A Differentiable Rank-Based Objective for Better Feature Learning
Authors: Krunoslav Lehman Pavasovic
Giulio Biroli
Levent Sagun
Venue: ICLR 2025 Poster (2025)
Keywords: ranks
statistics
machine learning
feature learning
rank ordering correlation
fairness
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning

Abstract: In this paper, we leverage existing statistical methods to better understand feature learning from data. We tackle this by modifying the model-free variable selection method, Feature Ordering by Conditional Independence (FOCI), which is introduced in Azadkia & Chatterjee (2021). While FOCI is based on a non-parametric coefficient of conditional dependence, we introduce its parametric, differentiable approximation. With this approximate coefficient of correlation, we present a new algorithm called difFOCI, which is applicable to a wider range of machine learning problems thanks to its differentiable nature and learnable parameters. We present difFOCI in three contexts: (1) as a variable selection method with baseline comparisons to FOCI, (2) as a trainable model parametrized with a neural network, and (3) as a generic, widely applicable neural network regularizer, one that improves feature learning with better management of spurious correlations. We evaluate difFOCI on increasingly complex problems ranging from basic variable selection in toy examples to saliency map comparisons in convolutional networks. We then show how difFOCI can be incorporated in the context of fairness to facilitate classifications without relying on sensitive data.

Published as a conference paper at ICLR 2025
A DIFFERENTIABLE RANK-BASED OBJECTIVE FOR BET- TER FEATURE LEARNING
Krunoslav Lehman Pavasovic* Meta FAIR, Paris
Giulio Biroli' ENS Paris
Levent Sagun' Meta FAIR, Paris
ABSTRACT
In this paper, we leverage existing statistical methods to better understand feature learning from data. We tackle this by modifying the model-free variable selection method, Feature Ordering by Conditional Independence (FOCI), which is intro- duced in Azzadiia & Chatterjee (20TT) While FOCI is based on non- con-maraettric coefficient of conditional dependence, we introduce its parametric, differentiable approximation. With this approximate coefficient of correlation, we present a new algorithm called difFOCI, which is applicable to wider range of machine learning problems thanks to its differentiable nature and learnable parameters. We present difFOCI in three contexts: (1) as variable selection method with baseline compar- isons to FOCI, (2) as a trainable model parametrized with a neural network, and (3) as generic, widely applicable neural network regularizer, one that improves feature learning with better management of spurious correlations. We evaluate difFOCI increasingly complex problems ranging from basic variable selection in toy examples to saliency map comparisons in convolutional networks. We then show how difFOCI be incorporated in the context of fairness to facilitate classifications without relying on sensitive data.
INTRODUCTION
Feature learning is crucial in machine learning (ML), enabling models learn meaningful repre- sentations of the data. It can improve performance, reduce dimensionality, increase interpretability. and provide flexibility for adapting to new data distributions and tasks (Bengio al., 2012 2013). However, increasing model transparency (Arrieta et al. 2020 Rauker et al. 2023). improving disen- tanglement and understanding architectural biases (Bouchacourt et al. 2021 Roth et al. 2022). as well as learning invariances to improve robustness (Arjovsky et al.) 2019) have proven to be challenging.
In this paper, we propose a new feature-learning approach that relies on ranks, notion seldom explored in the literature but thoroughly studied in statistics. The importance of ranks is evident, from independence tests (Bergsma & Dassios 2014; BBum et al. 1961, Csörgö 1985] Deb & Sen
2023 Drton et al. 2020) and sensitivity analysis (Gamboa et al. 2018). multivariate analysis (Sen & Purj 1971) and measuring deviation (Rosenblatt] 1975). However, most of these methods are nonparametric and, therefore, not easily extendable to feature learning with neural networks (NNs).
While there are handful of feature learning works that rely on rank notions (Kuo & Hsu 2017]
Wojtas Chen 2020 Fan et al. 2023 Li et al. (023b), these works do so indirectly and through reliance on two NNs; one that optimizes for a non-rank-based feature-learning objective and another
that learns how to rank those learned features according to some similarity measure.
To fill this gap. we propose difFOCL, a parametric relaxation of the nonparametrie, rank-based measures of correlation (Chatterjee 2020 Azadkia & Chatterjee, 2021). ). which generalizes the measure proposed by Dette et al. (2013), roots of the idea can be related to the Rényi correlation (Bickel et al.) 1993, Rényi 1959p. To the best of our knowledge, difFOCI is the first parametric framework that directly optimizes rank-based objective, making it directly applicable to numerous applications in machine learning, including end-to-end trainable neural networks. We demonstrate that difFOCI yields strong results in various areas, including (i) feature selection, (ii) domain shift and spurious correlation, and (iii) faimess experiments.
"Correspondence trunolp@meta.com. mema.com. Joint last author.

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Organization of the paper. In Section 2, we introduce the notation and technical background. In Section 3, we outline the main results this paper, explaining the proposed metric, and establishing its theoretical properties. We analyze it in toy examples that demonstrate solid performance. In Section 4, we extend difFOCI, showcasing its strengths in three examples. In Section 5, we highlight its wide applicability real-world data, showing it achieves state-of-the-an performance on feature selection and dimensionality reduction, and competitive performance in domain shift and fairness literature. Finally, Section 6, we conclude with few remarks on the potential future applications.
PRELIMINARIES AND TECHNICAL. BACKGROUND
2.1 NOTATION AND PRELIMINARY DEFINITIONS
We let la denote the x d identity matrix and [n] = {I....,n) n). We let = be the set of all permutations of A, with |A] all permutations its columns by 5(X) and by T!! (X), represent the i-th element of the j-th permutation. We denote its p-th through g-th column as Xp.q with > q. P. E N. We define the Hadamard product between vector a E 12 and matrix X T. (a ① X)i == X.x. We represent the scaled Softmax function with (I()), where az(2); = Sz, EE= G2, for x E Bd. E Rt. Finally, we use c(z,p) denote zeroing out any x; with |r;| S p. for x Rd and p R.
(A) For matrix X, we denote the set of
2.2 CHATERJEE'S COEFRICIENT
We present the novel rank-based estimator developed by Chattcrjee 2020), which is the first of two foundational works necessary for our approach. Consider random vector (X, Y) on a probability space (S2, F, P), with Y being non-constant and governed by the law The estimator approximates the following rank-based measure (Dette et 2013):
5(X,Y) = J Var (E (1(22)) | dp(t)
J'aar (1 (1(20) du(t)
Chatterjee (2020) establishes straightforward estimator for ①I that has simple asymptotic
theory, enjoys several consistency results and exhibits several natural properties; (i) normal-
ization: {XX,Y) Y) E 10. (ii) independence: E(X.Y) = O > Y 1L , (ii)) complete dependence: {(X, Y) = = Y measurable function of X a.s., and (iv) scale invariance: S(aX, = <(X, Y), Y), R*. To estimate &, consider i.i.d. pairs (X,, Y))
(X, Y), with 2 2.
Rearrange the data as (X(z). (X.)....,,, uniformly at random. Define r; as the rank (()) i.e. the number of j for which Y6) S M(j) and 4, as the number of such that Yo) Y(j. The estimator is then defined as:
...., (X(x), Y(m)) such that X(1) S .... I X(m), breaking ties
rit1 m
(x,
(2)
2 4
Furthermore, Chatterjee 2020 establishes the following consistency result for En:
Theorem 1. (Chatterjee 2020 Y is not almost surely a constant, then as + oo, En (x(X,Y) converges almost surely to the deterministic limit E(X, Y).
In simulations by Chatterjee (2020), this estimator demonstrates greater efficacy than most signal-detection tests. Its applications span diverse areas: approximate unlearning (Mehta al. 2022), topology (Deb et al. (020), black carbon concentration estimation (Tang et al.) 2023).
sensitivity analysis (Gamboa al.[[2022] and causal discovery (Li et al. 2023a). Extensive further research has been conducted: its limiting variance under independence (Han & Huang 2022).
permutation testing (Kim al. (2022), bootstrapping (Lin & Han 2024), rate efficiency (Lin Han
2023), minimax optimality (Auddy et al. (023) and kernel extension (Huang et al., (022); Bickel
(0022 analyzed it for independence testing, showing might have no power or prove misleading"
This does impact us, however, as do utilize for independence testing.
2

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2.3 EXTENDING THE COEFFICIENT FOR ESTIMATING CONDITIONAL DEPENDENCE
In a subsequent study, Azadkia & Chatterjee 2202 extend the coefficient I ε to measure T(Y, X). capturing the strength of the conditional dependence between Y and Z. given X. T can be interpreted as a non-linear extension of the partial R² statistic (Draper & Smith 1998). and reads as follows:
T T(Y X): = /E(Vaa 2 Z,X) X))du(t)
E (ar (1 (r21) X)) dy((t)
where denotes a random variable governed by μ. and (X1, (XI...., ,I,,).,pp)) Xp) and (Z), Eg) are random vectors, defined within the same probability space, with i.i.d. copies (X,, Zi, Yi) ~ (X, Z, Y), 2. Here, q 2 and p O, with p = 0 indicating X has no components.
The statistic T generalizes the univariate measure in (I). To construct its estimator, for each index define (i) the index j where X, the closest X,, and (i) the index j where the pair (X,, j. Zj) is closest to (X,, Z.) in RP+4 w.r.t. the Euclidean metric and resolving ties randomly. The estimate of given by:
T, = T,(Y, X) = E (min fri, (ro)) min frarwwo))
Σ (rs min {ri, (N()))
with Mi) denoting the index such that Zy is the nearest neighbor of Z, p 2 1 and rú, I as defined in Sec. 2.2 The authors establish the same four natural properties for as for the estimator in normalization. independence, complete dependence, and scale invariance:
Theorem 2. (Azadkia & Chatterjee 2021) Suppose that Y is not almost surely equal measurable function of Then T is well-defined and 0 < 1. Moreover, T iff Y and Z are conditionally independent given X, and T
Y is almost surely equal to measurable function of Z given X.
The authors further demonstrate that Tn is indeed consistent estimator T:
Theorem 3. (Azadkia & Chatterjee 2027) Suppose that Y is not almost surely equal to measurable function of X. Then as + x., Tn almost surely.
2.4 FOCI: A NEW PARADIGM FOR FEATURE SELECTION
Azadkia & Chatterjee (202)) utilize the estimator T, to propose a novel, modellindeepnndent bep-wise feature selection method. The method, termed FOCI: Feature Ordering by Conditional Independence, is free from tuning parameters and demonstrates provable consistency. FOCI is outlined in Alg. [ where observe its iterative nature: variables chosen one by until the estimator's value drops below zero.
