When majority rules, minority loses: bias amplification of gradient descent
Authors: François Bachoc
Jerome Bolte
Ryan Boustany
Jean-Michel Loubes
Venue: NeurIPS 2025 poster ()
Keywords: Bias
fairness training
gradient descent
unbalanced learning
Primary area: optimization
TL;DR: We analyze imbalanced training loss, showing that gradient descent dynamics can gradually reduce bias and recover minority-specific features with longer training.

Abstract: Despite growing empirical evidence of bias amplification in machine learning, its theoretical foundations remain poorly understood. We develop a formal framework for majority-minority learning tasks, showing how standard training can favor majority groups and produce stereotypical predictors that neglect minority-specific features. Assuming population and variance imbalance, our analysis reveals three key findings: (i) the close proximity between "full-data" and stereotypical predictors, (ii) the dominance of a region where training the entire model tends to merely learn the majority traits, and (iii) a lower bound on the additional training required. Our results are illustrated through experiments in deep learning for tabular and image classification tasks.

When majority rules, minority loses: bias amplification of gradient descent
François Bachoc
Jérôme Bolte
University of Lille
Toulouse School of Economics ANITI
Institut Universitaire de France (IUF)
francois bachoc@umiv-iille....
jerome bolte@tse-fr.en
Ryan Boustany
Jean-Michel Loubes
Toulouse School of Economics ryan. trra.booustanyttee--f
Université de Toulouse
ANITI & Regalia INRIA
jean-mi chel. a. www.ess@inria...................
Abstract
Despite growing empirical evidence of bias amplification in machine learning, its theoretical foundations remain poorly understood. We develop formal framework for majority-minet minority learning tasks, showing how standard training can favor majority groups and produce stereotypical predictors that neglect minority-specifie features. Assuming population and variance imbalance, our analysis reveals three key findings: (i) the close proximity between "full-data" and stereotypical predic- tors, (ii) the dominance of region where training the entire model tends to merely learn the majority traits, and (iii) a lower bound on the additional training required. Our results are illustrated through experiments in deep learning for tabular and image classification tasks.
Introduction
Imbalanced data are pervasive in machine learning, spanning rare-event detection, fraud, faults, medical anomalies, security, finance, and modern LLM pipelines with unequally represented subpop- ulations, see e.g.. [26] 29 for some references. A sensitive case arises in faimess-related applications, where decisions apply to human beings K TO 12 33 Addressing this issue increasingly impor- tant notably under regulatory frameworks such as the European Union's 'son's AI Act, which emphasizes non-discrimination and risk mitigation.
In all these settings, the goal is to learn predictors that genuinely capture minority structure.
Our focus is scenarios with two distinctive characteristics. First, the imbalance is typically significative, as we work directly with raw data without resampling or augmentation. Second, the imbalance is not corrected at the data level but addressed only through the training dynamics of gradient descent. An empirical fact, well known to practitioners, is that imbalance is not only preserved but often amplified by training: models initially align with the majority component and only later start to capture minority features. In the fairness literature, this is sometimes referred to as bias amplification 18 45 46 47 Related simplicity-driven behaviors have been observed in representation learning 6 16
22
39
This phenomenon has been documented since the 1990s in the class imbalance literature, [1] which motivated numerous work and heuristics: data-level remedies (oversampling, under-sampling. synthetic examples 99 43 algorithm-leve adjustments (15), cost-sensitive learning [14), focal and
reshaped losses 30 35 With the advent of deep learning, the issue became even more acute, as
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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high-capacity models and standard training budgets (a few hundred epochs) tend to privilege majority signals [26). Despite this history, the mathematical mechanisms of imbalance amplification are still poorly understood geometrically, especially in nonlinear nonconvex regimes 37 clarify these mechanisms which are essential for sensitive applications.
Our goal is to
Using Kantorovich-typp arguments, we develop a theoretical and a geometrical framework that explains why and how gradient-based training first produces stereotypical predictors, i.c. aligned merely with the majority, before catching-up and entering debiasing phase where minority features start to influence predictions.
Contributions.
They are as follows:
We first formalize the problem generic majoritt-minorii learning task min L := L1 + Lo- with Lo < 1 using second-order differentiability domination. We prove that each critical point of T., which corresponds to predictor, can be paired with a critical point of L., termed stereotypical predictor (Section 2.2). We bound their distance: it is what we call the stereotype gap.
The proximity of and Lj implies that the region where minimizing L is 'equivalent" mini- mizing L1 occupies nearly the entire parameter space (Section 3.2). This results in a close overlap between L1 training gradient path and the actual training path, illustrating how standard training neglects minority-specific characteristics (Appendix [.22. This proximity between learning curves, as the proximity between representative and stereotypical predictor, somehow deceptive, since the minority features lie precisely in what differentiates them.
We prove that gradient descent may require a fairly long training time to merely identify stereotypi- cal predictors, ignoring sinority-specifi aspects (Section 3.3). A common training failure is when a long initial training phase stalls at a stereotypical predictor. Although this predictor lies close to its corresponding representative predictor, escaping that neighborhood, and thus debiasing the model, often requires much more training because the gradients there are tiny. We derive a lower bound on this extra training duration. The corresponding ratio is called the catch-up overcost ratio. It is debiasing overcost. It quantifies the additional training time required to achieve unbiased predictions.
We illustrate our theoretical findings through numerical experiments on tabular and image- classification tasks with deep neural networks (Section Minority awareness emerges in preliminary experiments and appears linked to training duration; it also persists under alternative learning strate- gies as Adam W or XGBoost.
Related literature. The bias amplification phenomenon is well documented experimentally, see, for example, 6 18 37 and references therein. Yet few theoretical results clarify its causes. The earliest paper we are aware of that addresses the issue is []] via a diagnostic of an early-phase majority bias and via some algorithmic fix (bisect classwise gradients) with empirical speedups. In a Gaussian setting with ridge regression, (42) show that, for a single pooled model, the between-gree grropp gap in expected test risk can exceed the corresponding gap obtained by training separate models for each group. Leveraging the closed form of the ridge estimator, they analyze the asymptotic behavior of this bias ammlaiicaliiiinnn measure. In a related direction, [31) introduce a parametric Gaussian-mixture mixture framework with tunable imbalance and derive analytic ridge-regression solutions, comparing group- wise risks for a jointly trained model versus per- per-group models. All these results rely on analytic solutions and their asymptotics. Complementing this line, [15] analyze optimization dynamics and articulate theoretical conditions that clarify a phenomenon they term minority initial drop (MID) gradients. They also provide
an early deterioration of minority recall driven by majority-dominatee sufficient conditions for monotone per-class loss decrease and show that vanilla (stochastic) gradient descent can be sub-optimal under imbalance. Their perspective focuses on loss trajectories and per-class gradients, rather than the parameter-space geometry and time-to-learn bounds we develop below. In this sense, their results are complementary to ours, and a general, model-agnostic theory of bias amplification in modern ML remains largely open.
Notations. Notations on matrices, differential calculus and geometry, that are used throughout the paper, can be found in Appendix A.I
2 Predictions for majority-minority problems in machine learning
We first present our majority-minoris scenario in Section 2.1 as a minimization problem: min L == Tn + Log We aim at estimating the distance between a predictor obtained by minimizing the total loss L and neighboring majority-base predictor obtained by minimizing L1- In practice, the latter may
2

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represent a biased or stereotyped view that a user holds about the underlying problem. We show that small population and low variance for the minority group lead to proximity between the predictor and the majority-based predictor, making them difficult to distinguish. Our results are first presented for abstract equations (Proposition ( and general variational problems (Theorem I discussions on learning appear in Section 2.3 and in the Appendix.
2.1 The setting: majority-minority model and generic losses
model. We consider n observations of a variable = (X,Y) E R x x (d > 0) that can be divided into two groups following the values of a binary variable A E 10, In our scenario, the data unbalanced: there is majority group A (with cardinality denoted m1) and a minority group A = O (cardinality no), typically with no < n1- This heterogeneity, i.e., the variable A, may be unknown the user.
majority-minort
Consider a collection of models or predictors fo R + R indexed by parameters or weights 0 E Rd that are learned by minimizing some empirical loss function over the learning set. Given a discrepancy measure & IR? - R,+ we may define the total, majority and minority losses as: for A E Rd
1 n I(0)= = n
Σ e(fo(X.),
V.)
TI Σ (()(())) n
Σ ((fo(X,), VI).
n
(o)
In the training phase of a learning process, the parameters are often computed through first order methods and thus eventually through vanishing gradients. Assuming both E and fo are differentiable, we are thus led to consider equations of the form: VL(O)- = O, (L) = 0, for j E {0. 1}. In a strongly imbalanced scenario, Lo may become negligible with respect to L1 so that the equations VL, = 0 and VL = 0 have very close solutions. On the other hand, this proximity does not prevent solutions to the equation L1 (6)
from producing biased or stereotyped predictors they ignore, by definition, the influence of data underlying Lo-
The aim of the following sections is to study this phenomenon and provide set of assumptions for estimating the distance between full-data and stereotypical predictors.
Generic losses. For the rest of the article, we adopt genericity perspective on loss functions by assuming that their critical points are non-degenerated. For G F² E twice differentiable this means that
(G = - = V²G(0) invertible.
In other words, G is a Morse function. These functions are generic in the sense that they form an open dense subset in C* (k(Rd, .R) for the C2 topology whenever k 2 see e.g., (17].
In the machine learning perspective, this is not extremely demanding as, for fixed C2 function G, perturbations of the form E e + Gx,,(()) G(z) +-||æ ell with 7 are Morse for almost all couple (7, e) E 4 x " actually it holds true with linear perturbations, see c.g., [40). This approach aligns with statistical and learning practices, both through ridge regularization (pioneered in [23] whose use in data science developed for instance in 19 and references therein) and the weight decay approach in deep learning (8)
2.2 Perturbation results for critical points of generic losses
Assume L L1 + Lo is general cost. The spirit of the following results is that L1 corresponds to majority behavior while Lp is attached to minority features, for instance as in the scenario of Section 2.1 In an analytical setting. it translates into property of the type: Lo is negligible w.r.t L1 (see the assumptions below). We then aim at comparing crit L and crit L1; argmin-loc L and argmin-loc _ Note that the theorem below is a general-purpose purpose perturbation result, it is applied in machine learning setting in the remaining sections.
Theorem (Strong imbalance and critical points). Consider two functions L, and Lo from Rd to RR
that are two times continuously differentiable, and subset K C Rdd
Assume that there are strictly positive numbers c, M,T T such that
Recall that notations are provided in Appendix A.1
3

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Strong Morse property: For all 0 E K.
( S c = Pmin P (V2),(0)) 2 6,
(1)
Lipschitz regularity: for all 01, 02 E
Pmax (I)) LI M| 0211,
Puax (V2 (o22 - Mi 82ll,
Bounds on the 'minority loss':
sup VLo S T.
9EK
sup Pmax (a² I T.
(5)
BEK
Assume farther that
8²
c
T min
(6)
8'8'32M0
6r 5
dist и (crit L n K, bdry K) 2 67
dist и (crit L1 K, bdry K) 2
8
Then, for each 01 E crit n K (resp. 0 crit L n K) there exists unique corresponding 0 E crit L n K (resp. 0, E crit L1 n K) such that
4T
10 0 S
and 8, 0, have the same indexes, that is the same number of strictly negative eigenvalues of the Hessian matrices L2 and (().
Corollary (Distances between critical and local minimizer sets). In the context of Theorem 1 if crit L1 O is non-empty, then crit K is non-empty and we have
45 S
dist (crit L1 n K, crit L n K)
(8)
5
Also, if ararmmn--oo L1 n K is non-empty then argmin-loc L L K non-empty and we have
4T
dist (argmin-loc L1 n K, argmin-loe L n K) <
8
Finally, for each 0 E argmin-loc LA O K, there is 0' E aargmin-lo Ln K such that the ball B(0', (i))
contains 0, and I. is 5/8 strongly convex on this ball.
Comments on Theorem I and Corollary Assumptions 2
6 are warranted whenever V Lo is
Cm-small on K, i.e. small with respect the functional semi-norm
(L)
8 Lo)
Loll,,, max
max sup a= BEK 00,
max
sup
a,6=1..... I....... BEK
a0,00,
Assumption ( simply means that the critical sets are not too close to the boundary of K. the critical sets lie in a compact set, it suffices choose K large enough to satisfy the assumption.
simple reading of Theorem D is therefore that when and L1 are sufficiently close (on K), they share the same 'geometry' i.e.. they have the same number of local minimizers and, more generally, the same number critical points for a given index, with, in addition, corresponding points lying at small distance from one another.
Note that 32 establishes results similar to Theorem I and Corollary I but in a different setting: the comparison of theoretical and empirical risks under i.i.d. random data. In contrast, by relying on Kantorovich's 's method of proof, we impose no assumptions on the data. instance of Theorem for the special case of linear regression is provided in Appendix B.

