.004)
0.022
0.026 (0.004)
0.024 (0,007)
GA
Sun-REg
0,886 (0.000)
(1876 (0.053)
0.802
0.025
CMI-REG
0.892 (0.005)
0.567 (0.033)
(0.011)
0,818 (0.181)
(0.006)
0,021 00.5
0.025
0.021 (0.006)
Base
0.608 (0.062)
(0.063)
0,585 (0.072)
(0,012)
0.107 (0.06)
0,086 (0.011)
(0.011)
GU
Base Sun-Raa
0.6088 (0:074)
1.609 (0.099)
0.594 (0086))
0.572 (0.022)
0,644 (1)
0.107 (0018)
0.050 (1.011)
0,882 (0:029)
Base CMI-RRG
(0,054)
0.589 (0.067)
0.553 (0.091)
0.552 (0.094)
0. 102 (0.016)
0.105 (0.16))
0.101 (0.000)
0,057 (0.009)
crize
Base
0.676 125)
0,614 (0.084)
0,673 (0. (22)
0:612 (2,126)
0,77 (0.0..7
000600002
0.081 (0.014)
0 (0,027)
GA Bise SUF-Req
(..14
0.636 (0.124)
0,685 (0.102)
0.618 (0.119)
0,966 (0.0117)
0.082 (0027)
0.082 (0.017)
0,888 (0.024)
Rise CMI-Rnn
0.687 (0.155)
0:624 (0. 102)
0.677 (0.118)
0.634 (0.134)
0.070 (0.018)
0.092 (0023)
0.074 (0.021))
0,888 (0.021)
Rise
0.482 (0.123)
0.538 (0.038)
0.487 (9)1
0:442 (0.113)
0.994 (0.022)
0. (0,007)
0.095 (0.074)
0.05 (0.009)
GU Base SUF- Reo
0.509 (0.120)
0.518 (0.054)
0.466 (0.099)
0.539 (0.174)
0.997 (0.014)
0,109 (0.01))
0.094 (0.022)
0,082 (0.022)
Tawc
Bise CMI-Roo
0.559 (0.139)
0.529 (0.043)
01425 (0.174)
0.534 (0.167)
0.094 (0.019)
0,102 (O (0.008)
0.079 (0.000)
O 0,088 (C.013)
Rise
0.478 (0.116)
1.542 (0.034)
0.512 (L.068)
0,533 (0.042)
0.095 (0.021)
0.110 (0 006)
0.102 (0.000)
0.111 (G.036)
GA Base StF-ReG
0.594 (0.107)
0.536 (0.070)
0,514 (0.085)
0.523 (0.054)
0.065 (0.008)
0,108 (0 007)
0.101 (0.010)
1.009 (0.012)
C'M1RRG
(0.166)
0.538 (0.054)
0.442
(0.125)
0,107
0.084 (0:0E5)
(0.035)
Table 6: Clean test-set AUROC and sufficiency gap (Suf) of our regularizers when added to off-the-shelf loss functions, for tabular datasets with synthetic group-uniform noise at 1) = 0.6 and real vs. estimated transition matrices. Models are selected using the clean Brier score. The best method within each base loss is bolded; the overall best non-Oracle model for each metric/dataset is additionally underlined.
Real 7
Eatimated T2
income
adult
Sncooe
adult
AUROXT
Suf
AUROCT
Suf
ALROCT
Suf 1
AUROCT
Suf
CE
0.794 (0.000)
0.050 (0.003)
0.760 (0.001)
(0.003)
0.794 (0:000)
0.050 (0.004)
(0.001)
0.093
CE + SSff-Reg
0.792 (0.001)
0.054 (0.001)
0.782 (0.020)
0.020 (0.000)
0.781 (0:006)
0.054 (0.002)
0.779 (0.012)
0.043 (0.016)
CE CMI-Reg
0,802 (0.002)
0.043 (0.003)
0.878 (0.006)
0.022 (0.001)
0.782 (0.007)
0.025 (0.00.3)
0.782 (0.038)
0.061 (0.005)
Forward
0.797 (0.001)
0.041 (0.001)
0.799 (0. 00.008)
0.080 (0.002)
0.786 (0.006)
0.640 (0.002)
0.686 (0.029)
0.098 (0.008)
Forward+ Suf-RRg
0.782 (0.022)
(0.001)
0.854 (0.009)
(0.018)
0.768 (0.006)
0.048 (0.003)
(0.266)
0.067 (0.001)
Forward CMM-Rag
0.779 (0.007)
0.031 (0.003)
0.866 (0.009)
0.040 (0.015)
0.785 (0.011)
0.038 (0.005)
0.833 (0.042)
0.053 (0.037)
Backward
0.788 (0.002)
0.048 (0.001
0.795 (0.009)
0.062 (0.003)
0.769 (0.002)
0.015 (0.001)
0.687 (0.028)
0.098 (0.008)
Backward + Suf-Reg
0.790 (0.002)
0.048 (0.000)
0,818 (0.002)
0.073 (0.00-)
0.763 (0.001)
0.068 (0.000)
(0.00(
0.013 (0.001)
Backward + Reg
0.786 (0:002)
0.043 (0.001)
0,838 (0.005)
0.061 (0.004)
0.770 (0.008)
0.039 (0.000)
(0.012)
0.060 (0.005)
DMI
(0.122)
0.008 (0.001)
0.784 (0.004)
0.098 (0.002)
0.674 (0.
0,038 (0.001)
0.784 (0.005
0.098 (0.002)
DMI Suf Raf
0.624 (0.0155
0.037 (0.012)
0.791 (0.000)
0.091 (0.005)
0.625 (0.015)
0.038 (0.011)
0.783 (0.000)
0.097 (0.005)
DMI CMI-Rog
0.666 164)
0.037 (0.003)
0.791 (0.003)
0.095 (0.003)
0.628 (0.017)
0.038 (0.009)
0.784 (0.001)
0.097 (0.005)
E.R
0.811 (0:005)
0.015 (0. (0.00)
0.778 (0.000)
0.092 (
0.811 (0.00))
0005 (0.002)
0.778 (0.000)
0.092 (0.006)
ELR Reg
0.784 (0.001)
0.042 (0.001)
0.861 (0.002)
0.033 (0.009)
0.775 (0.002)
0.043 (0.001)
0.804 (0.025)
0.014 (0.006)
ELR + CMI Reg
0.785 (0.000)
0.036 (0.001)
0.815 (0.010)
0.036 (0. (0.000)
0.799 (0.007)
0.039 (0.001)
0.793 (0.015)
0.046 (0.014)
GCE
0.798 (0.001)
0.039 (0.000)
0.802 (0.001)
0.087 (0.002)
0.798 (0.002)
0.039 (0.000)
0.802 (0.001)
0.087 (0.003)
Sui-Reg
0.792 (0.001)
(0.000)
0.778 (0.054)
(0.008)
0.779 (0.007)
0.052 (0.001)
0.797 (0.045)
0.024 (0.001)
GCE CMI-Reg
0.821 (0.001)
0.020 (0.004)
0.852 (0.016)
0.053 (0.006)
0.792 (0.005)
0.029 (0.005)
0.864 (0.000)
0.043 (0.002)
Pler
0.792 (0.002)
0.045 (0.001)
0.756 (0.009)
0.082 (0.008)
0.792 (0.005)
0.045 (0.001)
0.756 (b.01 1)
0.052 (0.010)
Peer+ Suf-Reg
0.754 (0.015)
0 (0.002)
0.816 (0.02))
0.028 (0.007)
0.737 (0.017)
0.018 (0.008)
0.803 (0.007)
0.066 (0.008)
CMI-Reg
(0.01))
(0.000)
0,882 (0.005)
(0.000)
0.767 (0.016)
0.036 (0.011)
0,813 (0.005)
0.076 (0.008)
30

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Table 7: Clean test-set AUROC and sufficiency gap (Suf) of our regularizers when added to off-the-shelf loss functions, for tabular datasets with synthetic group-uniform noise at 1 0.8 and real vs. estimated transition matrices. Models are selected using the clean Brier score. The best method within each base loss is bolded; the overall best non-Oracle model for each metric/dataset is additionally underlined.