Algorithm FOCI
Input: n i.i.d. copies of X),, with the set of predictors X (Xx) jelp) and response Y Ji <- arg maxjejp) T, X;) if (Y, S then
5
else
while Tn (Y, X,
do jk++1 &- arg max (MlxxppMMi (1)) Ta (Y,X,
$ (ii,..,j') jr}
Output: Set S of chosen predictors' indices
"The expression for p is given Appendix A.1
3

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FOCI performs well on both simulated and real- datasets. In toy example with Y X1X2 sin(X, [xX),, where X; ~ N(O, o² Ip). o² = 1, and E [2000], p = 100, FOCI selects the correct subset 70 percent of the time. In contrast, popular scikit-learn feature selection algorithms (Pedregosal et al. 201T), explained in Sec. | almost never identify the correct subset (difFOCI, proposed in the next section, consistently selects the correct subset while preserving the same relative feature importance as FOCI during its correct runs). When applied to real-world datasets, FOCI matches the performance of established methods while requiring up four times fewer features.
2.5 EXTENDING T TO MACHINE AND DEEP LEARNING
From statistical point of view, both εn and Tn exhibit several strengths: well-established theoretical properties, are non- non-parametric, have no tunable parameters nor any distributional assumptions. Furthermore, simple application of Tf results in strong feature-selection baseline. However, the non-smooth nature of the objectives in and (3 renders them non-differentiable, and therefore not applicable to most ML application
In the following section, we make these objectives differentiable using straightforward, well-known tricks in the ML community. This allows us extend them to various ML and deep learning applications (as showeased in Sec. (5. Moreover, it also allows to account for interactions between all features simultaneously (rather than in a step-wist fashion in FOCT). Although FOCI could account for this in principle, as can be seen from Alg. [1] this would increase FOCI's complexity from Op²²) to potentially ((2)) thus preventing its practical use.
MAIN RESULTS
We now propose an alternative formulation to the estimator T, in (3). the objective of FOCL. As we will show later, this variation allows for the retention of FOCI's strengths well as the improvement of its shortcomings.
3.1 DIFFOCI: TOWARDS A DIFFERENTIABLE VERSION OF FOCI
The initial step involves making the objective T, (Y, Z differentiable w.r. inputs Z. Implementing this can be accomplished using straightforward techniques. We employ the following approach:
1. Compute the pairwise distance matrix M R where Mi.1 ||X; X,
2. Calculate S, E IR. such that S3 = a (M AI
3. Instead of indexing (N(() r[N(i)), utilize r S
Similarly, for Ug := as( (M + AI))), and M, I((,, Z.) - (X,, 2,)l-- This allows us to present difFOCI, a differentiable version of the estimator in (3)
= En (minfr;, Us, }} min(r;, }})
Tn,3 =
E min r S..
Using the following theorem, we establish that our new estimator theoretical properties as the estimator in (3):
enjoys the same limiting
Theorem 4. Let R+ Suppose that Y not almost surely equal a measurable function of X. Then, limi,-o gim To.s = T almost surely.
The proof's core argument (given in Appendix B is based on demonstrating that the quantities
r "U,,. and T " SS. converge to (s((s) and (N) respectively the inverse temperature parameter 33 approaches infinity, Once this convergence is established, the remainder of the proof follows easily
from Theorems 5 and 6 Azadkia Chatterjee outlined in Appendix
'Even if applicable, FOCT often not well-suited for deep learning applications, as shown in Sec. 5.2 "hhroughout the experiments,
max(1e'0 max,,; M,,j e).
4

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1.0
1.0
1.0
1.0
0.0
0.0
0.0
0.0
1.0
1.0
1.0
1.0
(a) Generating functions of functional process
Param. norms
[0]] - 0.27
[6:] 0.23
[6 0.18
[8 - 0.06
[x] 0.01
-10
(b) First plot: norms of 0. Remaining plots: features with largest param. norms (only first selected).
Figure I:
Synthetic dataset experiment, detailed in Sec. 3.2 Out of 240 total features, our vec- (dc(dFF)
selects three informative, yet diverse features (corresponding to norms 0.27, 0.23, and 0.18).
Making the estimator differentiable allows us to use Tu,6 in various ways. Considering the predictors X, response variable Y and potentially available sensitive attributes Xss or group affiliations Xc. parameterization fo. we highlight three ways to use Tn,s:
(dF1) To TY, fX)): as maximization objective, learning features that preserve ranks in the same fashion as the response
(dF2) ((Y. Y.Y. Y) + XTn,, (e( (X)): fo (X) for being dependent the protected groups Xg, where l denotes the standard loss used in machine learning
as a regularizer, penalizing the outputs (or learned features)
(dF3) nn fo, (x())(Xs))
conditioning objective, allowing learn features that contain information about the response only after conditioning out the sensitive information Xs
For instance, (dF1) can be utilized for feature selection or dimensionality reduction techniques. (dF2) can be employed to prevent the network from relying on spurious correlations when group attributes are available. (dF3 can be applied in fairness scenarios where we aim to avoid predictions based on certain personal information.
The remaining task is to select the parameterization fo(.). .). In the following sections, we will focus on two options: (i) vec a dot product parameterization fx(X) O X, or (ii) NN neural network parameterization, fe 5 Algorithm provides general outline for using the T, Th, with chosen parameterization, and specific instances of the algorithm are given Appendix
Algorithm 2 Differentiable FOCI (difFOCI)
Input: predictor "R response Y IR", and optional X {0, S, G). for sensitive ggn.d or group info. G' E gn.d d >
Input: parameterization fo E NN), objective choice T,,s E (dFl Initialize 0
(dF2) (dF3)
for 1. 1,..= Ther do
L (Y, Ja, (Z) X)
I/ Applying difFOCI
Update 0(+1 + Optim( C, 02) Output: parameterization fo
// Parameter update
We proceed by testing whether difFOCI performs well at FOCI's main application feature selection. begin with simulated dataset, followed by three experiments with increasing complexity.
$ Foor example, with (ec- we denote using (dF1) with vector parameterization.
WWe also tried vec-NW parameterization fx 0g ④ fo, (X), with 0 {01, 02) but it did not show any improvements over the NN parameterization.

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3.2 PRELIMINARY SYNTHETIC STUDY
To evaluate the feature selection performance of difFOCI, we utilize vec rec((F1) to obtain the objective Tn,a(Y, 0 @ X). Unlike FOCI, which returns a binary vector indicating whether a feature is selected or rejected, difFOCI's version 1ec yields real-valued vector with components (0:) representing the predictive informativeness of each corresponding feature (which allows taking into account feature variability). To perform feature selection, need to choose cutoff parameter and select the features with [ 2 v.
Alg. 2 therefore requires the following hyperparameters: softmax temperature 3, cutoff value v, and optimization parameters (e.g.. learning rate y, weight decay X, minibatch size b, etc.). Our experimental analyses show that / 5 and v 0.1 yield consistently good performance, so we set these as fixed" As a result, our algorithm simplifies solely to the hyperparameters used in conventional optimization methods, which are in Appendix foo all experiments.
Environment. As an initial example, we consider data-generation process ideal for FOCI: from a large pool of features, a handful is sufficient for strong performance with n p. The functional process is illustrated in Fig. Ia crafted to generate a diverse set of features: informative ones, such as straight lines, sinusoids, or parabolas, and functions individually uninformative, yet informative in muliidimensional contexts, e.g., ellipses, rotated parabolas, and more involved curves. This process includes 60 functions, each noised four times, resulting in p 240 features with = 100 points. Ideally, a feature selection method should pinpoint a small but diverse set of features 18
Baselines. For comparative analysis. we employ various feature selection techniques from the seikit-learn library (Pedregosa et al. 2011). These include: GenericUnnivariaeeeeeet (GUS) for univariate feature selection, SelectPercente (S.Per.), retaining only the top user- specified percentile of features, and statistical test-based methods: SelectFpr (FPR), SelectFdr (FDR), and SelectFwe (FWE) addressing false positive rate, false discovery rate, and family-wise error, respectively. Additionally, we employ Best (K.B) to select the best 25%, 50%, or 75% of features based on the ANOVA value test (Girden) 1992). We also benchmark against dimensionality reduction techniques including Linear Discriminant Analysis (LDA, Fisher (1936), Principal Component Analysis (PCA, Wold et al. (1987), and Uniform Manifold Approximation and Projection (UMAP, Mclnnes al. (2018). retaining 25%, 50%, and 75% of the features/principal components.
Throughout this and Sec. 4 we measure the performance by looking at the test error using Support Vector Regression (SVR, C = 1.0, e = 0.2) (Vapnik 1963 1964; [mool & Schölkopi 2004). For SeltctKBest, PCA, and UMAP, instead of reporting for 25, 50, and 75% of features/components separately, we only provide the results yielding the lowest mean-squared test error.
"A further discussion on this can be found in Appendix 'The exact data-generating process is given Appendix D.I
Feature selection benchmark results in terms of test MSE. Our algorithms consistently yield the most accurate predictions while selecting one of the smallest feature subsets (as seen in (Ta) With By we denote predicting the overall mean and with Full, regressing to the whole dataset.
Table I:
GUS
S.Per.
FPR
FDR
FWE
K.B.2
K.B. 50
K.B. 75
FOCI
Vec (dF))
(dF1)
# Feat Select. 1
24
112
95
53
2
50
100*
6
2
N/A
0.086
0.028
0.027
0.028
0.030
0.084
0.030
0.028
0.030
0.01 0.02
0.012
0.01
(a) Results from simulated data study, detailed in Sec. 3.2
Both (dF1) versions successfully inherit FOCT's
strengths: they select a small number features while exhibiting solid performance.
Ag
Full
GUS
S.Per.
FPR
FDR
FWE
K.B.
UMAP
PCA
FOCI
vec
(dF1)
NN (dF1)
Exp 1.
1,38
0.22
0.93
0.94
0.53
0.54
0.68
0.54
1.14
1.02
0.21
0.02 + 0.00
0.08 = 0.01
Exp 2.
0.49
0.58
0.53
0.58
0.58
0.59
0.58
0.58
0.55
0.52
0.53
0.24
0 0.00
0.02 0.01
3. 3.
0.35
0.31
0.32
0.32
0.34
0.34
0.33
0.33
0.33
0.34
0.30
0.23
0.00
0.18 0.01
(b) Results from three toy experiments, described in Sec. show that both versions of (dF [ FOCT's
strengths. In Experiments 2 and 3, they are the only methods that outperform regressing to the mean Ag.