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2.3 A machine learning view: the representative and stereotypical predictions
Let us interpret the above within a learning perspective. Under the premises of Theorem 1 we consider a machine learning model with loss L : 8 + L(0) decomposed into a sum L = L1 + Lo where L1 and Lo respectively correspond to some majority and minority phenomena.
critical point of I. is called a representative prediction, as it takes into account all available data encoded within I., i.e. both those in Ia and e In the majority
model, the critical points
of 1 ignore data corresponding to the case when A = 0, we thus call them stereotypical predictions. The quantity dist H (crit L K, crit n K) is called the stereotype gap.
Roughly speaking Theorem tells us, in particular, that each representative prediction corresponds to one and only one stereotypical prediction and that these predictions are close whenever the ratio
A Pmax (V² P (V² (1))
is uniformly small. This ratio is the key quantity that governs the stereotype gap.
The result is even more accurate, as Theorem I shows that the minimizers of L and L1 actually come by pairs as well, so that the stereotypical and representative predictors obtained in practice are 'dangerously' close in mijority-minority scenario. As we will see through theoretical and numerical experiments, this renders the training phase delicate and potentially biased. Using the well-known fact that gradient descent converges to critical points in the Morse case (see next section and Appendix [A.4), we may empirically estimate the stereotypical gaps and the associated 'debiasing training time' in our imbalanced setting (see also the following sections).
Protocol (Table T opposite): find a stereotyp- ical predictor 0, via the gradient flow - -V,, with Kaiming random initialization. Initialize from this predictor 0, and follow the flow of -V,, with the guarantee (see Corollary D of reaching the corresponding representative pre- dictor 0. Use these values to estimate the gap dist (crit L, crit L1) via proxies like 110 - 01ll. and to define debiasing time from 81 to its rep- resentative 9 using gradient descent on L with stopping criterion ||0++1 - 01 2 0.99| - -
Table 1: Stereotypical and representative predic- tions for imbalanced CIFAR-2 (na/n n 3%, Appendix [.I) with ResNet 18. We report the average and standard deviation over 30 runs.
Metric
Mean
+ Std
Debiasing time
469 epochs
9.4
| 01ll
0.6723
+ 0.0083
0 0 1l
0.0353
+ 0.0047
@ A
0.00602
+ 0.00007
[1
3 Learning unbalanced data with the gradient method
3.1 Gradient descent training
In this section, we study how gradient descent procedures may bias predictions in the sense that a 'careless training' may yield a stereotypical predictor rather than representative one. Gradient descent training C2 loss L modeled through the ODE (see Appendix for the representation of ODE curves):
d T. 0(1) - -0--VIL000 (((()) with 0(0) = Iiit E R.d.
(10)
The ODE solution is called training trajectory. In Appendix B.22 for the special case of linear regression, we also consider the counterpart 0, (t) of ((1) with L replaced by LI. We show that 0 (t) and 0(1) are close under strong imbalance.
3.2 The majority-traiee training and the majority-adverse zones
For C smooth losses L L1 + Lo. the sngjority-training zone is defined by
Zunj = {0 E Rd (V)(A), (L) > 0}.
II would be more natural to reserve that name for local minimizers, as those are generally obtained after training. but we do for simplicity.
5

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In this region, descending along the gradient of L also decreases L1, and vice versa. In other words, maj zone where training with gradient descent implies training the majority L1. The segjority-adversy zone is defined as
maj-ady (0 : (L) s 50 so that TRd Zmaj Zinaj-adv
(11)
We can similarly consider the minorityy-rrinnin and the minority-adverse zones. One easily sees that, under the Morse assumption, critical points of L or L1 lie in between Zuaj and Zmaj-adv (see Proposition in Appendix C.2 for details). In other words, the stereotypical and representative predictors lie on the boundary of Zmaj-
We now establish two major facts: first, the majority zone is typically large, meaning that training the entire model often results in learning only the majority traits (see also the illustration of Figure second, the majority- zone promotes the training of the minority loss.
Theorem 2 (Majority adverse zone). Let =
{0 E K; dist (0, bdry K) 2 }. Under Theorem assumptions:
Bast
Zia ude
01, 2T 8
mmaj-adv n nK= C
U
B
8 Ecrit Link
1
U
B 01:1
0 Ecrit LinK
Remark 1 (On the Under the assumptions of Theorem in the high dimensional regime the majority adverse zone has a volume lower than (44-4) -much lower in general as we have chosen a conservative bound. Note also that the stronger the imbalance, the more negligible it becomes, see the comments after Theorem[I
zone size).
The majority region (white) covers nearly the entire space, while the majority- adverse region (red) is small. With ran- dom initialization, training typically begins in white; hence majority features are learned during the initial phase, which accounts for most of the path length (though not the time). The gradient trajectory then enters a red zone, where minority features are improved. De- spite the short arc length in red, the time spent there may be very long. These transient pas- sages through red before convergence corre- spond 'unlucky' curves.
Figure 1:
Lemma 1 (The majority adverse zone favors mi- nority). For 0 Zmaj-adv. we have
(VL(O), (o)) 0.
Thus training trajectory 0 + Hd evolv- ing within Zmajaadv is such that Lo(O())) is non- increasing over the interval 1.
In other words, when the trajectory evolves within the majority-advend adverse zone, the dynamics learns minority features.
3.3 Lower bounds for debiasing duration and catch-up overcost
Although we cannot, at this stage, provide worst-case 'biasing' complexity bounds, we can obtain a lower bound by placing ourselves in setting with high risk of bias toward the majority. Consider an 'unlucky gradient curve' t + (tt) solving (10) that effectively ignores the minority until it meets majority predictor. On [0, stereotypel. the trajectory carries the initial condition Ons to critical point of L1, viewed as a stereotype and denoted
stereotype == O1 stereotyp))
Up to time i stereotype: it is 'as if' only L1 were trained - the minority is entirely ignored. Thereafter, (t)) moves toward critical point of the full loss L, denoted 0, which we interpret representative

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165 109
Figure 2: Training curves on 'Imbalanced CIFAR 2' with ResNet18 (see Appendix A.2 Left to right: unlucky curve with stopping rule based on minority recognition, i.e., Acco > 99%; random trajectory with the same rule; random trajectory with global accuracy stopping rule Acc 99%. Unlucky initialization has 600 epochs while 'careless training' (third one) needs 100 epochs and has much higher final Lg value. Middle; random initialization with minority aware stopping rule training has 140 epochs. In the real world Acco > 99% is not realistic criterion as we do not know the minority class. Conclusion: risk-averse training should rely on considerably longer training (here +500%), confirming the results of Section 3.3 For more confident training, substantially longer runs are still required (here +40%).
predictor. By the neighboring ones may be quite detrimental to fair predictions shown in Figure 2
existence theorem, this trajectory typically exists. This curve and
The next proposition shows that t stereotype is typically large as ||@ stereotype 0|l is typically very small (see Theorem and T)ble
Proposition 1 (Training duration). Assume that L is twice continuously differentiable Rd Con- sider ball B containing 0 and {0(t); 2 0}. Assume that for some M oc and for all 0 B.
Paaax
(V²L(0)) S M.
(12)
1 M
1
01
Assume ① 0 0 sterrotype then 'sterootype
log
01
sterrotyppe
Let us give simple yet illustrative example showing that the bound is tight and that training duration becomes rather long in the small step-size regime typical of large-scale deep learning problems.
Example (a) (The bound tight). Consider elementary but instructive model [1(x) x²² 2/2
Lo 6((z 2) with 0 0, ó being small imbalance factor that reflects the minority scenario.
Simple computations give the representative predictor T I+5C while the stereotypical predictor is 0. The time to reach the stereotype 0 from Finit < 0 is
T sseernttyp
T
Tinit
sttercotype 1 8
log
whence Proposition I is tight.
T
T. stereotype
(b) (Small steps yield long training duration). Consider now Tk++ Fy 7/ V with a short step 17 10-2, as it could be done in deep learning. Let Tinit -2c is multiple of for convenience. while remoteness from 0 reflects the ignorance of a blind user on the exact location the minimizer. Since [mii 2 2c and [ @Stereotype
3 x òc:
Finit T
log (2)
stercotype
1 log @stercotype
1 1
The discrete time when the stereotype is reached may be approximated by ksternotype R latereotype //n
We may provide a table for 77 = 10 2, Tinit = -2c.
t stereotype (bound)
kstereotype (bound)
10
log (200) & 5.25
R3 525
1,0I
10 10-3
log (2000)
7.59
759
1,p01
10
log(20000)
9.89
2 989
1.000T
Thus strong imbalance together with traditionally cautious DL steps give long training durations.

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Next, we provide lower bound on the extra-time i cattchup,e - E stereotype needed to achieve the relative precision, where (0, 1), catchup.c sterootype and
[0 catchup.c) 01
0 S E.
(13)
|10 stereotype
This extra time is interpreted a catch-up time for the algorithm to detect the minority with an acceptable precision. Indeed from time stereotype to catchhup.e> the trajectory 0(t) leaves "a stereotype' and becomes closer 'a representative predictor'. is a debiasing phase in which the algorithm progressively removes the bias it has itself created during the preliminary training phase.
Proposition 2 (Debiasing duration Assume that L is twice continuously differentiable and satisfies (2). for the same M and B. Assume that 0 + 0 stereotype- For 0 < E < 1, consider t catchup, such
that 13 holds. Then I catchup.c sterrooyp
Z M
log
Example 1 (continued). (On the length of debiasing duration) Back to the setting of Example[] from simple computations, the debiasing time needed to go from the stereotype Tstereotype 0 to relative precision E E (0, 1) around the representative F = 1+8
6
c is
1 log
t catchup.e stereotype
+ 6
so Proposition is tight. For gradient descent 2x+1
z V (æ+k), the error decays geometrically
with factor (1 8). Thus the number iterations to reach the same relative precision satisfies
log(1/e)
k catthhup (1,, 5,E) 2
log
log(1 n11 (1++))))) 8))
((1+6)
which is version of Proposition 2
Thus an approximate debiasing step count with 'standard' ML
learning rate v 10-2;
8
= 10
10-2
10-3
10-2
10
228
456
684
10-3
230
460
690
Hence, for strong imbalance (6 < 1) and small steps, debiasing typically costs a few hundred additional iterations even after reaching the stereotype.
Numerical experiments
We study the effect of subgroup imbalance in supervised deep learning using image (CIFAR-10 28
EuroSAT (2I)) and tabular (Adult (5)) datasets. Each dataset is denoted by D = [(X,, Y, A,) } where (X,, Y;) is an input-label pair and A, E (0. 1} is binary attribute is minority). While not used during training, it enables evaluation of model performance across imbalanced subgroups. In each experiment, we report the global loss L = Lo + LI, and average loss per sample in each group, i.e., (nLo)/ no and (nL1)/m, For details on the implementation setup, see Appendix F
predictions. We evaluate class-balance using training (and occasion- nally test) accuracy, denoted by Ace, Acco (see Appendix A.2 as our focus on optimization under imbalance. To measure the time cost of 'well balanced training', we track the number of epochs t needed reach a threshold accuracy level
Metrics for
[0,1]: min (A.) 2 k). Taebius rEN
Tearly
min (cc(O,)) 2 x), rEN
T Tinal
Tinul
T Tearly.
We define the catch-up up overcost the relative delay to reach good class- balance prediction, i.c., satisfying minority accuracy:
Tacbias
Catch-up Overcost
Tearly
See also Proposition 4 An Appendix B.3 for complementary results relative values of Lo.
8

