Fairness-aware Bayes Optimal Functional Classification
Authors: Xiaoyu Hu
Gengyu Xue
Zhenhua Lin
Yi Yu
Venue: NeurIPS 2025 poster ()
Keywords: Classification
excess risk
fairness
functional data
Primary area: social_and_economic_aspects_of_machine_learning

Abstract: Algorithmic fairness has become a central topic in machine learning, and mitigating disparities across different subpopulations has emerged as a rapidly growing research area. In this paper, we systematically study the classification of functional data under fairness constraints, ensuring the disparity level of the classifier is controlled below a pre-specified threshold. We propose a unified framework for fairness-aware functional classification, tackling an infinite-dimensional functional space, addressing key challenges from the absence of density ratios and intractability of posterior probabilities, and discussing unique phenomena in functional classification. We further design a post-processing algorithm Fair Functional Linear Discriminant Analysis classifier (Fair-FLDA), which targets at homoscedastic Gaussian processes and achieves fairness via group-wise thresholding. Under weak structural assumptions on eigenspace, theoretical guarantees on fairness and excess risk controls are established. As a byproduct, our results cover the excess risk control of the standard FLDA as a special case, which, to the best of our knowledge, is first time seen. Our theoretical findings are complemented by extensive numerical experiments on synthetic and real datasets, highlighting the practicality of our designed algorithm.

Fairness-aware Bayes optimal functional classification
Xiaoyu Hu
Gengyu Xue
School of Mathematics and Statistics
Department of Statisties
Xi'an Jiaotong University
University of Warwick
huziaoyy22199gmail.comm
gengyu. www.ee.wwwwwcc..cc ac.
Zhenhua Lin
Yi Yu
Department of Statistics and Data Science
Department of Statistics
National University of Singapore
University of Warwick
linzênus edu. ag
yd.yy.20warrick ac.
Abstract
Algorithmic fairness has become a central topic in machine learning, and mitigat- ing disparities across different subpopulations has emerged as rapidly growing research area. this paper, we systematically study the classification of fune- tional data under fairness constraints, ensuring the disparity level of the classi- fier controlled below pre-specified threshold. We propose a unified frame- work for fairness-aware functional classification, tackling infiniteddimensional functional space, addressing key challenges from the absence of density ratios and intractability of posterior probabilities, and discussing unique phenomena in functional classification. We further design a post- pess-processing algorithm Fair Func- tional Linear Discriminant Analysis classifier (Fair-FLDA), which targets ho- moscedastic Gaussian processes and achieves fairness via group-wise threshold- ing. Under weak structural assumptions on eigenspace, theoretical guarantees fairness and excess risk controls are established. As byproduct, our results cover the excess risk control of the standard FLDA special case, which, the best of our knowledge, is first time seen. Our theoretical findings are complemented by extensive numerical experiments on synthetic and real datasets, highlighting the practicality of our designed algorithm.
Introduction
Driven by technological advancements that enable high-resolution data collection and analysis, func- tional data analysis (FDA) has gained increasing attention over the past two decades. A wide range of statistical research has been carried out in this area, and we refer readers to Wang et al. (2016) for comprehensive review of recent developments.
Across a variety of statistical tasks, functional classification has emerged as one of the central fo- cuses, with applications in many areas including neuroscience (e.g. Heinrichs et al., 2023; Lila et al., 2024), genetics (e.g. Coffey et al., 2014), handwriting recognition (e.g. Hubert et al., 2017) and oth- Various classifiers have been proposed and thoroughly studied in the literature, among which are classifiers based on projection (c.g. Delaigle and Hall, 2012; Kraus and Stefanucci, 2019), Radon- Nikodym derivatives (e.g. Berrendero et al., 2018; Torrecilla et al., 2020), principal component score
"Equal contribution
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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densities (e.g. Dai et al., 2017) and partial least squares (e.g. Preda et al., 2007), to name but few. While these classifiers often perform well in terms of accuracy, our real data analysis on NHANES dataset in Section 4 demonstrates that applying standard functional classification algorithms can lead to substantial unfairness. Despite the growing recognition of fairness as a critical issue in machine learning and statistics, to the best of our knowledge, there is no existing work addressing unfairness in functional data classification. We are to bridge this gap by providing a principled framework for fairness-aware functional classification.
To date, fair classification for multivariate data has been extensively studied, including Yang et al. (2020); Jiang et al. (2020); Wei et al. (2021); Zeng et al. (2024a,b); Hou and Zhang (2024). These algorithms can be classified into pre-, in- and ods aim to modify training data prior to model training, allowing models to learn from debiased inputs (e.g. Calmon al., 2017; Johndrow and Lum, 2019). In-processing ones handle fairness con- straints during the training step. Common strategies include fairness-consraaiee Narasimhan, 2018; Celis et al., 2019) and fairness penalised objective functions (e.g. Cho et al., 2020). Post-processing one, by contrast, seek to reduce disparities by modifying predictions after training is completed (e.g., Kim et al., 2019; Li et al., 2022).
procedures. pre-processing meth-
optimisation (e.g.
However, the infinite-dimensional nature of functional data poses unique challenges, rendering the afformennioned fairness methods developed for multivariate data ineffective, see numerical results in Appendix A. To overcome this, we fully respect the functional nature of the data and derive the Bayes optimal fair classifier. Furthermore, we establish that the excess risk converges zero and provide an explicit convergence rate. The most related work to this paper is Zeng et al. (2024a), in which they develop a framework for Bayes-optimal fair classifiers under finteediimensional fea- ture spaces with a strong reliance on posterior probabilities, which are unfortunately intractable in functional spaces. This handicaps the direct application of Zeng et al. (2024a) to functional data. In this paper, we resort to the Radon-Nikodym derivative, which is used naturally as a substitute and plays a central role in characterising optimal decision rules under fairness constraints in the func- tional setting. Additional challenges, further comparisons and practical considerations are provided in Remark 1.
In fact, even without faimess constraints, the characterisation of excess risk for functional classi- fication is unresolved general settings with unknown eigenfunctions of the covariance operator. Addressing fairness in this setting is inherently more challenging, and quantifying the trade-off be- tween fairness and accuracy theoretically more demanding and remains largely unexplored.
1.1 List of contributions
In this paper, we study the problem of optimal binary classification for functional data under various fairness constraints. Specifically, focus the case when the sensitive attribute is binary and the probability measures of two classes of standard features within each sensitive group are mutually absolutely continuous. The main contributions of this paper are summarised follows.
Firstly, to the best of our knowledge, this is the first study to explore fair classification for functional data. We propose unified framework for constructing the fair Bayes-optimal classifier for func- tional data, providing a functional data-tailored treatment of fairnes- awwre classification problems. Our framework is sufficiently general to accommodate a wide range of extensions, including set- tings where the sensitive feature is unavailable at test time and multi-class classification problems. These extensions (detailed in Appendix B) highlight the flexibility of our approach and its potential to serve as a foundation for future advances in fairness-aware functional data analysis..
Secondly, when the non-sensitive features are assumed to be Gaussian processes, we introduce post-processing algorithm, the Fair Functional Linear Discriminant Analysis classifier (Fair-FLDA) in Algorithm 1, which effectively enforces faimess by group-wise thresholding. Our algorithm accounts for the most general setting where we assume all model parameters, including group-wise covariance functions and their eigenvalues and eigenfunctions, to be unknown.
Thirdly, we further establish the finite-sample theoretical guarantee for the proposed algorithm in terms of both fairness and excess risk control, ensuring our algorithm Fair-FL.DA not only adheres to the specified fairness constraint with high probability, but also achieves satisfactory classification performance, with the cost of faimess explicitly quantified. As a byproduct, our results cover the

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special case of functional classification without faimess, which serves as complement of Wang et al. (2021) under a more general setting when eigenfunctions are assumed to be unknown.
Finally, the proposed algorithm is validated through extensive numerical experiments on both sim- ulated and real datasets in Section 5 and Appendix A. The comparisons with existing multivariate fairness methods to functional data further highlight the superiority and practical necessity of our method for fair functional classification from a numerical perspective.
further supporting our theoretical findings and highlighting their practicality.
a}. For a, b E I, let a V b = max a, b). For two sequences of positive numbers {an and (b,). denote on S by (or on (b,,)) and an x bn. if there exists some constants c, C > 0 such that an/ an S C and c S an/b, - C. Write On & ba, if an /b, -> as n -> DC. For a sequence of random variables (X,) and positive numbers {an), denote X, = Op(a (an) if lim M- lim sup, P([Xal 2 Man) = 0. For any two a-finite measures f1 and v, denote μ < v if μ is absolutely continuous with respect to and write dy/dv the Radon-Nikodym derivative; write ~ if they equívalent. Let L (2) 1J) be the space of square-integrable funetiions on [o, 1]. For f L (10, 1]), denote Il So (2(s) ds. For f.g e ([0, 1)), denote the inner product by (f.g) = So (())((s) ds. For any bivariate kernel function K: [0, 1]2 - RR,, let H(K) denote the reproducing kernel Hilbert spaces (RKHS) generated by K. For f E H(K), denote Il.1l = Dラ= (f, o,) its RKHS norm, where {o, 1= and {>, are obtained by Mercer's decomposition of K: K (s, 1) = Σラ= (t (1), 8, E [0.
Notation. In this paper, for E N+, denote [a]
2 Fair Bayes optimal classifier
2.1 Problem setup
Suppose that we have n independent and identically distributed samples D {(X,, A,, Y.). e [n]], where X, E L²( 1]) is the standard (non-scnsitive) functional feature. A, E {0.1] is the sensitive feature (e.g. gender or race) and Y, E {0, 1] is the binary label. Let F be the class of measurable functions f (,² 1)) x {0,1} - [0. 1] and our goal is to identify a randomised classifier f E F, as defined in Definition 1, such that the misclassification error (X, A) + Y) is minimised subject to a specified fairness constraint.
R(S*) = P
Definition (Randomised classifier). For any x E [[, 1)) and E 10, a randomised clas- sifier f E F is a measurable function such that f(r, a) = P(Ý; 1|X x, = a), where Y, = 1y a) is the predicted label, i.e. Y,H[X z, A a} ~ Bernoulli (fx
For a,y E (0, 1}. let Pa. be the distribution of the random process X given (A,Y) (a. Assume that Pa.1 ~ Pa.0. thus the Radon-Nikodym derivative (X) i (X)/dPaa is well defined. Let Te,y IP(A = a, Y = y) and T. = P(A = a), then for each sensitive group, the Bayes rule that minimises the misclassification error is given by
f*(x,a)= = (X) 2 To.1/
where 1 denotes the indicator function (e.g. Berrendero et al., 2018). However, the above clas- sifier does not take fairness into account. To address this, in order to mitigate bias across groups, existing literature proposed various notions of parity, some of which are listed below.
Definition 2. A classifier f is said satisfy (i) equality of opportunity (e.g. Hardt et al., 2016) if the true positive rates are the same among protected groups, i.e. PP(; = 1|A 0, Y =1) = P(2 A = =1). (ii) predictive equality (e.g. Borbett-Davier et al., 2017) if false positive rates are the same among protected groups. i.e. IP(;- 1|A 0, Y 0) = P(Y, = 1|A= = 0), and (iii) demographic parity (e.g. Cho et al., 2020) if its prediction Y, is independent the sensitive attribute A, i.e. PÝ, = 1/A = a) = P(, 1), for E 10.1).
Enforeing exact parity may lead to a substantial loss in accuracy. In the literature, one popular approach is to instead control the disparity measure, D : F + 11, i.e. upper bounding the dif- ference in quantities of interest between the sensitive groups. The disparity measures corresponding to the notions of parity in Definition are presented in Definition 3 for completeness.
3

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Deluitian 3. For a given Clestite f E F, the disparity of Apporturt (DO), predictive disparity =
(PD) and Cempgrrranii
dispputtyy (DD) / are defined as (f( = (X, [] = 1|A = 1,Y
= 1| - = 0) -0-1101 - / =
1] 1 0)= = 1| - 0,Y = 0} and
= 1), PD(f) (D(f))=a = ( (x. 1) (x
(UDD DD(
P
P /
- 1|A 1 - 0}. 1) 13- - -
A - 0, Y,Y
1|A -
For any disparity M D. volerance level 6 2 0, we seek the ô-fair Bayes optimal classifier fö.8 auch that the missclssiification
dusclassifiiation error is intimissed over all classifiers that satisty the h-disparity b.dispantig i.e.
mngmin
f E arg min / {R(f) [Df) s 6}.
(2)
TEF
2.2 Unifiedd framework for fairness-aware awreess--wwree functional classification
To characterise the 8-fair Bayes optimal classifierr defined in (2), in this section, we apply the gen- Pearson lemma (see Lemma 43 in Appendix H) and exploit Rikon-Nikodym a
eralised Neymant derivatives, providing a unified framework for faimess-aware functional classification.
derivatiinn
Applyini the generalised Neyman-Pearson lemma is possible when both objective function and constratnt in (2) are linear in the classifier. For the objective function, it follows from fromiie Lemma 8 in Appendix C.4 that R(f) = Eue fx f(x a) TTa - Ta,1 af.. (x) (r:)ddPa.(8)) + P(Y = 1) is linear in f. For the constraint, we consider the class of bilinear disparity measures defined below.
dPa.1
Soures
Definition 4. For all probability measures P and f E F, a disparity measure D : F - [-1, I] is bilinear in the classifiers f and d /PP.,/dPap if there exist 8D,a, bD.a E R such that D(J) =
bD.a
(o.)) Jx f ((r,,a((D.... 2P.. aPa.(()) (z) + J
Definition 4 is also considered in Zeng et al. (2024a) and holds for many y commoniy used disparity measures in the existing literature, including those defined Definition 3.
Proposition The disparity measures DO, PD and DD defined in Definition 3 are bilinear with
8DO,a = 2a- 1, IDo,a 0; spD,a = O, bpn,a = 2a - 1; and SDD,a = -2a - 1)Ta,1 Ta. u for a E 10. bpD (2a- a)
With both misclassification error and disparity measures being linear in the classifier, the generalised Neyman Pearson Withaa--Peeade Jemma unlocks fō, a dlosed-form solution the 8-fair Bayes optimal classifier.
Theurem 2. Assume that d.a, dPa,/ddPan a.o is a continuous random variable for a E {0,1). For any E R and a given bilinear disparity measure D in Definition 4, denote the classifier r a) = dP (x)= 2 Ta,0 + +bb,, } and D(T) = D(gD.=). Then, for 5 2 0. the 8-fair [[ - TsD,a) af
fairrra Bayes ?
optimal classifier is fb,s = 9D.Th,s where
T = arg min {|r|: [(()) = 83.
FEB
The proof of Theorem 2 is in Appendix C.2. We remark that our analysis can be easily extended to the case when a.P / /dPa. is discontinuous, by including a handomised decision rule on the set where (Ta.1 - 75D,a d /dP = Fa,0 + TbD,a-
At a high level, Theorem 2 states that the 8-fair Bayes optimal classifier is shifted from the Bayes classifier (1) by linear factors 8D.a and bD.a. The linear shift is ensured from the bilinearity of the dispautt
measure D and the shift level 7 is further optimised in (3). The rntimisaatin in (3), at the prorra core is to minimise the misclassification eтror among all that satisfy the fairmess constra The intsclassooo
constratiie
sectioiiinn measure measure D ((DD,r) ) as functions of 7. We show in Proposition 7 in Appendix C.4 that the disparity is contimuuus and -an-increasidd while the misclassification error is non-decreasing in |T]. As a result, anding an optimal threshold that minimises the misclassification error reduces to minimising |T].
detailed form is a consequeens of behaviours of the misclassification error R(gD,r) and disparity formix
For D E (DO, PD, DD} in Definition 3, it holds that na.1 - TD, > 0 and T.,0 + +r > (see I Lemma 9 in Appendix C.4). Consequeent we rewrite the fair Bptesaoptimal classifier fi. as
B a) = 1 { dPP.1 (x)
Na.D + DD.
(4)
na.1 TD DD,
4