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Table Feat. selection and dim. reduction benchmarks in terms of logistic test loss. Reported are the mean and std. across five random seeds. Our algorithms yield competitive predictions.
GUS
S.Per.
FPR
FDR
FWB
K.B.
UMAP
LDA
PCA
FOCI
vec dF1)
Spambase
10.70
6.05
2.92
2.92
2.92
3.39
2.97
3.20
3.12
3.04
2.56 1 0.13
2.57 0.19
Toxicity
14.41
12.98
17.30
12.98
18.02
10.09
12.98
15.86
10.00
16.30
11.61
0.80
9.61
1.50
QSAR
2.88
3.16
3.16
3.76
2.92
3.52
2.32
2.16
2.16
3.44
2.54 + 0.07
2.11
0.11
Breast Cancer
4.66
1.69
0.42
0.42
0.00
2.48
1.42
1.24
0.62
0.00 0.00
0.00
0.00
Religious
0.84
0.56
0.65
0.57
0.56
0.48
6.63
1.61
0.60
0.53
0.48 = 0.03
0.56 = 0.04
Results. Our approach selects small, diverse, and informative set of features, resulting in good
performance and showcasing successful inheritance of FOCT's main strengths (see Table Tap. The norms of the selection parameter 0 are shown in Fig. [b] demonstrating the evident relationship between the predictive informativeness of the features and the corresponding parameter norms.
We have discussed the recent advances and methodologies necessary to introduce difFOCI, as well as provided experimental analysis on a synthetic examples. We now proceed to more challenging examples, and ultimately to real-world datasets.
FROM FEATURE SELECTION TO FEATURE LEARNING
With high-level understanding of difFOCI in place, we continue to assess its performance. We begin by highlighting two key observations we encountered during our preliminary experiments. We consider the following toy example:
sin(X, + sin (in) +3 sin( +G, where 6; N(0, ??).
i [n)), and X ~ (0, of Ip), with 2000, p 10, O. 0. 0.1.
Observation 1.(t) The objectives
2 and
consistently capture the correct feature func-
tional forms. Specifically, the values (i) T, Y.
Σ sin(m? (X13), simz(((Xa))))
E [3].
(ii) (Y sin(Xns)))))) and (ii) TY, sin X.)) are all significantly greater than (i) Tn Y, Di T. (X1s), mz (1))) (ti) (Y, X1s), and (iii) T, (Y,E X,) (as illustrated in
Figure 2a in the Appendix). Therefore, a more complex parameterization (than fo(X) might learn nonlinear transformation of the features, maintaining ranks in a manner more consistent with the true functional forms.
X)
Observation 2.()) The objectives (2 and consistently prefer correct, lower-dimensional bases of the features. Specifically, (Y, Σ i=1 sin (X,)) remains consistently greater than TY,Y, (Yy,sin(x, sin(X1,3)) Therefore, a more elaborate parameterization could learn an appropriate, possibly lower dimensional, basis transformation.
Motivated by these observations, we propose NN parameterizations to further explore the capabilities of difFOCI. We begin with simple hechhhdddddleee Multi-layer Perceptrons (MLPs) fo parameterizations. We set the output dimension to match the input, as this performed well across all experiments, though treating it as a hyperparameter might further enhance performance.
4.1 INITIAL ASSESSMENTS OF (F11)
We now evaluate both vec (F11) and NN (dF1) across three progressively challenging examples. We note that across all examples. FOCI selects the correct subset of the features more than 95 percent of the time. We set p = 10 throughout the experiments, and both a, o2 0.1. Full experimental details are given in Appendix
Toy example difFOCI successfully accounts for feature variability. Here, we test whether vec- (dF1) and NN [dFF on the following example, previously introduced in Sec. 3 Y (x)) + e, where e; N(0, o²), i E [n]. and ~ N of Tp), with 2000. In Table Ib we observe that (dF (dFFT)andd and NN-(FF]) successfully pinpoint the correct fea- ture subset and account for feature variability, resulting in improved performance to that FOCI. We observe the correct proportionality of the coefficients in the regression equation and the learned parameters 0,, 02 and 0g
sin Xi)+ + 2 (in
expand on this in Appendix, Fig. 2b for vec- (dF1) where
Note that this is already an improvement FOCI, cannot take into account feature variability.

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Toy example 2:
difFOCI can learn appropriate basis transformations. The goal of this toy example is to examine whether NN (dF1) effectively learns basis transformations. Data are generated as follows: Y sin(XX 2X2 + 3xaa) +c, where 6 N(0, i E [nl], and X [0, of Ip), with n. = 2000. We affirmatively demonstrate its efficacy by examining the test loss after fitting the SVR - the substantially lower test error can be observed in Table Tb
Toy example 3: difFOCI simultaneously addresses mutual interactions, basis, and nonlinear transformations. Our final example seeks explore the full capabilities (dF with NN param- cterization, examining whether it can simultaneously discern complex, interrelated relationships as well as multiple transformations, encompassing both nonlinear and basis transformations. The data generation process is as follows: Y = X2) + (X2-x)² + (X))Xa)²) +,, where 4 N(0, ).), [n], and X N( a Ip). with = 5000. As evidenced Table 1b (using a wwo-hidden-layer MLF we successfully learn effective transformations that result in strong performance.
Summary. Throughout the experiments, both dd1 and NN (dF1) yield strong performance, as seen in Table [b] The two penultimate examples emphasize the potential capabilities of difFOCI; not only can correctly identify the relevant subsets, but also learns useful transformation, yielding the only method that outperforms random guessing (see Byy column in Table Tb).
5 EXPERIMENTS
Having examined (dF1) on synthetic problems and toy datasets. we now proceed real-world datasets. We attempt to demonstrate the flexibility of difFOCI and highlight the promising potential of all three objectives: (dFT) Our aim in this section is not solely to outperform existing benchmarks, but rather to showcase difFOCI's broad applicability, inspire further investigation into these applications, and explorations new areas where the method can be applied.
5.1 REAL-WORLD DATA
In this section, we compare vec (dF) (rl))and NV NV- (W-(dF1) to feature selection and dimensionality reduction
methods using real-world datasets.
Environments. We evaluate our methods on five UCI datasets (Dua & Graff 2017): Breast Cancer
Wisconsin (Strect al. 1993), involving benign/mmalignant cancer prediction; Toxicity (Gul et al. 202T). aimed at determining the toxicity molecules affecting circadian rhythms; Spambase (Hopkins et al.) 1999). classifying emails as spam or not; QSAR (Ballabio et al. 2019). set containing molecular fingerprints used for chemical toxicity classification, and Religious (Sah & Fokoué, 2019), aimed at identifying the source of religious books texts. We perform Logistic Regression (Cox 1958) with default scikit-learn (Pedregosa et al., 2011) parameters (tol =1e-4,
C' = 1.0). Dataset information is provided in Appendix
difFOCI is competitive in feature selection and dimensionality reduction. As seen in Table 2 difFOCI achieves solid performance the experiments. For (dF) we use two-hidden-layer MLPs. The findings, which employ logistic loss, demonstrate that taking into account feature variability and using parameterization are crucial for improved performance compared FOCI.
5.2 DOMAIN SHIFT/SPURIOUS CORRELATIONS
Here, we investigate an application of difFOCI to deep learning in the form of NN NN-(FF2) The data consists of triplets (Y, X, Xx). denoting the predictor, response variables, and group attributes, respectively. In this context, difFOCI can be employed a regularizer to enforce the learning of uncorrelated features with respect to spurious attributes, thereby mitigating relying on spurious correlations and shortcuts the model (Kenney 1982).
For this example, we found one-hidden-llyer MI.P not be expressive enough.
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Table 4: Worst group accuracy across several datasets, difFOCI obtains competitive performance.
Dataset
diFFOCIERM
diFFOCI+RRO
ERM
DRO
JTT
Mixup
IRM
MultiNLI
77.6 0.1
0.2
66.9 0.5
77.0
0.1
69.5
66.5 + 1.0
CivilComments
66.32
70.3 0.2
641111
70.2 2
0.8
64.0
6..1
63.2 - 0.5
CelebA
0.4
8.9
65.0 2.5
0.6
0.5
63.1 5..6
NICO+
0.7
4.2
39.3
1.2
4..0
4..1
40.0
MetaShift
0.5
91.7
8..9
8.
8.66
84.0 80.9
CheXpert
54.42 3.2
75.3 0.3
50.1 3.5
0.4
6..
40.2
35.1
Environment. We use Waterbirds dataset (Sagawa et al. 2019), which combines bird pho- tographs from the Caltech-UCSD Birds-200- 2011 dataset (Wah al. 2011) with image back- grounds from the Places dataset (Zhou et al. 2017). The labels Y (waterbirds, landbirds are placed against G = {water, land} back- grounds, with waterbirds (landbirds) more fre- quently appearing against a water (land) back- ground (exact details given in Table 7 Appx. E}. Due to this spurious correlation, (Sagawa ct al. 2019 [drissi et al., 2022 Bell al. a (0224) observed that NNs (i.e., ResNet-50 (He et al. 2016), pre-trained on ImageNet (Deng et al. 2009)) tend to rely the background to infer the label, rather than solely focusing on birds.
Table 3: Average and worst group accuracies for the Waterbirds dataset. We compare to the ERM and DRO, where stands for early-stopping and for Ridge regularization. We can see that difFOCI performs comparably to state-of-the-ar spurious correlation methods.
Average
Worst group acc.
Train
Test
Train
Test
ERM
100
97.3
100
60.0
ERM (c.s. + strong (2)
97.6
95.7
35.7
21.3
ERM ROCI
99.9
77.8
ERM NN (1:2)
99.9
93.7
92.0
85.7
DRO
100.0
97.4
100.0
76.9
DRO (c.s. strong (2)
99.1
96.6
74,2
86.0
DRO + FOCI
99.5
74.5
6.1
3.9
DRO + NN (dF2)
80.1
93.5
99.2
87.2
Preventing reliance on spurious correlations. We investigate the potential benefits of employing [d-(FF2) as regularization technique, which penalizes the reliance of extracted features fre, on the spurious attribute XG (i.e., the background) via T,,a((.. frE, (X) Xa). From Tables 3 we can see that NN (dF) (applied to both ERM and DRO) compares competitively to tta methods. The exact algorithm is given Experimental details, reported average accuracy and further examples are in Appendix G
difFOCI increases worst group accuracy while maintaining solid performance. We can observe in Table 3 that NV)( successfully prevents the network from relying on the spuriously correlated background while improving worst group accuracy for both ERM and DRO. Apart from Waterbirds dataset, we also tested difFOCI additional datasets: two datasets: MultiNLI (Williams et al.