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Imbalanced CIFAR-10. We investigate the effect of class imbalance CIFAR-10 using models from 100K 25M parameters (see Table 2 The original dataset has 10 classes with 5000 samples each. To create imbalance, we subsample one class (denoted A 0) to retain
samples, and keep the others (A 1) unchanged with 71 x 5000. As in 26 we define the imbalance ratio as Ç Ro/5500000 /5000, which gives group proportion no/ (Mg + n1
Ç/(C + 9), and consider four imbalance levels: Ç E {1%, 10%, 30%, 80%). Figure we show the results for ResNet- 18 (see also Appendix D for more). For Ç 1%, Acco remains close to zero for about 60 epochs, following a stereotypical training curve (see Appendix).
19%
C= 10%
[ 30%
100
Train accurscy (96)
80
40
20
20
20
60001
A-0
A-1
Global
<theeshold
Figure 3: Training accuracy for different subgroup imbalance scenarios (1%, 10%, and 30%) using R on CIFAR-10 and threshold 90%. Greater imbalance delays the learning of minority features: their accuracy reaches k later.
Table 2: Catch-up overcost (in %) for each model across imbalance levels Ç & (1%, 10%, 30%, 80%).
We report means over 3 runs with thresholds (90%, 99%). ). and model parameter counts.
Models
Number of parameters
K
90%
K: 99%
1%
10%
30%
80%
1%
10%
30%
80%
MobileNetV2
543K
450
275
166
62
52
42
0
SqueezeNet
727K
270
203
150
0
55
53
32
0
VGGI1
41
9M
291
171
114
0
53
44
25
0
ResNet18 20
11M
292
164
113
0
61
49
31
0
VGG19
20M
152
112
0
70
65
50
0
ResNet50
25M
157
86
68
0
50
37
37
0
ResNet101
42M
145
90
64
0
34
36
26
0
EuroSAT. We use a ResNet18 model and evaluate its behavior on binary classification task derived from the EuroSAT dataset. Images are labeled according to binary attribute A, where A = 0 corresponds to bluish images (no/(no- + n1) Av 0.03) and class proportions and use the imbalance present in the original dataset. Figure&ddsppla accuracies for both subgroups evidencing a catch-up overcost of 45% for a threshold k = 90%.
to all others. We do not modify
losses and
Adult income census. We train a TabNet classifier on binary task from the Adult dataset [5]. The minority group (A = 0) includes high-income women, representing only 3% of the training set. The majority group (A = 1) includes all others. We preserve the original class distribution and train with cross-entropy loss, tracking subgroup metrics. Results are shown in Figure and we have catch-up overcost 416% for a threshold 90%.
Results and discussion. Fairnsss under imbalance requires much longer training: the minority group (A4 = 0) consistently reaches к much later than the global accuracy. The catch-up overcost particularly high under strong imbalance, exceeding 400% on Adult and CIFAR-10. Empirically, larger models reduce this overcost but do not eliminate the necessity of longer well-tailored train- ing. These results support our theoretical findings on debiasing duration imbalanced settings (see Section 3.3), the overwhelming dominance of the majority-training zone, and the difficulty of

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240
42
Train accuracy 1%6)
Test accurary (36)
30"
72
less
30
350
630
150
250
250
302
360
100
20
150
152
200
250
100
350
Epech
Epoch
Epech
Figure 4: Training and test loss/accuracy for ResNet- 18 on EuroSAT (mean of 3 runs). Minority classes exhibit delayed learning their accuracy improves substantially only in later epochs.
7.0
Train loss
Train accuracy
207
40
320
450
360
200
562
640
Epoch
Epoch
Epoch
A=
A=1
Global
threshold
Figure 5: Loss and accuracy with TabNet Adult (mean of 3 runs). Under strong imbalance, the catch-up overcost is substantial - around 400%.
distinguishing representative predictor from a stereotypical one. We also ran preliminary experi- ments with AdamW and XGBoost ggradientbboostee decision trees). In both cases, we observe the same qualitative phenomenon. Irrespective of faster or slower absolute training, attaining minority awareness requires a comparable relative increase in training: extra epochs/steps for AdamW W and extra boosting rounds for XGBoost; see Appendix D.I and Appendix
Conclusion
Although our goals primarily theoretical and future research should explore more refined training protocols, we can draw several conclusions supported by both theory and numerics -ours and the community's 'sssi well, e.g., [ [2]] These conclusions may also practitioners. Two key quantities emerge as critical in our study: the stereotype gap and the training duration. Additionally, have empirical evidence that the model size may be determining factor in achieving budget frugality.
recommendations for
In majority-minority scenario, population and variability imbalance determining factors influencing the stereotype gap (Theorem [ and the subsequent subsections). This gap, between stereotypes and representative predictors, can be very small in severely imbalanced cases.
For convex or deep learning problems, gradient training generally leads to 'satisfying predictor' the sense of low-value loss L, see e.g., 4 [3]. However, in our majority-minority scenario, the action of Lo is generally almost indetectable, as shown in Figure I and Section 3.3 thus early
stopping and under-dimensioned models are prone produce stereotypes.
To obtain representative predictor, it is advisable to use larger networks and extend the training duration, as supported by Propositions [ and 2 and the numerical section. The corresponding catch-up overcost ratio can take considerable values, e.g., from 25% to 450% for the imbalanced
CIFAR-10. However, this must be mitigated view of possible spurious correlations that arise in overparameterized regimes (37].
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Acknowledgments and Disclosure of Funding
The authors are grateful for the feedback by the anonymous reviewers, that led considerable improvement of the paper. This work was supported by the ANR project Regul IA and by the Chairs TRIAL and UQPhysAI of the Toulouse ANITI Al Cluster. JB acknowledges support from the Air Force Office of Scientific Research, Air Force Material Command, USAF, under grant number FA8655-22-1-700 and TSE-P. Access to MesoNET resources Toulouse was granted under allocation m23038.
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This is the appendix for 'When majority rules, minority loses: bias amplification of gradient descent'.
Contents
Notations and auxiliary results
21
A case study: linear regression
22
Proofs and extra results
23
D
Additional experiments on CIFAR-10
34
Additional experiments on Adult
37
Experimental details
38
A Notations and auxiliary results
A.1 Notations.
For matrix A, we write Pmiin (A) and (A) for its smallest and largest singular value. If the matrix is square symmetric, we write Amia (A) and Amax (A) for its smallest and largest eigenvalue. Given Rdd write || for its Euclidean norm and for 0, write B(x, c) {y vll E).
R4; ||
For function / Rd + Rd and for x E Idd, we write Jac (xx for the Jacobian matrix of f at x. For function f Rd + R and for æ e Rd, we write V fz()) for the gradient vector of f at æ and 22 fx: for the Hessian matrix
æ. For a function G Rd R, we write
G = (0 e Rd
VG 0}
argmin-loc G (0 Rd 0 is local minimizer of G over Rd).
For a non-empty subset A dd and for x E IRd we let dist (x, A) = inf || yll. For two non-empty subsets and of Rd the Hausdorff distance between A and B is denoted by
dist (A, B) max
sup inf ||x - yll, sup inf || - w/l A yEB
veB #EA
When A and B are non-empty and bounded this quantity finite. The topological boundary of A is written bdry A.
A.2 Training metrics
Let fo Rd RC be neural network, parameterized by that maps an input E Rd to vector of C' class scores. We denote [C] = [1....,C, C} the set of class indices, and define the predicted label as (xx) arg max fo (x)e. We compute accuracy separately for each group j E (0, 1} as the proportion of correct predictions in A= and define the global accuracy as the weighted average across groups For each E 10.
Acc Nj (EDA)
Σ
(p((,) yıl, and Acc(O) Acco (0) AC),
where [,;))) = yi] denotes the indicator function, equal to the predicted label matches the true label and 0 otherwise.
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A.3 Lemma
The next lemma is well-known but stated here for convenience.
Lemma 2. Let E be an open set of g& for some & E N. Let f E - 12 have Jacobian Jac f. Let
2, E so that the segment between and y E. Then
(v) - [(x)ll S
sup Pmaxx (max [(u))) 11 æ01-
NEE
A.4 Discretization of ODE curves
In various parts of this paper, we refer to represent ODE curves in our experiments, Unless otherwise specified, this refers discretization of the ODE using small step sizes. For instance, given the dynamics
((t) F(O(t)), ((0) Omit.
with F: IR Rp locally Lipschitz field, the discretization we use is of the form
0k+1 0g 5y. F(
where the step size 8n < 1: in practice, we typically use 8 0(10~)).
Note however that for the numerical section, we proceed differently as our objective rather training through the gradient method. We thus use larger steps and mini-batches.
B A case study: linear regression
To illustrate further our results Sections and with loss
consider multidimensional regression model
L(O)
1 2n
X r11² 2n
1
1 -
1 2n
- Y
(14)
(.)
La(
where X is d with rows XT ..... XT and where X (respectively X contains the of X from the majority (respectively minority) class. Similarly, Y n-dimensionnl with components Ii,..., Yn and y (respectively yo) contains the components of Y from the majority (respectively minority) class. Letting X = (X*) for
0, we define the corresponding empirical covariance matrices S = X X/n,, So = X x0/no: S = I xTTx'/n1 Assume that the covariance matrices are invertible, which may be granted through ridge regression model in the generic/regularized spirit presented in Section 2.1
B.1 Distance between minimizers
In the setting of linear regression, Theorem D in Section becomes the following (simpler) theorem. Theorem (Representative-sttrron sttroooyysstttctcticcllllll gap: linear regression case). Assume that So and S₁ are invertible, and denote by 0, 0,, and 0o. respectively, the unique global minimizers of L, L1, and Lo, respectively (on Rdd), Then
2Pmaxx Sg) (1 (1+ 110 -
@ 011l
Pmin (n St)
The key quantity behind the stereotypical gap is the ratio
Prnaxe (o) Pmaxx (2T.o)
Pmin (n S1) Pmin (V2
where we have omitted the dependence on the Hessians, which are constant. Two statistical effects drive this ratio:
Population size ratio: when the majority is much larger than the minority, then no small, this tends to increase the risk of stereotypical predictions.
Min-max variability ratio: the smallest variability of the majority much bigger than the largest variability of the minority group then stereotypical predictions more likely.
22

---

B.2 Stereotypical and representative training curves
In this section, we compare training L as in (IO), which provides a representative training curve. with training on the majority group, which provides stereotypical training curve as if the minority did not exist. The stereotypical training curve
_e = -V (0)
Our estimate depends once more on ratio A - Pmax S Pmin S1)
Proposition 3 (Distance between stereotypical and representative training curves). We have, for any t t> 0,
[ (1) S 100 011 + tpmax
110 5o 7
PPmin (1))
(10:1 [|@mitll)
110 + 1l Pmax (no
||0 - 0:113 := sup |l@(t) (t)ll - ||0 01l t>0
Pmin (nin(n11 (1])))
B.3 Catch-up overcost as measured with the minority loss Lo
Proposition measures the catch-up overcost as the time needed to get close to as measured by the distance between parameters. The following proposition shows more qualitatively, for the linear model, that the catch-up overcost goes to infinity, when it is defined as the time needed to get close to O as measured by the minority loss Lg-
Proposition 4 (Catch-up overcost measured with the loss Lo). Assume that O, Oo and 0n are two- by-two distinct. Consider a representative training curve - 0(1) as in (10p. such that for some t stereotype. (f stereotyy) = Q Assume that for 2 tstereotype. 0(t) E Zmaa-adv.
For 0 < < 1, consider t' catchup. such that
Lo(O(t' ((((t'aacchup.e)) Lo(0)
(o(O)) - (0) S.
(15)
Then we have
t catchuu.e
Estercotype 00.
B.4 The minority-adverse zone can be large
The midority-adverse zone is defined as
Zmin-adv = fOER" (VL(O), (o)())) S 0}
and is the counterpart to the majority-adverse zone in (1I The next proposition exhibits ball of radius R that is contained in the minority-advers zone. This radius R is large whenever |0 0oll is large and S and So are well-conditioned. Hence, roughly speaking, while Theorem Saremmmssttar states that the anders zone is always small, the next proposition states that the minority-adverse -avvere zone can be large. Hence, gradient descents on may not decrease Lo over long training times, which is conclusion of our numerical experiments in Section 4
Proposition 5. Assume i Gv. Let
R Punin (So) (S)
11- 0oll-
(16)
33 (o) Praxx (S)
Then there exists E Rdd such that B(0, R) C Zmin-adv-
C Proofs and extra results
C.1 Proofs and extra results of Section 2.2
Kantorovich theorem. great part of Section 2.2 relies on a theorem Kantorovich type for Newton's method [1] Theorem 5] whose proof is based on [27]. This result is recalled below:
23