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Compared to the Bayes classifier without faimess constraints in (1), mitigating disparity is achieved by adjusting the classification thresholds, from / Ta,1 to (Ta.0 + " T (T sD.a)) with the shift determined by the chosen disparity measure and the underlying population distribu- tions. When Ti. = O, i.e. [(O)] I 6, we recover the Bayes classifier f (zz a) in (I), which is, in this case, automatically fair. We write the classifier in (1) as xD in the rest of the paper.
Remark Our framework is inspired by the one in Zeng al. (2024a), which also an application of the generalised Neyman-Pearson Pearson lemma. The key difference between Zeng et al. (2024a) and Theorem 2 lies in the generalisation of the classifier dinnnie-ad spaces, the absence of default base measure leads to the use of the Radon- Nikodym derivative i (1) /dPa.o. It is a more natural functional used for functional classifica- tion, rather than posterior probabilities IP(Y = 1|4 a, X = x) considered in Zeng et al. (2024a). In particular, in important cases such as when functions are Gaussian processes, aPa,, 1 (rr)/dPa. Pa.o analytically tractable but not the posterior. In the cases when d&P,, (z)/PPo o is not tractable, there are ample tools for its approximation (c.g. Bongiorno and Goia, 2016; Dai et al., 2017).
functional feature space. When moving to
2.3 Fair functional linear discriminant analysis classifier for Gaussian processes
As a concrete and important example, we focus on a specific setting when the functional features are Gaussian processes. We propose the Fair Functional Linear Discriminant Analysis classifier in Algorithm 1, featuring a plug-in estimator built on S in Theorem 2 and, specifically, in (4).
Additionally to the problem setup in Section 2.1, for any collection of (X, A, E D. we assume that the functional feature X a Gaussian process, i.e. {X|A = a. = y} gP(pay Kay)), where (t) = A
a.Y = y) and ts, Ma.p (t)]() = a, a.Y = y) are mean and covariance functions, s,t t [0. 1] and a,y E (0.1).
[s) - (
For simplicity. we consider a homoscedastic setting within each group, i.c. Kn,0 = Ka Ka. The covariance function Ka. consequently, admits the spectral expansion Ka(s, Ej=1 (8)),) where Aa,1 2 Aa,2 2 ... 2 0 are eigenvalues and (0o. IjEN, are eigen- functions. Without fairness constraints, the functional linear discriminant analysis (FLDA) classifier in (1) is known to be optimal to minimise the misclassification error (e.g. Berrendero et al., 2018).
To account for fairness and construct a plug-in type classifier for SD,s- it is essential to evalu- ate the Radon-Nikodym derivative dl',, dLPa./dPPa. a.o. which plays central role in the decision rule and analytically available under Gaussian settings. By standard results of Gaussian measures (e.g. Theorem in Berrendero et al., 2018), the distributions Pa.o and Pa. are mutually con- tinuous if and only if the mean difference Ma.1 Ma.0 belongs to the RKHS space H(Ka). - (0a.1.) -0a }. where Ao, Sa.j = (X, &n.g) is the principal component scores of X and la.g.j = (Hagg Pag) 12 are the coefficients of the mean functions projected onto the eigenfunctions of Kg-
then holds that dfa.
TP.. (X) = exp (Ca.)- 0000.1100.111 -0..0..
To estimate d.P., d.Paa/PPa.o in practice, we assume the availability of an additional training dataset, D D= ((X,, A,, Y.), which is drawn independently from the same distribution as D and used to estimate Ta in the initial classifier. We refer D as the calibration data, subsequently used to post- process the initial classifier by selecting the adjusted threshold 7D.6.
We decompose the calibration data as D Do.1 U Do,o D1,1 D1,.. where for a, E {0, 1}. let Do.y = {(X;, A; = a, Yi })}, Ra,y = |Da.vl and n = En.y Pa,yy For notational clarity, we denote the i-th feature in Da.y as X a.y. for i E [na.y]- The notation for DD follows similarly. The resulting classifier is detailed in Algorithm 1, with its theoretical guarantees discussed in Section 3.
Remark 2 (Perfect classification). For functional classification without fairness, the intrinsic infinite- dimensional nature of functional data gives rise to vanishing misclassification errors under certain scenarios. This is first discussed in Delaigle and Hall (2012) and known as perfect classification in the existing literature. As discussed in Berrendero et al. (2018), for homogeneous Gaussian pro- cesses. perfect classification arises when the class distributions Pa.ı and Pa.u are mutually singular, i.e. W1x,114a.9 & H H(.K),, in which case the Radon-Nikodyn derivative d.Pa. //d.//P.... a..o does not exist. In this paper, we restrict our theoretical analysis to the more challenging regime of imperfect clas- sification with non-vanishing classification error. Notably, in the perfect classification regime, the optimal FLDA classifier is automatically fair when the disparity measure D (DO,PD). When
5

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Algorithm Fair Functional Linear Discriminant Analysis classifier.
INPUT:
Training data 1 UDoooDDo, 1 calibration data
disparity
level 6, level of truncation J. S1. Estimating Radon Radon-Nikodym derivatives Ta and class probabilities Ta,y using training data.
For a E {0,1), 1), calculate Fa.y Ha.x (1a 1 Eier X (!) and Ra(s,1) =
1:
Raay 1 E Di=l (X ( (s) Pa,y (s) ( (t,((t))
yve(0,1) +h.0+ -a.u-1
2:
Estimate eigenvalues (Na, )/= eigenfunctions { of Ka by spectral expansion.
3:
Estimate projection coefficients Oz.g. -oo Ma.g 00), (1) dr. For any function X, denote
Na (X) = ex (0.2,,- -0. o.,,, s x(t)).,(t)dlt dt) -0..)))))))
ビ 1011100 a. }.
X=.; S2. Estimating the optimal threshold using calibration data.
Xa.
Let ÿD, (x, a) = [ (ft.1 TTD, (x) > aT.00 -bD,o) Calculate ((T) = Eae(o,)) (D, a)sD,a dP), fx ID. (2, a)Da di dP),)())), where fx (ax, a) dPa.. = may EiEr f(Xa.u a.y. a). Set TD = arg min [17] (D(T) s 8). OUTPuT: fD, with fD. a) 1f (D. ao.0 + TD.bDD.).
4:
5:
D DD. (1) is automatically fair if [P(Y- 1|A = 1) P(Y 1|A = 0) < 8. Further insights into the phenomenon of automatic fair are supported by numerical experiments in Appendix A.4.
3 Theoretical Properties
For general bilinear disparity measure (Definition 4), we provide the theoretical guarantees on the fairness and excess risk control of the Fair-FLDA algorithm (Algorithm 1) in Theorems 3 and 5, with a special case regarding the disparity of opportunity in Corollary 6. We start with assumptions.
Assumption (Class probabilities). Assume that there exist absolute constants O < Cp S C, < 1,
such that the class probabilities satisfy 0 < Cp I Tn,y I S. < 1, a. Y. (0.1).
Assumption 2 (Gaussian processes). Assume that the standard features (Xay)ie U (eEn,nl are collections of Gaussian processes GP(Ha.y Ka) with continuous trajectories, y. E 13. In addition, assume the following holds for any {0, 1}: a. (Covariance function) The covariance function Ka is continuous and there exist absolute constants C22 C > O such that the eigenvalues of the covariance operator are decreasing with j-a 2 Cxdan 2 C ja
for a and j N4: b. (Signal-to-nois ratio) There is an absolute constant CK 0 such that ||Ha,1 Mo, 2 Cg; and c. (Mean difference) There exists a constant Cp > 0 such that for any j E N., it holds that [Ma,1 Ma.0, La.i) < C,.j with 3 > (ax 1)/2.
In Assumption 1, assume that the class probabilities are bounded away from 0 and 1, essen- tial to ensure that a sufficient number of samples for each group can be observed. Assumption characterises the properties of the functional features. Assumption 2a specifies the decaying rate of eigenvalues and quantifies the spacings between two consecutive eigenvalues. This is commonly seen in the FDA literature (e.g. Hall and Horowitz, 2007: Dou al., 2012) involving eigenfunc- tion estimations. Assumption 2b imposes a lower bound on the magnitude of the signal-to-noise ratio ||Ma.1 ll We thus exclude the trivial classification regime and preclude the clas- sifier from degenerating into random guessing. Assumption 2c enforces the alignment between the mean difference and eigenspace, with larger values of 3 indicating better alignment. As- sumptions 2a, 2b and 2c jointly imply that ||Pa,1 ol k. = 1 with the tail sum satisfying Ej=111 (0.,1.5 ou,,.j))/Xx ///; S Ja-25+1, J N4. Note that decay rates and 3 are assumed to be invariant for {0, 1}, but this can be easily generalised to different decaying rates between groups.
Theorem 3 (Fairness guarantee). For any 8 > 0 and bilinear disparity measure in Definition 4, let Jo.8 denote the classifier output by Algorithm 1, we have, for any E (0, 1/2), that P P(D((fo))) < T7, where C > 0 is an absolute constant.
6 + C. Coooo(L/m))//