2017), CivilComments (Borkan et al. (2019), and four image datasets: NICO+ (Zhang et al. [2023]
CelebA (Lang Zou 2022), MetaShift (Liang & Zou 2022) and CheXpert (Irvin al. [019 Full experimental details (including average accuracy performance) can be found in Appendix G We experimented with various architectures: in addition to the ResNet-50, we used BERT and ViT-B with pretraining strategies like DINO and CLIP. Furthermore, we compared to Just Train Twice (Liu et al.) 202T), Mixup (Zhang) 2017), and Invariant Risk Minimization (Arjovsky al. 2019) as baselines. As shown in T5 difFOCI demonstrates competitive performance in terms of both average and worst- worst-group accuracy.
5.3 FAIRNESS STUDY
Finally, we explore NN (FF3) This section, while not the primary focus of our contribution, offers a complementary illustration of the difFOCI objective's versatility through a heuristic example. We found that this form (F3) preserves the performance of the chosen parameterization while significantly reducing its predictivity of the sensitive attribute.
Environments. We utilize classification datasets with interpretable features and sensitive attributes: (i) Student dataset (Cortez & Silva, 2008), aimed at predicting if a student's 'ss performance surpasses a specific threshold (sex as the sensitive); (ii) Bank Marketing dataset (Moro et al.) 2014) with predicting if customer subseribes bank product (marital status the sensitive); and two ACS

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Table 5: NN- (dF) allows preserving predictivity of y while significantly reducing predictivity of X
Dataset
Features
Train acc:
Val. Acc:
Test acc:
Train acc: X,
Val. Ace: X,
Test ncc: X,
Stand. data
91.32 + 2.3
93.27 1.2
90.05 + 2.0
89.09 + 1.2
72.26 + 1.5
70.93 + 0.9
Bank marketing
(dF3) features
90.81 # 1.8
92.13
2.6
89.35 + 1.1
63.12 + 2.8
62.24 = 0.7
63.81 = 2.1
Stand. duta
88.35 1: 1.7
79.63
0.9
76.67 1: 1.3
95.68 1 2.1
72.16 1 2.4
71.21 1 1.5
Student data
dF:3) features
80.18 + 2.9
72.16
1.6
72.73 + 1.7
59.47 + 1.1
58,95 = 1.0
48.89 = 1.1
Stand. data
83.49 1. 2.4
85.10
2.1
81.30
2.7
68.97 1. 1.6
67.67 1 2.6
66 00 1 0.7
ASCI Income
dF33 features
82.80 + 0.8
81.99
1.5
82.95 + 0.9
56.58 + 1.2
55.01 + 2.0
52.73 + 2.0
datasets (Ding at. status and whether their income exceeds threshold, with sex and race as sensitive attributes both datasets. Exact experimental details are provided Appendix
2021 (ii) Employment and (iv) Income, for predicting individual's employment
Findings. Leveraging the conditional dependence expression in (3), our method flexibly incor- porates sensitive features facilitate fairer classification without exploiting sensitive data. Using (d33 we optimize nn,p( (Y, , (X) X,) to learn features that are informative about Y, offering an optimization that heuristically seems to favor solutions less predictive of X,. Specifically, we train two NNs to predict y: the first NN trained on X (without X,), while the second NN was trained on features (X) obtained using (dF3) We then used the final layers of both NNs to predict the sensitive X,, As can be observed from Table difFOCI (dF3) significantly reduces the predictability of X, (sometimes to chance level) without significantly impacting accuracy on in some cases even slightly improves it.
Despite conditioning out sensitive information, difFOCI delivers solid performance. From T we see that (d3) demonstrates (as it cannot predict the sensitive X, well), while keeping informativeness regarding y- In Appendix [ we conduct another experiment with similar findings showcasing the promising potential of (dF3)
demmdmmsstttaes strong performance by effectively debiasing the network
CONCLUSION
In this paper, we discussed two recent advancements in rank-based measures of correlation, critically examining the proposed estimators, including the FOCI algorithm and its barriers to adoption in machine learning. Leveraging these advancements, we introduced three enhanced and more adaptable versions of FOCL. We conducted several studies showcase the retention of FOCT's strengths and the improvement of its weaknesses. We evaluated difFOCT's capabilities from toy examples, where our method was the sole exceeding random guessing, to comprehensive real-world datasets involving feature selection and spurious correlations, where it demonstrated state-of-the-art performance. Finally, proposed direct application of algorithm in fairness research, showcasing that difFOCI successfully debiases neural networks on several datasets.
ACKNOWLEDGEMENTS AND FUNDING
This work has received funding from the French government, managed by the National Research Agency (ANR), under the France 2030 program with the reference ANR-23-AACL-000S We extend our thanks to Samuel Bell, João Maria Janeiro, Badr Youbi Idrissi, Theo Moutakanni, Stéphane d' Ascoll and Timothée Darcet for feedback and support. Finally, we also thank Carolyn Krol for extensive consultation and support throughout this project.
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APPENDIX
The organization of the appendix is as follows:
Section provides foundational material necessary for understanding proof methodology.
Section presents the technical results that support the conclusions drawn in our work,
particularly in relation to Theorem
Section gives further insights into the Toy Experiment 1.
Section D offers details regarding the feature selection datasets, which include both the
synthetic dataset and the UCI datasets.
Section contains information about the Waterbirds dataset utilized for feature learning.
Section includes additional comments on the fairness experiments, which involve two UCI datasets and American Community Survey (ACS) data, made available through the Folktables package.
Section elaborates the experimental analyses and the configuration of hyperparameters.
Section gives three concrete examples of the pseudocodes required for Alg.
Section provides experimental details regarding experiments in Section 5.3
Section J
gives empirical evidence for the choice of fixing the parameter 8 = 0.2 by
analyzing the performance of difFOCI for different values of
Section K
presents experimental evaluation of robustness domain shift for feature selec-
tion using (FT)
A ADDITIONAL TECHNICAL BACKGROUND
A.1 AUXILIARY RESULTS REQUIRED FOR THE PROOF OF TIM.. 4
In this section, we provide the required results for the proof of Theorem the full forms of the estimator, including the case O.
We begin by providing
p 2 the estimate of reads as follows:
T₁ = TY, X) = Ei (min fri, TMG) min fra (ro)))
(5)
En (r: min (F.TN())
And if p 0. we obtain:
E (nmin (M()) 12)
Tn (Y,
(6)
En 4 (n 14)
with (i) denoting the index j such that Zj is the nearest neighbor of Z,, and ris l as defined in Sec. 2.2
We then proceed by defining:
1. (Y,
Ex (r: min (x())) and
2. Qn (Y, 17 X = v2 Ee (min
(ri
These two quantities are important for following theorems:
Theorem Azadkia & Chatterjee 2021 Suppose that p 2 1. As - 0o, the statistics QY Z X) and P, X) converge almost surely to deterministic limits. Call these limits and b, respectively. Then
1. 0 S
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2. Y is conditionally independent of Z given X if and only if 0.
3. Y is conditionally a function of Z given X if and only if b.
4. Y is a function of X if and only if b 0.
Explicitly, the values of a and b are given by
[(Var(P(V22 21 (, X))dj((t)
and
b E (Var (L,224 X)) du(()
E(P(Y 2 t X1 P(Y 2t X)))d((t).
Next, suppose that = O. Define Qn = _/ Σi= (min (rorMMe) Li and PY(Y)
1 Ei=1 La (n L.), where L, is the number of j such that Y, 2 Y... Then, show the following:
Theorem 6. (Azadkia & Chatterjee 2027) Asn + oo, Qn Z) and PY) converge almost surely to deterministic limits c and d. satisfying the following properties:
1. 0 < sd
2. Y independent of Z if and only if = 0.
3. Y a function of Z if and only if =
4. d >> 0 if and only if is not a constant.
Explicitly,
c= / 2t Z))d
Var((P(Y
and
d = Var (11r20) dp(t)= PPY t) P(Y 2 t))dt(t).
The two aforementioned theorems serve as the key ingredients to Theorems and as well as the proof of Thm. which is given in Sec. B
A.2 WORST-GROUP-ACCUR.G. (WGA) METHODS
Below, we mention the two most- popular methods appearing in the literature on WGA maximization:
ERM. Empirical Risk Minimization (ERM), proposed by Vapnik (0066). chooses the predictor minimizing the empirical risk " (f (I:)) ys). ERM does not use attribute (group) labels.
Group DRO. Group Distributionally Robust Optimization (gDRO) proposed by Sagawa al. (2019) aims to minimize the maximum loss across different groups. The objective is formulated as:
nl (f (1))
f
sup AEAio
g=1
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where = A represents the set of all groups, Aga denotes the dimeenssil simplex, and Ng is the number of examples belonging to group E G within the dataset. As a result, gDRO incorporates attribute labels. Specifically, gDRO assigns dynamic weight q to the minimization of the empirical loss for each group, which is proportional to its current error rate.
Other methods. The body of literature on robust, worst-group optimization is rapidly expanding, making it infeasible to compare all available methods thoroughly. Additional examples of robust learners that do not utilize attribute information (like ERM) include Learning from Failure (Nam al.
2020 the foo-Good-to-be- bruT prior (Dagaev et al. 2023), Spectral Decoupling (Pezeshki 2021> Tast-Train-Twice (Liu et al. (021), and the George clustering algorithm (Sohoni et al. 2020p. Conversely, methods that incorporate attribute information (like gDRO and difFOCI) include
et al.
Conditional Value at Risk (Duchi et al. 2019), Predict then Interpolate (Bao et al. (202T). Invariant
Risk Minimization (Arjovsky et al. 2019), and wide range of Komain--ennralizatte algorithms
(Gulrajani & Lopez- Paz 2020).
B PROOF OF THEOREM 4.
In this section, we re-state Theorem 4 and prove it.
Theorem 4. Let B IR.. Suppose that Y is not almost surely equal to measurable function of X. Then, lim lim Ta,3 T almost surely.
Proof. Let Y be random variable and X (X1,..., Xp) and Z (Zi.... Zg) be random
vectors, defined on the same probability space. Here g 2 1 and p 2 0. The value p = 0 means that X does not have any components. By μ. we denote the law of Y.