---

Theorem 4 (Newton-Kand Kantorovich Theorem "with only one constant' (existence)). Let 0" E B and R. > O. Let S2 be open containing the closed ball B(0", R). Let G: Ω + Rd be continuously differentiable mapping, Suppose that the following conditions are satisfied:
(KI) Jac G(0*) is invertible with |JacG((0))+G((0)]] A
(K2) For all 0, 0' E =B(0", (),, Pmax (Jac G(0)) G(0) Jac G(O))) - 18-9'11
Then there exists unique E B(0 R) such that G(O)) 0.
We need beforehand abstract results on equation perturbations. Let 0* E Rd, We consider a function F Rd - Rd such that F((*) 0. Let p: R - F and consider the equation defined for E Rd
F(0) p(0).
(17)
If the function p is negligible, in certain sense, with respect to the dominant term F(0). [7] becomes perturbed version of equation F(0) = 0. Its solution will be close to the solution of the non perturbed equation, 8* Proposition @ quantifies partly this phenomenon.
Proposition 6 (Distance a perturbed solution). Assume F and p are continuously differentiable and that there are strictly positive numbers 6, M, r such that:
Conditioning of and
Puin (Jac F(A*)) 8 and Punin (Jac (F-P)(8 2 6,
(18)
Differential regularity of the nonlinear equation
2T
Pmax (Jac F(0) Jac Jac F(8")) _ M| 8'11,
0,0' E
H+,
(19)
27
Pmax (Jac Jac J(')) S M|10 0'1,
0,0'
0r,
(20)
Perturbation bounds
|p((0))| ST,
(21)
Pmax (Jacp(O") S
(22)
If the perturbation ratio T /8 satisfies
7/8 4M'
(23)
then, there is unique Op solution to F(O,) p(0)), which is close to the solution of F(0*) 0, in the sense that
8* 2T
6p E B
8
Proof of Proposition 6 For 0 E Rdd let G(0) = F(0) p(0). We apply Kantorovich's Theorem
above (Theorem to the function G. The quantity It is taken as
A 2T
Let us check Assumption (K1). We have, using (18)
||Jac G(00) <
|G((=)]|
[p((0))
T
Pmin (Jac G(O*))
8
Hence (K1) holds since R 4.
24

---

Let us check Assumption(K2). For all 0 E B(0", R). we have
1
Pmax (Jac G( (Jac G(0) Jac G()))) <
Pmin (Jac G(B+)) Piaxx (Jac G(Ø) Jac G(
2M
from 18
19
and
20
10
0'||-
6
From 23 we have
and thus
Hence (K2) holds.
4M
2r
R²'
Hence we can indeed apply Theorem 4
and we obtain that there is unique 0p E B(0", "/) such that
G(Op) 0, that F(Bp)= = p(6p)
Proof of Theorem 7
Set 6* crit Ln K. We apply Proposition with F VI VLo and with 8*, M,, 7 there given by the same notation here. We take ö there as 6/2 here. Then indeed F(Ø*) 0.
We have Pain L1 2 8 from I Hence, using (5) and (6).
5
Pmin V (L1 + Lo)( 2 -
9
2
Hence
18 holds.
The conditions
19
to 22 hold by the assumptions
to 5 For
19 and
20
note that
B(0", 8/2 c from
The condition (23) holds from
Hence all the assumptions of Proposition are verified. We conclude that there is unique Op E B(0", 4T) such that F(Op) = p(0p), that V(L, + (o)
From 7 0p E B(0", 4 K.
Assume now that there a different number of strictly negative eigenvalues between L2 (0*) and L(,,). Write X (Q) < ... S (m()) for the m ordered eigenvalues of a symmetric m X m matrix Q. The eigenvalues of V (0*) are in IR) [ 6) since we have observed that Pmin P L, ( 2 5. Hence, if I, (0*) and (T
(Op) do not have the same number of strictly negative
eigenvalues, there would exist i {1..... ...,, d) such that ( La (8*) (L + (6p))
However from Problem 4. in (24] we have
IA. (v (L1 + (o)(),)) S Pmax (0") V2(LI + Lo)
Spmax
( - (²L)(0,)) + Pmax (V Lo
4T
from 2 and 5
+T
8
from 6
8
<6.
This is contradiction and thus (L and (2 (O,) have the same number of strictly negative eigenvalues.
Consider now 0+ E crit (L1 + Lo) K. We will apply Proposition 6 with = VL1 VL and
p= Lo. In Proposition 6 we will take for 7 the same value as here. The quantity M in Proposition 6 will be taken 2M here. The quantity 8 in Proposition 6 will be taken as 5/2 here. Let us check the conditions of Proposition 6
We have, from and since (LL, + Lo)(0)) =0 0.
(0 S 2
Hence from
1 Prinn (V*I((0)) 2 8. Hence, from 55)
Pmin ( (V2 Lo)
T
(24)
because by assumption 7 S 5/8. Hence 18) holds (with 5 in (18) taken as 8/2 here). Next, for all 81, Og B(#",
6/2) C K (by (7). from and (3)
27
Pmax (V + (0) ( Lo) (02)) M| 021 + +M 021 2M 1|0
0211.
(25)
25

---

Hence 19 holds (with M in 19
taken as 2M here).
The conditions 20
21 and
22
hold by assumption from 14 and 5
For 20
note again that
B(0", 2r
c K. Equation 23 in Proposition 6 holds from (6 in Theorem
8/2
Hence the conclusion of Proposition hholds and there is Op B(0", + such that V(L, + Lo) (O,) that (0,) 0. Also 02 E (300, CK
Similarly as above, assume now that there is a different number of strictly negative eigenvalues between (o)(0) and ((pp)) Since have established Puin (V2 Lo) 8/2
from (4), there would exist i {_,.... d} such that (L, (o0)) - - (1 2
6/2. However from Problem 4.3.P1 in (24), we have
(L (o)( As L.
Spmax
(V2 + Lo)(o)) (OL2(O,))
Pmax
(V²Lo(0")) + Pmax (0")) - LI
4rM
from 2 and (4)
+
5
from
6
8
2
This is contradiction and thus (0*) and (2 Lo) have the same number of strictly negative eigenvalues.
Hence have established the theorem.
Proof of Corollary 7 From Theorem 1 for A crit L1 n K, there is 0' crit Ln K such that
||0 - 81l < 8
4r Conversely for E crit L K, there 0' E crit L2 n K such that ||0 0'l S
Hence dist (crit L1 7 K, crit Ln S 4r
Also from Theorem for argmin-loc! L1 n K, since 0 crit L1 n K, there is 0 crit L K such that 0
<
2r Also (0) has no strictly negative eigenvalues since 0 E argmin-loe L
Hence from Theorem 1 (('' has no strictly negative eigenvalues. As observed in 24 in the proof of Theorem [ V L(0" has no zero eigenvalues. Hence 6' argmin-loc L n K. Similarly,
for 0 argmin-loc L O K, can show that there is 0' argmin-loc L1 K with |00 0' 4.
Hence
4r
dist (argmin-loc L, O K, argmin-loo L nn) < 5
Finally, from ②). for each 0 E argmin-loc L n K, there indeed 0' E argmin-loc L K with
|10 - 0'll < 4r For each 0 E B(0', (F) C K, we have, using 24 and (25 from the proof of Theorem and then (6).
Amin (VLL(0)) 2 Amin (V²L(8') Pmax (V²L(0) (L)0))-9((8)) 2
8 12T
This concludes the proof.
Corollary 2. In the context of Theorem assuming further that dist н (crit L K, bdry K) 2 and dist h (crit L1 K, bdry K) Z
5 m we have the following additional conclusions.
(i) For each pair of distinct elements 0, 0' E crit. L1 O K (resp. crit L K), we have
|10 0'|l > 32M
(ii) The sets crit L1 O K and crit L n. K are finite.
26

---

Proof of Corollary
Proof of the first conclusion. To establish the first conclusion, consider 6* E crit L K. Let us apply Proposition 6 with F = VIn taken the zero function, 6 there equal to I here, M there taken as M here and r there taken as quantity that we write 7' and that arbitrarily close to but
strictly smaller than With similar arguments as in the proof of Theorem we can check that
(T8) holds, and that 19 holds. Trivially, (20). (2T) and
22 hold. Finally (23 holds because
82 8
4M6
4M
Hence the conclusion of Proposition is that for 0' E crit L1 n nKK 0' + 0", we have
2T'
100 0'1 2
Thus, letting r' arbitrarily close to
4M we get
82 8 4M
8
8
100 - 0 2
2
2M
32M
Conversely, consider 0* crit (L1 + Lo) n K. Let us apply Proposition 6 with F V(L1 + Lo), taken as the zero function, 8 there equal to 6/2 here, M there taken as 2M here and 7 there taken as a quantity that we write r' and that is arbitrarily close but strictly smaller to 8437 With similar arguments as in the proof of Theorem we can check that (18 holds, and that (19) holds. Trivially,
.²
(20). and 22 hold. Finally (233 holds because
T'
2
8²
5
64M
32M
4(2M)
2
Hence the conclusion of Proposition
6
is that for 0' E crit (I1 + Lo) with 0' + 0", we have
1 0'
2
Hence, letting 7' arbitrarily close to
d² 64.77
we have
2 82
0 0'
8 64M
32M
Proof of the second conclusion. Since crit L, n K is bounded, and since to each 0 E crit L2 n K we can associate a ball of fixed radius containing no other points of crit L n K (second conclusion). we deduce that crit L1 O K is finite set. Similarly, crit (L1 + Lo) O K is finite set.
C.2 Proofs and extra results of Section 3.2
Proof of Theorem Consider 0 E Zmaj-adv K 27. We have
(VL(O), (O) = VL (0)
VL)(O)
> VI, (0)] (VZ,)= [VL,(( = VL -
Note that, by 4 [L)( S Hence, since 0 Zuajjadv we have [[)]] S..
We then apply Theorem 4 with 8* there equal to 0 here, with G equal to VL, and with R equal
25 Note that B0, R) C by assumption. Since 7 S c by (6). then from ( we have to 8
Paiin ( (0)) 2 8. Hence
(V) S
and thus (K1) holds in Theorem
Also. for all 0', H" B(, R), have from (
M|
0"
00' 8"1
PIax (12221))) -22
8
27

---

because :/4 < since o/ S
8 M
since 8 2N
from 66 Hence (K2) holds in Theorem 4
Hence Theorem 4 implies that there exists @ such that (L) = 0 and |@ 01 s R = 2z. Hence we have
OE
U
B
1 2r
8, Ecrit LinK
which concludes the proof.
Proof of Lemmal /
We have
(VI(0). (L) -VI(()) VI(()) - (VI(0))VI100)
20,
because 0 E mmmj-aav means by definition that (VI(0), ()) S 0. The rest is the classical Lyapunov computation.
Proposition 7. Consider that Lu, L1 Rdd - R twice continuously differentiable with the properties that
V(L](0)) = = Pmin (v²)((0)) > 0, e R
(26)
and, using L = Li Lo.
V(I(0)) = Pmin (2I(())) 0, 0 ER..
(27)
Then, recalling the symmetric difference notation
crit L1 crit = (crit L, crit L)rit L, n crit L),
we have
crit I, A crit C bdry Ziaj
Proof of Proposition 7 Consider 0, E crit L, /erit L. Then VL(0,) * O. By continuity, there exists
co O such that for 0 0 S Co.
|VL(0) - VI((I) |V((,)) |
Consider 0 cg. From 26 and from the local inversion theorem, there are neighborhoods U of 0, and V of 0 Rd such that U ((),,e) and such that Vi bijective from U to V.
Hence, there 0 (small enough) and there Ø E U such that
(0) = t, V.
(28)
Hence
(L) VLO)) =t. VLOO)) VLO)
+t (L)) VL(O))) +t. (Z)
2t. (2 /2 - |
VL(0) -
2/. (L(0))
/- -- |LO0O||
2
>0.
We proceed similarly from
28 but this time with 0' U, < 0 and
VL, (VL()) EV.
This yields
(L) (L(O)) <0.
28

---

Since this holds for any > O there are two sequences (0x)a and (0%) that converge to 0 with Og Z maj-ady and 02 E Zmaj-adv . Hence 0, bdry Zaajaddv Hence
crit In Ii'eriit I. bdry ady
We can show symmetrically
crit A L1 bdry mmaj-adv
C.3 Proofs and extra results of Section 3.3
Lemma 3 (Duration for proximity to local minimizer). Consider a function L R + R that is twice continuously differentiable. Assume that there exists trajectory [0, 00) 3 ++ 0(t) satisfying
d 0(1) = -VVV))(((()) with ((0) = Oinit E TRd
dt
Consider critical point 0 of such that Ō + Oinit. Consider a ball B containing Ō and (0(1); 0}. Assume that, for some M- < oo and all E B.
Pmaax
(V))(0)) M.
(29)
Consider E 10, and te E (0, 00) satisfying
|(((t.) 0| S E Eliiiiit A||
Then we have
1 M
4 2
log
Proof of Lemma
Without loss generality, we can consider that
inf { 2 0: (10(t) - 0| s el|Omit - -01} co.
Consider the function t0, 1 g(u) 110(te u) -Øl Note that this function is strictly
positive and differentiable on [0, t.] (since 110(1e - u) - Ōl 2 <||Oinit - 0 > for [0, L.). The
derivative at [0, te] satisfies
d du
0(t u). Oll. - u) 0
g'(u)
11((1, - u) - 01
(VL) - w)), O(te 1)
110(te u) 011
(I((t)) w))l
=I(( u)) - (VL)
Lemma
<M]l0(t. u) Ø
=Mg(u).
Hence can apply Grönwall's inequality, yielding
g(t,) I (000e Clit atte
On the other hand g(tz) 110 01 and thus
cll@.uit Alte 2 || 01|-
This yields
te log ! M
1
This concludes the proof.
29