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Theorem 3 is proved in Appendix D and shows that with probability at least 1 - 77, after calibration step, the disparity level of fD.s does not exceed the pre-specified level 8 by a small offset term up to (1))///)/)). The magnitude is quantified by the high probability upper bound on (i(T),s) ((TD.s)). measuring the deviation of the empirical distribution from its population counterpart.
If insisting on controlling the population unfairness D(fr.s) below 6, then provided that Vlog(1/n\/n X 6, it suffices to adjust the input 6 to CVlog(11n)/ i.e. set 7D = arg mrg [ [D(T] < d - C CV/oo((/n)/m/ in S2 of Algorithm 1. We refer to the re- sulting method as the Fair Functional Linear Discriminant Analysis classifier with Calibration (Fair-FL.DA.). Numerical results comparing the performances are presented in Section 4.
To present the excess risk control of our method, preliminary step, we establish the misclassifi- cation error of the oracle classifier in Proposition 4.
Proposition 4 (Misclassification error). Under the model setup in Section 2.3, for any 8 2 O and bilinear disparity measure D such that Ta.1 TD. > and Ta,u + TD.
the corresponding misclassification error for Jb, defined in Theorem 2 logi(zo.o .t. Tb. +
given by R(fb,s) = E.E(o,1) To.0 li -e Eae(o.1) Ta,1 [i1_-llo.1---0 11l..,--u..0 Na + log(( bb. (F. sw.)) where the cumulative
2
1 na.o Ka
I| - a0.0 Ixa
distribution function of the standard normal distribution.
See Appendix C.3 for the proof of Proposition 4. Compared to the standard excess risk without fairness constraints, R(fD,>c) (e.g. Theorem 2 in Berrendero et al., 2018), the cost of faimess con- straints is quantified by the logarithmic term involving DD.5 resulted from the adjusted group-wiss thresholds in (4). As shown in Proposition 7 in Appendix C.4, a decrease in 6, i.e. stronger fairness constraints, leads an increase in [b. hence a larger misclassification error R(fD.s). s). When fū, is automatically fair, i.e. TD.5 = 0, Proposition 4 recovers Theorem 2 in Berrendero et al. (2018).
Moving towards excess risk controls, as discussed in Zeng et al. (2024b), for any f E F, the traditional excess risk R(f)) R(DDss 8) may be negative as fD.s does not necessarily minimise the excess risk. To make a meaningful control of the misclassification error, we resort the quantity [R(()) - R(fB,s),, which can be further decomposed as the non non-negative fairness-awan aware excess risk de(f.fD.b (defined in Definition 5) and a disparity cost (D...D - -(ff), that R(f)- R(fb,a) = (J, (b,s) + Tb, (D(fi.s) - D( S de(S. + ITb,sIDD(fbs)) -D())f The derivation above is directly via the definition below.
Definition 5 (Faimess-awaar excess risk). For 5 2 0, let fü.s be 8-fair Bayes optimal classifier defined in (2), recalling TD.8 in (3) and SD.w, bD.a in Definition 4. For any classifier f X x (0.1) + [0, define the fairness-aware excess risk dE(S, f5.5) = Eeetony Sxlf(I,a) aPa. (x, bD,) (Ta,1 TD. aP,.0 (x)) aP d.P.
We are now ready to present the excess risk control for Algorithm 1 in Theorem 5.
Theorem 5. Denote Cr. Cn and cD the estimation error related to Ra.g. na and Π) i.e. for any small E 1/2), y.y 10,1] and X ~ GP K.), it holds with probability at least 1 - n/3 that [a..y To,y S en, (i
log no [X) S En and supTER (D(T) D(7) ED-
Suppose that Assumptions and hold; and for 5 2 0, bilinear disparity measure D (Definition 4), (ii) Na,1 &D, C1. Na,u t DD.DD C1. with max max{|sD,el) |bb S cz and (ii) (DTT.8) D(T + + CDlEI for any in the small neighbourhood of 0 and some > 0, where CI, C22 CD > Oare absolute constants.
satisfying that (i) D(0) (D- 8] - -
Then, it holds with probability at least 1 - 1- n... E (0, G + G, + c7), that the classifier PD. output by Algorithm I satisfies that
[B(Po.s) -R(F.)))(((E.)) S (F. + | (lo((1/))/
(5)
where de (D..s fb, S (Ex + en + ez with e, PiD T 1 S # 0}.
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Theorem 5 provides a general characterisation of fairness- aware excess risk and traditional excess risk when (0) not too close to i.e when fD is either sufficiently fair or unfair. We impose two extra assumptions on D, with the first one controlling the behaviours of Th. near the bound- aries. This condition is used ensure that 7D lies within the range of its estimated counterparts, i.e. aa,1 - & > 0 and a0.0 + FD, > O. This guarantees that the misclassification error R(fD,s) retains well-structured form, analogous that in Proposition 4. The second assumption controls the steepness of D in a small neighbourhood of TD.6 with larger values of 7 corresponding to greater steepness. In the fair-immn impacted case, i.e. DD. * O, if D is potentially very flat near DD. i.e. y small, the estimation problem becomes more challenging. resulting in a larger estimation error E, (as illustrated in Figure 1). We remark that when D is explicitly given, both of the above assumptions can be verified in most of the cases, see Corollary 6 as an example.
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Figure 1: Effects of steepness of disparity levels on the estimation error of T.s Left and right panels illustrate steep and flat D(-). Red solid line: D(.). Blue dotted line: D(.).
The disparity cost in the right-hand side of (5) is a direct consequence of the faimess guarantees in Theorem 3. The fairness-aware excess risk, de(fD, fb,8). consists of two components, with (ex + =2,)² capturing the intrinsic excess risk and 2 reflecting the cost of fairness. This is the first time seen in the FDA literature, echoing the same spirit in the mnite-dimensiona classification literature (Zeng et al., 2024b; Hou and Zhang, 2024). When D is chosen as one of the disparity measures Definition 3, eD can be explicitly controlled as éD = 6x + En + (log(1/)))
n
To further illustrate Theorem 5, we apply the framework when D = =DO, the disparity of opportunity. The corresponding results up to noly-logarithmic factors are summarised in Corollary 6.
Corollary (Excess risk under DO). Under Assumptions and for any 5 2 0, B 2 3a+2 it holds
2
for any J satisfying J2a+2 S T. that d Sibo,s = =pp + Ja-2841 + 1(7,, ). if we additionally assume that DO(0) & (5 - (4 Ja-22411 v x+) + -0-6 - G 20-22+1- 1 Additionally, if we further assume n.
N and pick x No a it holds de [,Do.s f
Op(N
11 } and (R([ro.s - R(Sio,)) x) S dno,s,, S + | (N-
Corollary 6 is shorter version of Corollary 11 in Appendix E.2, presenting only the case when 3> 22 Detailed results for the case when a+1 < B < 30+2 are deferred to Corollary 11. 2
At a high level, Corollary 6 shows the upper bound is of the form de(fDo.s fño.s) < variance to estimate 7/2 + squared bias due to truncation + cost of fairness, which highlights the role of truncation parameter J in determining the final convergence rate through the underlying bias- variance trade-off. Moreover, shown in Corollary 11, as the mean difference aligns more strongly with the eigenspace, i.c. as increases, a smaller estimation variance is observed. Note that the cost of fairness is masked when n x ñ x N, which shares the same finding as Hou and Zhang (2024).
Corollary 6 is the first time providing samile guarantees for functional classification under fairness constraints. There is no existing work in FDA with fairness constraints, and even for the FDA work without fairness, such detailed characterisation is novel in the literature. Without any predecessors regarding the former, we list some relevant works with the latter point.
Excess risk control for functional classification has previously been considered in Meister (2016) and Wang et al. (2021), with the former relying on smoothness assumptions of functional densities and decay rates of the metric entropy of functional space, and the latter assuming that eigenfunctions
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of the covariance operator are explicitly known. Instead, our result in Equation (5) is established requiring only weak structural assumptions on the eigenspace. When D.S = 0. our result in Corol- lary 6 recovers the excess risk rate established in Wang et al. (2021), provided that f20+2 2 n. We would like to remark that the upper bound on J arises from estimating eigenfunctions; similar con- dition is also in Hall and Horowitz (2007) and Dou et al. (2012). Another relevant work is Cai and Zhang (2019), studying the linear discriminant analysis (LDA) classifier for J-dimensional Gaus- sian random variables. Our key variance term 1.2 resulting from estimating the first J leading terms in nas aligns with the minimax optimal result derived in Cai and Zhang (2019). At a high level, fō. can be viewed as applying LDA an infinite number of principal component scores. By focusing only on the first J scores in Fair-FLDA, we recover the variance term in J-dimensional LDA.
4 Numerical experiments
In this section, we demonstrate some key numerical evidence via simulated and real data analysis.
Simulation. Generate (Y, E {0, 1) according to the distributions P(A 1) = 0.7, P(Y A = 0) = 0.4 and P(Y = 1|A = 1) 0.7. Given Y = y and A = a, generate the functional covariate (t) as Xa, (f) = (o.g(t) + >x=
k= Sa (t), where Φx (t) = v2 cos(ktt), Sa,k ~ [(, )a,k), A0,1 = k \1.k = 2k - and the mean functions are specified as follows,
30
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10,0 = 11,0 0, 1o.1 (t)
Σ (08(-1)*--0o66) (1) = Σ 1
Let 8 = 1.5 and n = 1000. To conserve space, we defer detailed settings, implementation details, more numerical results on effects of sample sizes (n2,, alignment of mean difference (8), model mis- specification tmon-Gaussianiyg and perfect classification, and comparisons with multivariate base- lines to Appendix A.
Figure 2 reports the medians across 500 times of the classification error and disparity measures of our proposed methods Fair-FLDA and its calibrated version FLir-FLDA, (in Section 3), with the classical functional linear discriminant classifier (FLDA) and oracle Bayes classifiers as competitors.
The classification errors of Fair-LDA and Fair-FLDA, exhibit a non-increasing trend as å grows, mirroring the behaviour of the oracle Bayes classifier. The error of FLDA remains constant across different values of 6, as it does not incorporate any fairness constraint. When 8 is small, correspond- ing to fair-impacted regimes, the classification errors of our methods decrease with increasing ò, due to the less stringent fairness constraints. Once 5 exceeds certain threshold, the faimess constraint becomes inactive, and the classification errors of the fairness-aware classifiers converge to that of the unconstrained FLDA. As for the disparity control, the FLDA consistently fails to meet fairness requirements, but the Fair-FLDA maintains the desired median disparity level and the Fair-FLDA, typically achieves median disparity levels strictly below 5. As established in Theorem 3, the em- pirical 95% quantile of disparity of Fair-FLDA may slightly exceed 6, whereas the Fair-FLDA, effectively corrects for this offset, achieving probabilistic control of the disparity below 5 with prob- ability at least 95%. As 8 increases beyond a critical threshold. both hairness-aware classifiers grad- ually reduce to the unconstrained classifier FLDA, and their corresponding disparity levels stabilise accordingly. Both Fair-FLDA and Fair-FLDA, satisfy their respective fairness criteria without sig- nificant compromise in classification accuracy.
Real data. For the real dataset, we use the 2005-2006 National Health and Nutrition Examina- tion Survey data (CDC, 2006), where the sensitive attribute is race and the classification task is to determine if an individual is under 20 or over 50 years old based on the quantile function of intensity values. Results are plotted in the 4-6th columns in Figure 2, with details and additional experiments presented in Appendices A.1 and A.5. It shows that the FLDA exhibits substantial un- fairnsss, whereas the Fair-FLDA effectively controls the median disparity. In terms of probabilistic disparity control, the 95% empirical quantile of the disparity for Fair-FLDA. A, slightly exceeds with the default choice of the calibration constant under DO. We recommend tuning the calibration parameter to achieve more reliable probabilistic disparity control (Appendix A.5). The classification errors of Fair-FLDA and Fir-FLDA. remain comparable to that of the FLDA. demonstrating that

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Figure 2: From left to right: medians of classification errors, medians and 95% quantiles of the disparity measures, in the simulated (Ist-3rd columns) and real data (4-6th columns). Orange dots: FLDA; blue stars: Fair-FLDA; pink triangles: Fair-FLDAc: red solid line: oracle Bayes classifier; grey dashed line: y =
our fairness-aware classifiers effectively mitigate unfairness while maintaining competitive classifi- cation accuracy.
Discussions
In this paper, we use the publicly available NHANES dataset as a representative data example to demonstrate the effectiveness of our method. It is worth noting that our approach is broadly appli- cable to wide range functional data classification problems where fairness a concern. For example, the Siena Scalp EEG dataset in the PhysioNet database contains EEG recordings from male and female subjects and can serve as a natural application of our fair functional classifier, where the response variable can be defined the occurrence of seizure. Overall, method offers principled and broadly applicable tool for fair classification in functional data analysis.
We envisage several potential extensions. Firstly, our framework and Fair-FLDA algorithm depend on the availability of the sensitive features, which may be restricted in certain practical settings due to privacy concerns. When sensitive features A are available during training but not available during prediction, we provide necessary steps to extend our algorithm in Appendix However, when A is not available even during the training process, more refined methods for inferring the sensitive features would be necessary, see Appendix for additional discussion. Secondly, in many scenarios, the Radon-Nikodym derivatives d.P.. d /d are not explicitly known and easy to work with. To address this, a natural strategy during implementation is to approximate it using the density ratios of projection scores (c.g. Bongiorno and Goia, 2016; Dai et al., 2017). Thirdly, in reality, functions can only be discretely observed over sampling grids. Investigating the effect of sparsity on the excess risk under fairness constraints remains intriguing area.
Acknowledgments and Disclosure of Funding
Yu's research is partially supported by the EPSRC (EP/Z531327/1) and the Leverhulme Trust (Philip Leverhulme Prize). Lin's research is partially supported by the Singapore MOE AcRF Tier 1 grant.
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NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification: In the abstract and introduction, we claim that we provide fair Bayes opti- mal classifier for functional data, with a by-product being some finite-sample analysis for the functional classification problems without fairness constraints. Our methods are pro- posed Section 2.3, with theoretical justifications Section and numerical evidence in Section 4 and Appendix A.
2. Limitations
Question: Does the paper discuss the limitations the work performed by the authors?
Answer: [Yes]
Justification: We explained assumptions right after the assumption statements in Sec- tion 3, with discussions on some simplification due to notational simplicity and how to relax them. A main limitation of our method is the reliance tive. We explained how our method proceeds when it is not tractable in Remark and demonstrates the numerical performances in Appendix A.3.
the Radon-Nikodym deriva-
3. Theory assumptions and proofs
Question: For each thcoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
Answer: YYs]
Justification: We collect assumptions Assumptions and and the proofs for all theo- retical results are presented in Appendices C H.
4. Experimental result reproducibility
Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclu- sions of the paper (regardless of whether the code and data are provided or not)?
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Justification: In this paper we proposed classifier, with numerical justification done in a wide range of simulations and real data analysis. The code producing all numerical results included in the submission. In particular, both the paper and the code include clear description on how to reproduce the algorithm.
5. Open access to data and code
Question: Does the paper provide open access the data and code, with sufficient instruc- tions to faithfully reproduce the main experimental results, as described in supplemental material?
Answer: [Yes]
Justification: We have submitted code including those generating all the numerical results in this paper. The real dataset is obtained from https : /wwww//wwaa/////////////////// wwc.cddc.oooooooooeee www.:/ooo///hoooeeeeee Contimuosshaanes/ DDefst/t..attaaaa wtw.ffuu.tss.xeiiiiie.... Assx=BeginYear=2000
6. Experimental setting/details
Question: Does the paper specify all the training and test details (e.g., data splits, hyper- parameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
Answer: [Yes]
Justification: See Section 4 and Appendix A.
7. Experiment statistical significance
Question: Does the paper report error bars suitably and correctly defined or other appropri- ate information about the statistical significance of the experiments?
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Answer: [Yes]
Justification: To conserve space, we report the medians over 500 Monte Carlo trials in each simulation settings, rather than mean accompanied with error bars. However, all results can be obtained from the submitted code.
8. Experiments compute resources
For each experiment, does the paper provide sufficient information on the com- puter resources (type of compute workers, memory, time of execution) needed reproduce the experiments?
Question:
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10. Broader impacts
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Answer: [Yes]
Justification: This paper proposed a fairness-aware classifier, which aims to tackle the issues when the classification is partially based sensitive traits, leading to inequality in society. This paper however stays as a methodological and thcoretical paper, we hence do not envisage negative societal impacts.
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Technical Appendices and Supplementary Material
All technical details and additional numerical results are collected in the Appendices. Detailed simulation settings and additional experimental results are collected in Appendix A. We present proofs and properties related the Bayes optimal classifier f in Appendix C. The proof of Theorem is collected in Appendix D, with the proofs of Theorems 5 and 6 presented in Appendix E. All results related to class probability and eigenspace estimation can be found in Appendices F and G. For completeness, necessary technical lemmas are included in Appendix H.
Throughout the appendix, with a slight abuse of notation, unless specifically stated otherwise, let C1, C1. C22 C2.... 0 denote absolute constants whose values may vary from place to place. For a, b E R. write a A b = min(a, b]. For an R-valued random variable X and k E [2], let [X|[lv denote the Orlicz-v/ norm, i.e. || ll.. = inf (1 > 0 [lexp([[X[/44)) s 2}.
A Further details of numerical experiments
The figure labels are consistent with those used in Figure 2 in the main text, unless otherwise spec- ified. We implemented all methods R (version 4.3.1). Experiments were conducted on a server equipped with an Intel(R) Xeon(R) Platinum 8280 CPU @ 2.70GHz (28 cores) and 503GB of RAM. The source code is provided in the Supplementary Material.
A.1 Detailed setup of Section 4
Simulation. For the simulation results, we generate AY {0, 1) according to the distribu- tions P(A 1)
1|A 0) 0.4 and P(Y 1|A 1) 0.7. Given Y y and A = a, generate the functional covariate (t. as Xt, = Ma.y (t) >E>e (a, where (t) = /2 cos(k)t), Sa.k (0,, Aa,x), Aao = k- AI.E = 2k 2,- and the mean functions are specified as follows,
50
N0,0 11,0 00 0, (1) = Σ
(b)), m1)
Σ
(t)
k=1
Let denote the size of the training sample. We implement the proposed fairness-aware classifier under two calibration settings,
Fair-FLDA: calibration constant set to O;
* Fair-FLDA.: calibration constant set to min 2 log(l1/))/n// 8), with p = 0.05.
Truncation levels are selected via 5-fold cross-validation, specifically by minimising the average classification error associated with the unconstrained classifier.
During implementation, the training set D is randomly split into equal- subsets. D U D2- One subset is used to estimate 9a and Ra while the other is used to estimate the threshold Let Ji denote the classifier estimated using D; for model estimation and D2 for threshold calibration, and let /2 denote the classifier constructed with the roles of DD and D2 reversed. To mitigate the randomness caused by random splitting, we adopt cross-fitting approach and define the final probabilistic classifier as the average = (f1 f2)
Real data. For the real data analysis, we apply the proposed method to data from the 2005-2006 National Health and Nutrition Examination Survey (CDC, 2006). Further details about this dataset can be found in Lin al. (2023). Following the preprocessing steps in Lin et al. (2023), we exclude observations with questionable data reliability according to NHANES protocol, remove observations with intensity values higher than 1000 or equal to and retain subjects with at least 100 remaining observations. The response variable is whether an individual is under 20 or over 50 years old, with quantile function of intensity values as the functional covariate. The sensitive attribute refers to race, categorised non-Hispanic white and non-Hispanic black. The final dataset consists of 3252 instances, which we randomly split into equal-sized training and test subsets. The methods are implemented following the same procedures described for simulation experiments.
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A.2 Additional simulation results under Gaussian models
Results under varying sample sizes. We evaluate the model from Appendix A.1 under vary- ing sample sizes. As shown in Figures 3-5, the excess risk of both Fair-FLDA and Fair-FLDA decreases with increasing n. Moreover, with larger n, the difference between Fair-FLDA and F in disparity control becomes less significant.
Error-unfairness trade-off. We illustrate the error- eror-unfairness trade-off in Figures 6-8. There is only a single point in each figure for FLDA, as it does not incorporate any faimesss correction. Both Fair-FDA and Fair-FLDA, demonstrate comparable trade-offs between classification error and unfairness, since they are both derived from the Bayes optimal fair classifier.
We evaluate the methods under the Gaussian model with , = 2, while keeping other model parameters consistent with those in Appendix A.1. The patterns of disparity control are similar to those observed under 3 1.5. As illustrated in Figures 9-11, the classification errors are generally higher compared the case of / 1.5, due to the lower signal-to-noie ratio. Nonetheless, the excess risk decreases more rapidly with increasing n for larger values B, aligning well with our theoretical results.
Results under 3
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n 2000; bottom: n = 5000.
2000; bottom: n = 5000.
A.3 Simulation results under non-Gaussian models
Although the proposed fairness-awar classifier is established based on the explicit form of the Radon Radon-Nikodyne derivative under Gaussian assumptions, it remains applicable in more general sce- narios. However, when the Gaussian assumption violated, the proposed classifier may no longer be Bayes optimal. To assess its performance beyond the Gaussian setting, we generate non-Gaussian stochastic processes by sampling Ça.k: 1/2 rameters identical those in Section A.
X Unif(- v3), while keeping all other model pa-
The results are presented in Figures 12-14. Despite the lack of Bayes optimality guarantees this setting, the Fair-FLDA and Fai-FLDD, continue exhibit effective disparity control and satisfac- tory classification accuracy. This robustness highlights the practical utility of our approach in more general, non-Gaune non-Gaussian scenarios.
A.4 Results under perfect classification
Functional data exhibit unique property known as perfect classification, where the classification error can vanish, phenomenon that does not typically arise in multivariate data. To evaluate the
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middle: n 2000; right: n 5000.
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sian model, 5 2. Top: n 1000; middle: sian model. 3 2. Top: n 1000; middle:
n 2000; bottom: n = 5000
n 2000; bottom: n 5000.
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dle: n 2000; bottom: 5000.
performance of our algorithm in such scenarios, we consider the Gaussian model in Appendix A.1 with 3 = 0.5, under which the signal-to-noise ratio is sufficiently high to mimic the perfect classifi- cation regime. For class probabilities, we examine two settings: (I) P(Y = 1|A = 0) 0.4, P(Y
1 = 1)= 0.7; and (II) P(Y 1/A 0) PPY 1|A 1) 0.5, while keeping all other model parameters in Appendix A. unchanged.
As discussed in Remark 2, under setting (I), the classical unconstrained Bayes classifier is automat- ically fair with respect to DO and PD. In setting (II), it is automatically fair with respect to all the three disparity measures DO, PD and DD. To visualise this difference, we plot DD function of T in Figure 15.
The results are presented in Figures 16-19, Under the disparity measures DO and PD in setting (I), the classification errors of the fairness-aware classifiers are nearly zero, and the median disparity levels converge to zero across all values of 5 as increases. This confirms that our approach naturally reduces to the classical FL.DA classifier in such automatically fair cases. In contrast, under DD in setting (I), the fact that DD(r) = 0.3 indicates that it is infeasible to achieve lower disparity levels
20