Recall that we denote with r; the rank of Y(e)- i.e., the number of j for which Yo) S Y(s). and with 2 the number of j such that Yi) 2 Y(e)- For each index i, (ii) is the index j where X, is the closest to X,, and (t) is the index j where the pair (X, is closest to (X,, Z,) in RP-9 w.r.t the Euclidean metric and resolving ties randomly.
The two quantities Q. Z X) and P. (Y, YYY, X) and their respective limits a and (see Theorems and (6 are key proving Theorems 2 and In order to prove that Tn,s converges to the same limit. as Ty we have to introduce the following two quantities:
1. (Y, = #/2 Eim (ra T Sp,, H), and
2. Qn,a(Y; ((, + Σ (min(ra,r UR min(ri, T (p,
with Sş = - U-= 0r( (--(((1) [M AIm)), and Mr,; = 11x,, 2,) - (X,, Z;) with 0g the softmax function as defined in Sec.
Now, define % == Ir; rM(ol and 8; = Ir; - rNel Let = min(? Yn ,,)).
Then, by the continuity properties of a=(+) and setting À max(1e20, maxi, M, + e). we have lim SB..
1i; = arg maxje(n]); M. = 1fi = arg min je[m |X X,1I} = N(1). One can similarly show that limg- Up.. = lim 0a a- (1-(0M (M+ + AL,),,,
lim d-(-(M + AL,)),,-
M().. Therefore, we can choose n N max NN N2). such that Vn > N1. max, [NN()) - r S < e, and Vn > N2, max, |r r UU, s,. < Then, can easily show that (Y, X) converges to the limit with c lim (Y,
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(X,p(Y,Z(X) cl
miinfre Us Us, min(ra, Sp) " cl
112
i=
2ne
(min (min(ri, (M(())) minffr,rN(0 - + n
12
==1
2nc
(Y, -c+ -/+
where both terms go zero as we take infinity (for the first term, see Thm. straightforwardly show that Pa.p (Y, X) converges to the same limit 6 as P. (Y,
5 One can also
Finally, we can closely follow Sec. 10 Azadkia Chatterjee (202 conclude; For case p 2 we recall the quantities and from the statement of Theorem and notice that T = a/b. By Theorem 5 Qn - and S, + in probability. Thus, T, - a/b in probability. This proves
Theorem when 2 I. Finally, for case p 0, here T c//d, where and d are the quantities
from Theorem Note that T, = Qn/Sn, where Qn = QY, Z) and S, = (Y). ). By Theorem
On and Sn - d in probability. Thus, Tn c/d T in probability. This proves Theorem when p = 0.
C CONTINUATION OF TOY EXPERIMENT 1.
We present plot for the two observations discussed in Section 5 well Toy Example 1. The left plot shows that the differences between the three observations are all statistically significant, while the right plot highlights two key strengths of method: it quickly stabilizes, and the parameter norms reflect the variability of the features.
H
param.
3
Obs. 1(a)
+
I
Obs. 1(b)
Param.
2
I
Obs. 1(c)
0.1
Obs. 2
0-
0.0
2500
5000
7500
0
Iteration
(a) μ4 1 or for Obs
(b) |0,,, || in Toy Exp. 1.
Figure 2: Left: Mean and std. across 50 random inits. All expressions yield values significantly greater than zero. Right: Development of the first five parameters in Toy Exp 1.
D EXPERIMENTAL ENVIRONMENTS
In the following, provide details the environments used in experiments in Section 5 We list the number of features, samples, and classes in each UCI environment in Table 6
Spambase
Toxicity
QSAR
Breast Cancer
Religious
4601
171
8992
569
8265
57
1203
1024
30
590
classes
2
2
8
Table 6: Feature Selection Dataset details
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D.1 SYNTHETIC ENVIRONMENT
Here, we briefly describe how we generate the synthetic environment depicted in Fig. D the synthetic dataset that is created using trigonometric transformations and permutations parameters.
Let æ: be linearly spaced vector defined as linspace(- 6, 6, 100). Define the parameters a, b, and e as: a = linspace(O.1, 2),, b = linspace (0.1, 2, 15), and = linspace 1, 4), where linspace(a.
b. n) represents uniformly spaced points in the interval (a, bj. Features are then generated using the formula:
f(r)a,bec a sin +
where b, and are elements from the Cartesian product of the parameter sets b, and The features are stored matrix X where each column represents feature vector. For each feature vector, transformations are applied as follows: Xnow (-1) [ 15 + j] for E {0, and j E {O..... 14). Additional transformations are applied based on a permutation of parameters c, a, and b. The transformed features are: Xina -1 Xeew [ i 15 j] for selected indices i and all j. The final dataset is obtained by concatenating all transformed features and adding Gaussian noise: 240. The predictor variable is calculated as sin(x) we do not add further noise here as the features already contain noise.
X X (O, 0.1) four times, yielding 100 and p 4 * 15
D.2 UCI DATASETS FEATURE SELECTION
Below, we briefly describe the five UCI datasets (Dua Graft 2017) used our feature selection comparison.
D.2.1 SPAMBASE
The "Spambase" dataset (Hopkins et al. 1999) is designed for classifying emails as spam non-spam. consists of 4,601 email instances with 57 features, characterized by both integer and real values. The dataset is multivariate and is often used in computer science, with classification as the primary task.
The dataset includes diverse types of spam, such as product ads, money schemes and chain letters. The goal is to identify whether an email spam, with some non-spam indicators like the word "george" or code "650" reflecting personalized filters.
D.2.2 TOXICITY
The "Toxicity" dataset (Gul et al., 2021 contains data on 171 small molecules designed for the functional domains CRYI, core clock protein involved in circadian rhythm regulation. Of these molecules, 56 are toxic, while the rest non-toxic. The dataset is tabular, with 1,203 molecular descriptors per instance. The primary task classifying molecules as toxic or non-toxic.
D.2.3 QSAR
The "QSAR Oral Toxicity" dataset (Ballabio et al., 2019) consists of 8,992 chemical compounds represented by 1,024 binary molecular fingerprint attributes. These attributes are used to classify the chemicals into two categories: very toxic (positive) or not very toxic (negative). The dataset is multivariate and is often used in physics and chemistry, with classification the main associated task.
D.2.4 BREAST CANCER
The "Breast Cancer Wisconsin (Diagnostic)" dataset (Street et al. 1993) used for classifying breast cancer diagnoses based data from fine needle aspirates (FNA) of breast masses. It consists of 569 instances with 30 real-valued features that describe characteristics The dataset is multivariate and is often used in the field of health and medicine, with classification as the primary task. The features were created through an exhaustive search using the Multisurface Method- Tree and linear programming techniques to create decision tree.
cell nuclei in digitized images.
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D.2.5 RELIGIOUS
The dataset, "A Study of Asian Religious and Biblical Texts." (Sah & Fokoué 2019) primarily consists of texts sourced from Project Gutenberg. It includes a collection of key religious and philosophical texts, such the Upanishads, Yoga Sutras, Buddha Sutras, Tao Te Ching, and selections from the Bible (Books of Wisdom, Proverbs, Ecclesiastes, and Ecclesiasticus). The dataset is multivariate and analyzed in Social Science contexts, with associated tasks including classification and clustering.
E WATERBIRDS DATASET FEATURE LEARNING
Group Counts
Dataset
Target
Water
Land
Class Counts
Land bird
1057
1113
Waterbirds
Water bird
3498
184
3682
Table 7: (Sub)group counts for the Waterbirds Dataset
The Waterbirds dataset consists of images of birds that have been digitally cut and pasted onto various backgrounds. The objective is to classify the specimens as either water birds or land birds. The group attribute indicates whether the bird is depicted in its natural habitat. The details of class counts given in Tab. 7 While performing hyperaparameter search, each experiment is run on one Nvidia Tesla V V100 GPU.
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Bank Marketing
Student Performance
ACS Employment
ACS Income
41,188
395
3,236,107 107
1,664,500
20
30
17
10
# classes
2
2
2
2 Race, Sex
Protected attributes
Marital Status
Sex
Race, Sex
Table 8: Number of samples, parameters, classes, and sensitives for each dataset
F FAIRNESS EXPERIMENTS
F.I UCI DATASETS FAIRNESS
The UCI datasets DDu & Grafi 2017) used in our fairness experiments are briefly described below.
F.1.1 BANK MARKETING
The "Bank Marketing" dataset (Moro al. 2014) contains data from direct marketing campaigns (phone calls) conducted by a Portuguese bank. The goal is to classify whether a client will subscribe to term deposit (variable 'y'). The dataset is multivariate, with 45,211 instances and 16 features that are either categorical or integer. The ratio of sensitive marital status is 60-30-10% (married, single, divorced).
The marketing campaigns often involved multiple contacts with the same client to determine if they would subscribe the term deposit. This dataset is used business applications, with classification being the main associated task.
F.1.2 STUDENT PERFORMANCE
The "Student Performance" dataset (Cortez & Silva 2008) aims to predict the performance of secondary education (high school) students. consists of 649 instances and 30 integer features, and the associated tasks include classification and regression.
The data collected from two Portuguese schools includes student grades, demographic, social, and school-related information. Two separate datasets cover performance in Mathematics and Portuguese language. The target variable, G3 (final grade, whether 2 12 or not), is strongly correlated with GI and G2 (grades from earlier periods), making it more challenging but useful to predict G3 without using GI and G2, as we do in our experiments. This dataset supports educational performance modeling in the Social Science domain. The ratio of sensitive sex is 50-50%.
F.2 FOLKTABLES DATASET FAIRNESS
The two datasets below taken from the Folktables package (Ding et al. 2021 designed to provide access to datasets derived from the US Census. features a range of pre-defined prediction tasks across various domains, such income, employment, health, transportation, and housing. Users can also create new prediction tasks within the US Census data ecosystem. Additionally, the package facilitates systematic studies the impact of distribution shifts, allowing each prediction task to be applied to datasets covering multiple years and all states in the US, We use the Alabama data from 2018 with the 1 year horizon.
F.2.1 INCOME
The task to predict whether an individual's income exceeds $50,000 based on filtered sample of the ACS PUMS data. The sample includes individuals aged 16 and older who reported working at least hour per week over the past year and carning a minimum income of $100. The $50,000 threshold was selected to make this dataset a potential replacement for the UCI Adult dataset (Kohavi & Becker 1994). although the original paper provides additional datasets with different income thresholds, as detailed in their Appendix B. We use the California data from 2018 with the I year horizon. The ratio of sensitive for sex is 50-50% and for race 62%-17% 5% for White, Asian, Black (other minorities include American Indian, Hawaiian, etc.)