---

Proof of Proposition L We apply Lemma 3 with L in the lemma equal to L here, with M in the lemma equal to M here and with in the lemma equal
| stereotype 0
0mit
0|1
here. Then we have
00 stereotype 011 eelOmit 01
and so the lemma yields
1
t steerootpe
2
log
M
c
which concludes the proof.
Proof of Proposition We apply Lemma which directly concludes the proof.
C.4 Proofs and extra results of Appendix B
Proof of Theorem 3
Let us apply Proposition 6 to the linear model. We let, for 0, f:(0)
||Y
x We apply Proposition to
Vı. p = Voo 0" = 01
and with constants 6, M, to be specified later.
We have
V(i(0) = -x 2x X'O
and
fi =2X" X².
Hence taking
8 = 2pmin (X' x!)
we obtain that 18 holds in Proposition 6 Furthermore, the Hessian matrices of fo and fi are constant and thus we can take 0 in Proposition while still having that (19) and (20) hold. Next,
Vo
2x0T + -X X
-x0 yo - X0 + 2x x² 0o)
-2x x² -0))
Hence we take
7 = 2pmax X X (1 + 100 Gull
to ensure that Pmax 2
2 fo())) S and / fo))ll) )l - Thus, 21 and
(2
hold in Proposition 6
Hence, apply Proposition that yields
2r
2pmax (nt Pmin (n) S1)
1 - 0 S
1 110 0o1l
8
Note that the constraint 23 becomes vacuous since M = 0.
The following lemma provides the expression (well-known) solutions
di -0())=VL((0))
d
0,(()) -VL,( i 1,2, 0(0) (o(0) (0) Omit.
Note that, with the uniqueness assumption, have θ = (nS)- y and 0, (n,S,) -
30

---

Lemma 4. We have, for t 2 0.
((1) = 0 T (Onit 0
for = 0, 1 and 2 0.
02(1) = 0, + -(n/nn)s (Omit 0.)
Proof of Lemma 4
We have
d
8(t) = (VL)
dt
x x8(t) + Y
n1
n
1 -x" x0(t) - x XXXXXXXXXXXX
= St(t)) - sO.
Hence,
((()) 0) = S ((((t) 0)
dt
and (()) - -iit
0. Hence
(()) = 1 -e -15 (mmi - 0)
We then provide a similar proof for (t). We have
(t( (O())
X x'O(t) 1 X
n
n 1
1xT x'0(1) + xx xxx
Ti s,0(t) + S
Hence,
d (0.(1) - 0.) - s (0()) 0.)
di
and 0,(0) 0, =0iii- 0, Hence
0c(1) = 0; + e -1 Omii - 0.)
This concludes the proof.
Proof of Proposition
We have, using Lemma 4
[(O(t) (1)( < (la e 15 0- (la e 1 0i + |1(
et8 -t Omni
(Ia E et)) (o- 01) + ( -t
(0-1)
1 -e5 03 + ||(e ( -e- Omni
SPmax (Lu - e 15) 110 01ll + Pmax
(e-is-e -1 (10:11 + ||9 Il)
Since Ia 15 has eigenvalues between and and from
34
Lemma 3.24], we obtain
[|O(t) - 01 (t) < 0 0ill + tpmax
(S- n1 St
Tmmin (155)) (10h1) + [[0mite]] II)
1 011 + tpmax
nu So mpoin (1)))
[1011 || + [10maell)
31

---

The maximizer (over 1) of te 1pmmin S:) is mmax 1/ Pmin (i) which yields
(0(() (1(1) - 110-011 + Pmax (no 011l + |10 II)
sup t>0
e Pmin (7) Si)
This concludes the proof.
Proof of Proposition Without loss of generality, we can consider that t steerootpp = and mit bj. Because for t 2 0. (t) Zmaj-adv the function t + (0(tt) is non-decreasing. Assume that there exists 1 < oo such that Lo(O(L)) Lo(O). Then L(O(1)) S L(0) and thus 0(1) = 0. From Lemma 4 this is a contradiction because Oinit & Hence, because (o()))) Lo()), the function + Lo( - Lo (0) is strictly positive [0, oo) by continuity.
Finally, for simplicity, write & catcchup.e Assume that & does not go to infinity as Then there subsequence (Ce)REN going to zero and constant T oo such that t, T. By compacity, we can extract further convergent subsequence (lreg )KN with leeg + l' E [0, T). We have
Lo(O(tes,) )) (o)
Lo(0:) - Lo S Cfa k
and thus by continuity (o((t))) = (0). This is contradiction, which concludes the proof.
Proof of Proposition
Let
0(1) ④+0 -e 18 (0o-0)
Then as in Lemma 4 have
Lo()(t)) - (VLO) (0(t)) 4 000)
-( (VL
= S(O(t)),
(30)
defining
S(0) (VL)(0), VLO)).
Let
T
Punnn (S)
Then
- 100
(I((T) - 0 S e Pmi (5) (S)
2
Then, using Lemma 2 for any 0 in the segment between ((T) and
[V)o(
=1
ng
So σn)l
n
(convexity of Euclidean norm:)
Tig
Pmax (Sn) (II0(T) - 0oll + 110 - 0b1l
n
71
Pmuux (So) (I0(T)) 0 2110 0u1l
33
n20
Pmnax (So)ll -boll-
Then, by convexity,
Lo(O(T)) - Lo(O))
> 71
Pmin (So)]10(T) 0.122
2n
7g
Pmin (So)
-0:10--6111 0011
8n
32

---

since ((((T) - -0l - 18ol.11. Also, using (3
8oll/2.
Lo(((T)) Lo(D Lo dLo (0(t)) dt /
/T
-S(O(t))dt S - min S(O(t)). [E[0,]]
Combining the two last displays,
min S(O(t)) - Lo Lo(On) (o())))) sE(U,2]
T
no Pmin
(So) 0oll
<
8T
no
Punn Punn 1 00112.
8n
Let = 0(t) with E argmin S(O(t)). For E B(0, R), we have
rE(0,T]
[S(0) S(0)
V(()), VLO(0)) - (VL(0)VVLO()) VLO(O))
VL(O), VLu(O) - VL (8) + - (
(8)
(Lemma 2
[L( no
Pmax [So)]| B| + Pmax (S) B L 0|
n10
$ Pmax (So) (0) Apmax (S) V La
n
(Lemma 2
4 R Puax So Pimax (5)] @ + RP (S) ?00
110
Praax ( (So) 1 0
n1
R + | 01 110 8u Bull)
< R
Pmax Su) (S)
We recall
((t) = 0 + -t (80 0)
Hence, [|0(t) - 0| s 110 - Ø| and [[0(t) 0o1 ll s 21100 Ø1.
Thus we have
((( ( = R no
Pmax (Su (S)
R + 3 3|0 01)
Hence, let us take as (6.. with in particular R < 1100 -1.. Then to satisfy (VL(O), (l) so for all f E B R), it is sufficient that
R no Pmax (So) (S) R (R+310 3100
011 <
1 no
Pwii PSonn (5)1 0.112.
For this it is sufficient that
32R 140
Paax (So) Prnax (S)10 0| < n
11
Pmnn (So) (S) 0b11
n
which is implied by
Pmmin (So) Pronn (S)
R < Pm (So) Preaxe (S) 10 0oll
This concludes the proof.
33

---

D Additional experiments on CIFAR-10
In this section, in Figures 6 provide complementary results for additional architectures on the Imbalanced CIFAR-I0 benchmark. We report detailed training and test dynamics (loss and accuracy) across different subgroup imbalance levels (ç E (1%, 10%, 30%)) and threshold 90%. These figures illustrate that the qualitative behavior observed in the main paper is consistent across models of varying depth and capacity.
(- 1%
2 10%
[= 30%
Tain loss
10"
10"
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140
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120
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00
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Eporh
A-O
A-1
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K thresholdd
Figure 6: Training and test loss/accuracy dynamics for ResNet50 CIFAR-10 10 across imbalance scenarios (C (1%, 10%, 30%)) with threshold = 90%. Minority accuracy lags behind global and majority early, then catches up both train and test. Compared to VGG-19 the delay to reach r is shorter, hence lower catch-up overcost. Capacity mitigates, but does not remove, the extra training required for minority awareness.
34

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10%
c=
Thain loss
100
10P
0
1.5
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6.
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baa
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205
120
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72
Epoch
Eprch
Eparcs
threshold
Figure 7: Training and test loss/accuracy dynamics for VGG19 on CIFAR- across imbalance scenarios (C E (1%, 10%, 30%}) with threshold 90%. VGGI9 is much bigger than ResNet-50 and does faster minority learning. However one sees a higher catch-up overcost (e.g. 280% VGG19 vs 157% ResNer50 for the 1% imbalance scenario) underscoring the role of architecture in the extra training needed to achieve minority awareness.
35

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10%
(=
Thain loss
20e
123
10-1
10-1
60
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103
120
140
30
50
10
1.00
100
00
20
3x.
2x
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342
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Epoch
A=0 Am1
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Uhreshold
Figure Training and loss/accuracy dynamics for VGG11 on CIFAR-10 across imbalance scenarios (C E (1%, 10%, 30% }) with threshold к 90%. Training and test dynamics display clear minority
0) delay the к threshold, longer than with ResNet-50 and comparable or slightly worse than VGG-19. Consistently with Table 2, the catch-up overcost high (e.g., 291% %. 171%, 114% for ( 1%, 10%, 30%), underscoring the role of architecture the additional training required to attain minority awareness (cf. Figures 6 and 7
90% is
36

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D.1 Impact of the optimizer: AdamW
While the main experiments in the paper use standard SGD without momentum, we also investigated the impact of adaptive optimizer. In particular,
repeated the same Imbalanced CIFAR-10 protocol using A with learning rate = 1 x 10- weight decay 1 x 10-2, and a cosine annealing scheduler. The loss function was standard cross-entropy.
Minority accuracy (A 0) shows the same early delay observed with SGD: global and majority (A = reach first, and the minority catches up later train and test. Adam often reaches the global threshold sooner, yet the catch-up overcost remains of comparable magnitude llrge under strong imbalance (about 400% at 1%) and decreasing Ç grows. As reported in Table 3 the overcost remains substantial across imbalance levels; changing the optimizer does not eliminate bias amplification, while its magnitude vary with model capacity (see, e.g., [2
Table 3:
Catch-up overcost (in %) with AdamW on imbalanced CIFAR-10. Reported values are means over 3 independent runs across imbalance levels Ç E {1%, 10%, 30%}.
Model
Parameters
1%
10%
30%
MobileNetV2 38
543K
326
310
214
VGGI1
4
9M
401
244
169
ResNet18
11M
465
342
209
ResNet50
25M
369
327
219
ResNet101
42M
220
192
172
E
Additional experiments Adult
To complement our deep learning results, we also ran an XGBoost logistic regression on Adult. The model was trained incrementally by adding batches of 10 trees at each iteration (using the xgb_model argument to continue training from the previous booster). Each step records cumulative training time and subgroup accuracies for A = 0 (minority) and A (majority). The optimizer and objective are casall_mtrii==ooooll
handled internally by XGBoost
Despite the very different model class, we observe the same qualitative behavior: a pronounced minority delay to the к threshold. followed by a late catch-up visible on train and test. Changing the learning rate, tree depth, or regularization alters the number of boosting rounds needed to reach the threshold, but the relative catch-up overcost remains large under strong imbalance (about 400% at 1%) and declines as Ç grows. This shows the phenomenon is model-class robust, extending beyond deep networks.
Table 4: Incremental training with XGBoost logistic regression on Adult. We report global accuracy and subgroup accuracies for A (minority) and (majority) on both train and test sets as the number of trees increases.
Train accuracy
Test accuracy
Training time (s)
Global
A =
A =
Global
0
A = 1
0.47
0.8772
0.5530
0.8893
0.8679
0.5263
0.8807
7.70
0.8989
0.6734
0.9074
0.8679
0,5996
0.8781
16.36
0.9098
0.7235
0.9168
0.8668
0,5940
0.8771
26.45
0.9190
0.7623
0.9249
0.8649
0.5921
0.8752
38.03
0.9261
0.7987
0.9308
0.8624
0,5977
0.8724
51.11
0,9312
0.8205
0.9353
0.8606
0.5996
0.8705
65.64
0.9359
0.8464
0.9393
0.8589
0.5959
0.8688
82.94
0.9385
0.8529
0.9417
0.8579
0.5959
0.8678
37