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in this setting. By comparison, in setting (II), the unconstrained Bayes classifier is automatically fair under DD, and the empirical results exhibit similar pattern to those observed for DO and PD in setting (I).
Overall, in perfect classification cases, our proposed algorithm continues perform comparably the oracle fairnes--wware Bayes optimal classifier, further highlighting its effectiveness and adapt- ability.
8
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1000; middle: 2000; bottom:
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5000.
A.5 Additional results for real data
In practice, we recommend tuning the calibration parameter in Fair-FLDA, to achieve more re- liable probabilistic disparity control. Specifically, we select the smallest value of such that the empirical P quantile of the disparity remains below the pre-specified threshold 5. To estimate this empirical quantile, we resort to random splitting. The data are randomly divided into two sub- sets, with one used to estimate the fairness-aware classifier, and the other to evaluate the resulting quantile
disparity. We repeat the process multiple times, e.g, 100 times, and the empirical then computed from the empirical distribution of observed disparities.
The results obtained using the tuned calibration levels reported in Figure 20. As shown, the tuned Fair-F LDA. consistently maintains disparity below with probability at least 1 except for slight violation under one small ò under DO. This demonstrates the overall effectiveness of the proposed tuning strategy.
A.6 Numerical comparisons with multivariate baselines
To the best of our knowledge, our work is the first to derive the Bayes optimal fair classifier and establish explicit convergence rates in the context of functional data. There are no existing fair functional classifiers available for direct comparison.
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For comparison, we have incorporated additional baselines following a "dimension reduction stan- dard fair classification" strategy. Specifically, we first apply functional principal component analysis to extract features, and then employ fair classification methods designed for multivariate data. These include three post-processing methods FPIR (Zeng al., 2024a), PPF (Chen al., 2024) and PPOT (Xian et al., 2023), and one prepprcesssing approach FUDS (Zeng et al., 2024a), with default pa- rameters in the open source code of Zeng et al. (2024a).
Results in Tables and 2 show that our proposed fair classifier, Fair-FLDA, consistently achieves the lowest classification errors while effectively controlling disparity under the pre-specified levels. In contrast, the other four baseline methods exhibit higher classification errors. In particular, PPF shows poor disparity control when 8 0. and FUDS fails adequately control disparity the NHANES dataset. These extensive numerical results highlight the superiority and practical necessity of our method for fair functional classification.
A.7 Results on effects eigenspace estimation and data splitting
Eigenspace estimation. Eigenspace estimation plays fundamental role in FDA. We justify the performance of eigenspace estimation from both theoretical and numerical perspectives. our paper, the theoretical guarantees for eigenfunction and eigenvalue estimations have been established
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Table 1: Median classification error and DD over 500 runs under Gaussian / = 1.5, n = 1000.
0.00
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Err
UD5.50
Err
UDD,50
Err
UpD
Err
Dpp.50
Err
Dp,,5
Fair-FLDA
0.234
0.022
0.227
0.048
0.219
0.098
0.213
0.149
0.208
0.197
FPIR
0.276
0.024
0.269
0.039
0.264
0.080
0.258
0.122
0.253
0.168
PPF
0.240
0.323
0.269
0.039
0.264
0.080
0.258
0.121
0.253
0.167
PPOT
0.275
0.023
0.269
0.040
0.264
0,080
0.258
0.122
0.253
0.168
FUDS
0.276
0.030
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0.040
0.263
0.087
0.256
0.142
0.251
0.195
Table 2: Median classification error and DD over 500 runs under NHANES.
6
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Er
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Er
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Fair-FLDA
0.314
0.021
0.305
0.048
0.297
0.099
0.289
0.149
0.285
0.200
FPIR
0.385
0.016
0.377
0.050
0.369
0.101
0.360
0.151
0.354
0.200
0.343
0.722
0.378
0.047
0.369
0.097
0.361
0.144
0.355
0.193
PPOT
0.385
0.016
0.377
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0.369
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0.360
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FUDS
0.403
0.157
0.371
0.189
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0.351
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0.349
in Lemmas 35 and 36, respectively. These results are optimal, matching the minimax rate established in Wahl (2022) up to poly-logarithmic factors.
To support our theory, we provide further simulation results in Table 3, where Fair-FLDA refers to the proposed classifier in Section 4; Truth uses true eigenfunctions and eigenvalues; Fourier replaces estimated eigenfunctions with Fourier basis and eigenvalues with covariance projection scores. Overall, disparity control is comparable across methods, with all meeting their fairness criteria. Comparing the results of Fair-FLDA with Truth, we see that the misclassification errors of the proposed classifiers even smaller than those obtained without eigenspace estimation. This improvement is attributed to the data-adaptive nature of the estimated eigenfunctions, which captures more variance than the fixed true basis. Substituting the estimated eigenfunctions with the pre- specified Fourier basis leads to a noticeable increase in misclassification errors.
Data splitting. The calibration data are primarily introduced for technical convenience bring independence among samples in our theoretical studies. In practice, our Algorithm 1 can be imple- mented by executing both steps and $2 on the whole dataset. numerical experiments in Section 4, we mimic the effect of sample splitting via a cross-fitting approach detailed in Appendix
Table Median classification error and DO over 500 runs under Gaussian 3 1.5, n = 1000.
Fair-FLDA
Truth
Fourier
Error
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Error
UD0,50
Error
UDo.50
0.00
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0.235
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0.274
0.027
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0.213
0.049
0,228
0.055
0.266
0.053
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0.207
0.099
0,223
0.101
0.261
0.101
0.15
0.203
0.150
0.220
0.140
0.256
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0.200
0.191
0.218
0.172
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0.200
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Table 4: Effects of data splitting for Fair-FLDA in the real and simulated (Gaussian with n = 1000, 3 1.5) datasets. Results are reported as the median over 500 iterations. NoSplit: the results of Fair-FLDA applied without data splitting.
Fair-FLDA
NoSplit
Error
D0050
Error
DDO.50
Simulated data
0.00
0.221
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0.213
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0.203
0.059
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0.207
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0.198
0.110
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0.203
0.150
0.195
0.164
0.20
0.200
0.191
0.193
0.208
Real data
0.00
0.291
0.034
0.284
0.039
0.05
0.286
0.054
0.281
0.056
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0.284
0.099
0.278
0.101
0.15
0.282
0.148
0.277
0.152
0.20
0.281
0.196
0.276
0.205
A. where two classifiers are trained by alternating the roles of data used for model estimation and threshold calibration, and then averaged. To further illustrate the effect of sample splitting, we in- clude additional numerical results on both simulated and real datasets Table below. Although the reduction in sample size from data splitting slightly increases the misclassification error, the un- fairness measure under NoSplit is usually higher than the threshold due to the dependence of the data used in training and calibration.
B Extensions of our method
Our framework in Section 2.2 can be naturally extended to settings where the sensitive feature unavailable at testing, as well to multi-class classification problems.
Extension missing sensitive attrihutes. The extension missing sensitive attributes during testing stage consists of three steps.
Step 1. A key ingredient in our framework, when sensitive features available, is that both the misclassification error and disparity measure D are linear in classifiers f X' A - [0, 1]. To extend the framework to settings where sensitive attributes to show that R and remain linear to classifiers f X > [o, 1]. which is solely defined on Following similar idea the proof of Proposition it can be verified that DO, PD and DD still linear and of the form D(J)= = fx weight function.
unavailable at testing, it is necessary
(x) where wX.D X + is
Step 2. By the generalised Neyman-Peante proof of Theorem 2, we can derive an explicit formula for 8-fair Bayes optimal classifier of the form d' dfxxx=1 (x) To 2 7 (.,,
lemma and a similar argument to the one used in the
f (x) 1 |
xPx
Step 3. Construct a plug-in classifier. Further to the estimation in Algorithm 1, an additional non- parametric estimator can be used to estimate the probability of A given X and
When sensitive information not available in training time, the fair classification is challenging problem because, without direct access sensitive information, it is difficult leam and correct for