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F.2.2 EMPLOYMENT
The objective of this task is to predict whether individual employed, using filtered sample from the ACS PUMS data. This sample has been carefully curated to include only those individuals who are between the ages of 16 and 90. The ratio sensitive for sex 50-50% and for race 12%%-7%-59 for White, Asian, Black (other minorities include American
etc.)
Both tasks contain codes regarding the selected features Sec. 5.3 The codes are explained below.
DEMOGRAPHIC VARIABLES
1. OCCP:
(a) Person's occupation
(b) approximately 500 categories (management, business, science, arts, service, sales, office, construction, maintenance, production, transportation, material moving, etc.)
2. COW:
(a) Class of worker
(b) 10 categories (e.g.. employee of a private for-profit company, local government em- ployee, state government employee, federal government employee, self-employed. working without pay, etc.)
3. POBP:
(a) Place of birth
(b) approximately 300 categories (countries/states of birth), range of values includes most countries and individual U.S. states
4. SEX:
(a) range of values: Male and Female
5. MAR:
(a) Person's 's marital status
(b) 5 categories (married, widowed, divorced, separated, never married, under 15 years
old)
6. ANC:
(a) Ancestry
(b) 5 different categories (single, multiple, unclassified, not reported, suppressed informa- tion)
7. AGEP:
(a) Age, Range of Values: 0-99
8. ESP:
(a) Employment status parents
(b) 9 different categories (Living with two parents both in labor force, living with two parents father only in labor force, living with father father in labor force, living with father father not in labor force, etc.)
CITIZENSHIP AND MIGRATION
1. CIT:
(a) Citizenship status
(b) categories (Bom in the U.S., Bomn abroad of American parent(s), U.S. citizen by naturalization, Not citizen of the U.S., Born in Puerto Rico, Guam, the U.S. Virgin Islands, the Northern Marianas)
2. MIG:
(a) Mobility status (whether the person lived at the same location year ago)
(b) 4 categories (N/A if less than year old, Yes same house, No outside U.S. and Puerto Rico, No different house in U.S. Puerto Rico)
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EDUCATION
SCHL:
(a) Amount of schooling completed
(b) 24 categories (No schooling completed, Nursery school/preschool, Kindergartenn Grade Grade 2...., Regular high school diploma, GED or alternative credential, Bachelor's degree, Master's degree, etc.)
RACE AND ETHNICITY
NAT:
(a) Whether native or foreign born
(b) 2 categories (native or not)
2. RACIP:
(a) Recorded detailed race code
(b) 9 categories (White alone, Black or African American alone, American Indian alone, Alaska Native alone, Asian alone, Some Other Race alone, etc.)
3. MIL:
(a) Military service
(b) 5 Categories (Less than 17 years old, Now on active duty, On active duty in the past but not now, Only active duty for training, Never served in the military)
DISABILITY AND SENSORY IMPAIRMENTS
1. DIS:
(a) Disability recorded: With or without
2. DEAR:
(a) Hearing difficulty: Yes or No
3. DЕYE:
(a) Vision difficulty: Yes or No
4. DREM:
(a) Cognitive difficulty: Yes or No
For further explanation on the codes, we invite the reader to see Appendix B. and B.4 in the original paper (Ding al., [221). Below, in Table 9 attributes removes additional features, highlighting the potential of the vec (dF3 method to examine interactions between several sensitive attributes, well as several features simultancously.
that conditioning multiple sensitive
Data
Race
Sex
Both
Employment
OCCP
COW
OCCP, COW, POB
MAR, ANC, CTT, MIG, SCHI., NAT
Income
MAR, ANC MAR, ANC, CIT, MIG
Table 9: ACS dataset features which were not selected when conditioned on race, sex or both, represented in first, second and last column, respectively.
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Dataset
Year
Size
N. Classes
Modality
Architecture
MultiNLI
2017
300k
Text
BERT
CivilComments
2019
250k
Text
BERT ResNet-50 w. ImageNet
CelebA
2015
200k
Image
NICO+
2022
90k
60
Image
ViT-B w. DINO
MetaShift
2022
3.5k
2
Image
VIT-B w. CLIP
Table 10: Dataset Overview for experiments performed in Section 5.2.
Dataset
difFOCI-BRM
disFOCI-DRO
ERM
DRO
Mixup
IRM
MultiNLI
81.9 : 0.2
81.800
$1.4
80.2
81.2 0.4
8.17
77.7
CivilComments
86.3 0.1
819 0.3
8..7
0.4
84.3
0.5
8.3
85.4 0.2
CclebA
9..4+11
92.9 2.1
9.2
93.1
92.4 1.6
0.2
94.5
NICO++
85.7 + 0.3
85.8 + 0.5
84.7+ 0.6
83.0 +.1
45.3+
0.1
84.2+0.4
84.7 + 0.5
MetaShift
92.1 0.2
91.8 + 0.3
91.3 0.5
0.2
90.7
+ 0.2
91.2
91.5
0.6
CheXpert
87.1
0.5
86.5 0.3
0.4
75.7
1.7
82.2
90.0 0.2
Table 11: Average accuracy for benchmark methods on various datasets performed in Section 5.2. We can see that on almost all datasets, diFFOCI performs competitively.
G EXPERIMENTAL CONFIGURATIONS
For the first two toy examples, we use one-hidden-layer MI. with a configuration of 10-20-10 10 neurons. In contrast, the third example employs swo-hidden-layer MLP structured as 10-20-20-10 neurons. For all benchmarks using vec((dF)) and (dF3) we initialize the parameter 0 from I ~ N(1, Ip), with o² 0.1. In the of dd we design one-layer MLP where the hidden dimension is double that the input layer, and the output layer has the same number of neurons the input layer. The ReLU function is used the activation function. All input data is standardized, and across all benchmarks, we perform a (75-15-10)% train-validation
split. For the regression experiments (toy examples), we employ SVR (C 1.0, 0.2) along with the Adam optimizer (Kingma & Ba 2017). For classification (UCI, Bank Marketing, Student and ACS Datasets), we employ Logistic Regression (Cox 1958). For the Waterbirds dataset, we train ResNet-50 models pre-trained on ImageNet (Deng et al. 20099) using the SGD optimizer with the PyTorch (Paszke al. 2017) implementation of BCEWahLogitsLoos which combines a Sigmoid layer and the BCELoss in one single class.
We adjust the learning (10-4, 1000 1.10-,10-2200 100-210011 10-1,,-3 5,,,,555555 3,3,5 1-5-22) II The number of epochs is optimized within the range {10, 20, 50, 100}. and batch sizes are chosen from {8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096). Notably, we train the Waterbirds dataset for 360 epochs, in line with previous research. As mentioned carlier, we keep the clipping parameter v 0.1 and the softmax temperature parameter 3 = 5 consistent across all experiments. The value of r) for gDRO is set 0.1. Each combination of hyperparameters is executed three times to compute the average and standard deviation of the chosen loss metric. We select the best models (in terms of hyperparameter combinations and epochs) based on the lowest MSE/logistic loss observed on the validation set, and for Waterbirds, we choose based on worst-group accuracy. We provide the hyperparameter configurations used to obtain our results in Table
rate
and weight decay from the set
Finally, for the experiments on NICO++ Zhhag et al. 2023), MultiNLI (Williams et al. (017).
CivilComments (Borkan et al.] 2019) and CelebA (Liang & Zou 2022) we follow the experimental configuration from Yang et al. (023)), who provided very useful codebase for benchmarking various methods, which hyperparameter joint distribution is taken directly from their codebase. For each algorithm, we perform thorough hyperparameter tuning process. This involves conducting 16 random searches over the entire range of hyperparameters. We then use the validation set to identify the optimal hyperparameters for each algorithm. With these hyperparameters fixed, we repeat the experiments three times with different random seeds and report the average results along with their standard
thankful for. We briefly describe these datasets below, where the
We also experimented with É1 regularization, but it yielded poorer performance.
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Dataset
diFFOCI+ERM
diFFOCI+DRO
ERM
DRO
JTT
Mixup
IRM
MuluNLI
77.6
7..
0.5
77.0
69.6
69.5 1.4
66.5 -:
CivilComments
66.32402
7.
64.1111
70
63.2 65.1 640
CelebA
39.32 0.4
8.
888 65.0
57.6 0.5
63.1
NICO+
47.10
4..
20.
38.3#12
4..0
43.1
40.0 0.0
MctaShift
83.1
91.7 0.2
86.2 0.6
82.6
=
0.6
80.9 =
84.0 = 0.4
Table 12: Worst group accuracy for benchmark methods on various datasets performed in Section 5.2. We that almost all datasets, diFFOCI performs competitively.
deviations. This approach ensures fair comparison between algorithms, where each is evaluated with its best possible hyperparameters, allowing for reliable assessment of their performance. We provide brief dataset information below and in Table
binary classification image dataset comprising over 200,000 celebrity face images. The task is to predict hair color (blond vs. non-blond), with gender serving as a spurious correlation. We employ standard dataset splits from prior work (Idrissi et al.) (022) and note that the dataset is licensed under the Creative Commons Attribution 4.0 International license.
CelebA (Liang Zou 2022):
MetaShift (Liang & Zou 2022): A dataset creation method leveraging the Visual Genome Project (Krishna et al. 2017). We use the pre-processed Cat vs. Dog dataset, where the goal is to distinguish between the animals. The spurious attribute is the image background, with cats likely to appear indoors and dogs outdoors. We utilize the "unmixed" version generated from the authors' codebase.
CivilComments (Borkan et al. 2019): A binary classification text dataset aiming predict whether an internet comment contains toxic language. The spurious attribute is the presence of references to eight demographic identities. We adopt the standard splits provided by the WILDS benchmark (Koh) et al. 2021)
MultiNLI (Williams et al. 2017): A text classification dataset with three classes, targeting natural language inference relationships between premises and hypotheses. The spurious attribute is the presence of negation in the text, which is highly correlated with the contradiction label. We use standard train/val/test splits from prior work (Idrissi et al.) 2022).
NICO+ (Zhang et al. 2023): a large-scale dataset for domain generalization. Specifically, we focus Track 1, which invol ves common context generalization. Our analysis is based the training dataset, comprising 60 classes and shared attributes: autumn, dim, grass, outdoor, rock, and water. identify all pairs of attributes and labels with fewer than 75 samples and exclude them from our training data, reserving them for validation and testing purposes. For each attribute-label pair, we allocate 25 samples for validation and 50 samples for testing, while using the remaining data as training examples.