Real 7
Eatimated T
income
adult
income
adult
AUROC |
Suf
AUROC
4
AUROCC
AUROCT
Suf
CF
0.755 (0.001)
0.045 (0.0000)
0.476 (0.067)
0.090 (0.011)
0.755 (0.001)
0.04% (0,000)
0.476 (8.182)
0.090 (d.014)
CE Suf-Reg
0.681 (0.021)
0.026 (0.011)
0.759 (0.060)
0.051 (0.008)
0.678 (0.020)
0.021 (0.007)
0.596 151)
0,886 (0.005)
CE CMI-Rey
0,767 (0.008)
0.025 (0,008)
0.831 (0.063)
0.047 (0.015)
0.669 (0.037)
0.058 (0.027)
0.632 (0.11))
0.101 (0.015)
Forward
0.753 (0:002)
0.043
0.473 (0.077)
(0.012)
0.756 (0:000)
0.043 (0.001)
(6.091)
0.089 (0.016)
Forward
Suff-Rg
0.710 (0.0600
0.015 (0.001))
0.845 (0.000)
0.040 (0.014)
0.639 (0.008)
0.049 (0.009)
0.361 (0.033)
0.037 (0.118)
Foroard
CML.Reg
0.759 (0.012)
0.052 (0.006)
0,889 (0.0033
0.021 (0.005)
0.598 (0.047)
0.015 (0.014)
0.500 (0.005)
0.093 (0.004)
Backward
0.755 (0.000)
0.041 (0.002)
(0.07)
0.090 (0.013)
0.684 (0.028)
0.011 (0.002)
0.469 (0.091)
0.089 (0.016)
Backward
Suf-Reg
(0:003)
0.040 (0.001)
0.680 (0.008)
(0.002)
0.750 (0.014)
(0.001)
0,407 (0.026)
0.063 (0.003)
Backward a CMi-Reg
(0.000)
0.840 (0.0011)
0.791 (0.030)
0 (0.006)
0.653 (0.019)
0.058 (0.002)
0.545 (D.009))
0.087 (0.005)
DMI
0.649 (0.071)
0,940 (0.002)
0.411 (0.085)
0.099 (0.008)
0.649 (0.067)
0.040 (0.00.)
0.411 (O. (08)
0.099 (0.010)
DMI Saf-Reg
0.585
0.052 (0.003)
0.631 (0.012)
(0.006)
0.589 (0.009)
0.051 (0.008)
0.423 (0.095)
0.099 (0.008)
CMI-Reg
(0.)
0.041 (0.002)
0.411 (0.103)
(0.010)
0.567 (0:015)
0.047 (0.006)
(0.059)
0.096 (0.006)
ELR
0.755 (0.001)
0.043 (0.001)
0.476 (0. (02)
0.088 (0.018)
0.755 (0.001)
0.043 (0.001)
0.476 (0. (2)
0.088 (0.023)
ELR + Suf Reg
0.757 (0.002)
0.045 (0.001)
0.658 (0.182)
0.079 (0.025)
0.662 (0.040)
0.643 (0.029)
0.597 (0.178)
0.056 (0.031)
FLR CMI-Reg
0.759 (0.004)
0.041 (0.001)
0.726 (0.0003
(0.006)
0.671 (0.048)
0.046 (0.028)
0.545 (0.106)
0.084 (0.008)
GCE
0.753 (0.001)
0.013 (0.001)
00.72(0.078) (0.078)
0.090 (0.013)
0.753 (0:000)
0.643 (0.001)
(0.096)
(0.016)
GCE Suf Reg
0.756 (0.029)
0.038 (0.006)
0.759 (0.006)
0.061 (0.013)
0.640 (0.010)
0.048 (0.006)
0.394 (0.084)
0.050 (0.040)
GCE + CMI Reg
0.759 (0.007)
0.037 (0.004)
0.892 (0.000)
0 (0. (0.002)
0.661 (0.015)
0.066 (0.002)
0.569 (0.097)
0.082 (0.017)
Peer
0.755 (0.004)
0,042 (0.002)
0.720 (0.004)
0.108 (0.002)
0.755 (0.005)
0.042 (0.003)
0,500 (0.000)
0.093 (0.004)
Suf-Reg
(0.036)
(0.004)
0.771 (0.013)
(0.010)
0.612 (0.068)
(0.011)
(0.000)
Peer + CMI-Reg
0.699 (0.003)
0.017 (0.001)
0.858 (0.005)
0.041 (0.008)
0.640 (0.002)
0.043 (0.005)
0.500 (0.000)
0.093 (0.004)
no clearly labeled data is available, our regularizers consistently achieve lower sufficiency gaps than the base losses, and the selected models with the lowest sufficiency gaps across losses correspond our regularizers in six of eight combinations of datasets and noise types.
E Broader Impacts
In this work, we explored unfairness induced by label bias, and introduced regularizers so that models stay calibrated across groups with respect to the true labels. Though we evaluate on real-world data, we do not advocate for the blind deployment of ML models in any real-world safety critical setting (such as health, credit, or hiring). There are myriad of other factors that should be taken into account before their deployment (e.g. privacy, regulation, interpretability, and technicll considerations). Misuse of such models could lead to real harm.
31

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Table 8: Clean test-set AUROC and sufficiency gap (Suf) of our regularizers when added to off-the-shelf loss functions, for tabular datasets with synthetic group-asymmetric noise and esti- mated transition matrices. Models are selected using the noisy Brier score. The best method within each base loss is bolded; the best non-Oracle model overall for each metric/dataset is additionally underlined.