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the potential biases. Relatively few works have studied this issue. Existing approaches (Lahoti et al., 2020: Zhao et al., 2022; Veldanda et al., 2024) generally rely on the assumption that standard features X are sufficiently informative, allowing bias mitigation through indirect inference for protected attributes. In Kallus et (2022), is further shown that various disparities remain unidentifiable when X lacks sufficient information.
In our work, focus developing principled framework for achieving fairness in functional data classification when sensitive features are available, either during both training and testing or at least in the training stage. Extending our framework to settings where sensitive attributes are completely unavailable is an important direction for future research.
Extension to multi-class classification. Consider predictive disparity an example. For a multi- class sensitive attribute E AA = [l.....: [A]] motivated by proof of Theorem our paper and Theorem 4.8 in Zeng et al. (2024a), we conjecture that the generalised Neyman-Pearson lemma leads to f'p (z,, a) 1 P.. (x) 2 oo.0+Ta where the threshold T is selected way similar to Equation (3).
=a.1
C Proofs for Bayes optimal fairness-aware classifier
C.1 Proof of Proposition
Proof of Proposition 1.
For DO, we are to show that spo,a = 2a - and bpo,a = 0. It can be seen from the following that
DO(S) = -11
= -,,
= =/ f(z, 1)
dP1.1 (x) dL1.(x)) - / fzz 0)
dPo.1
(x) dF).o(z).
dPi,o
dPo.0
For PD, to show that sPD,a 0 and bPD,a = 2a 1. It can be seen from the following that
PD() = - -(()))))))))))------00)))) = 1/A - 0,Y = 0}
= f(z,1) d - / f(x, 0) d.o.o(z).
For DD, are to show that SDD.a (2a a / Ta and bDD (2a 1)TT, Ta. It can be seen from the following that
DD(f) = P() = 1/A (X,)
0}
/ f(x T1,1 dl
(x) dP + / [(x, 1) TL.0 (
Ti dP1,0
-1 f(x, 0)
T0.1 dPo.
To,0 dPo.( (x) dPo.o(a) - -f f(x,
To dPo.o
no
- f(0,1)
T1,1 dP
T1a0
(x) +
dP(u(z))
T dP1.0
-1 f(z,0)
To.1 dPu,
To.0
(x) +
d.o.o(r).
To
dpo,0
to
25

---

C.2 Proof of Theorem 2
Proof of Theorem 2. If [((]] I 6, the unconstrained Bayes optimal classifier satisfies the fairness
constraint. Therefore, we have DDS = 0 and the 8-fair Bayes optimal classifier is given by fD,
9D.0-
If D(0)= > 6, by Proposition 7, we have D(Tb,s) = 5. Moreover, Th,s > 0. By Lemma 10,
9D.rb.s arg R(f) D(f) Tb,DD T
arg min JEF
R(f) [P(])
JEF
Analogously, we can establish the claim when D(0) < 6. This completes the proof.
C.3 Proof of Proposition
Proof of Proposition 4. Let
(Sa. - 0a.0,) 0a.0.s)
Aa (X Mo,0s (a,0) K.
Aaj
Then, by standard properties of Gaussina processes, we have that
(0a.1.) - la.0.)3²
A1((= - 03 0, j=1
Aaj
(Gn.1 on.0.)
(Ba.1 - Ba..)
a
Aa.i
Xa.j
j=1
The proposition then follows by similar argument as the one used in the proof of Theorem 2 in Berrendero et al. (2018) and the format of fbs in (4).
C.4 Auxiliary results
Proposition Recall that D(T) = D(gD,=), where gD is defined as
gD. (z, a) 1 (Ta,1 T8D.a)
dPa.1
(x) na,0 + TbD
aPa.o
Then, under the assumptions Theorem 2, the following properties hold.
(i) The disparity D(T) is continuous and non-increasing.
(ii) The misclassification R(gD.r) non-increasing on (-o0,0) and non-decreasing on (0, +oo).
Proof of Proposition 7.
ii) Note that by Definition 4,
Σ (0
(x) dPa.1 (x) + dD.a dPP.(x))
D(r)
$D.a
dP..
aE(0.1)
is a continuous random variable given y E {0, 1), we have that the function 7 -> I PX (Ta,1 TSD. ddp. Tn.0 + rbp.a) continuous for a E {0,1} and y E {0, Thus, the function 7 + D(7) is continuous.
Since
E [0,1) and Y
Define
dP.
(x) + .D > >0)
Ea,
x Ex
X: :
8D aP.o
26

---

and
dPa.1 (x) + bD.a <0.
E,.- x E X: SD.0 a
Let Ti < T2. For a E {0,1] and x E Ex,
dFa.l (a) Ja.0
Fa,1 aF
rl
a (a)+bp,a
S T2
I E Eaiti
JP
a
(D, - (D,
Fa,1 aFa.
(a) Ta.0
71
I E Ei.-i T2..
$0,0 aP
(x)+b
0,
otherwise.
We then have
D(Ti) (TT2)
dPa1 (x) + bD. dPau(x)
Σ
(0, 8D,a dla.0
eE(0.1)
x
df (x) Ta,0
Σ
1 TI
Ta,1 aP, 6
S
eE(0.1) JEE.,
8 D dPe.0 PpaiPPe. (x) + bD,a
dP
dPa;1 (x)
bn.a Jonndddaaoo dPaolv)
8p, aPa.0
1 -
-E (E(1) /DEE,
Σ
Na,1 aPa.
dP..1
(I) Ta,
< 52
$D.0 a
dP..
(x) + bD.0
dPa.1 (x) + bD.a dP
$D.a dPa.0
2 00.
Consequently, the function 7 + D(F) is non-increasing.
We first consider TI < T2 < <. If E Ea.t.
Ta,1 dfP.0
dP
(I) Ta.0
V
dP aa.l
1
dP
(x) + bDDa
I T2
Na.0 Ta,1
dPa.0 (x)
D,a TP..0
dPo.1 (x) Na.0
dPa,1
Na,1 a
2 T2 8D.0 dPa0
(x) + .D, } 1
S
2 0.
8D.0 aP..a
dPa1
(x) + bD.a
If x E Ea.-,
No a7',
dPa.1
(x) To.0
V
dP
1
$D.a dP,.0
dPa.1
(z) + bD.a
< T2 {
Tap0
Na.1 dP
(x)
dP..1 (x) - Ta.0 < 50.
dPa,1 (x) + bb,a 1 T
n. na.1 aP
1
< TT T2 8D.0 dPa.0
$D aP, (x) + bp
dP.,
Then, by Lemma 8, it holds that
R(gD.)) - Rz
27

---

=E
/
(00,n(0,0) 9D (2.0)) (00t0))99.,.2000000
dPa.1
(0) dp
dPa.(())
na.0 Ta,1
oE(0,1) o
dPa.0
d (x)
E
1
n aPo.0
Ta,0
S T2 l
aE(0,1) 2EE
DD aPa.o
dPa.1
(2) +DD,a
dPa.1
(x) dPao(x)
Ta.0 To,1 dPa.0
E
Σ
AfT1
na.1 aP.
dPs.1
(x)
Ta.00
dP..1
<
aE(0.1) /EEE,-
$D00 dP..
(x) + .DDa
dP
(x) d dPao(x)
Ta,0
na.1 dPa
20. Therefore, - R(gD,r) is non-increasing on (-00, 0).
Consider 0 T 72. If x E Ea
dP.. (x) - Ta,0
To,1 oP..
dP a.
1 T1
8 d.a.
d...
(z) + bD,a
S T2
Ta,0 - Ta,1 dPa.0
(x)
dPa.1 (z) Ta.0
dP
Ta.1 aP,.0
< T1
DD. aPa.0
(z) bD.a 1 TI
8D dP,
dP..1
(x) + bD.a
T2 S
If x E Ea,-,
dP. (I) Ta.0
No,1 A...
dP
1
$D aP..
dP_
(z) + bD,a
T2
Na.0 Ta,1
dP 0,0
(z)
dPa,1 (x) + +b,a 1 TII
Ta,1 aP..o (x) - Ta.0
dP
1
2 -
8D.0 dPa.0
<T2 2 0.
SD.0 aP.,
(x) + bD
Then, we have
-
E
())
ID (x,
dPa.
(x) (dPau(a)
Ta,0 Taall dPa.
x
(E(0,1)
=E
Σ
1 177
T na aP.
dP..1
(Ir) 7a.0
$ T3
aEf0,1) IxEEatt
DD, T...0
dPa.1
(x: + bn,a
dPa.1
Ta,0 - Ta.! dPa.
(x) dPa.(I)
Σ
1/ms R
Na.1 ar.
dP.
(I) Ta,0
<T² T2
aE(O.1) DEE..-
$D.a dP
dP..
(x) + +bD
dP 0. (x) dP dPa.ox))
Na,0 - Na.1 dP.o
28

---

0.. Therefore, T + R(gD.r) is non-decreasing on [0, +oo).
Lemma 8. For any classifier f : x A + [0, we have
RU) Σ
/ x
[(x, [( Ta.00 Ta.1 dP..0 (2)
d.a.1
P(V + P(V = 1).
aE(0,1)
Proof. By definition, we have that
R() E
a.yE
Σ [ aE(0,1)
+ = a
[L f(e,
/ f(I, f(z.
E aE(0,1}
=
+
Σ
P(A = a, a.Y=
ac aE(0,)
Σ f0
/
(( Ta,00 (aa
dPa. na,1 aPa.0 (x)
ddPao(o)++ + P(Y 1).
Lemma 9, For the bilinear disparity measures DO, PD and DD, it holds that a.. DD > O, and na,0 + TD.DD > 0,
with DD and bp defined in Definition 4.
Proof.
1. DO: In this case, it suffices to show that -To11 Too.s < M1.1. If M1, then
gDo.m,, (x, 1) = for all x E X, and,
dPo. (x) > na A = 0, = =1] so. =1}50
DO(m..) (O = -Pf (TO. + +1.1)
doo.0
I = 1, then 9Do (z, 0) = for all x E X. Then
DO(gDo,-m.)) = Pf (TL,1 + mo.1)
dPi.1
(x) 11.0 1 = 1 20.
dPi.o
Note that if DO(0)) < 5, then Tho,s = 0. By Proposition 7(), if DO(0) > 5, then 0 Tho. T1,1- If DO(0) -ō, then T0,1 < Tho, < 0. Therefore, we conclude that To,1 < <bos < M1.1.
2. PD In this case, it suffices show that - < <TiD, No,0. Note that if - -T1,0
have gPD (z, 1) for all X. Then,
d.o (z) > 0o,0 + 11. A= 0, = =0} 2 o.
PD(gPD.-sLo) = P T..1 aPo.o
29

---

If = Tu,0, we have o (z, 0) = for all x E X. Then,
dPi, (x) > N1,0 + T0,0 A A= 1 - 150.
PD(grD,so.)) P 111 dPi.0
Note that if |PD(0) 6, then TPD 0. By Proposition 7 (i), if PD(0) 6, then 0 < TDD.8 00,0- If PD( ò. then 11 TPD, O. Therefore, we conclude
that 11,0 TPD.6 m0,0-
3. DD: In this case, suffices to show that bDD, < minfTo: x1). Note that if = T2 then JDD,a (z, 1) = 0 for all x E X. Also,
DD(=T) -P (mU11 +
To
dPo.1 (1) ooo.
A - 08
NT,0
n0
T0
= =-1, T 2 n0,
5 00 T < Nu.
If = To. then 9DD (z,0) 0) for all X. Then
DD(()) = I [1,11 1,
dPi.1
(x)
T1.0
A=
1]
1 s O.
T
dPi,
Moreover, if = TTo then IDD,-no (x, 0) = for all x E X. And,
DD(m)o) P 11, n
dP1. 1.1
(x) 11,0
To*T10
A 13
dP
{ 2 0, 2 no:
1, πt πo.
If = -1,, then (DD,- (x, 1) = for x E X. Then,
DD(-TI) = 1
-PPPP.. Ppf To,1
#1 #1
dPo.1
To.0
A A 0) 0.
dPo.o
To.0
To
If (D) < 6, then TDD,8 = 0. By Proposition 7 (i), if (D > 8, then 0 < TDD < min ( To, T1}. }. If DD( < -5, then max(-nu To. -x1 < TDD 0. Therefore, we have Tool < minn m1).
Lemma 10. Recall that yo,r is defined in Theorem 2 and D(T) = D(gn,)), For any fixed R,
DD(f)
TD(T)
9D.F = arg min
R(J):
<
|
|
JEF
Moreover, for all classifiers f E arg min {R()) +D()/17| S 7D(7)/7l)) I' gD.
almost surely with respect to PX.A. In addition, i E (min(0,7b.o)). max(0, -b,))
TD(T)
gD.r = arg min
feF
Proof. If = 0, then the result follows because gn.o the unconstrained Bayes optimal classifier.
T * 0, take @o(a,, a) = Ta, /dPa//PPa va,0 and oi (x, a) sp.ad Pa../dPaoo0 + bp.e in Lemma 43.
Write
0D, a) = 1 do 6o(x, a) > |7| TØI (2,
[
30

---

Define
Acc(f) =1 -
E
/
f(x,a)
Na.1
dPa.1
(x) a0 ) ( =0),
dPa.0
aEfo,1)
D(()= E (
[(e a) (7 (1((x))))Pao(())
aEf0,1}
Let
Fr.= [EF: (,(1) +D(T)
FrHli = (EFF:IDA00S +D(T) | i Frg {EEF: ((() D.(f)
[
-D(r) ) |r|
Since |r| 2 by Lemma 43,
9D.r arg max Acc(f). IEFr,s
Moreover, since /Pa.1/dPa.002 is continuous random variable given A {0, and Y = prd.sp.dPP..ddP... (x))bbaa)
y E {0, 1}, we have Px Ama,Y-gl (T...dPa.//dP..oll O. Thus, for all f' arg max Ace(f) 9D.r almost surely with respect to PX.A-
By Lemma 43, we have @D.r E arg max Ace(J). By result (i) of Proposition 7, if 2 O, then have D(r) when [min(0, Thoo): max (0, Tb,)l. we have TD(T) 2 0. Consequently, gD.r E Fr,= S Fr.... S. Fr.s, we have
for (b, If D S then D(r) s for b 0). Therefore,
Acc(f) Acc(gD.)) max Acc(f) FEF,ss
Acc(f) I max Ace(f). JEF,s
fEF,--
JEF+.
Thus, we conclude that
[(J): ID()) TD(T)
9D.T arg Acc(f) arg min JEF....s
|r|
fEF
D Proof of Theorem 3
Proof of Theorem 3. Conditioning on the training data D, by the Dvoretzky-Kiett in- equality (Dvoretzky et al., 1956; Massart, 1990), we have, for any E {0, 1), that
P sup / "0, (2, sD.
dPa, (x) boma dPa.(11) dPa.o
dPa. (z) + bp.a} d dPao(a)) 2 S eexp(-(na A )a,)))
ØD.+ $D.a dPa.
where ID.r given in Algorithm Thus, by taking e / (lo((1/)) (og((/n)/(r (na,1 (n))//n.n A na,p) and applying union bound argument over {0, 1} and the event in Lemma 32, the theorem holds by taking
another expectation with respect to D.
E
Proofs for excess risk control
To simplify the notation, we write = tD, -"d == = this section. For a E (0, 1). denote
(So.j On 0n.0.s) - 0a.0.s) (0a.1.) 0a.05)21
Ta(X)=
log
Ta.0
1=1 Aa.j.
2da.j
Na,1 7* 7*(2 1)
(6)
31