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Experimental details
This section provides full details to ensure reproducibility of our experiments. We describe hardware specifications, training hyperparameters, implementation details, and evaluation protocol for each dataset and model used.
F.1 Datasets
We conduct experiments mix of image and tabular datasets with varying levels of class imbalance. Below, we describe the construction and preprocessing steps for each dataset used in our study.
CIFAR-10. We use the standard CIFAR-10 dataset, consisting 60,000 color images (32x32 pixels) in 10 classes, with 50,000 training and 10,000 test samples. To induce group imbalance, we define a binary sensitive attribute A E 0, 1, following the approach detailed in Section 4
CIFAR-2. We consider binary classification task derived from CIFAR-10 10 by selecting the two vehicle-related classes "automobile" and "truck". We refer to this subset as CIFAR-2. To simulate a highly imbalanced scenario, we drastically reduce the number of "automobile" (car) samples to small fraction of their original count (e.g., retaining only 3%). while keeping all "truck" examples. This creates pronounced majority-minort setting, suitable for studying bias amplification under imbalance.
EuroSAT. EuroSAT a land use and land cover classification dataset based on Sentinel-2 satellite images. We use the RGB version comprising 27,000 labeled images across 10 classes. For our binary classification task, we select two visually distinct classes: Highway and River. The input images are resized 64x64 pixels and normalized. We define binary sensitive attribute A by thresholding the average blue-channel intensity to distinguish between "bluish" and thon-bluist images, following the approach of (6]]
The Adult dataset is a standard benchmark for fairness and tabular learning. It contains approximately 48,000 examples with demographic and income information. We treat the binary income variable as the label and use "gender" (male vs. female) as the sensitive attribute A.
Adult.
F.2 Hardware and runtime
Experiments were conducted on a computing cluster equipped with NVIDIA A100 40GB GPUs. Each experiment single GPU unless otherwise specified. Average runtime per training is reported in 5
Table 5: Average training time per across datasets and models.
Dataset
Model
Runtime (h)
GPU
CIFAR-10
ResNet- 18
0.26
A100
EuroSAT
ResNet- 18
0.1
A100
Adult
TabNet
0.16
A100
F.3 Optimization and training
We use SGD with a constant learning rate for image models and tabular data. In order to match our theoretical setting, no weight decay or learning rate decay schedule was applied. Models were trained from scratch without pretaining. Refer to Table for more details.
F.4 Evaluation and reporting
All results are averaged over 3 random seeds. We report mean and standard deviation of accuracy and loss metries across groups. Class imbalance ratios are detailed in the main text (Section 4).
38

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Table 6: Optimization hyperparameters for each task.
Dataset
Model
Optimizer
Learning rate
CIFAR-10
ResNet-18
SGD
x 10-2
CIFAR-10
VGG19
SGD
X 10-2
EuroSAT
ResNet-18
SGD
X 10
Adult
TabNet
SGD 2 10
F,5
Reproducibility
All code and configuration files (including seed control, training logs. and plotting scripts) are available at httts:///gith/: com/ conm/ryanboustnny//eeo hataaa//bassaallliiiaaaaaaeeeeee amplification We follow best prac- tices for reproducible research and ensure all experimental figures can be regenerated with a single command.
39