To adapt this dataset for attribute generalization,
CheXpert [Irvin et al. 2019:: a collection of chest X-ray images from Stanford University Medical Center, consisting of over 200,000 images. In this study, we use "No Finding" as the label, where a positive label indicates that the patient does not have any illness. Following previous research (Seyyed- Kalantari (Seyyed-Kalantari et al., (021), we use the intersection of race (White, Black, Other) and gender as attributes. The dataset is randomly divided into
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Dataset
Method
Batch size
l.r.
epochs
w.d.
Val. loss
Test loss
Synth. Dataset
vec- (dF
full
5e-
2000
1e-1 1
0.02 0.01
0.02 + 0.02
Toy Ex.
5e-3
1000
1e-4 4
0.01 = 0.00
0.02 = 0.00
Toy Ex. 2
vec- (dF1)
full
10e-1
1000
1e-4
0.24 0.00
0.25 = 0.00
Toy Ex. 3
50-2
1000
5e-4
0.23 0.00
0.24 + 0.00
Toy Ex.
3
1000
le-2 2
0.08 0.01
0.08 + 0.01
Toy Ex.
NN (dF1)
full
5
1000
le-2
0.02 = 0.01
0.02 = 0.01
Toy Ex. 3
5e-4
1000
le-2
0.18 0.00
0.18 = 0.01
Spambase
2048
le-2
100
le-5
2.24 0.14
2.56 + 0.13
Toxicity
4096
5e-2
50
le-1
9.23 = 1.96
11.61 + 0.8
QSAR
vec (dF1)
512
le-2
10
5e-2
2.16 0.14
2.54 0.07
Breast Canc.
2048
le-3
50
le-4
0.00 0.00
0.00 + 0.00
Biblical
8
50
1c-4
0.36 0.02
0.48 0.03
Spambase
64
1e-4
10
1c-2
8.65 = 0.91
9.61 = 1.50
Toxicity
512
50-4
50
5e-2
1.97 0.13
2.11 = 0.11
QSAR
NN (dF1)
512
1e-2
le-1
2.52 = 0.16
2.57 = 0.19
Breast Canc.
2048
5e-3
1e-6
0.34 0.32
0.00 + 0.00
Biblical
128
1e-4
20
5e-4
0.69 = 0.17
0.56 + 0.04
vec- (dF1)
64
1e-1
100
5e-4
8.03 1.07
8.41 + 0.82
Student
vec
(dF3)
256
5e 2
5
7.65 0.56
8.52 0.89
vec (dFF)
2048
5e-3
le-5
2.61
0.02
2.68 - 0.04
Bank
(d133
256
5e-3
50
5e-4
0.06 2.59
2.90 + 0.07
vec- (dF1)
64
5e-2
50
5e-3
7.65
0.08
7.81 + 0.03
ACS Empl.
vec- (d133)
256
30-3
50
le-5
7.81
0.01
7.97 - 0.02
vec- (dF1)
1024
le-2
10
le-4
7.65
0.01
7.65 0.01
ACS Inc.
vec- (EF3)
256
5e-2
100
5e-3
7.90
0.01
7.92 0.01
Table 13: Hyperparameter configurations used throughout the experiments.
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Published as a conference paper at ICLR 2025
Method
Batch size
I.r.
reg.
w.d.
Val. Acc.
Test Acc.
Val. WGA
Test WGA
ERM
8
le-5
1e-3 3
5e-2 2
91.2
93.7
84.2 85.7
85.7
gDRO
32
1e-5
le-1
1e-5
92.1
93.5
87.2
Table 14: Hyperparameter configurations for Waterbirds experiment with NN NN- method.
H ALGORITHMIC EXAMPLES
In this section, we give three concrete examples of Alg. 2 used in Sections 5.1 for completeness: using the vec-(dFI) MV- (F2 (22))0))) and vec (dF3) versions respectively.
Algorithm 3 difFOCI: version vec v (dF1)
Input: Standardized input X E R' Y E ER
Input: learning rate 7, weight decay parameter batch size cutoff parameter v, softmax parameter /
init. N(1, 0² Ip). with o² 0.1
for t = 1,.... Thtee do
L - nn, Y, 0g ② X)
Differentiable objective // Parameter update // Parameter clipping
O2+1 4 O2 AdamwD,, (C)
Oanal c(fmam: v) Output: parameter 0inal
Alg. is version of difFOCI used in Section 5.1 for feature learning and domain shift experiment.
Algorithm 4 difFOCI: version NN (dF2)
Input: Standardized input X E R'-P, Y RR. group attribute XG
Input: learning rate y, weight decay parameter X, regularization strength 17, batch size b, softmax parameter 3, neural network (.) = fLLo (-)), where fLLO and fFE, denote the last layer and the feature extractor respectively, BCEWithLogits loss {(:,))
init. NN parameters 0 for t 1,... Thiter do
C1 t E(YY,(Y,,, (FFEO (X)))
// Standard BCEWithLogits Loss // difFOOI regularizer Total loss calculation I/ Reweighting (in case of DRO) // Parameter update
L2 <- (X,, J (X)) L + L1 C122
L + WeDRO( (C) or WERM(D)
0.+1 0r SSGDwD (C*)
Ornal cOO 0.1) Output: neural network parameters Onnal
|/ Final parameter clipping
Alg. version of difFOCI used in Section 5.2 for feature learning and domain shift experiment.
Alg. version of difFOCI used in the fairness Section 5.3
I FAIRNESS EXPERIMENTS
I.1 EXPERIMENT IN SECTION 5.3
Here, we provide experimental details regarding the experiment in Section 5.3 In this study, we employed data splitting approach where the dataset was divided into training, validation, and testing sets in a ratio 75For our first network, we implemented a multi-layer perceptron (MLP) with three hidden layers, each featuring ReLU activations. We employed the BCEWithLogits loss function from Py ToTcch, along with the Adam optimizer as our optimization algorithm. The learning rate and weight decay
hyperparameters for the Adam optimizer. To predict sensitive attributes, we leveraged the last layer of the MLP and trained an additional three-layer MLP on top of it, again utilizing ReLU activations and the BCEWithL Logis loss function from PyTorch. The Adam optimizer
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Published as a conference paper at ICLR 2025
Algorithm 5 difFOCI: version vec (F33
Input: Standardized input X E R" Y R", sensitive attribute(s) Xs Input: learning rate 7, weight decay parameter X, batch size softmax parameter / init. 0 ~ N11, _²
Ip). with o² = 0.1
for = 1,.... Rlrer do
L o (Y, 0. X X3) 0++1 f 02 AdamwDa, (C) Onnal cOn 0.1)
I/ NN-based diffcrentiable objective // Parameter update // Final parameter clipping
Output: parameter Orinal
GUS
S.Per.
FPR
FDR
FWE
K.B.
UMAP
LDA
PCA
POCI
vec
[dFT)
Yec (dF3)
Student 13.30
14.14
11.36
11.64
11.36
10.53
8.87
8.31
8.59
10.25
8.41 + 0.82
8.52 = 0.89
Bank
3.45
3.19
3.19
3.19
3.01
3.32
2.90 0.07
Income
10.39
10.39
7.62
7.62
7.62
7.71
7,49
7.86
8.02
7.96
7.65
- 0.01
7.92 0.01
Employment.
12.09
11.23
8.31
8.31
8.31
8.41
14.60
8.67
9.01
8.43
7.81
+: 0.03
7.97 0.02
Table 15: Fairness experiments and test log-loss. vec (dF1) achieves best performance by not conditioning sensitive attributes, though vec 3d33 remains competitive even though it conditioning out the information regarding the sensitive data Xs-
was used with learning rate and weight decay parameters, and the hidden dimensions had sizes of 128. When training using the joFe)tbjective of diffoci, we employed a three-hidden-layer neural network, where all layers were of size 128. Throughout all experiments, the beta parameter was consistently set to 5.
1.2 ANOTHER STUDY ON FAIRNESS
In this section, we perform similar experiments to Section 5.1, however we use the vec-(F33) rather than (dF1) We note that here we just experiment whether, by conditioning on X,, we still achieve good performance, which is affirmatively confirmed.
Environments. As in Section 5.3
we use Student dataset (Cortez Silva 2008), Bank Marketing
dataset (Moro et al. 2014); and two ACS datasets (Ding et al. 2021). (iir) Employment and (iv) Income. Again, the performance is measured using Logistic Regression Cox 1958).
Despite conditioning out sensitive information, difFOCI delivers solid performance. From T5] we see that demonsstttss strong performance, regardless of whether we condition on the sensitive data not. Both algorithms outperform other methods, and, expectedly, we observe slight decrease in performance when conditioning on the sensitive attribute(s). For the Student dataset, conditioning on sex leads to exclusion of seven additional features (a total 11 out of 30), while for the Bank Marketing dataset, conditioning on marital status results in the exclusion of one additional feature (a total out of 20).
difFOCI might be useful in intersectional fairness. In both ACS datasets, conditioning on both sensitives led to the exclusion of previously included features (when conditioning on just one sensitive), as shown in Table Appendix This reveals that the additional features, excluded only after considering both sensitives, might contain intertwined relationships with the two sensitives, providing an interesting avenue for intersectional fairness research (Gohar & Cheng 2023). We leave this as future work.
J CHOICE FOR THE REGULARIZATION PARAMETER
In this section, provide empirical evidence of for parameter 3 choice (we fixed it to 5), highlighting that although we might observe minor improvements by tuning the parameter, the performance is consistent.
We observe that difFOCI exhibits robust performance across a range of values for the hyperparameter 3. As long as B is set within reasonable range, avoiding extreme values that either out gradients
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Published as a conference paper at ICLR 2025
A:
1e-5
1e-3
5
100
1e5
1e7
Standard
Avg. Acc.
97.1 + 0.4
97.0 0.7
9..2+
08.7+00.11 0.1
94.6
07.40001
97.4 +
0.2
97.3 0.2
(dF2) ERM
WGA
61.0 -: 0.2
62.3
0.8
84.8 0.8
85.7
85.7 0.8
63.9
0.900..
61.2
1 1.0
0.5
Avg. Acc.
97.2 +.12+-0.11 0.1
97.5
0.7
93.9
+
0.2
98.5
+ 0.5
97.5
0.3
07.2+-01
97.4+0044
(dF2) DRO
WGA
75.7
97.2
90.0
0.4
87.2
77.1 8.30
0.3
7..9
Table Results for various 3 on Waterbirds dataset. The results for reasonable values of 3 yield similar performance, and very large or small values result the performance falling back to the standard ERM DRO performance.