income
adult
1sac
crime
AUROC
Suf
AUROC
Suf 1
AUROC 1
Suf J
AUROC
Suf
IM682 (0.001)
0J065 (0.002)
(0.00)
0.026 (0.004)
(0.077)
0.109 (0.066)
0.740 (0,023)
0,072 (0.901)
CE 1. SUP-RIG
0.679 (0.00))
0.076 (0.006
0.854 (0.0035)
0.028 (0.001)
0.477 (0.109)
0.092 (0.007)
0.722 (0.042)
0.074 (0.021)
CE + CMI-REG
0.663 (0.003)
0.079 (0.0035
0.592 (0.000)
0.023 (0.002)
0.479 (0.021)
0.102 (0.010)
0.731 (0.013)
0.078 (0.010)
Forward
0.633 (0.000)
0,109 (0.00)
0.886 (0.001)
0.029 (0.000)
0.446 (0.065)
0.095 (0.001)
0.689 (0.057)
0.057 (0.003)
Forward+ SSU-REG
(0.030)
0.043 (0.015)
(0.008)
0.028 (0.00))
(0.059)
0.060 (0.028)
0,596
(0.009)
Forward+ -MMI-RG
0.609 (0.09)
0.103 (0.007)
0.890 (0.001)
0.026 (0.002)
0.554 (0.169)
0.092 (0.019)
0.637 (0.114)
0.084 (0.002)
Backward
0.607 (0.001)
0.092 (0.006)
0.886 (0.0022
0.029 (0.001)
0.445 (0.061)
0.091 (0.002)
0.564 (0.039)
0.098 (0.013)
Backward SUF-REG
(0.02(00000)
0.091 (0.005)
0,856 (0.0022
0.027 (0.001)
0.506 (0.09K)
0.090 (0.001)
0.526 (0.001)
0,111 (0.022)
Backward CMI-REG
0:601 (0.005)
0.106 (0.000)
(0.0022
0.027 (0.000)
(0.093)
0.084 (0.001)
0.563 (0.073)
0.099 (0.009)
DMI
0.590 (0.094)
0.47 (0.013)
0888 (0.001)
0.024 (0.004)
0.477 (0.138)
0.000 (0.001)
0.475 (0.027)
0.108 (0:020)
DMI SUP-RRG
0.590 (D: 113)
0.146 (0.016)
0.56 (0.0035)
0.017 (0.022)
0.503 (0.130)
0.99 (0.017)
0.459 (0.029)
0.101 (0.097)
DMI CMI-REG
0.587 (0.114)
0,046 (0.017)
0.890 (0.00ES)
0.024 (0.006)
0.480 (0.170)
0.089 (0.001)
0.453 (0.038)
0.097 (0:034)
ELR
(0.06()0.00)
0.017 (0.0033
0,887 (0.001)
0.026 (0.001)
0.520 (0.043)
0.101 (0.011)
0.656 (0.072)
0,088 (0.002)
SUF-REG
0.677 (0.001)
0.066 (0.002)
(0.002)
0.026 (0.000)
(0.065)
0.088 (0.011)
(0.051)
0.092 (0.038)
ELR REG
(0.003)
0.080 (0.003)
0.883 (0.001)
0.020 (0.002)
0.375 (0.051)
0.095 (0.006)
0.650 (0.094)
0.098 (0.007)
GCE
(0.7500.005)
0.070 (0.006)
0.884 (0.001)
0.029 (0.001) 0.491 (0.051)
0.100 (0.001)
0.743 (0.025)
0.083 (0.020)
GCE SUF REG
0.677 (0.003)
0.078 (0.007)
0.854 (0.005)
0.027 (0.001)
0.467 (0.118)
0,089 (0.010)
0.719 (0.019)
0.075 (0.027)
GCE CMI-REG
(0.005)
0.082 (0.007)
(0.001)
0.024 (0.004)
(0.066)
0.098 (0.000)
(0.018)
0,084 (0.064)
Poer
0.730 (0.018
0.030 (0.000)
0858 (0.000
0.025 (0.001) 0.566 (0.033)
0.112 (0.005)
0.696 (0.041)
0.080 (6.001)
Poer SUP-RIG
0.644 (0.046)
01000 (0.05)
0883 (0.001)
0 (0.007)
0.280 (0,03))
0.053 (0.317)
0.498 (0.0664
0.085 (0.1166
Poer CMI-REG
0.589 (0.002)
0,054 (0.006)
0.892 (0.0011
0.021 (0.001)
0.481 (0.097)
0. 1010 (0.000)
0.663 (0.061)
0.100 (0:011)
GraupPeer
0.8% (0.002)
0.023 (0.005
0.659 (0,011)
0.094 (0.018)
Oracle
0.854 (0.001)
0.003 (0.000)
0.919 (0.002)
0.013 (0.001)
0.838 (0,01)
0.018 (0.0099
0,825 (0.013)
0.046 (0:010)
Table 9: Clean test-set AUROC and sufficiency gap (Suf) of our regularizers when added to off-thc-shelf loss functions, for tabular datasets with synthetic group-uniform noise and estimated transition matrices. Models are selected using the noisy Brier score. The best method within each base loss is bolded; the overall best non-Oracle model for each metric/dataset is additionally underlined.
income
adult
leac
crine
AUROCT
AUROC
AUROC 1
Sul 1
AUROC 1
Suf
CE
0.630 (0.004)
0.053 (0.006
0.634 (0.019)
0.108 (0.001
0.568 (0.050)
0.111 (0.006)
0.609 (0.015)
0.090 (0.003)
CE + SEP-REG
0.651 (0.027)
0.042 (0.022)
0.573 (0.026)
0.109 (0.029)
0.464 (0.083)
0.093 (0.008)
0.647 (0.005)
0.090 (0:020)
CE + CMI-REG
0.681 (0.003)
0.034 (0.008)
0,593 (0.000)
0.494 (0.036)
0.101 (0.012)
0.650 (0.057)
0.092 (0.005)
Forward
(0.00)
0.047 (0.001)
(0.009)
0.107 (0.004) 0.456 (0.080)
0.095 (0.000)
0.609 (0.014)
0.103 (0.009
Forward SUP-REG
0.692 (0.005)
0.027 (0.007)
0.552 (0.352)
0.013 (0.004)
0.417 (0.131)
0.089 (0.020)
0.435 (0.037)
0.102 (0.003)
Forward + CMI-REG
(0.19)
0.034 (0.007)
0.592 (0.010)
0.095 (0.011)
0.453 (0.140)
0.087 (0.0144)
0.655 (0.068)
0.084 (0.013)
Backward
0,44 (0.003)
0.048 (0.001)
0.699 (0.008)
0.106 (0.004)
0.454 (0.074)
0.0% (0.001)
0.531 (0.084)
0.09-1 (0.009)
Backward+
(0.014)
0.054 (0.03)
(0.141)
0 (0.018)
(0.003)
0.091 (0.009)
Backward CMI-RIG
0.659 (0.008)
0J041 (0.007)
0.530 (0.002)
0.0%6 (0.044)
0.485 (0.100
0.095 (0.002)
0.538 (0.094)
0.089 (0.001)
DMI
06660 (0.06))
0.034 (0.009
0.641 (0.05)
0.677 (0.02)
0.478 (0 144)
0,888 (0.000)
0.489 (0.045)
0.109 (0.016)
DMT Sur- RBG
0.643 (0.09))
0.051 (0.0035
0,573 (0.026)
0.078 (0.020)
0.472 (0. (7.1
0.089 (0.002)
0.634 (0.087)
0.081 (0:023)
DMI CMI-REO REG
0.642 (0.042)
0.025 (0.007)
0.604 (0.020)
007 (0.004)
0.482 (0.177)
0.088 (0.001)
0.488 (0.054)
0.110 (0.017)
(0.000)
0.056 (0.002)
(0.010)
0.112 (0.005)
0.523 (0.042)
1.i (0.011)
0.608 (0.027)
0.088 (0.005)
ELR+ SUP-REG
0.642 (0.012)
0.043 (0.013)
0,290 (0.026)
0.029 (0.015)
0.330 (0.077)
0.083 (0.013)
0,459 (0.049)
0.108 (0.007)
ELR + CMI- REG
(0.01)
0.029 (0.008)
0.105 (0.002)
0.066 (0.002)
0.375 (0.052)
0.091 (0.007)
0.645 (0.065)
0.096 (0.004)
GCE
66616
0.052 (0.005
0.629 (0.001)
0.112
0.458 (0.062)
0.097 (0.000)
0.506 (0.016)
0.090 (0.007)
SUF-REG
(0.04))
0.026 (0.020)
0.124 (0.019)
(0.086)
0.092 (0.008)
(0.0066)
0.086 (0.001)
GCE CMI-RBG
0.637 (0.026)
0.043 (0.026)
0.607 (0.000)
0.101 (0.009)
0.614 (0009)
0.056 (0.012)
0.651 (0.068)
0.093 (0.004)
Pler
0.767 (0.000)
01145 (0:00)
0.640 (0.020)
107 (0.001
0.573 (0,044)
0.110 (0.017)
0.68 (0.028)
0.085 (0.0)2)
Poer SUI-RIG
0.581 (0.028)
IJ04 (0.014)
0.559 (0.282)
0,034 (0025)
0.389 (0 100)
0.095 (1.126
0.501 (0.074)
0.198 (0.010)
Peer CMI-REG
0.654 (0.007)
0.042 (0.009)
0.513 (0.011)
0.092 (0.005)
0.457 (0.097)
0.099 (0.003)
0.657 (0.064)
0.096 (0.008)
GroupPeer
(0.095)
0.088 (0.005)
0.646 (0.007
0.098 (0.024)
Oracle
0.854 (0.001)
0.003 (0.000)
0.919 (0.002)
0.013 (0.001)
0.838 (0.011)
0.018 (0.009)
0.625 (0.013)
0.0066 (0.010)
32

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Reviews:
[{"review_id": "PArv9NlPxU", "reviewer": "Reviewer_xUeM", "summary": "The paper introduces a method for ensuring per-sensitive-group calibratedness in the presence of noise. Under the assumption that one knows the per-group transition probabilities for the noisy labels, the paper provides a two-step algorithm to calculate a calibration gap using a mutual information between the ground truth label and the group membership condition on the classifier output. In the first step, the mutual information is estimated, while in the second step, the model is trained using a regularizer informed by the mutual information bound. Empirical results are presented comparing the method with a baseline where the calibration is directly regularized.", "questions": "- Q1. On page 7, I think the transition matrix for Eq. (4) should probably be $T^{-1}$, not $T_{g_{i}}^{-1}$. No? Same for Eq. 6 and onwards, as none of these probability estimates have a conditioning on group $G$.\n- Q2. Please discuss multi-calibration.", "limitations": "As also stated in the paper's limitation section (thanks for making this very clear), the assumption that we have the transition matrices as inputs to the algorithm is a rather strong one that limits the applicability of the algorithm as it is. I think the paper would have benefited greatly from an estimation step using a small, clean dataset (for example).", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Overall, despite the interesting problem, this paper falls below the acceptance threshold for me.  I will enumerate **S**trengths and **W**eaknesses.\n- **S1. The problem of fairness calibration under noise is relevant.** Given how much noise real-world data has and the fact that many risks of systematic discrimination are attributable to noisy data, this is a valuable problem to tackle.\n- **W1. Lack of meaningful baselines.