---

and
Ta(X)= = Σ
20
Co.s 0n0.s) - -daos)
(a..s - Danid log
22a 2xas
Ta.0
An.i
Ra.1
7(2a
j=1
We further let
20 Σ
(Sout 00.0,)) 0o.0.)
(0a.1. ba.r))
Ha(X) =
Ang
2Aau
i=1
and
Gas 0,,0,) Ba,os)
(Ba.1. Ba.us)
Hx(X)= Σ
(8)
Xaj
2)as
With the above notation, we can rewrite Ta (X)) and Ta (X) as
Tx(X) (x(X) log
Ta.0
TT.1 - Tx T²(2a 1)
and
Ra,0
(X) lx(() log
Ta,1 =(2n 1)
E.1
Proof of Theorem 5
Proof of Theorem 5. Theorem 5 is general version of Corollary 6. Most of the proof follows from a similar argument to the one used in the proof of Corollary 6. We only include the difference here.
Upper bound on F T- Consider the following event, Ed = fsup,eR (Ī(T) D(F) < eD). Then, condition En happening, by the argument in the proof of Lemma 14, we have that with probability least 1 - that
ColF-r*1f Ennllr" #0}.
Thus, it holds that 1- 1- 6 (Tb 03.
Upper bound d=(fD,s fb.b). The proof follows from similar argument leading to (11) and it suffices to verify (EEE)) ) P(E n& 1 1), with &T and εr, defined in the proof of Corollary 6. To control E,, since by assumption, it holds that n.,1 - SDo 2 and to 2 c, have
0 - -FDD.a 2 Na.1 En - -T* 8p,a - |S 2 c
where the first inequality follows from the fact that 7* and 7 share the same sign, and the last inequality follows from the fact that [sD,al 1 and Gx 1. Similarly, we verify that Ta.0 + Fbp,a > c/2. To control εr, for E 1). by similar argument as the one in Lemma 12, pick eT. S tex ter, then by union bound argument, have that (PP(T, (A, n 1-1).
(11) thus leads to de (fD.6, sb,s) S (c,, + 6. + 2
Upper bound on R(fD.s) (Kffb,))) This follows directly from the fact that
[R()) - R( d= + (D)
D(NH
log(1/n)
S dd fb,8) 17"|1
71
32

---

E.2 Proof of Corollary 6
Corollary 11 (Excess risk control under disparity of opportunity). Suppose that the training and calibration data 'D U 'D are generated under Assumptions and For any 2 0, the classifier foo.5 output by Algorithm satisfies the following properties.
1. For any truncation level J E .84 in S1 in Algorithm such that
z (J) and j2a+2 log (J) log(ūi/)) S
(9)
and any arbitrarily small constants n E (0,n 1/2 A (2 (000022+1/(222--2 ). denote
no- Ja-25-4 log (ū/)) log(1/m) + Ja 23+1 log(1/n),
2 <B< a+2
En
J3 log(R//)) (og (1/())))
V Jo 20+1 log(1/1)
a+2
< $ at3
2
2
/log(i/v) (1/(1/n) J 28+1 log (g/()))).
B> a+3
1
Then, it holds with probability at least 1 1) that
log(1/n)1(TD,6 + 0}
dglfDo.s. fo, S Sc +
n
if we additionally assume that DO & (8 6n
Vlg((1 /n) 6) U (-o, -5 + 6n
log(1/1) (og()/n)/n)
2. If we further assume that ni tion level J in Algorithm as
x
N up poly-logrithmic factors and select the trunca-
N
{{ ja+1
2
3
3x + 2
2 ]
+
N
1 .132
3x + 2
then it holds that
Op N 2
2011 2u12
n+1
< 3
3re+2 2
de de(Foos:
2
=
0
N 20
9-28+1
3
3x+2 2
and
[R(Dooss R(fro,s) S dg (fDo.s. (Do,)) + 10p
Proof of Corvllary 11. For any classifier f : X x {0.1) + [0, 1]. by Proposition I, the fairness- aware excess risk under DO is defined as
Σ / (f(z,a)- - fb,((z, eE(0.1)
dB (f.
d dP. dP.o
Ta.0 + (2a (2a-1) Ta,1)
(x) dP, (x))
With the notation in (6), we have that
dPa.1
Ta.0
exp (Ta())
dPa.0
Ta.1 72a 1)
In addition, consider the following event & = (7 E
When & holds, we can then
control the fairness-wware excess risk d0o (f, ft]
(P.(
=/6 7 11,0 + (7' m1,1)
dPi. (z) d.)
dP
33

---

-Inieizo T1,0 + (7* (r"-T1,1) -1,1
dP1.1 (x) (Z)
T1(1)20
1P10
+ $ T0,0 + (- T0.1)
dPo.1 (x) dFo.(2))
Ta(x)20
dPo.o
-S To,0 + (- -T01))
dPo.1 (a) (F)
-
dPu.o
dP (a) dP dP),)(z)
+1 (f" -1,1)
dPi.
dPo.1 (z) d.).o(z))
T0,0 + ( (-r' -To,1)
dPo.
1100
1-
ele ((()
dP),(())
/ J 10000)-
+ T0,0
1 eola) To(a) dlo,o(.)
=110 Ep (1- 2 (T())
< <0]]
{1 ) To()) < <0]]
+ +Top Epo.o
(1 (1-e (1x)) ((x) sT()) < (0)1(((())))) s em)
=1,00- Ep,,
{1 - (10=))00 (2(x)) < =1000))((((0))) en }
+T1100 Ep,
- -((())
[To(x) Fo(x)) $c,o }
+ To.0 EPo.
{1 2 To(x) STo()) (
{1 {1- (1-Ebb(x)]1 1 TT() - Fo(x)) S To() 10 Fo(x)) C0O
+ To.0 Ero,
For any a E {0, 1}, write Xa ~ @P((Ha, Ka). We further denote
Er. = (Xa)) Ta (X.)l S }
(10)
Consequently, we can further upper bound the fairness-aware excess risk by
do (P. ( 11,0 . cT Er. [1 - (xx) <3]
+ TOO ET..EP. [1( eT To(X) <0]]
+ ((... +1,
S sup
fo.T. (t) + eτ
(_uup sup Sersioo
(t) + +P(())) + P(E)))
Sersrco
- -M,,I-Hi,lll V1
S 2
exp
|1111 11,llx
1 M1,11 12
1
|| - -uuaal V1
+ +
+ P(S.) +P(E))
exp
| Mo.oll
|| - -0.001
34

---

5 + P(ET, UE,,).
(11)
where for E {0, 1}, Sa,T, denotes the density for Ta given Y 0, the first inequality holds as 1 - exp(Ta(x)) I -Ta(X) I Ta(X) (x(X) I ETo and the last inequality follows from
Lemma 17 and the fact that
Ta(Xa)lY =0 N(
|| Mo.0 Ma.olik.
log
Na.1 7* TJ
Ta.0
IlMa,1 - Ha,0 Ha.oll
In the rest the proof, it suffices to control the probability for E,, Er, Ed, happening. By Lemmas 14, 17, 31 and union bound argument, it holds with probability least 1 n//4 that
{v
log(L/n) tes + log(L/n)
c
1 ~1, -
2 c
"
2
Analogously, we can also show that with probability at least
1//4 that, when 7
<
0,
{v log (1g)
1og(1/n)
.o + 7 2 TU,1 + T
ñ +C +
2 2
Morcover, taking ET EH Vlo((I/n)/n ([)DO(0) > 8 EH v Vlog(L///))
(//n/// in (10), by
Lemma 12 and an additional union bound argument, holds that P(E, A IPETT, N 2 1 Thus, conditioning on & happening, we have with probability at least 1 - 1 that
2
po (f.)))
log(1/n)
if DO(O) ó - cH
log(1/n)
eH cu +1
+
)²
2
x EH + log(1/n) 1f |O(0)) --
log(1/m) 1/n)
n
whenever (0, 7( 2(a-2043)/28-00))
E.3 Control of [7,(X) - Tx(X)
Lemma 12. Suppose the training and calibration data DD U D are generated under Assumptions and 2. Then for any {0, 1), J E N such that
JZ log (J) and J2a+2 log (J) log S n and any constant 7) E (0, 1/2), holds that
P. Fa(x) Tx()) S /log(1/)) ((DO((0) 5 cH V (10((1/m)) } }
2 1 -
where eh is defined in (12).
Proof. It follows from the triangle inequality that
[fa(X) - Tx(XI (X) Hx(X)) + (log(Ha,o) log(TT,o))
+ loo f =(2a - 1)} ( 7"(2a - 1)}}-
For X ~ GP(Ha, Ka), consider the following events,
Ex [ir. Tayl 2 log(1/n) a,y E (0.1)).
EH { [[Ha(X) Ha(X)) S en. a a (0.1)) }.
and
E, = 77 1 -- 7-| fEH log(1/n)
1 [D(o))] -cH
log (gg(1/))
n
n2
35

---

By Lemmas 13, 14, 31 and a union bound argument, we have that for any E {0, 1), Pa,o(E, n
En E,) 2 7. The rest of the proof is constructed conditioning on & En n & happening. Therefore, we have that
[T(() - (x)) [xx(Xx) (x) + aa, - Ta.ol + | - -T,11 + 17 +*1
log(1/n)
[DD(0))> 6-en -1 log(11 (1/n)
S E +
E.4
Control of [X) -Xx)))
Lemma 13. Under the same condition of Lemma 12, for any a, y E {0, 13 and small constant
" E (0, 1/2), it holds that
Pa. -
- Ha(x)) S CH 2 21-1
where
Ja-25-44 25-44 log(ū/n) log(i/ v Ja jaa-2. 23+1 log(1/1)
when
g+1 < a+2
12
2
6H
log((/n)) log(1/n) + v Jo 28+1 (og(1/1).
when
a+2 <
a+3
(12)
2
J log(h//) log(1/)) v ja 28+1 log (1/n). when
8> a+3
Proof. Note that
[xa(X) Hx(X))
(Ga.s o, (0.1,)(0u111 0u.0.s)
(5) (a.0,) Baoss
Σ
Aaj
Aa.
(Sa.j Oa (0,0,) - Oa..,)
Aa.
(0a.1.) 02,0.5)
(0a,1.3 0a,0.s2
2)a
2das
(0a.1.) 0²
2\as
In the case when X GP(Ma.o Ka), E (0, 1}, the lemma thus follows by applying a union bound argument to the results in Lemmas 19, 20 and 30, together with the fact that under Assumptions 2a and 2c, have
(0..1.j 0,,,,)
S J
2%..1
J+1
The case when X GP(Ma, K.), a E 10,1} can be justified similarly, hence is omitted here.
E.5
Control of | -*
Lemma 14. Under the same condition as Lemmna 12, for any small constant E (0, 1/2), it holds with probability at least m that
F F--TS f(n 1 [og(L//)
log (1/))
DO(O)) --6H
n
n
where eØ is explicitly defined in (12).
36

---

Proof. Denote EDo {sup/eR [DO(T) - DO( - <Do}. Taking
log(1/n)
EDO x EH +
it holds from Lemma 15 that Po)) - 1). The rest of the proof is divided into four cases.
Case 1: When DO(O)) - 5 <po. In this case, we have that T 0. Moreover, under the event Epo, [DO()] 6, then F 0. Consequently, | --1==0 0.
Case 2: When n" = and 8 EDo < <DDO(O) < In this case, we have that P(7 r" > <) P( > E)
P(f E, EDE P(Ebo)
Pff > e, (0 8,E P(Ebo)
((D((s) 6, 6,EDo) + P(E)o)
$ ((D((()) - () > 6 - DO(0) + (i,,Eo) + PPESo)
P(EDO),
where the second inequality follows as 7 > 0, the third incquality follows from Lemma 18, the fourth inequality follows as DO(e) I DO C1 by Lemma 16 and the last inequality follows by taking & > <DO/c1 C1. Analogously,
Pff < -) IP( < -s)
I P(f ED,EDO + P(EDo)
F(f -E, (0 < E + P(EDo)
< <P(D0(-6) -8, + P(EDO)
S <P(DO(-E) - ( < - (O C1E, EDo) + P(EDO)
<((EO)):
where the second inequality follows from Lemma 18, the third inequality follows from the fact that DO(-s) DO(O) + 0i by Lemma 16 and the last inequality follows by taking EDO/CI.
Case 3: When DO(7*) = å and 7* > In this case, by Proposition 7, it holds that DO(0) > 8
Hence,
P(F > T" + e) s P(f
+ EDO) P(ED))
= =P(f>>*+0 + (0) > 6, 50E + POEDO)
< <P(DO(T +5) 8, 800 + P(ED))
P(D00r7*+ +6) DO +8) DO(T*) DOOT" +8),Eno) P(EDo)
S P(eno > cIe) + P(E = P(EDo),
where the first equality follows as P > from Lemma 18, the third inequality follows from Lemma 16 and the last equality follows by taking EDo/C1. Similarly, by taking E > epo/cs /c, it also holds that
DO < 8) 0, the second inequality follows
P(F < t" -6) E)
< PP < - - Epo) + P(ESo)
< PD0((( =) 2,666 EDED) + POE)
37

---

=P(DO(T' =) DO -)) DO -) - Eo() + P(EDO)
P(CDO aE) P(EnO)
S P(EDO))
Case 4: When DO(+*) - and r- O. In this case, it holds that DO(0) -ô. The rest of the proof follows similarly to the proof for Case 3, we include them for completeness. For > epo/es.
P(f > T +
SP > T" +E, ED) + P(EDO)
SP(DO(TT +&) -à, Eo) + P(E
=P(DO(T' +2) + + 2 DO(r*) DO - -E),EDo) + P(E
S sP(E)))
Since P(f T* E, (0 6) = 0, for eno/cI,
PPF T* - <) s P(F <T* E0) P(E
P(f < <T E, DO -5, EDo) + P(E)o)
< <P(DO - =) < 8, E0))) + P(EDO)
= =P(DO(TT - 5) - DO - 8) < - --)()-O)))())))
- E))ED)) + P(EDo)
< P(ETO).
The lemma thus follows by combining results from all cases together.
E.6 Control of
Lemma 15. Under the same condition of Lemma 12, for any small constant 1) (0, 1/2), holds with probability at least 1 1i that
sup (DO(r) DO(T)) - en + log(1/n)
TER
where en is explicitly defined in (12).
Proof. Note that, by triangle inequality,
sup DO(T) DO(())
TER
sup DO(T) B(DO(T))HH + sup [(DO(r) - DO(T))
rER
rER
= (7) + (TT).
(13)
Step 1: Upper bound on (I). To control (I), by the (Dvoretzky et al., 1956; Massart, 1990), have that
inequality
P sup TER
Σ
( (X, =0,0 - -P(( 7) (X
n. D) 2 2
11,1
1=1
$ 1oxpl12n1,,12)
38