Standard
(dF2) ERM Avg. Acc.
0.7
9.9
0.310.2
9.
9..7
914
91.3
0.5
WGA
8.2
8..2
8.2
0..
8..1
8..6
81. 0.3
8.3
Ace.
0.2
9..0
0.99033
018103
918401
8..7
8..0 85.9
8.
(dF2) DRO
WGA
86.1
8.4
9.3
91.7 919
86.8
0.6
Table 17: Results for various & on MetaShift dataset. The results for reasonable values of , yield similar performance, and very large or small values result the performance falling back to the standard ERM DRO performance.
or result in a uniform distribution from the softmax function, difFOCI consistently delivers robust results. This evident in Tables 17 and 16 which present results on the MetaShift and Waterbirds datasets, respectively. Our experiments show that tuning 3 leads to only minor performance improve- ments, which are largely statistically insignificant. Furthermore, setting extreme values causes the estimator T(Xc: (X)) to degenerate to a constant, effectively reducing difFOCI to standard ERM performance.
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Published as a conference paper at ICLR 2025
Method
Train Accuracy
Test Accuracy
OOD Accuracy
Difference
Standard
85.20 + 2.0
82.75 + 1.9
70.2 + 1.6
12.55
difFOCI with 75% feats.
82.66 + 1.2
81.7 2.7
68.95 0.8
12.22
difFOCI with 50% feats.
80.19 + 2.4
79.4 + 1.0
67.9 1.2
11.5
difFOCI with 25% feats.
79.55 + 2.1
78.72 + 1.1
65.40 + 2.8
13.32
Table 18: Difference between standard predictive accuracy using ResNet-50 on CIFARIO and CIFAR10.1
Table 19: Difference between standard predictive accuracy using Resnet-50 on various DomainNet datasets
Dataset
Method
Train Accuracy
Test Accuracy
OOD Accuracy
Difference
Standard
88.72 + 1.7
78.93 1 0.6
29.34 +: 2.1
49.59
Real vs. Sketch
difFOCI, clip at 0.1
85.58 + 1.3
77.50 + 1.3
27.58 + 1.5
49.92
Standard
89.98 + 2.1
61.85 + 0.6
39.43 2.1
22.42
Clipart vs. Sketch
difFOCI, clip 0.1
88.95 + 1.6
61.96 1.3
40.34 + 1.8
21.92
Standard
66.51 1.0
53.17 + 0.9
7.01 1.9
46.16
Sketch vs. Quickdraw
difFOCI, clip .1
65.54 1.7
52.97
0.8
6.98 1.2
45.99
K DIFFOCI'S S ROBUSTNESS TO DOMAIN SHIFT
This section presents experimental results on feature selection using difFOCI's objective (dF1) on CIFAR10/10) (Recht et al. 2018) and DomainNet datasets (Peng al. 2019), specifically examining the Real vs Sketch, Clipart vs Sketch, and Sketch vs Quickdraw domain shifts. The results, summarized in Tables 18 19 (for CIFARIO CIFAR10.1 and DomainNet respectively) demonstrate that difFOCI maintains consistent performance across distribution shifts, with the selected features exhibiting similar performance differences as the full dataset. This consistency highlights difFOCI's ability to effectively handle distribution shifts.
31


Reviews:
[{"review_id": "RQ01suX0aF", "reviewer": "Reviewer_ZQah", "summary": "The paper introduces a new method called difFOCI, which bases on an existing approach called FOCI. FOCI is a tool for choosing important features from data based on their statistical relationships, but it is not easy to use in some deep learning tasks because it does not adapt well to different models.  Motived by this, difFOCI creates a flexible, learnable version of FOCI that can be used in more situations.\n\nThis submision uses difFOCI in three ways: as a tool to pick important features, as part of a model that learns using a neural network, and as a method to help models learn better by reducing irrelevant correlations in the data. To validate this, this submission tests difFOCI on a range of tasks, from simple examples to more complex problems like understanding which parts of images are important in neural networks. Also, difFOCI is shown to be helpful in making fairer predictions by letting models classify without depending on sensitive information.", "questions": "- Please discuss the scalability of difFOCI on large-scale datasets and its applicability to more challenging tasks.\n- Additionally, please clarify the purpose of the visualization in Figure 2.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "- The motivation is clear to me. FOCI is a tool for selecting important features from data based on their statistical relationships. However, it is not differentiable, which hinders its use in deep neural networks. To address this limitation, this submission proposes a differentiable, parametric approximation.\n- This submission provides a clear definition of difFOCI. The toy examples offer some intuition into how difFOCI works.\n- The three applications are well-chosen. They effectively demonstrate the usefulness of difFOCI.", "weaknesses": "- While some real-world datasets are used in the experiments to demonstrate the effectiveness of difFOCI, it is unclear if it can be extended to large-scale datasets. Specifically, the datasets in Section 5.1 are small-scale, and the neural networks or learning algorithms used are relatively simple. The Waterbird task, for example, is simpler compared to multi-class tasks. Please discuss the scalability and generalization potential of difFOCI.\n- The fairness study is interesting; however, the dataset and task here are relatively simple. Can difFOCI be applied in large-scale settings?\n- The visualization in Figure 2 is not very clear. What is the main takeaway from this figure? Additional visualization examples would be helpful.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "328dHe59sw", "reviewer": "Reviewer_cLoX", "summary": "This paper uses existing statistical methods to better understand feature learning from data, by modifying an existing model-free variable selection method to a trainable version. Experiments on toy examples and real world datasets show the effectiveness of the proposed method.", "questions": "See weakness.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The motivation of the paper is well stated.\n2. The paper is well-structured and well-written.\n3. Providing results on both toy experiments and real world datasets makes the paper more solid.", "weaknesses": "1. The real world datasets seem to be out-dated, where the latest one was released in 2019. It would be more convincing to presents results on more recent are more complex datasets, such as those in WILDS benchmark.\n\n2. This paper only considers one model architecture, i.e., ResNet-50. With the increasing usage of Transformer-based models, it is also important to show the effectiveness on more complex models.\n\n3. Simply showing the improved performance on worst group accuracy does not sufficiently suggest the decrease reliance on the spurious features. Note that, the overall performance of the proposed method shows consistently lower performance \\(2 - 4%\\) for average accuracy. It would be suggested to conduct the synthetic experiments in [1].\n\n4. Another concern is that the experiments are conducted on the case where the train test set share the same distribution. Given that the feature selector is a trained NN, it is interesting to show if such a method maintains its performance of feature selection where the train test set have a different distribution.  For example, CIFAR10 vs CIFAR10.1, DomainNet-real vs DomainNet-Sketch? as in [2].\n\nI would be happy to increase my score if some of my concerns are resolved. \n\n[1] The Pitfalls of Simplicity Bias in Neural Networks, NeurIPS, 2020.\n\n[2] A CLOSER LOOK AT MODEL ADAPTATION USING FEA- TURE DISTORTION AND SIMPLICITY BIAS, ICLR, 2023.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "uwWZvAbvOF", "reviewer": "Reviewer_uzzo", "summary": "The paper introduces an algorithm (difFOCI) that gives a measure of correlation based on Feature Ordering by Conditional Independence (FOCI). The method is differentiable and can be used in ML methods. It is used in 3 ways: \n\ndF1) maximize T(Y,fθ(X)); \n\ndF2) minimize loss + T(X_G,fθ(X)) such that the features are independent of group features\n\ndF3) maximize T(Y,fθ(X)|X_S) such that features are correlated with Y when conditioned on X_s\n\nSynthetic and more real data experiments are performed for all cases.", "questions": "See weak points.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "* S1. The proposed method is sound and has good potential for feature selection and feature debiasing \n\n* S2. The method has mainly good results in synthetic and some real datasets.", "weaknesses": "* W1. The experiments on real datasets are not that strong. \n\n* W1.1 For the spurious correlation experiments, Waterbirds dataset is a small and simple dataset and a successful method should be tested on other datasets besides it. How many seeds were used? Was the same protocol used to select hyperparameters for the baselines and the proposed method? The benchmark used in [A] can be used to evaluate the proposed method more rigorously. The results of the method will be more reliable if multiple datasets from [A] were used, together with similar hyperparameter selection and evaluation.\n\n\n* W2. For the fairness experiments, some additional explanations are needed. \n\n* W2.1 Are the explicit features Xs a subset of X? \n\n* W2.2 The authors state that “the predictor should be independent of one or more sensitive features”. How is this enforced? By maximizing T (Y , θ ⊙ X | Xs ) the model would learn to predict the target when we know Xs. Is this right? Can the authors explain this?\n\n* W2.3 Is there a measure to evaluate if the model relies on sensitive features? At the moment the results only report performance, and it is not clear if these results incorporate sensitive features or not.\n\n\n \nMinor: \n* In the context of Waterbirds, group refers to the combination of class and spurious variable (background). So X_G should be denoted as a spurious variable, not a group variable.\n\n\n[A] Yang et al. \"Change is hard: A closer look at subpopulation shift.\" (2023)", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "knyn5K1A7i", "reviewer": "Reviewer_P7Dz", "summary": "This paper introduces Chatterjee’s coefficient into feature learning by proposing a differentiable formulation to estimate it. The conditional dependence metric proposed by the authors is applicable to feature learning and enhances performance in downstream tasks.", "questions": "1. The definition of the rank \\( r \\) in Line 096 is unclear. What is meant by \\( Y_{(j)} \\leq Y_{(i)} \\)?\n2. Line 076 declares \\(\\sigma\\) as a Sigmoid function, yet it denotes the Softmax function. Could the authors clarify this discrepancy?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. The proposed correlation estimation term is highly versatile and “plug-and-play,” making it adaptable for feature selection, predictor training, and as a regularizer to mitigate spurious correlations.\n2. The paper is well-organized, with clear sections on methodology, relevant theorems, and extensive evaluation experiments that strengthen the contribution.", "weaknesses": "1. There is a disconnect between the theoretical foundation and its practical application, leading to a potentially vacuous theorem. The assumption that the coefficient \\(\\beta\\) of the Softmax function is infinite is impractical, as it is only set to values like 5 in the experiments.\n2. Symbol definitions are ambiguous, as discussed in the questions below.\n3. The comparison of the proposed method focuses on conventional methods such as LDA and PCA. It would be beneficial to include comparisons with more state-of-the-art methods.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)