** The baseline discussed in the paper adds its regularizer to prior noise-robust methods to adapt them to the multi-group calibration problem with noisy data. This is useful and does show the versatility of the method, and allows judging the improvement that the use of CMI-REG and CMI-SUF brings to their respective methods. However, these are not competitive baselines for the problem. I understand that the authors claim that no prior work has tackled this exact problem, but it behooves them to construct reasonable baselines for comparison as a result. Here is my suggested baseline: \n  \n  Since the transition matrices are given to CMI-REG, why not transform the noisy data using the transition matrices $T^g$ and $T$ as a pre-processing step and then use any standard multicalibration algorithm?  \n  \n  On the latter note, I find that lack of any discussion or reference to the (now) vast literature of multi-calibration is a miss.  \n- **W2. Novelty and contribution are not clear.** Methodically, the paper's approach can be summarized as adding a calibration regularizer with corrections to prediction rates due to noise, which is completely specified via transition matrices. Furthermore, the robustness to noise seems to be more a function of the base loss $\\ell_{pred}$ in Eq. 2 than a contribution of the presented methods. Unfortunately, the experiments do not convince me otherwise, either due to small relative improvements, many of which fall within error margins.\n- **W3. Uneven presentation.** The level of detail in the presentation is uneven. As an example, I was very surprised to find SUF-REG being presented in the experimental section (Line 254) in 2 lines with minimal details. The authors emphasize that this is a novel contribution. If so, why treat it as an afterthought? Especially given how competitive it can be with the main method, CMI-REG. This left me with more questions than answers, such as: If we can regularize sufficiency directly, what is the point of all the mutual information estimation?\n  Similarly, backward and forward corrections in Lines 206 and 208 are presented with only a reference and no details. \n- **W4. Empirical results are not very convincing.** I touched upon this briefly in W2. Looking at Figure 2, I cannot confirm that CMI-REG and SUF-REG indeed Pareto dominate in every case. In lsac+DMI and lsac+Peer that does not seem to be the case; also, Figure 3 in the appendix seems to be a similarly mixed set of results. Also, overall, the Pareto fronts are rather sparse. I understand that training and hyperparameter tuning may be difficult, but it's really hard to compare a single point to a full curve in a Pareto front setting.", "was_revised": true, "final_justification": "I have adjusted my score to a weak accept post rebuttal following additional experiments and discussions around multicalibration.", "rebuttal": "Thank you for the insightful review and constructive feedback!\n\n> W1. Lack of meaningful baselines. Here is my suggested baseline: Since the transition matrices are given to CMI-REG, why not transform the noisy data using the transition matrices $T^g$ and $T$ as a pre-processing step and then use any standard multicalibration algorithm?\n\nThank you for the suggestion! There are two primary issues with the proposed baseline. First, the inverse of the transition matrix $(T^g)^{-1}$ is in general not row-stochastic, and may contain values outside of $[0, 1]$. Thus, it cannot not be directly used in a _pre-processing_  step to transform the observed labels into unbiased true labels. Second, applying multicalibration during test-time requires knowledge of the group attributes during inference. None of our methods or baselines require this assumption. \n\nRegardless, we have adapted the proposed baseline to our setup in the following way:\n- We apply a softmax to each row of $(T^g)^{-1}$ so it is row stochastic.\n- We randomly flip observed labels according to this matrix, and then train an ERM model on this processed data.\n- We apply Algorithm 1 from [1] for multicalibration, specifically using the implementation from [2], using the single attribute to define $\\mathcal{C}$.\n- We vary all hyperparameters (including $\\alpha$ and $\\lambda$) using a random search, and select the optimal model using the same setting as Table 2 of the paper.\n\nWe present results for AUROC and Sufficiency Gap below. We include only the Forward loss for our regularizers for brevity. We find that sufficiency gaps for this baseline remain substantially higher and AUROC is often worse.\n\n\n|                   |         |                 income  |          |                 adult  |           |              lsac     |         |                crime    |\n|-------------------|--------------:|------------------:|--------------:|------------------:|--------------:|------------------:|--------------:|------------------:|\n|                   |         AUROC |               Suf |         AUROC |               Suf |         AUROC |               Suf |         AUROC |               Suf |\n|           Forward | 0.801 (0.002) | **0.038 (0.002)** | 0.887 (0.001) | **0.024 (0.004)** | 0.436 (0.006) |     0.097 (0.001) | 0.788 (0.011) |     0.056 (0.010) |\n| Forward + Suf-Reg | 0.812 (0.002) |     0.059 (0.003) | 0.898 (0.000) |     0.032 (0.001) | 0.690 (0.011) |     0.091 (0.016) | 0.789 (0.006) |     0.053 (0.006) |\n| Forward + CMI-Reg | 0.801 (0.002) | **0.038 (0.001)** | 0.888 (0.002) | **0.024 (0.004)** | 0.704 (0.172) | **0.065 (0.070)** | 0.802 (0.015) | **0.046 (0.011)** |\n|         GroupPeer |             - |                 - | 0.898 (0.001) |     0.035 (0.004) | 0.664 (0.010) |     0.095 (0.023) |             - |                 - |\n|                Multicalibration | 0.561 (0.019) |     0.078 (0.010) | 0.787 (0.004) |     0.017 (0.001) | 0.500 (0.000) |     0.107 (0.007) | \t0.508 (0.013) |     0.104 (0.012) |\n|            Oracle | 0.854 (0.001) |     0.003 (0.000) | 0.919 (0.002) |     0.013 (0.001) | 0.838 (0.011) |     0.018 (0.009) | 0.825 (0.013) |     0.046 (0.010) |\n\n\n\n> W2. Novelty and contribution are not clear. Methodically, the paper's approach can be summarized as adding a calibration regularizer with corrections to prediction rates due to noise, which is completely specified via transition matrices. Unfortunately, the experiments do not convince me otherwise, either due to small relative improvements, many of which fall within error margins.\n\n\nWe argue that our contributions extend beyond simply combining noise-robust losses with a heuristic regularizer. First, we propose and formalize this particular problem, which, to our knowledge, has not been studied in prior work. Second, we theoretically characterize this problem, showing necessary & sufficient conditions showing that label bias, even when classifiers are calibrated on noisy labels, inevitably induces a sufficiency gap on the true labels (Theorem 4.1 and Corollary. 4.1.1). Third, though our experimental gains are modest, they do seem to be consistent, and statistically significant in many cases.\n\n\n> W3. Uneven presentation. As an example, I was very surprised to find SUF-REG being presented in the experimental section (Line 254) in 2 lines with minimal details. Similarly, backward and forward corrections in Lines 206 and 208 are presented with only a reference and no details.\n\nDue to space issues, we were not able to describe SUF-REG thoroughly in the main paper and had to relegate it to the appendix. We focused primarily on CMI-REG because it performed marginally better, and because the techniques used in CMI-REG (e.g. linear heads for posterior estimation) are common across the two methods. The backward correction used in L206 and L208 are applications of a well-known result from prior work [3], and so we did not describe it in detail. We will provide more exposition to this in the revision.\n\n\n> W4. Empirical results are not very convincing. Looking at Figure 2, I cannot confirm that CMI-REG and SUF-REG indeed Pareto dominate in every case. Also, overall, the Pareto fronts are rather sparse. \n\nWe note that expecting our regularizer to improve every single base loss is a lofty goal that may not be realistic, as some losses may have optimization landscapes that naturally work better with our regularizer. We note that overall, either CMI-REG or SUF-REG achieves the lowest sufficiency gap in 8/10 experimental settings (Tables 3 and 4 in the appendix).