---

no.1 f + (Xo.)) m0.0} - P( 7To $0,0 D
P sup LEE
2 2
1
S 2 -2
Thus, by union bound argument, holds with probability least 1/22 that
log(1/n)
log(1/n) no,1
log(L/n)
sup
FR (O(T) B(DOG))) x
(14)
101,1
71
where the last inequality follows by further conditioning on the event in Lemma 32 and union bound argument.
Step 2: Upper bound on (II). Note that
B(DO(T))D)
( - (X) t1,o|D} -oo, ( + (X) Ro.olD)
P1,1 log (x)) > log(
$1.0 - | - Po.1 log (X)) > log( Ro.o
#o,1
E (- 11,1),
P1.1 [oo > log( --,1-r
$..0
) [D]
T S 1
-Po,1 log((7 > log(
7o.o
|D],
No.1
7 2 11,11
We denote II = (- min( 1) min(m., In the rest of the proof, to control (II), will consider various cases.
Step 2-Case 1: r E II. Note that for 7 E II,
B(DO(T)(5
= (((81117))) -T) > R1. D) - P( + (No,.)) > To.olD}
= =Pi, - T) (X) D) Po,1 ( ( + T)) T0,o|
= - (
Joao:)) dPo,(1),
with H, defined in (8). Similarly, rewrite (T(()
DO(T)= - 1 D (Tt()))-o dPr,(
(P),
Hoa(z)-lo log +T))>0
with Ha defined (7). For a E 1) and gP(Ha,11 Ka); consider the following event,
En.- = { reil
fx((X) - logg
na.1 (1 -207T)
(x)X)+ log
Ta,0
Na,1 + +(1 (1-2a)7
S SeH, a 10.
Note that
sup
(x) -log
Na.1 + (1 - 2a)T)
Na.0
Ha(X) + log
Na,1 + (1 - 2a)T
ka.0
rell
(1Aa(X) HX) + log( log(Ta,o)l
sup reil
39

---

llo TT,1 + (1 (a) log + (1 -2a)r)]
2 1 Ha(x)) + Ta,0 - To.ol + | nT,1
Thus, picking CH as the one in Equation (12), by Lemmas 13, 31 and a union bound argument, we have that Po + P1, 2 n/2.
For any 7 II, holds that
B(DO(T))D) DO -0)())
-Incco- J1 ( f /
/1(00)-oo((T,o/(t1)--))) dP),1())
 -J dx) - (
+
/ooo)-bogfto///8ot111100 (x)
4ot.::/a.o00ooo.110//4/////0aaa0 dP),1())
+ / JocHo(o)
d2)
|11,1 11,,ol v
exp {
I CH
|| M1
|| - 11
1
|| 10,0 k1 v11
+ Mo.1 M0,ol.Ko
exp
(15)
|| - No,0l22.
X C.:
(16)
where the last inequality follows from similar argument as the one leading to (11). Using a similar argument, we can also achieve same (up to constant order) upper bound for DO(r)-B(DO(r)D)) as the one in (16).
Step 2-Case 2: When T0,1 < < - Note that in this case, by standard calculation, we have that
B /otte) dP),
DO(T)= - dp),1 -
and
B(DŌ(T)) J J / (
Consequently,
E -)
sup To.1KFSf
sup -oo
dPi,(x)) -HAcco--oogtro/////----)) dPin(z))
| omm (x)
sup
(A) (A)+(B).
Note that (A) can be controlled using a similar argument as the one used in Step and we only present the upper bound on (B) in the sequel. Consider the event Ex {1TT,11 fo.1l S
40

---

/log(1/n)//) By Lemma 31, it holds that P(Ex) 2 n/2. The rest of the proof is constructed conditioning on the event & happening. Therefore, we have that
(B) sup
( Po.1
{ Ho(()) log
To.1 + T +)}
$0,0
T0.1KTS-
< Po.1 { (Ho(X) > log
T0,0
No,1 To.1
< <Po.1
<Po,,(Ho()) (Ho(X) > log (v [og((11))
= Po.1 (x) EP, (Ho())) > log (v log(1/m)
|| 140.0
1220 }
2
S exp
log² (v. log(l/n)
ni
) S (17) S EH.
(17)
| po,oll P..
where the third inequality follows from the standard property of Gaussian random variables.
The proof for the other cases follows similar argument as the one used in Step2-Case 2, hence it is omitted here. The lemma thus follows by substituting the results (14) and (16) (or (17)) into (13) and union bound argument.
E.7 Behaviour of DO around T*
Lemma 16. Recall DO given in Definition 3. Under Assumption 2. for any in small neighbour-
hood of O, there exists absolute constants Cj. C2 > O such that
C1E S (DOD) - DO + <) S o2e, C1E S DO -e) - DO S c2e.
Proof. Note that for any E R, DO(7)
=P(FL,m(( -i,
1, =}
- -P(To,1700 To,00 > -X) AA 0, Y. 1}
P1,1 (og( > log( #1.0 - Po.1 log (og()To())) > log(
mr.1+r)l )]
To.0
-u0,1 < T < T1,1
P1,1 log m (X)} > log( 11.1
11.0 }].
T -T0,1
- Fo.1 (oggtno(X) log( -
=0.0 -o.1t7 )},
2 M1111
1 R1. oll
log [T.00/ (T.o/((L,11))
I1
No.0 180
log Fo.0/ ( 1T))
2
11 Ma,all
2
| 10.0 Ago
To,1 < T < 11.1
log { (m.
||1
S
2
Ma.1 -,,,ll,,
T TS -
|o.1 pp.lll
log f (0. No.1 Ao,ollKo
7 2 -1,
By standard calculation, the derivative of DO with respect to is
1
|1,1 N1 K.
log { /1 ( 7)
DO'(T)= = ||i, - 11,0l (TI,1 - 7)
2
| - M1,0lkk,
41

---

Ma.1 Po,ollKo
log a
(To (TO, 7)}
Φ
(18)
M Ho,ol Ko (To,1 + r)
2
Mo. Mo,o Ko
where Φ is the standard normal pdf. For any in small right neighborhood of O, by Lemma 17, it holds that - + c < T² ", + < #1,1 - G, where e > 0 is small constant. Thus, by (18), we have that -02 S DO -)), (O"(T"), (T' ++)= S -1 for some universal constants Ci, 02 > O. Hence, by the mean value theorem, we have that
Cz 2 -OO(7)))---) DO e) 2 eie, 02 2 DO - e) - DO((=) 2 cre.
E.8 Auxiliary results
In this subsection, without loss of generality, assume that X GP(Na, K) for a {0, 1). The case when X ~ (P(pa,1 K) can be justified similarly.
Lemma 17. Recall the definition of T" in (3) with D chosen as DO. Under Assumptions and 2, there exists small absolute constant 0 < min (1,1) such that
No.1 + S S M.,
Proof of Lemma 17. By the definition of DO, we have that DO(T)
= =P(ML,77(X) T1. ( =}
- -P(TO.10(X) T0.0 > -7)((1 0, 13
PI,1 [lg(mn > log( 11.1
m1.0 - Po.1 log(<no > log(
50.0
)}
No.1
To,1 < < 11,1
P1, log m (X)) > log( T1.1 )}
7 T 70,1
- Foo.1 log (X)) > log( 3o.1
N0.0
2 T1,1
|| P
log { (110/(
MM - Ho.oll +
log no.0 (To, +r)
M1.1 11,001
M Mo,0
-To,1 < 11,1
|| Mioll
log N1.0 /
2
|| M1.0
T T -
|1 Ho.al Ko
log No.0 o0.0// (T0,
2
M -Moool No
T 2 T1.1
In the case when [OO(O) away from the boundary by a small constant. We will divide the following proof by conditioning on different assumptions.
< 6, the lemma holds trivially as r* 0. and it is automatically bounded
Let To denote the value such that DO((T) 7" > 0 and DO(r*)= = 8. By Proposition and the fact that DO(m).) < 0, it must be the case that 0<T' < To < TT,1- Consequently, we write To N1, where < 61 < Ti,1, and in the rest of the proof it suffices to prove that x 1. By the fact that DO(TT) = 0 and & is a strictly increasing function, holds that
Then in the case when DO(0) > it holds that
|1,1 - M1 log { (T. //.../////// (TI,) - m)) I| /o,oll Ko
log ATo.o/(TO,) + To)]
2
1 M1,ll,k
2
||40,1
M0,ol Ko
42

---

hence implies
|1,1 /1,,ll.,
log (11.0/()
| No,ollko
log { (Fo.a((TO, + 11,1 -1,1 -1) (19)
2
| - 11,0ll.
2
| - H0,0llKo
Suppose that is a function of n and 7i such that < 1. However, in this case log (T1.0/e) > log { (To.o//oo
x 1, hence (19) never holds when || -oo,o- Ko = |l 1 -1.0ll X 1. Thus, we achieve contradiction, and must be the case that Ei = 01 where Ci > is an absolute constant. Therefore, we have that 7* > 0 and TI,1 - > TI, To E1 -1. Ct.
Similarly, when (O) < -6, we have T* 0 and DO = -. With the same notation as
above, by Proposition 7. it must be the case that -T,11 < To.1 + 62 = To < < < 0. where <<< 62 < ~oo,1 Using a similar argument as above, it holds that
|1,11 /1.0ll,, log ( (TL./(TT,1 + TO,1 - =2) M Mo.o
log (TO.O/E2) (20)
2
|| ML,l
|
Mo,ol|Ko Ko
Suppose that 62 is function of and ñ such that 62 < 1. However, in this case, have that log /e)) >> log [11,/ + To.1 - Thus (20) never holds and we reach a contradiction. Hence, we conclude that 62 C2 and r* + TT,1 2 To + To,1 E2 Cz. The lemma thus follows by combining the results in the three cases above.
Lemma 18. Recall the empirical estimator for Disparity of Opportunity
(T(T)=
1
Σ
n1.1
no,1
1 (ffo,1 + (No,())) > mo.0}-
n1,1 1=1
n0,1
i=1
holds that DO is a non-increasing function. Moreover, 10 S and DO((-oo.1) 2 O.
Proof of Lemma 18. Recall that $D0 -2a and e = 0. Then {i E R Aa. + TbD,a 2
0, Re.1 TSD,a 2 0, Va E {0, 1}} = {i E R a.,1 S r S 1 The estimated disparity
function is
1 1111
no.1
DO(G)= = 11.1
(( - -) () > mo.
Σ
( + (Xaoo)) > oo.0
No.1
1 E
m1 1 (x(()11,)) >
11o0
Dio 1
Ro.0 Poo >
-1,1-
n0,3
No,1
TE 1-,, 11,1);
1
Di=l 1
n/
Ri.0 (X1,1)))) >
11.1
#1.1-7
T (-00, -
Eme 1
(oo(oo.1))
Rd.0
00,1
#0.1
T E t00,
For -oo S T1 < T2 S 11
D
- (T(2))
11
$1.0
$1.0
1
< (X111) 5
T1,11
M1,1
Ti
T1.1
T2
1=1
No.1
1
To,0
<
(o( 5
10.11
T
+ T2
To,1 + m
1=1
0.
43

---

For T1 < T2 -
DO(n)) -7(2))
1
11,0
$1.0
(X1,11) S
2 0.
1,, i=1
11,1 71
111
72
For 11,1 < < 72,
DŌ(n) - (T
Σ 110.1
0.00
(x),))) 5
To.0 T0,1
2 0.
NO.1
+ 72
71
Hence, (T is non-increasing function. Therefore, if [DO()) < 6, then = If DO(0) > 6, then F 2 0. If DO(0) < - O. Moreover, straight calculation leads to
1) SO and DO(- 2 0.
Lemma 19. Conditioning on the training data D. under the same condition as the one in Lemma 12, for any small constant 1 (0. 1/2). holds with probability least 17 that
(Ca 0..0.i)(0..1. Daos)
(Co.s 0 00.0,5) (0.0,))(0.11 0g,0,)
Σ
Aa.i
Na.
22--264 log(ū/n) log
when a+1 < <
log(n/n) /n) log(1/n)
when
a+2 < 3 S a+3
12
2
/lggt//)) log(1/v)
when
B> a+3
Proof. Note that
Cas (a.s,)(aass - Duo.s)
(Saut (0,0,) - 0o,o,)
Aas
Aaii
1=1
j=1
(Gas (0,,)) Daoos)
(So.r 00.0.)) - Canss)
Aaj
Aaaj
j=l
(Souf 00,0,) Da.0s)
(Sa.s 02.0,)) lo,0.))
+
Ao.
Aa.j
1=1
Ca. -Dau, Sans + (a.0)) - 0o.0.))
Aa.s
j=1
Da,1. - -
0a.1.1 0,,0.0
+
Σ
(Sa - -aa0a)
Aaj
j=1
(1) + (1I).
(21)
When X ~ GP(Ma.o: K), by standard properties of Gaussian process, it holds that (Sa,1..... Sa.j) ~ N(O, where A = diag(Aı,.... >a.j) and for any j. E [J] such that j * k
var(Ga.) >aaj.
var var(Ga.) Pa. Sai + 0..D.)) = - ka(s, - (o.((s)) - (t,(t)) ds dt.
44