Reviews:
[{"review_id": "hlzBsyZ3vI", "reviewer": "Reviewer_DEPM", "summary": "This paper investigates how gradient descent amplifies bias when training on imbalanced data with majority and minority groups. The authors develop a theoretical framework for majority-minority learning tasks where the total loss L = L1 + L0, with L0 (minority) being negligible compared to L1 (majority). They prove that critical points of the full loss L (representative predictors) are close to critical points of L1 (stereotypical predictors that ignore minority features), with distance bounded by a \"stereotype gap\" that depends on population and variance imbalance ratios. The analysis reveals that the parameter space is dominated by a \"majority-training zone\" where minimizing L essentially minimizes only L1. The authors derive lower bounds on the additional training time required to achieve fair predictions and validate their findings through experiments on CIFAR-10, EuroSAT, and Adult datasets.", "questions": "1. The paper uses \"SGD with a constant learning rate... no weight decay or learning rate decay schedule was applied\" to match the theoretical setting. How would the results change with other optimization practices (Adam, learning rate schedules, weight decay, pretrained models)? Would the bias amplification be worse or better?\n\n2. Can the theoretical framework extend to settings with multiple minority groups?", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "**Strengths:** \n- The paper is well written and easy to understand.\n- The concepts of \"stereotype gap\" and \"majority-training zone\" provide clear geometric intuition for why standard training amplifies bias in imbalanced settings.\n- The lower bounds on debiasing time offer concrete guidance on how much additional training is needed to achieve fairness.\n- The experiments are validated on different domains (vision and tabular datasets).\n\n**Weaknesses:**\n\n- Theorem Statement: For each pair of distinct elements θ,θ′∈crit L₁ (resp. crit L), we have ∥θ−θ′∥≥δ/32M. \nHowever the Proof Issue: The proof derives two different bounds for critical points of L1 and L:\n\n      - For crit L₁: ∥θ−θ′∥≥δ/2M\n\n      - For crit L: ∥θ−θ′∥≥δ/32M\nHowever, the theorem claims the same bound δ/(32M) for both. This contradicts the proof, which shows a stricter bound δ/(2M) for L1 and a looser bound δ/(32M) for L. This error affects the interpretation of critical point separation and needs correction.\n\n- While the paper identifies the problem and quantifies training time, it doesn't propose algorithmic solutions beyond \"train longer,\" missing opportunities for more efficient debiasing methods.\n\n- The figures lacks consistency and clarity. Several figures (e.g., Figure 1 and Figures 3–5) are not accompanied by sufficiently descriptive captions, and the figure references in the main text are very short.  This makes it difficult for the reader to connect theoretical claims with empirical evidence. For example, the figure showing the majority and minority training zones (Figure 1) is central to the paper’s geometric intuition, but its caption is concise and does not adequately explain the experimental setup or relevance\n\n- The paper lacks a dedicated related work section.", "was_revised": true, "final_justification": "The theoretical framework is solid, the experimental validation supports the claims, and the identified issues are addressable in revision. The core insight about the dominance of the majority-training zone and the quantification of fairness overcost provides valuable theoretical grounding for understanding bias amplification. The authors have satisfactorily addressed my concerns about the proof, committed to improving figure captions and adding a proper related work section. While the paper could benefit from more algorithmic solutions beyond ‘train longer’.", "rebuttal": "**General comment.**\n\n\nWe greatly thank the referees for their useful comments, we believe that they will considerably improve the quality of the paper. We would like to make preliminary statements that will also impact the presentation of our revision.\n\n**A)** We will certainly include all the missing references —we apologize for the oversight, and we acknowledge their importance. A dedicated section on previous work will be added.\n\n**B)** Our contribution is principally theoretical and aims to explain the majority/minority scenario in the nonconvex setting. The empirical part is merely intended to illustrate our theorems, which is why it focuses solely on raw SGD. Our approach via the Kantorovich theorem is both non-trivial and notably robust to changes in assumptions, as it applies to nonlinear equations and admits well-known generalizations to Banach spaces and Riemannian manifolds. While we have opted for a reader-friendly presentation by placing most of the mathematical intricacies in the Appendix, these should not be overlooked, as they form the core of our contribution.\n\n**C)** Through our theoretical contributions, we also provide a framework for analyzing fair training by introducing several notions: the stereotypical gap, stereotypical predictors, fairness overcost, and a geometrical framework for understanding \"stereotyped training,\" which can readily be adapted to other methods.\n\n\nWe now proceed to answering point by point.\n\n\n\n**Reviewer DEPM:** \"*Theorem Statement: For each pair of distinct elements θ,θ′∈crit L₁ (resp. crit L), we have ∥θ−θ′∥≥δ/32M. However the Proof Issue: .... This error affects the interpretation of critical point separation and needs correction.*\"\n\n**Authors:** We respectfully disagree and we think there might be a confusion of some sort. Indeed, there is no mathematical issue with the proof of Theorem 1. We have just chosen to provide only one lower bound (the smaller of the two) for concision in the theorem statement.\n\n(Details: As pointed out by the reviewer, the proof states\n\n- For crit L1, $|| \\theta - \\theta' || \\ge \\frac{\\delta}{2M} $. Since $\\frac{\\delta}{2M} \\ge \\frac{\\delta}{32M}$, we also have  $|| \\theta - \\theta' || \\ge \\frac{\\delta}{32M} $\n\n- For crit L, $|| \\theta - \\theta' || \\ge \\frac{\\delta}{32M} $.\n\nSo it is indeed true that for each pair of distinct elements $\\theta,\\theta'$ in crit L1 (resp. crit L), we have $|| \\theta - \\theta' || \\ge \\frac{\\delta}{32M} $. Note that it is  interesting to see in (ii) that the critical points of L are well-separated while the matching pairs of critical points of L and L1 are close. We shall explain this more clearly.)\n\n\nThe referee's remark is nonetheless useful, as it highlights the need to place greater emphasis on point (i) of Theorem 1, which is the fundamental fact. To avoid confusion, we propose relocating point (ii) and underscoring point (i), which conveys the core result.\n\n\n**Reviewer DEPM:** \"*While the paper identifies the problem and quantifies training time, it doesn't propose algorithmic solutions beyond \"train longer,\" missing opportunities for more efficient debiasing methods.\"*\n\n**Authors:**  We have two answers to this:\n\n- The paper is theoretical: it highlights the key role of the Kantorovich theorem, the importance of second-order proximity, and, beyond the \"train longer\" recommendation, offers a flexible framework for measuring stereotype gaps both numerically and geometrically. We believe these properties and metrics are robust across different settings and provide a solid foundation for a general framework and further studies on other methods. \nLet us also mention, that while we agree that alternative methods may ultimately achieve greater fairness, our study makes us think that long training remains unavoidable when relying on first-order methods. \n\n\n- We shall also provide further references towards recent empirical/theoretical references that we missed and that will open perspectives towards other settings; [1,2,3] of Reviewer 48E9, and also Kunster et al: *\"Heavy-Tailed Class Imbalance and Why Adam Outperforms Gradient Descent on Language Models\"* .  \n\n**Reviewer DEPM:** \"*The figures lacks consistency and clarity ... but its caption is concise and does not adequately explain the experimental setup or relevance.*\"\n\n**Authors:**  We totally agree and will improve the presentation. In particular, we will explain that within the majority training zone on Figure 1, learning is almost blind to L0, making little distinction between training with L1 and L. The learning of L0 is mainly activated outside or near the boundary of this zone.  Here is the addition to the caption that we propose: \"The majority training zone, where only L1 is effectively learned, covers almost the entire space.\"\nFor Figures 3 to 5, the referee is perfectly right again, we will clarify that prolonged training is fundamental to learn minority traits — and the smaller the minority, the longer the training required.\n\n**Reviewer DEPM:** \"*The paper lacks a dedicated related work section.*\"\n\n**Authors:** Definitely, we will add such a section including the references provided by Reviewer 48E9 and new ones that were communicated to us. We will also make more explicit our current references. We refer to the answer to Reviewer 48E9 for an example of a new content of this related work section.\n\n**Reviewer DEPM:** \"*Question 1: The paper uses \"SGD with a constant learning rate... no weight decay or learning rate decay schedule was applied\" to match the theoretical setting. How would the results change with other optimization practices (Adam, learning rate schedules, weight decay, pretrained models)? Would the bias amplification be worse or better?*\"\n\n**Authors:** This is indeed an important question that we plan to study into detail in the future. A comprehensive study of the many adjunction used in classical deep learning deserves to be understood independently, at least empirically, not to speak of the fact that some of them are probably extremely complex to study simultaneously.  In the spirit of the current paper, we could nevertheless include some precise references and a few new experiments to lay the groundwork for future work. For instance we ran the same experiments as Imbalanced CIFAR-10 with AdamW. In the table below we display the fairness overcost (in %) for $\\kappa = 90%$ across imbalance levels  $\\zeta \\in$ [1%, 10%, 30%]. Means over 3 independent runs. Experiments were conducted on a single NVIDIA A100 80GB GPU with a total runtime of approximately 10 hours.\n\n| Model                               | Parameters | 1%  | 10% | 30% |\n|------------------------------------|------------|-----|-----|-----|\n| MobileNetV2  | 543K       | 326  | 310  | 214  |\n| VGG11 | 9M         | 401  | 244  | 169  |\n| ResNet18        | 11M        | 465  | 342  | 209  |\n| ResNet50                           | 25M        | 369  | 327  | 219  |\n| ResNet101                          | 42M        | 220  | 192  | 172  |\n\nFairness overcost increases with imbalance, especially for smaller models. Larger models like ResNet101 are more robust, showing lower overcosts. This suggests that higher capacity helps reduce performance gaps under imbalance.\n\nNote that the overall recommendation \"train longer\" remains and proportions are similar to those of SGD.\n\n\n\n**Reviewer DEPM:** \"*Question 2: Can the theoretical framework extend to settings with multiple minority groups?*\"\n\n**Authors:** Our theory analyzes bias amplification under a binary partition of the data: a majority component and a single  minority subgroup. The loss is decomposed accordingly, and the proof techniques rely on this two‑group structure. The creation of the two subgroups is agnostic. As a result, any subgroup can be studied via a one versus the remaining part split, by repeating the analysis for each subgroup in turn (e.g., on CIFAR‑10, treating each class as the minority in separate runs after inducing imbalance).\nYet the limitation is that  the current theory does not characterize joint interactions among multiple minority groups considered simultaneously. This is an interesting topic that will be the subject of a future work."}, {"review_id": "YJmzi39xeN", "reviewer": "Reviewer_4k56", "summary": "This paper introduces a theoretical framework for analyzing majority–minority tasks. It shows the proximity between “full-data” and stereotypical predictors, analyzes the dominant region of the parameter space, and provides a lower bound on the additional budget required to “unlearn.” The authors also perform experiments to validate their theoretical results.", "questions": "Although the theoretical framework appears model-agnostic, the paper discusses only deep-learning models—why exclude other model classes?\n\nQuestion on Line 99: Why does ∇L₁ = 0 imply ∇L = 0 even when L₁ is much larger than L₀?", "limitations": "See weaknesses.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n\n**Theoretical novelty**: The analysis of the distance between critical points in Theorem 1 is both novel and intuitive.\n\n**Empirical validation and reproducibility**: The authors provide experiments on distances to critical points and include the linear regression case for clarity.\n\n**Weaknesses**\n\n**Unclear scope**: In the introduction, the authors situate this work in the bias literature, yet most experiments use an imbalanced-learning setup. It therefore remains unclear whether the results extend to learning from genuinely biased data.\n\n**Restrictive binary setup**: The assumption of a binary majority/minority split is overly limiting, since real-world settings often involve intersectional minority groups.\n\n**Missing test-set generalization**: It is surprising that test errors are not reported for the experiments presented in Table 1 and Figure 3.", "was_revised": true, "final_justification": "The authors have replied to most of my concerns regarding the paper. My score remains unchanged as the reasons to accept outweigh the reasons to reject.", "rebuttal": "**General comment.**\n\n\nWe greatly thank the referees for their useful comments, we believe that they will considerably improve the quality of the paper. We would like to make preliminary statements that will also impact the presentation of our revision.\n\n**A)** We will certainly include all the missing references —we apologize for the oversight, and we acknowledge their importance. A dedicated section on previous work will be added.\n\n**B)** Our contribution is principally theoretical and aims to explain the majority/minority scenario in the nonconvex setting. The empirical part is merely intended to illustrate our theorems, which is why it focuses solely on raw SGD. Our approach via the Kantorovich theorem is both non-trivial and notably robust to changes in assumptions, as it applies to nonlinear equations and admits well-known generalizations to Banach spaces and Riemannian manifolds. While we have opted for a reader-friendly presentation by placing most of the mathematical intricacies in the Appendix, these should not be overlooked, as they form the core of our contribution.\n\n**C)** Through our theoretical contributions, we also provide a framework for analyzing fair training by introducing several notions: the stereotypical gap, stereotypical predictors, fairness overcost, and a geometrical framework for understanding \"stereotyped training,\" which can readily be adapted to other methods.\n\n\nWe now proceed to answering point by point.\n\n**Reviewer 4k56:** \"**Unclear scope**: *can the results extend to learning from genuinely biased data?*\"\n\n**Authors:** Many thanks, we will indeed clarify this important point and emphasize it further in the introduction. Our goal is to explain how data imbalance leads to bias amplification in learned models. The theory is agnostic to why the imbalance arose; it requires only the imbalance conditions formalized in our theorems (i.e., a majority component and a minority component inducing asymmetric risk contributions as modeled by the ratio term after Theorem 2, that mixes the sampling effect and the variability effect). We shall provide a more direct and clear statement in the introduction, note however that the second paragraph came with a clear description of what could be the different notions of bias as well as the one we study in the paper.\n\nIf we understand the referee correctly, \"genuinely biased data\" refers to bias introduced during the data collection or construction process, rather than bias arising from the learning algorithm itself. As such, it falls outside the scope of this paper.\n\n\n**Reviewer 4k56:**  **Restrictive binary setup**:  \"*The assumption of a binary majority/minority split is overly limiting, since real-world settings often involve intersectional minority groups.*\"\n\n**Authors:** The subject with two groups is not new, however the bias amplification phenomenon was not  explained in the nonconvex case. Along our theoretical perspectives, we believe it is the first natural and substantial step before moving to more complex ones. We also draw the reviewers's attention to the strong theoretical nature of the paper and to the substantial technical background required to understand this majority/minority scenario.  We also point out that our theory singled out several metrics/objects that were not used previously and that will be needed to understand more complex scenarios: stereotypical predictors, stereotype gap, \"fair training=long training\", fairness overcost ...\n\nHowever, we fully agree that intersectional minority groups are fundamental, and we hope to contribute to the development of a theory addressing this case in forthcoming papers. \n\nPlease also refer to our response to Reviewer DEPM, which raises a similar concern.\n\n**Reviewer 4k56:** **Missing test-set generalization.**\n\n**Authors:** We greatly thank the reviewer for this remark. We will add test errors in the main paper and provide a Table 1 with test errors in Appendix C.\n\n**Reviewer 4k56:** *Question: \"Although the theoretical framework appears model-agnostic, the paper discusses only deep-learning models—why exclude other model classes?\"*\n\n**Authors:** While our fairness analysis is model-agnostic, deep learning was indeed our main target. Besides in DL, frameworks like PyTorch are well suited for tracking training dynamics (e.g., mini-batch updates, subgroup metrics). This is mainly why we focused on neural networks, including TabNet, a strong baseline for tabular data.\n\nStill, you will find below an XGBoost experiment on a logistic regression for Adult dataset. We train a single model incrementally: at each iteration, we add 10 more trees (using xgb_model=... to continue training from the previous booster), and evaluate performance on both train and test sets. We track global accuracy and subgroup accuracy (A = 0: minority, A = 1: majority) at regular time intervals.\n\nTraining Time (s) | Train global | Train A=0 | Train A=1 | Test global | Test A=0 | Test A=1 |\n|-------------------|------------------|------------------|------------------|------------------|------------------|------------------|\n| 0.47 | 0.8772 | 0.5530 | 0.8893 | 0.8679 | 0.5263 | 0.8807 |\n| 7.70 | 0.8989 | 0.6734 | 0.9074 | 0.8679 | 0.5996 | 0.8781 |\n| 16.36 | 0.9098 | 0.7235 | 0.9168 | 0.8668 | 0.5940 | 0.8771 |\n| 26.45 | 0.9190 | 0.7623 | 0.9249 | 0.8649 | 0.5921 | 0.8752 |\n| 38.03 | 0.9261 | 0.7987 | 0.9308 | 0.8624 | 0.5977 | 0.8724 |\n| 51.11 | 0.9312 | 0.8205 | 0.9353 | 0.8606 | 0.5996 | 0.8705 |\n| 65.64 | 0.9359 | 0.8464 | 0.9393 | 0.8589 | 0.5959 | 0.8688 |\n| 82.94 | 0.9385 | 0.8529 | 0.9417 | 0.8579 | 0.5959 | 0.8678 |\n\nAs in Deep Learning, we observe that longer training progressively improves the minority group’s accuracy, reducing the gap with the majority group. This effect is visible both in train and test performance. While the global test accuracy plateaus early, the test accuracy for A = 0 continues to increase with training time.\n\n\n**Reviewer 4k56:**  *\"Question on Line 99: Why does $\\nabla L_1 = 0$ imply $\\nabla L = 0$ even when $L_1$ is much larger than $L_0$?\"*\n\n**Authors:** We are confused as we did not write this implication on Line 99. We meant to informally refer to conclusion (i) of Theorem 1, that states that when there is a local minimizer for the majority loss $L_1$, then there is another local minimizer of the total loss L that is close in the parameter space. Of course, these two local minimizers are not exactly equal in general, so we did not mean to say that $\\nabla L_1 (\\theta) = 0$ exactly implies $\\nabla L(\\theta) = 0$.