\n\nRegarding the sparsity of the Pareto plots, we emphasize that all methods were given the same budget of 20 hyperparameter from a random search, from a fairly wide range grid shown in Appendix B.3. The reason why some Pareto curves collapse to a single point is because there is no observed trade-off between fairness and performance in those cases. In general, such a trade-off may not exist, e.g. the group Bayes optimal score with respect to true labels has maximum performance and zero sufficiency gap.\n\n\n> Q1. On page 7, I think the transition matrix for Eq. (4) should probably be $T^{-1}$, not $(T_g)^{-1}$. No? Same for Eq. 6 and onwards, as none of these probability estimates have a conditioning on group.\n\nIn our class-attribute-conditioned noise model (Assumption 3.1), the backward correction always involves $T_g$ because noise depends on $(Y, G)$, which leads to $\\\\mathbb{E}_{\\\\tilde Y \\\\mid Y = y,\\\\, G = g} [(T_g^{-1})\\_{\\\\tilde{Y},\\\\cdot}] = e\\_y^{\\\\top}$. Using the marginal matrix $T$ would result in a biased estimator, as it does not correspond to the actual probability of corruption for that sample.\n\n> On the latter note, I find that lack of any discussion or reference to the (now) vast literature of multi-calibration is a miss.\n\n> Q2. Please discuss multi-calibration.\n\n\nWe will add a paragraph on multicalibration to the related works in Section 2.2, and add the above experiment with the multicalibration baseline to the main paper. Please let us know if this is satisfactory!\n\n\n> As also stated in the paper's limitation section (thanks for making this very clear), the assumption that we have the transition matrices as inputs to the algorithm is a rather strong one that limits the applicability of the algorithm as it is. I think the paper would have benefited greatly from an estimation step using a small, clean dataset (for example).\n\nThank you for the suggestion! We note that we do experiment with using Confident Learning [4] to estimate the transition matrix (only from noisy data), though this generally results in worse performance than using the true transition matrix. We emphasize that as better transition matrix estimations methods are developed, our method will improve in this setting as well.\n\nIf we were instead given $n$ iid samples with labels for whether they are mislabeled or not as the reviewer suggests, we could then use the empirical estimator for each element of $T$. We have theoretically characterized the behavior of this estimator, and its impact on sufficiency gap estimates, in Appendix A.5. \n\n\n\n[1] Multicalibration: Calibration for the (Computationally-Identifiable) Masses. ICML 2018.\n\n[2] When is Multicalibration Post-Processing Necessary? NeurIPS 2024.\n\n[3] Making Deep Neural Networks Robust to Label Noise: a Loss Correction Approach. CVPR 2017.\n\n[4] Confident Learning: Estimating Uncertainty in Dataset Labels. JAIR."}, {"review_id": "UPODkUC5mW", "reviewer": "Reviewer_fbDW", "summary": "This paper studies the group sufficiency fairness metric in the presence of group-dependent label noise. The authors propose a regularization-based fix for such label noise and perform well-designed experiments to demonstrate the effectiveness of their approach.", "questions": "I like the problem, but the two severe limitations overpower the theoretical analysis performed by the authors. I would want to ask the authors the following to reconsider the rating on this paper:\n1. In the noisiest of settings where you can only estimate Brier scores and transition matrices with noisy data (perhaps with the help of Clusterability or some anchor points), will the proposed CMI-REG show a lot of significant improvement?\n2. Can the authors comment on the convergence of the proposed mutual information estimators with finite samples? Off the top of the head, they can either argue that it minimizes a relaxed version of the sufficiency gap or assume a parametric distribution on data (Gaussian) to show that it at least works as intended on those kinds of distributions.", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. The problem of studying group sufficiency with label noise surprisingly flew under the radar. Many fairness metrics have been studied with label bias except this one, and thus, this work identifies an important gap.\n2. Theorem 4.1 is interesting as it precisely gives how worse can sufficiency gap can get with label noise.\n3. The experiments are comprehensive (however, there are some doubts, which we will discuss next).\n\nLimitations:\n1. The proposed method has some dominance over others (like vanilla noise correction, since unconstrained clean ERM is sufficient to ensure group sufficiency according to Liu et al. 2019). However, a case where the method can shine out and establish clear dominance is when the transition matrix is estimated and the Brier score itself is noisy. I couldn't find that set of results anywhere (please do point me to them if I have missed those). The supplementary does show real-world label noise results (Table 5), and the CMI-REG doesn't seem to beat vanilla CE, both in Suf-gap and Acc. Because of this, I am skeptical of the empirical advantages\n2. I am also skeptical of the finite sample estimator of the CMI regularizer. Since the regularizer depends on mutual information and conditional entropy, and estimators for both of them have been known to be very tricky and have convergence issues with finite samples,  I am not sure whether the loss function would do its intended job. Had the experimental results outperformed everything else, I could have believed that the estimators are doing their job, but due to the last point, I am not sure.", "was_revised": true, "final_justification": "The authors have provided results for the setting provided and even managed to supply some analysis for their estimator.", "rebuttal": "Thank you for the insightful review and constructive feedback!\n\n> The proposed method has some dominance over others. However, a case where the method can shine out and establish clear dominance is when the transition matrix is estimated and the Brier score itself is noisy.\n \n> In the noisiest of settings where you can only estimate Brier scores and transition matrices with noisy data (perhaps with the help of Clusterability or some anchor points), will the proposed CMI-REG show a lot of significant improvement?\n\nThank you for the suggestion! We note that the setting where we only have estimated transition matrices, as well as no clean examples for model selection, is the most difficult setting for our problem. We have provided results for this setting below. We observe that across the 24 combinations of losses and datasets, CMI-REG achieves lower or equal sufficiency gap than the base loss in 20 such combinations. \n\nFurther, expecting our regularizer to improve every single base loss is a lofty goal, as some losses may have optimization landscapes that naturally work better with our regularizer. Instead, looking at the best achievable sufficiency gap for each dataset, we find that CMI-REG achieves the lowest sufficiency gap across all four datasets. \n\n\n\n\n\n\n|                    |               |            income |               |             adult |               |              lsac |               |             crime |\n|--------------------|---------------|------------------:|---------------|------------------:|---------------|------------------:|---------------|------------------:|\n|                    |         AUROC |               Suf |         AUROC |               Suf |         AUROC |               Suf |         AUROC |               Suf |\n|                 CE | 0.631 (0.005) |     0.052 (0.009) | 0.634 (0.023) | **0.108 (0.001)** | 0.565 (0.061) |     0.111 (0.008) | 0.609 (0.018) |     0.090 (0.003) |\n|       CE + CMI-Reg | 0.681 (0.003) | **0.034 (0.008)** | 0.593 (0.000) |     0.112 (0.019) | 0.494 (0.036) | **0.101 (0.012)** | 0.650 (0.057) | **0.092 (0.005)** |\n|            Forward | 0.764 (0.003) |     0.047 (0.001) | 0.656 (0.012) |     0.107 (0.005) | 0.456 (0.097) |     0.095 (0.000) | 0.609 (0.018) |     0.103 (0.011) |\n| Forward + CMI-Reg  | 0.649 (0.019) | **0.034 (0.007)** | 0.592 (0.010) | **0.095 (0.011)** | 0.453 (0.140) | **0.087 (0.014)** | 0.655 (0.068) | **0.084 (0.013)** |\n|           Backward | 0.747 (0.004) |     0.048 (0.001) | 0.669 (0.010) |     0.106 (0.005) | 0.454 (0.091) |     0.096 (0.001) | 0.531 (0.103) |     0.094 (0.011) |\n| Backward + CMI-Reg | 0.659 (0.008) | **0.041 (0.007)** | 0.530 (0.002) | **0.096 (0.004)** | 0.485 (0.100) | **0.095 (0.002)** | 0.538 (0.094) | **0.089 (0.001)** |\n|                DMI | 0.660 (0.076) |     0.034 (0.011) | 0.641 (0.079) | **0.077 (0.025)** | 0.478 (0.177) | **0.088 (0.000)** | 0.437 (0.172) |     0.095 (0.014) |\n|      DMI + CMI-Reg | 0.642 (0.042) | **0.025 (0.007)** | 0.604 (0.020) |     0.107 (0.004) | 0.482 (0.177) | **0.088 (0.001)** | 0.435 (0.173) | **0.084 (0.008)** |\n|                ELR | 0.630 (0.001) |     0.056 (0.002) | 0.632 (0.012) |     0.112 (0.006) | 0.523 (0.051) |     0.101 (0.014) | 0.608 (0.033) | **0.088 (0.006)** |\n|      ELR + CMI-Reg | 0.694 (0.001) | **0.029 (0.008)** | 0.405 (0.002) | **0.066 (0.002)** | 0.375 (0.052) | **0.094 (0.007)** | 0.645 (0.065) |     0.096 (0.004) |\n|                GCE | 0.646 (0.005) |     0.052 (0.005) | 0.629 (0.004) |     0.112 (0.008) | 0.488 (0.076) |     0.097 (0.000) | 0.606 (0.020) | **0.090 (0.008)** |\n|      GCE + CMI-Reg | 0.637 (0.026) | **0.043 (0.026)** | 0.607 (0.000) | **0.101 (0.009)** | 0.614 (0.029) | **0.096 (0.002)** | 0.651 (0.068) |     0.093 (0.004) |\n|               Peer | 0.767 (0.000) |     0.045 (0.004) | 0.640 (0.024) |     0.107 (0.001) | 0.573 (0.049) |     0.110 (0.009) | 0.618 (0.034) | **0.085 (0.015)** |\n|     Peer + CMI-Reg | 0.654 (0.007) | **0.042 (0.009)** | 0.513 (0.011) | **0.092 (0.003)** | 0.487 (0.097) | **0.099 (0.003)** | 0.657 (0.064) |     0.096 (0.008) |\n\n\n\n\n> I am also skeptical of the finite sample estimator of the CMI regularizer.\n\n> Can the authors comment on the convergence of the proposed mutual information estimators with finite samples?\n\n\nWe will add the following proposition to the paper, characterizing the finite sample convergence of the MI estimator.\n\nProposition: Let $\\\\hat{I}\\_n$ be the CMI estimator defined in Eq (7), estimated with IID samples $(x\\_i, g\\_i, \\\\tilde{y}\\_i)_{i=1}^{n}$.\n\n Under Assumptions 3.1 and 3.2, and that $\\forall x_i \\in \\mathcal{B}: h_{\\\\phi}(f_{\\\\theta}(x_i)) = \\\\mathbb{P}(Y \\\\mid f_{\\\\theta}(x_i))$ and $ h_{\\psi}(f_{\\theta}(x_i)) =  \\\\mathbb{P}(Y \\\\mid f_{\\\\theta}(x_i), g_i)$, we have that, with probability $1 - \\\\delta$,\n\n$|\\\\hat{I}\\_n - I(Y; G \\\\mid f(X))| \\leq \\sqrt{\\frac{k \\log (2/\\delta)}{n}}  = O(\\frac{1}{\\sqrt{n}})$,\n\nwhere $k$ is a constant that depends only on $K$ and $|G|$.\n\nFurther, combining this with Proposition 5.1, we have that\n\n$Suf_g(f) \\\\leq \\\\sqrt{\\\\frac{2}{K}} (\\\\hat{I}_n +  O(\\\\frac{1}{\\\\sqrt{n}}))^{1/2}$.\n\nWe will include the full proof in the appendix in the revision, but it involves (1) showing the Lipschitzness of the mutual information in the total variation distance, (2) applying the Bretagnolle–Huber’s inequality on the two $P_{Y,G}$ distributions, and (3) plugging (2) into (1)."}, {"review_id": "UpFBclCJX4", "reviewer": "Reviewer_NBv5", "summary": "The paper investigates the challenge of sufficiency in datasets that contain label bias. Similar to previous work that focused on demographic parity and equalized odds the paper tackles the problem first from a theoretical point of view. The authors introduce the sufficiency gap as way of measuring the discrepancy between the sufficiency measure on the biased and unbiased labels. From this they introduce two methodologies to calculate the sufficiency gap without having access to the unbiased labels, which can be used to introduce an additional regularization term to the training loss in order to reduce this sufficiency gap.", "questions": "If the camera-ready allows for an additional page then I would ask the authors to increase the size of the plots in Figure 2. Pareto-fronts are still the standard and easy to interpret model capabilities. For the labels I would suggest to replace l_pred with naïve model as I assume this is the model with no bias mitigation method effecting it. It is also quite confusing how this model can exhibit a curve. It is better to plot points that have the same hyperparameters but run for several seeds be 1 point that has a confidence ellipse around it. This shows volatility of a method better. Also if the plots are larger I would suggest having the x-axis to be the same for all plots on the same dataset as this is better visualization for the reader. \n\nThe paper would also improve if the authors could provide results comparing their method with other methods, however I can see how this would fall out of the scope of the rebuttal. While I appreciate it if the authors would do this, I do not expect it.", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "The paper starts out from a strong theoretical basis. In order to come to these theorems, certain assumptions are needed. These assumptions are logically sound and do not effect the usability of the method and are in line with the promises that the paper made in the beginning with relation to the capability of the method. \n\nThe methods are tested on a wide set of datasets with different data types such as tabular, image, and text. All information is provided in the appendix of the paper in order to reproduce the experiments conducted in the paper. \n\nThe value of the novel methods proposed in this paper would have been greatly enhanced if the authors would have used a different method to compare their method against. Fairret is an example of a method that can function for predictive parity (the simplification of sufficiency when dealing with binary classification) and also works as an added loss term. These types of method would allow the authors to show that their specific focus on label bias is warranted as this cannot by done by traditional methods not assuming this level of label bias. \n\nThe paper uses the adult dataset, while still popular in the research area of fairness in AI, it has been shown that this dataset should not be used anymore for showing the capabilities of methods (Fabris et al., Algorithmic Fairness Datasets: the Story so Far). The paper contains datasets with both simulated and real noise with regards to the labels. This is already commendable but the authors could have also used the dataset of Lenders and Calders (Real-life Performance of Fairness Interventions-Introducing A New Benchmarking Dataset for Fair ML) which provides a dataset with these biased and unbiased labels on a dataset where biases are more inherently present. The dataset however is very small. \n\nThe interpretability of figure 2 could be improved; the plots are very small and the legend is not very easy to understand without reading the entire paper.", "was_revised": true, "final_justification": "My opinion of the paper remained unchanged based on the rebuttal provided by the authors. The paper works on a rather niche but interesting topic and the problem is tackled from a good theoretical point of view and validated by experiments.", "rebuttal": "Thank you for the insightful review and constructive feedback!\n\n> The value of the novel methods proposed in this paper would have been greatly enhanced if the authors would have used a different method to compare their method against. Fairret is an example of a method that can function for predictive parity (the simplification of sufficiency when dealing with binary classification) and also works as an added loss term. These types of method would allow the authors to show that their specific focus on label bias is warranted as this cannot by done by traditional methods not assuming this level of label bias.\n\n> The paper would also improve if the authors could provide results comparing their method with other methods, however I can see how this would fall out of the scope of the rebuttal. While I appreciate it if the authors would do this, I do not expect it.\n\nWe have added a multicalibration baseline from [1] following a suggestion from Reviewer xUeM. Specifically, we pre-process the data using the $T_g^{-1}$ matrix, learn an ERM model, then apply Algorithm 1 from [1] for multicalibration, using the single attribute to define $\\mathcal{C}$.\n\nWe present results for AUROC and Sufficiency Gap below, using the same setting as Table 2 of the main paper. We include only the Forward loss for our methods for brevity. We find that sufficiency gaps for this baseline remain substantially higher and AUROC is also significantly worse.\n\n\n\n|                   |         |                 income  |          |                 adult  |           |              lsac     |         |                crime    |\n|-------------------|--------------:|------------------:|--------------:|------------------:|--------------:|------------------:|--------------:|------------------:|\n|                   |         AUROC |               Suf |         AUROC |               Suf |         AUROC |               Suf |         AUROC |               Suf |\n|           Forward | 0.801 (0.002) | **0.038 (0.002)** | 0.887 (0.001) | **0.024 (0.004)** | 0.436 (0.006) |     0.097 (0.001) | 0.788 (0.011) |     0.056 (0.010) |\n| Forward + Suf-Reg | 0.812 (0.002) |     0.059 (0.003) | 0.898 (0.000) |     0.032 (0.001) | 0.690 (0.011) |     0.091 (0.016) | 0.789 (0.006) |     0.053 (0.006) |\n| Forward + CMI-Reg | 0.801 (0.002) | **0.038 (0.001)** | 0.888 (0.002) | **0.024 (0.004)** | 0.704 (0.172) | **0.065 (0.070)** | 0.802 (0.015) | **0.046 (0.011)** |\n|         GroupPeer |             - |                 - | 0.898 (0.001) |     0.035 (0.004) | 0.664 (0.010) |     0.095 (0.023) |             - |                 - |\n|                Multicalibration | 0.561 (0.019) |     0.078 (0.010) | 0.787 (0.004) |     0.017 (0.001) | 0.500 (0.000) |     0.107 (0.007) | \t0.508 (0.013) |     0.104 (0.012) |\n|            Oracle | 0.854 (0.001) |     0.003 (0.000) | 0.919 (0.002) |     0.013 (0.001) | 0.838 (0.011) |     0.018 (0.009) | 0.825 (0.013) |     0.046 (0.010) |\n\n\n\n\n\n\n> The paper uses the adult dataset, while still popular in the research area of fairness in AI, it has been shown that this dataset should not be used anymore for showing the capabilities of methods (Fabris et al., Algorithmic Fairness Datasets: the Story so Far). \n\n\nWe agree with the issues with the adult dataset, and have included it here solely due to its popularity in other fairness literature. However, we have actually also conducted experiments on the ACS Income dataset from [2], which has been proposed as a drop-in replacement for adult which addresses some of its issues (age, arbitrary threshold, outdated feature encodings).\n\n\n> The paper contains datasets with both simulated and real noise with regards to the labels. This is already commendable but the authors could have also used the dataset of Lenders and Calders (Real-life Performance of Fairness Interventions-Introducing A New Benchmarking Dataset for Fair ML) which provides a dataset with these biased and unbiased labels on a dataset where biases are more inherently present. The dataset however is very small.\n\nThank you for pointing out this dataset! We have conducted an experiment on it, using the biased human labels from the _ranking_ strategy as observed labels, and the eight categorical features from the Kaggle version of the dataset. \n\nDue to small sample size (n=856), and the fact that even the oracle (training on clean labels) has quite high Sufficiency Gap, we expect the gain of our method in this setting to be limited. However, we do find that (1) in all cases, applying CMI-Reg reduces the sufficiency gap relative to the base loss, and (2) applying CMI-Reg does not significantly change overall AUROC when the confidence intervals are taken into account.\n\n\n|             Method |         AUROC |               Suf |\n|-------------------:|--------------:|------------------:|\n|                 CE | 0.681 (0.043) |     0.050 (0.028) |\n|       CE + Suf-Reg | 0.682 (0.055) | **0.047 (0.008)** |\n|       CE + CMI-Reg | 0.683 (0.049) | **0.047 (0.030)** |\n|            Forward | 0.689 (0.048) |     0.055 (0.021) |\n|  Forward + Suf-Reg | 0.688 (0.060) | **0.052 (0.025)** |\n|  Forward + CMI-Reg | 0.674 (0.044) |     0.054 (0.014) |\n|           Backward | 0.692 (0.032) |     0.078 (0.002) |\n| Backward + Suf-Reg | 0.691 (0.053) |     0.063 (0.016) |\n| Backward + CMI-Reg | 0.688 (0.058) | **0.051 (0.007)** |\n|                DMI | 0.688 (0.044) |     0.061 (0.005) |\n|      DMI + Suf-Reg | 0.688 (0.056) |     0.051 (0.008) |\n|      DMI + CMI-Reg | 0.708 (0.036) | **0.048 (0.003)** |\n|                ELR | 0.667 (0.035) |     0.060 (0.012) |\n|      ELR + Suf-Reg | 0.669 (0.043) |     0.067 (0.022) |\n|      ELR + CMI-Reg | 0.666 (0.040) | **0.057 (0.009)** |\n|                GCE | 0.683 (0.026) |     0.073 (0.017) |\n|      GCE + Suf-Reg | 0.688 (0.056) |     0.054 (0.018) |\n|      GCE + CMI-Reg | 0.681 (0.053) | **0.048 (0.009)** |\n|               Peer | 0.681 (0.035) |     0.067 (0.020) |\n|     Peer + Suf-Reg | 0.660 (0.006) |     0.062 (0.011) |\n|     Peer + CMI-Reg | 0.692 (0.043) | **0.053 (0.018)** |\n|          GroupPeer | 0.687 (0.046) |     0.069 (0.012) |\n|             Oracle | 0.713 (0.075) |     0.050 (0.009) |\n\n\n\n> The interpretability of figure 2 could be improved; the plots are very small and the legend is not very easy to understand without reading the entire paper.\n\n> If the camera-ready allows for an additional page then I would ask the authors to increase the size of the plots in Figure 2. \n\nThank you for these suggestions! We will make these changes for the next revision if additional space is available.\n\n\n[1] Multicalibration: Calibration for the (Computationally-Identifiable) Masses. ICML 2018.\n\n[2] Retiring Adult: New Datasets for Fair Machine Learning. NeurIPS 2021."}, {"review_id": "RNUM96mfSB", "reviewer": "Reviewer_HAo2", "summary": "The paper studies the problem of mitigating bias in the presence of noisy labels. It specifically considers the fairness notion of **Sufficiency**, where the true outcomes are independent of the group conditioned on the prediction. First, the authors provide a theoretical analysis demonstrating that enforcing fairness with respect to noisy sensitive attributes necessarily results in a non-negative group sufficiency gap with respect to the true labels. They then propose a regularization based on mutual information that can upper bound the sufficiency gap, assuming access to noisy labels and a transition probability matrix $P(y_{\\text{noise}} \\mid y, \\text{g})$. The paper also presents an extensive experimental study on multiple datasets with both synthetic and real noisy data to evaluate their proposed regulation method.", "questions": "**Q1.** My first question concerns the entropy term in Equation (4). Specifically, the entropy $H(Y \\mid F(X))$ is not conditioned on the sensitive attribute $G$, so the use of  $T_{g_i}$ is a bit confusing to me. This also appears in the appendix (lines 1049–1055), particularly in line 1052, where the definition of mutual information appears to differ from the one presented in line 202 of the main text. Could you clarify this discrepancy?\n\nThe rest of my questions concern the experiments.  \n**Q2.