\nWe will clarify this sentence, as it importantly relates to one of the core conclusions of the paper."}, {"review_id": "9kTsWdyrMd", "reviewer": "Reviewer_48E9", "summary": "This paper investigates theoretical causes of *bias amplification*, specifically in settings where class imbalance leads models to favor majority groups over minority ones. The authors formalize a majority–minority learning scenario in which the empirical loss decomposes into majority loss $L_1$ and minority loss $L_0$, with $L_0 \\ll L_1$. They prove a perturbation theorem: every critical point of the full loss $L = L_0 + L_1$ is close to a critical point of $L_1$ alone. These paired solutions are termed “representative” (trained on all data) and “stereotypical” (majority-only), and their distance, the “stereotype gap”, is bounded by a quantity that depends on the imbalance in population size and variance. This implies that most of parameter space is dominated by stereotypical solutions. The authors also derive a lower bound on the time required to escape these solutions during training. Experiments on tabular and vision tasks show that typical training procedures converge to stereotypical predictors, and that long training or dataset rebalancing is needed to recover features specific to the minority group.", "questions": "* Please expand the related work on the theory of bias amplification, including how this work connects to and extends prior theoretical and empirical studies.\n* Please include a brief description of the main assumptions (e.g., in Theorem 1) and discuss when they are expected to hold in practice.", "limitations": "The authors adequately addressed the limitations.", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n\n* The paper is generally well-written and clearly structured.\n* It tackles an important and under-explored theoretical question: why models trained on imbalanced data tend to amplify bias against minority groups.\n* The theoretical results are not purely abstract; they have practical implications, for instance, highlighting the potential risks of early stopping when validation metrics are dominated by majority-class performance.\n* The experimental evaluation spans a diverse range of settings, including multiple model architectures, various imbalance ratios, and both image and tabular datasets.\n\n**Weaknesses**\n\n* Although the paper claims to be among the few theoretical treatments of bias amplification (lines 28–30), it overlooks several relevant prior works that directly address this issue from a theoretical standpoint \\[1,2,3].\n* Key assumptions underlying the theoretical results (e.g., those in Theorem 1) are not discussed in terms of their realism or the conditions under which they are likely to hold in practice.\n* While the main theoretical insight is sound, the underlying intuition is fairly well-established in the field, so the paper may not significantly advance the state of the art.\n\nReferences:\n\n[1] Francazi, E., Baity-Jesi, M. and Lucchi, A., 2023, July. A theoretical analysis of the learning dynamics under class imbalance. In International Conference on Machine Learning (pp. 10285-10322). PMLR.\n\n[2] Subramonian, A., Bell, S.J., Sagun, L. and Dohmatob, E., 2025. An Effective Theory of Bias Amplification. In International Conference on Learning Representations.\n\n[3] Mannelli, S.S., Gerace, F., Rostamzadeh, N. and Saglietti, L., Bias-inducing geometries: exactly solvable data model with fairness implications. In ICML 2024 Workshop on Geometry-grounded Representation Learning and Generative Modeling.", "was_revised": true, "final_justification": "I believe the proposed theoretical framework offers a valuable perspective for understanding bias amplification. While the rebuttal provided a rough characterization of the main assumptions and included a commitment to incorporate the suggested related work, a clearer overview of all core assumptions is still lacking, and it remains unclear how the contribution fits within the broader context of existing literature. Overall, my recommendation is a borderline accept.", "rebuttal": "**General comment.**\n\n\nWe greatly thank the referees for their useful comments, we believe that they will considerably improve the quality of the paper. We would like to make preliminary statements that will also impact the presentation of our revision.\n\n**A)** We will certainly include all the missing references —we apologize for the oversight, and we acknowledge their importance. A dedicated section on previous work will be added.\n\n**B)** Our contribution is principally theoretical and aims to explain the majority/minority scenario in the nonconvex setting. The empirical part is merely intended to illustrate our theorems, which is why it focuses solely on raw SGD. Our approach via the Kantorovich theorem is both non-trivial and notably robust to changes in assumptions, as it applies to nonlinear equations and admits well-known generalizations to Banach spaces and Riemannian manifolds. While we have opted for a reader-friendly presentation by placing most of the mathematical intricacies in the Appendix, these should not be overlooked, as they form the core of our contribution.\n\n**C)** Through our theoretical contributions, we also provide a framework for analyzing fair training by introducing several notions: the stereotypical gap, stereotypical predictors, fairness overcost, and a geometrical framework for understanding \"stereotyped training,\" which can readily be adapted to other methods.\n\n\nWe now proceed to answering point by point.\n\n**Reviewer 48E9:** \"*Although the paper claims to be among the few theoretical treatments ... that directly address this issue from a theoretical standpoint [1,2,3].*\"\n\n **Authors:** We thank the referee for pointing us these recent papers. The reviewer is perfectly right on their relevance: we apologize. We will fix this issue carefully and mention these contributions in a section on related works. We will discuss several aspects of these works, as for instance, the quadratic cases and the phenomenon of Minority Initial Drop (MID) which clearly resonates with the majority-training zone effect.\n\n**Reviewer 48E9:** \"*Key assumptions underlying the theoretical results ... under which they are likely to hold in practice.*\"\n\n **Authors:** We will clarify that point. At this stage we may say that when the proportions are considerably imbalanced, \\$L\\_0\\$ becomes a perturbation of \\$L\\_1\\$, and our assumptions are very likely to hold. Note also that we discussed the conditions in the linear regression case in Theorem 2 - a precised version of Theorem 1. In that case they have an appealing statistical nature as we see that  the majority/minority ratio of \"variabilities\" is also an important factor. Data with low variability are more likely to be \"ignored\" by standard training. This will be commented in the new version.\n\n**Reviewer 48E9:**  \"*While the main theoretical insight is sound ... may not significantly advance the state of the art\".*\n\n **Authors:** We are unsure of how the statement of the reviewer should be understood, but for us the objective of this paper was to provide mathematical grounds to intuition and empirical observations of bias amplification. From that viewpoint we believe it is successful. Moreover, note also, that as mentioned in some of our other answers, we wish to emphasize that we also bring several novelties that allow for a real understanding of the issues at stake: metrics (gaps, ratios), geometrical insights, recommendation of longer training. See also general comments.\n\n**Reviewer 48E9:**  *Question: Please expand the related work on the theory of bias amplification, including how this work connects to and extends prior theoretical and empirical studies.*\n\n **Authors:** We will definitely do this. In particular, we will add and comment the following content to the new related work section: \n\n[1] Francazi, E., Baity-Jesi, M. and Lucchi, A., 2023, July. A theoretical analysis of the learning dynamics under class imbalance. In International Conference on Machine Learning (pp. 10285-10322). PMLR. \n\n[2] Subramonian, A., Bell, S.J., Sagun, L. and Dohmatob, E., 2025. An Effective Theory of Bias Amplification. In International Conference on Learning Representations. \n\n[3] Mannelli, S.S., Gerace, F., Rostamzadeh, N. and Saglietti, L., Bias-inducing geometries: exactly solvable data model with fairness implications. In ICML 2024 Workshop on Geometry-grounded Representation Learning and Generative Modeling.\n\n[4] A systematic study of bias amplification, Hall, Melissa and van der Maaten, Laurens and Gustafson, Laura and Jones, Maxwell and Adcock, Aaron, arXiv preprint arXiv:2201.11706, 2022\n\n[5] An investigation of why overparameterization exacerbates spurious correlations, Sagawa, Shiori and Raghunathan, Aditi and Koh, Pang Wei and Liang, Percy, International Conference on Machine Learning, 8346--8356, 2020.\n\n[6] Kunster et al: \"Heavy-Tailed Class Imbalance and Why Adam Outperforms Gradient Descent on Language Models\" .\n\n**Reviewer 48E9:** *Question: Please include a brief description of the main assumptions (e.g., in Theorem 1) and discuss when they are expected to hold in practice.*\n\n **Authors:** Yes, we will definitely do that. When the proportions are considerably imbalanced, \\$L\\_0\\$ becomes a perturbation of \\$L\\_1\\$, and our assumptions are very likely to hold. Other possibilities include differences in variability, as in the linear regression case, which have a more statistical nature (see the ratio after Theorem 2). See also our answer to Reviewer 4HDc."}, {"review_id": "t7t2LZb7rQ", "reviewer": "Reviewer_4HDc", "summary": "The paper develops a framework for majority-minority learning tasks, going beyond empirical evidence of bias amplification to provide theoretical foundations. It shows how standard training can favor majority groups and produce stereotypical predictors that neglect minority-specific features.", "questions": "- In the experiments on the imbalanced CIFAR-10 dataset, the paper states that one class (A=0) is subsampled while others (A=1) remain unchanged. Which specific class is chosen? Does the choice of class influence the results?\n- Does Theorem 2 hold only under the linear regression model assumption? Why was linear regression selected as the case study?\n- Although the paper states that “assessing fairness on the test set would require extensive tuning and is left for future work,” and evaluates fairness only using training accuracy, are there any preliminary insights into the gap between training-set fairness and test-set fairness?", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 3, "originality": 2, "strengths_and_weaknesses": "**Strengths**\n\n- The paper conducts various experiments across image and tabular datasets to validate the effectiveness of the proposed method and introduces the \"fairness overcost ratio\" to quantify the additional computational cost required for fair training.\n- Based on both theory and experiments, the paper offers practical advice: extend training duration, use larger models, and carefully monitor minority-group performance to avoid stereotype gaps.\n\n**Weaknesses**\n\n- The paper lacks a discussion of the related works. Please add a section on related works in the main paper.\n- The theoretical part relies on relatively restrictive assumptions, such as in Theorem 1, which assumes, which assumes $\\\\tau < \\\\min\\\\left\\\\{ \\\\frac{c}{2}, \\\\frac{\\\\delta}{8}, \\\\frac{\\\\delta^2}{32M} \\\\right\\\\}$. Do the real-world models, like the models tested in the experiments, meet these assumptions?\n- The theoretical results are derived primarily for convex/quadratic losses, while the experiments are conducted using non-convex deep networks. Is there a theoretical gap between these settings?\n- The presentation could be improved.\n    - For example, Figure 1 could be better illustrated with a more detailed caption or elsewhere in the paper, better pointing out what Figure 1 wants to show.\n    - Section 2.4 (linear regression) appears abruptly after the general theoretical framework without a clear rationale. The authors don't justify why linear regression was chosen as the case study or explain what insights it provides beyond the general theory. This also creates poor logical flow: general theory → sudden linear case → back to general gradient dynamics.", "was_revised": true, "final_justification": "The response addressed my concerns and helped me better understand the work. After reading the reply and the comments from other reviewers, I lean towards acceptance.", "rebuttal": "**General comment.**\n\n\nWe greatly thank the referees for their useful comments, we believe that they will considerably improve the quality of the paper. We would like to make preliminary statements that will also impact the presentation of our revision.\n\n**A)** We will certainly include all the missing references —we apologize for the oversight, and we acknowledge their importance. A dedicated section on previous work will be added.\n\n**B)** Our contribution is principally theoretical and aims to explain the majority/minority scenario in the nonconvex setting. The empirical part is merely intended to illustrate our theorems, which is why it focuses solely on raw SGD. Our approach via the Kantorovich theorem is both non-trivial and notably robust to changes in assumptions, as it applies to nonlinear equations and admits well-known generalizations to Banach spaces and Riemannian manifolds. While we have opted for a reader-friendly presentation by placing most of the mathematical intricacies in the Appendix, these should not be overlooked, as they form the core of our contribution.\n\n**C)** Through our theoretical contributions, we also provide a framework for analyzing fair training by introducing several notions: the stereotypical gap, stereotypical predictors, fairness overcost, and a geometrical framework for understanding \"stereotyped training,\" which can readily be adapted to other methods.\n\n\nWe now proceed to answering point by point.\n\n**Reviewer 4HDc:** \"*The paper lacks a discussion of the related works. Please add a section on related works in the main paper.*\"\n\n**Authors:** We definitely agree, a section on related work will be added. We refer to our responses to Reviewer 48E9.\n\n**Reviewer 4HDc:** \"*The theoretical part relies on relatively restrictive assumptions ... meet these assumptions?*\"\n\n**Authors:** It seems to us that our assumptions are in line with common standards in Deep Learning, which often offer only qualitative insights into key quantitative aspects. In contrast, our assumptions are rather concrete: they amount to say that variability (embodied in second-order dominance) and population ratios must be unbalanced. This is further illustrated in the convex case (Theorem 2), which is of course more accessible. On this occasion, we should also mention that applying Theorem 1 to the convex case yields estimates very similar to those in Theorem 2—except for some boundedness constraints, which can be easily lifted by leveraging convexity.\n\nNote as well that, as explained to the other referees, strong imbalance is likely to provide the assumptions required by Theorem 1. \n\nWe will add comments on our main assumptions and concrete unbalanced examples in which our setting makes sense concretely.\n\n\n\n**Reviewer 4HDc:** \"*The theoretical results are derived primarily for convex/quadratic losses ... between these settings?*\"\n\n**Authors:** We respectfully disagree with this statement, though we will revise the order of presentation to avoid such confusion. Our theoretical results are stated for general nonconvex losses—this includes Theorem 1, Corollary 1, Theorem 3, Lemma 1, and Propositions 2 and 3 (which require minor adjustments as noted in Remark 2 and the Appendix).\n\nThe only results that are specific to convex or quadratic losses are Theorem 2—an “augmented” and specialized instance of Theorem 1—and Proposition 1, which are quadratic by construction. As clarified in Remark 2, Propositions 2 and 3 rely on Lemma 4, which remains valid in the nonconvex setting; they therefore hold under the usual localization arguments.\n\nIn summary, we do not see a theoretical gap between the nonconvex and convex-quadratic settings. We have emphasized the convex-quadratic case simply to provide a more concrete and interpretable illustration, especially given the familiarity of many readers with that framework.\n\n**Reviewer 4HDc:** \"*The presentation could be improved. For example, Figure 1 ... what Figure 1 wants to show.*\"\n\n**Authors:** We agree with the comment on Figure 1. As also replied to Reviewer DEPM, we will explain that within the majority training zone on Figure 1, learning is almost blind to $L_0$, making little distinction between training with $L_1$ and $L$. The learning of $L_0$ is mainly activated outside or near the boundary of this zone. Here is part of the updated caption that we propose: The majority training zone, where only $L_1$ is effectively learned, covers almost the entire space. Other captions will be clarified.\n\n**Reviewer 4HDc:**  \"*The presentation could be improved. Section 2.4 (linear regression) appears ... sudden linear case → back to general gradient dynamics.*\"\n\n**Authors:** As stated above, our motivation for including linear regression is theoretical (it is the “linearized model” in the Morse case), it is an illustration of our theorem (improved through convexity), and it is also simpler and easily interpretable. But this comment, along with your earlier remark on the distinction between the nonconvex and convex-quadratic settings, prompts us to give less prominence to linear regression. We will isolate the results specific to linear regression in a dedicated section. \n\n**Reviewer 4HDc:** \"*Question: In the experiments on the imbalanced CIFAR-10 dataset ... Does the choice of class influence the results?*\"\n\n**Authors:** Thank you for your question. In our experiments on imbalanced CIFAR-10, we subsampled the class “cat” as the underrepresented group (A = 0), while keeping the other nine classes unchanged (A = 1). We chose “cat” arbitrarily to ensure reproducibility across runs, but we emphasize that the qualitative conclusions of our paper do not depend on this specific choice, as confirmed by our preliminary tests. We will clarify this point in the revised version.\n\n\n**Reviewer 4HDc:** \"*Question: Does Theorem 2 hold only under the linear regression model assumption? Why was linear regression selected as the case study?*\"\n\n**Authors:** Theorem 2 is indeed specialized to linear regression, but it is an instance of Theorem 1, which holds in the general nonconvex setting. The benefit of Theorem 2 is that it requires the definition of fewer quantities than Theorem 1 and offers a more accessible interpretation for a broader audience.\n\nBeyond the familiarity of linear regression to most readers, this model has a deeper theoretical significance: by Morse’s lemma and linearization theorems, the local geometry of Morse functions (our nonconvex assumption) is essentially quadratic—hence, locally equivalent to the linear regression case. The theoretical rationale is thus one of *local linearization*, which naturally leads to convex models, and in particular to linear regression.\n\n\n**Reviewer 4HDc:** \"*Question: Although the paper states that ... between training-set fairness and test-set fairness?*\"\n\n**Authors:** While our main results focus on the training set, we did run preliminary tests on the test set. The trends are similar overall, but test set disparities tend to be slightly larger, likely due to generalization gaps, especially for the minority group. A full analysis would require careful tuning, which we leave for future work."}]

Author rebuttal:

Meta-review: This paper studies bias amplification, and proposes a theoretical framework for majority-minority learning tasks, showing how standard training can favor majority groups and produce stereotypical predictors that neglect minority-specific features. The rebuttal successfully addressed most of the initial concerns (clarifications of the contributions, stronger empirical evidence, and related work analysis), leading to two reviewers raising their scores. After careful deliberation, the decision was made to accept this paper - congratulations to the authors! It is crucial, however, that all provided clarifications are incorporated in the final version of the paper.
Decision: Accept (poster)