** Regarding the \"Noise Types,\" I would like to ask why the authors add noise only to one specific sensitive group. Performing additional experiments with the same level of noise applied to different groups seems straightforward and would provide a more comprehensive evaluation.\n\n**Q3.** Regarding the additional experiments, as mentioned before, further evaluation—especially studying the effect of different noise levels in the dataset—would be beneficial. Combining this with experiments that use estimated transition probabilities would provide a more thorough evaluation.", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Strengths**  \n1. The paper is well written, easy to follow, and presents a comprehensive literature review.  \n2. The concept of noisy labels is particularly interesting and effectively captures many practical challenges.  \n3. The authors contribute by focusing on the notion of *sufficiency*, which has received less attention compared to other fairness notions in the literature.  \n4. They provide a theoretical analysis that highlights the importance of their proposed regularization approach.  \n5. Extensive experiments are conducted to evaluate the performance of the method.\n\n**Weaknesses**  \n1. **Dependence on transition probabilities**: The method requires knowledge of the transition probabilities. While the authors acknowledge this limitation, and I understand that estimating such quantities may require additional information, there are existing methods that attempt to relax this assumption (e.g., reference [48]). The claim in line 126 that the method requires only noisy labels seems overstated, as the transition probabilities are also needed.\n\n2.  **Entropy term and conditioning**: My second concern is about the correctness of the proposed regularization method. I found it somewhat difficult to follow the derivation of the entropy term in Equation (4), particularly the use of \\( T_{g_i} \\) as the entropy is not conditioning to the sensitive group. See also question Q1.\n\n3. **Evaluation**: Although the paper includes many experiments, additional evaluation—especially studying the effect of different noise levels in the dataset—would be beneficial.", "was_revised": true, "final_justification": "The paper is strong overall, with its main weakness lying in the presentation of the core concepts and the experimental evaluation.\n\nWhile the authors clearly explain their motivation behind the proposed formulation, I believe the paper could benefit from a clearer presentation of the backward correction approach and the role of the regularization.\n\nRegarding the experiments, I would have expected a more thorough evaluation in the main part of the paper, including:\n1. Results under different noise levels (also small noise levels),  \n2. A comparison between the estimated and true transition matrices,  \nalong with a more conclusive analysis of the impact of both regularization methods.", "rebuttal": "Thank you for the insightful review and constructive feedback!\n\n> W1: Dependence on transition probabilities. The claim in line 126 that the method requires only noisy labels seems overstated, as the transition probabilities are also needed.\n\n\nWe note that we do experiment with using Confident Learning [1] to estimate the transition matrix only from noisy data, though this generally results in worse performance than using the true transition matrix. We emphasize that as better transition matrix estimates are developed, our method will improve in this setting as well. We will clarify the claim in L126 to reflect this.\n\n\n\n> W2: Entropy term and conditioning.\n\n> Q1: My first question concerns the entropy term in Equation (4). Specifically, the entropy H(Y | F(X))  is not conditioned on the sensitive attribute G, so the use of T_{g_i} is a bit confusing to me. In line 1052, where the definition of mutual information appears to differ from the one presented in line 202 of the main text. Could you clarify this discrepancy?\n\n\nIn our class-attribute-conditioned noise model (Assumption 3.1), the backward correction always involves $T_g$ because noise depends on $(Y, G)$, which leads to $\\\\mathbb{E}_{\\\\tilde Y \\\\mid Y = y,\\\\, G = g} [(T_g^{-1})\\_{\\\\tilde{Y},\\\\cdot}] = e\\_y^{\\\\top}$. Using the marginal matrix $T$ would result in a biased estimator, as it does not correspond to the actual probability of corruption for that sample.\n\nAfter this backwards correction, $−\\\\log \\\\pi(Y \\\\mid f(X)) $ is independent of $G$. Thus, the two definitions on L1052 and L202 are equivalent. We will modify L1050 to the following to make it more explicit:\n\n$\\\\mathbb{E}\\_{\\\\tilde Y \\\\mid G, Y=y}[(-T_G^{-1}\\\\log\\pi(\\\\cdot\\\\mid s))\\_{\\\\tilde Y}] = -\\\\log\\\\pi( y\\\\mid s)$.\n\n\n\n\n> Q2: Regarding the \"Noise Types,\" I would like to ask why the authors add noise only to one specific sensitive group. Performing additional experiments with the same level of noise applied to different groups seems straightforward and would provide a more comprehensive evaluation.\n\nWe add noise only to one specific group as it is the worst-case scenario. Because the entire disparity is concentrated in one group, the resulting sufficiency gap is maximal for a fixed average noise rate. Further, if both groups were noisy, as one group's noise matrix approaches the other, training a naive calibrated classifier with respect to noisy labels is sufficient to remove the sufficiency gap as shown in Corollary 4.1.1, and thus the problem becomes trivial.\n\nWe appreciate the suggestion to add noise _solely_ to the _other_ protected group. In our current experiments, the noisy group was selected largely for historical reasons, and we did not experiment with noising the other group due to computational considerations. We expect our method to perform similarly with noise in the other group, and will verify this in a future revision.\n\n\n\n\n\n\n\n[1] Confident Learning: Estimating Uncertainty in Dataset Labels. JAIR."}]

Author rebuttal:

Meta-review: The paper tackles the important problem of achieving group sufficiency (equal calibration across demographic groups) under label bias, where observed labels differ systematically from true labels at different rates across groups. The authors provide theoretical analysis showing that enforcing fairness with respect to biased labels necessarily results in group miscalibration with respect to true labels, and propose a conditional mutual information-based regularizer (CMI-REG) to minimize an upper bound on the sufficiency gap. They evaluate their approach across eight datasets with both synthetic and real label noise, demonstrating consistent improvements in sufficiency gap reduction of up to 9.1% without significant accuracy loss.

The reviewers and authors engaged in substantive technical discussions that strengthened the paper considerably. Reviewer xUeM raised concerns about the lack of meaningful baselines and suggested implementing a multicalibration baseline, which the authors successfully added along with related work discussion. Reviewer fbDW questioned the method's performance when transition matrices must be estimated rather than known, prompting the authors to provide extensive additional experimental results for this challenging setting. Reviewer HAo2 identified confusion in the derivation of Equation 4 regarding the entropy term conditioning, leading to detailed mathematical clarification from the authors about why group-specific transition matrices Tg are necessary for unbiased estimation despite the entropy not explicitly depending on group membership. Reviewer NBv5 suggested additional datasets and experimental comparisons, resulting in the inclusion of results on real-world biased label datasets. Through this iterative process, all reviewers moved toward acceptance, with the final paper incorporating multicalibration baselines, additional noise levels, finite-sample convergence guarantees, and clearer mathematical exposition.
Decision: Accept (poster)