---

and
cov(Ca. - -D,us - - + 00.0.11a. - Dau. St + 0g,0,))
/ ()(88,)((an)()) - [.,]()) x1)) )a,x(1)} ds dt.
Therefore, holds that
Gas 0a,0,1 So.t
-Sa, + B (0..0))00a11 Ga.us) - (2)
Aa,
and
J
Da.1. - Oao.
O0,1,5 - Do0is) ~ (0.00a)
Σ (Sas 0a,0s)
das
Aaj
jm1
where
j
(Ba1. - Bauis)) /
Q1 = Σ
- - (o., dt,
Xas
j=1
aa.1. 0a / (a(s, [ (Par(t) Oas(t)} ds dt
Q2
22
j=l
0a.15 - Baod Da,1K -a.0.k
+ 2
Σ
Xeid
Na.
15jckgJ
(a(s, (a(e,t)(0as(()) - (s)(s)) (a)( ds dr,
and
Q3 Σ No. 0 0a,u.
l 0 8g.0.1
da.j
Aaj
Consequently, by standard Gaussian tail properties (e.g. Proposition 2.1.2 in Vershynin, 2018), it holds with probability at least - 7/ that
((I) S 1Q11 + loglnn)) and ((I)) log(L))
(22)
Substituting (22) into (21), the lemma thus follows by applying a union bound argument to results in Lemmas 21, 23, 24 and 27.
O
Lemma 20. Under Assumptions I and 2, if we assume that X ~ GP(Po,o for a E (0, 1), then for any small constant 1 (0, 1/2). holds with probability least 1j that
(So. a2.0. (e.0.))(a,.1. - Baoos) S / JJ Ja-2881 28+1 log(1/n).
Add
j=1+1
Proof. When X GP(Ma, K), it holds that
(Sa. - 00.0,) Oani) N(
(0a,1 - 00.0,))
Σ
Aag
Aa.
J=J+1
J=J+1
Consequently, by standard Gaussian tail properties (e.g. Proposition 2.1.2 in Vershynin, 2018),
have with probability at least 77 that
(Sas Pn 0u,0,)) 02.03)) S log(1/n)
(0a,1, 0a,0,))
Σ
das
Ae.5
J=141
j=J+1
Ja Ja+28+1 log(1/n),
where the last inequality follows from Assumptions 2a and 2c. The lemma thus follows.
45

---

Lemma 21. Under the same condition of Lemma 12, for any small constant E (0, 1/2), it holds with probability at least 1) that
(a.11 Banns) ʃ (pa,o(t) Pla,((t) dt
Q1
Ras
==1
J log(ū/n) n
2+1 when
2 < S o+2
log(A/n) 7
when 8 > a+2 2
Proof. By Lemmas 30 and 22, the event in (44) and union bound argument, we have with proba- bility at least 1 - 7 that
(a.1.. Daoss)
/ (mmol()) fi,, dt
Xas
0 1f Pos0u
.11 (Ha,(()) ( o(t) dt
Σ
Xas
/=1
Da.1. - Da.o. 1
(nn)
Vdoii
Vda.
10.1. - Ta.o.s
/log(m/n)
V
(23)
Vai
To further control (23), we will consider three different cases.
Case 1: When (a + 1)/2 < 3 < (a + 2)/2. In this scenario, by union bound argument, we have
with probability at least 7) that
(a..1.) 8 02.0.)22
log(n/n
(23) S
Ras
ñ
1=1
J llo((a/1)
log"" (J) log(n/ (7/1)) 1 $}
3 2A
ñ
[log(n/n)
7i where the first inequality follows from from Lemma 30 and the last inequality follows from the fact that
inequality, the second inequality follows
2 log" 2 (J) log(ū/n) 1.
Case 2: When (a ( 2)/2 < < (3a + 2)/2. In this case, by Lemma 39, have with probability at least 1 - 112 that, for any j E [J]. 10, 0,,0.5 S z- Therefore, we have that
(23) $ (og( 8 log(ū/n)
j=1
where the last inequality follows as a/2 - 3 < 11
Case 3: When 8 > (3a + 2)/2. In this case, by Lemma 39, we have that with probability at least n/2 that, for any j [J].
log (7/n)
- 18 +
71
46

---

Therefore, we have that
log(n ñ m)
J
log (7/n)
(23) S
log( 11)
log(ū//n)
log (ng(
where the second inequality follows as a/2 - -1 and the last inequality follows as J log(n/n) n.
The lemma thus follows by combining results in three cases together.
Lemma 22. Under Assumptions and 2a, for any small constant n E (0, 1/2), it holds with proba- bility at least 1 that, for any a.y [0,1] and j E [J] such that J2n log(n//) S n,
I -en) Pau (a dt S
j log log(n/n) n/n)
Proof. Consider the following events,
& = {IIPau(tt (t)(())l
log(1/n) ñ
for a. E [0,13].
j2 plog(m//n) a (0, E [J1].
&2 {lius Doollee S
and
1 na.y
/
log(n/n) for a E {0.1).j [J]]
Eg
(a. dt S
Nayy
By Lemmas 32, 33, 35 and 40, holds from union-bound argument that (& n& (8)) The rest of the proof is constructed conditioning on the events happening. Note that
n
/ (Ha,(()) - (t) (1 dt
= =1/ (Tav(()) -0,, ( (1)] - -a,((0} dd + 1/ (Ta.(1) - (a. di
= (7) + (IT),
(24)
and in the rest of the proof, we will control (1) and (II) individually. To control (T), it holds from Cauchy-Schwarz Schwarz inequality that
(1)] V / (paw(()) Pa)(t))² dt / (0a.(t) ( (t) dt
log(1/1) j2 (2log(m/n) log((i/n) log(1/n)
(25)
7
n
n
where the last inequality follows from &, and E2. To control (II), it holds that
Dayy
/
((IZ)
(Xay(1) pa (aa di
47

---

j 1
(og(m//)
(26)
where the inequality holds from Eg. Substituting the results in (25) and (26) into (24), we have that for any j E [J].
If (Mam(()) - ((ay(()) dil x
1a log((//n)) (1g(1/n)
a log (7/1) n))
n
7
log (ū/n)
n
whenever Jotz log(1/n) S n. Thus, the lemma follows.
Lemma 23. Under the same condition of Lemma 12, for any small constant (0, 1/2), it holds with probability at least that
(On.1. Dn
/ (aul)))) (un))) (t)(t)) ds dt
Xas
ja 20+- log(F//))
when
a-1
<
< a-2
i
2
2
log(A/v)
when
2-2
< $ 0-3
2
log(is/))
when
>> a-3
Proof. Consider the following event
& = {
S 3Na,/2, for e [J], E 10.1)]
(27)
By a similar argument as the one used to control (44), we have that P(E1) 2 1 - n. The rest of the proof is constructed conditioning & Note that, by triangle inequality, we have that
(0,11 1J 02 [ [ ( - Φas(t)} ds de (0.11. 2e0 /
Xa.
(00 (t,) ds de). (28)
X2
We divide the following proof into three cases depending on the value of B. For any a > 1, by Lemma 25, we have with probability at least 17 that
/ʃʃ Ka ka(s, Se, (s) (a)( ds di x 1 f2 log(ū/n)
Case 1: When (a+1)/2 8 S (a+2)/2. In this scenario, by Lemma 39, we have with probability at least 1 - n/2 that, for any [J], 0a, 0o.0,s S j_ Therefore, have that
(28) log(n/n)
28-20+220 log(ū/n) j=1
-2444+2 Ja-28+3 log (o((ū/))
70
"
where the last incquality follows from the fact that a 23 + +220.
Case 2: When (a + 2)/2 < 3 S (a 3)/2. In this scenario, by Lemma 39, we still have with probability least 1 n/2 that, for any [J], (a,13 Danos S j- Therefore, have that
log(ū/n) 7
Σg j 28+a+22 < /log(//n)
(28) S
7
j=1
48

---

where the last inequality follows from the fact that -1 -23 +a+2+ 0.
Case 3: When B (a 3)/2. In this case, by Lemma 39, have that with probability least
1 - n/2 that, for any j E [..
Pa.1 10o + j log(n/n)
n7
Therefore, have that
(28) x 71
log(7/n) J Σ
28 (j
log(ū//) v
J
1 5
log(nz/ 17)
2342
Σ
"
71
j=1
log(ū/n) m)
J log(m))
log(ū/n)
where the third incquality follows from the fact that 23 + inequality follows from the assumption on J.
< -1 in this case and the last
The lemma thus follows by combining the results for all six cases above.
Lemma 24. Under the same condition of Lemma 12, for any small constant (0, 1/2), holds with probability at least 1 17 that
Da.1. - -Danss
0a.1.k Da,0.k
Σ
das
Xa,k
ISJCKEJ
/ (K((st)(@aast (ou(s)) (ax(t)) ds dt
Ja 221 log(i//)
when
a+1
< 8 a+2
2
J2 (og(ni/))
when
a+2
< 5 a+3 2
2A
n
2
log (1//1)
when
a+3
< B at4 2
2
when
8 a+4
Proof. Note that by the triangle inequality, have that
1a.1. ia...;
Σ
das
Ra.k
I3jCkEJ
1/ [ (,) ds dt
1o.1. Banosl
1a.1A= - Da.a.K
Aas
18jcke
//
Ka ()) - (s) (o (1)( ds dt
J-1
0a.1.1 Da.usl
- -
Ze.s
Na
k=1/=
11 ka(s, (ay( ()))))))))))))))))))) (aa((1)) ds de
(29)
49

---

The rest of the proof then follows a similar argument as the one used in the proof of Lemma 23. The proof below is constructed conditioning on the event in (27) and we will divide the proof into various cases. For any a >> 1, by Lemma 26, we have with probability at least 1 1y that
/ Ka ()
(t)( ds dt - 12 log (m/1)
(5)(())
Case 1: When 1)2 2~ 3 S (a+2)/2. In this scenario, by Lemma 39, we have with probability least 1 n that, for any j E [J]. |0,1. 00,0.5 2 j_ Therefore, we have that log(n/n) (29) 5
و 0a.1. - 0a.
J
10,1 0a,0.
Σ k=j+1
k' k' a
71
Aas
Aa,
log ( m)
log(n//n)
J-1
8+1
Σ
k
12-8+1
S
g -8-1 J2-
n
n
k=j+1
Ja-20+1 20+ log (7/(7/))
where the third inequality follows as -B++++
Case 2: When (a + 2)/2 < 3 v (a + +3//2. In this scenario, by Lemma 39, we still have with probability at least 1 - 2 that, for any j E [J]. Ba.. G,,u,l 2 5. Therefore, we have that
log (i/n))
J-1
J
J [log(m/n)
J-1
J (olog(n/n)
(29)
aa-2842
72
72
n
k=j+1
jm1
as 1 1 where the second inequality follows as Σ _kmj+1 ka/2-841 8+1 S Jja/2- B+1 and the last inequality follows as -1 S a - 2B + <.
<<0
Case 3: When (a + 3)/2 3 (a 4)/2. In this scenario, by Lemma 39, we still have with probability at least 1 - n/2 that, for any j E [J], 0a,1s - Dn,osl $ Therefore, we have that
(29)= log(n//) n
J-1 Jllo(m//n) b - 3+1 <
Jllo((/n) ñ
Σ 8-
Σ
2842
n
k j+1+1
where the second inequality follows as -1 S a/2 3 ++11 1 < 0 and the last inequality follows as 233 + 2 < -1
Case 4: When 8 > (a + 4)/2. In this case, we have that with probability at least 1 - 11/2 that, for any [J],
log (n/))
0a,13 Geoss155 8 +
Therefore, we have that
log bog(ū/n)
Plog(in/n n))
122 log((n/n) /n)
(29) <
1 15--01 8+1 V
k2 14- B+1 V
π
ñ
1=1
k=j+1
J-1 log(ū/n)
j2 log (og(
J log(ū/n)
q 9-1811 8+1
log(5/n (7/n)) m)
J-1
j22 log(ū/n) /n)
log (ū/m)
Σ
72
72
j=1
where the second inequality follows as a/2 assumption of J.
8 + 111- -1 and the third inequality follows from the
The lemma thus follows by combining results for all cases together.
50

---

Lemma 25. Under Assumptions 1 and 2a, for a E {0.1} and any small constant 17 E (0, 1/2), it
holds with probability at least -11 that, for any j E [J] such that J2 log(1/n) S n,
|
5 log n/n)
(u,
(t)(t)} ds dt x
ñ
Proof. By Lemma 46, we have that
1 Ka(s, Ka(s,t)(oasste Des(t)) (oa))))) ds
(aa)(mm)-a.hfl < // (0ns(m)- 2a. (o. ah dm / Ka(s, (())) (sa)(s) (as(t)} ds de (O)
Eass- / Ka(s, D0a:10)) - ( ds dt
/ (Ra(m Ka )u. (m)) dm de.
(30)
and for any E N+, it holds that
/ Ka(s, Ka(ss - (s)(s)} ds dt
/ [ ()(ss ds dt / (Dns(m)- - )a.( (mm))mmmmmm))))) (m) dm
+ Σ (Ns - -aar) / Ka Ka(e)))0aa ds dt
/ (Ra(m, Ka(m, 2)) ( (2) dm de
(Das - )ay) Aan J J (Ra(m,)) Kalm,
dm de,
when 1 j.
Nas S (mas(m)) Φagl (m)) Pa.sm)) dm
when / = j.
(31)
Substituting the results in (31) into (30), we have that
If
0n, (t) (s) (s) ds dt
< (ano (m) (a. (a) dm) }
Σ
Aas - - -
(m,,)) - Ka 0)) (m (m))0. dm di !
= =(I) + (II).
(32)
Consider the following events.
ε, = { Aax)) < 2(Nas >2, for for k 1 JE a (0,1)},
E3 E2= ff - Kallez S log(1/n) a (0.11)
51

---

Eg = = 1laas Daslles & / ( for j [J], a (0.1)].
and
E4 = [1/1
Ra((m, C) - Ka(m, 8) (m) (1) dm de E
o k (og((//)
71
k. E [J], a (0.1))
By Lemmas 32, 34, 35, 37 and similar argument as the one leads (51), holds from union- bound argument that P& n& n (&) 2 1 The rest of the proof is constructed conditioning on these events happening. To control (T), we have that
(1) و f
((an)(m)
Dun (m)) dm f (m) dm 12 log((i/n)
(33)
where the first inequality follows from the triangle inequality, we have that
inequality. To control (11), note that by
Pass >a.)) -axs)=(() {f ( (R)(m, Kalm,
(IT)
dm de)
Saii(has - Aa,1)
+
/ (Ra(m, Ka E)} (m)) Po. (m) Jou dm de)
=II), + (1T)..
To control (II) in separate large probability event, by similar argument as Steps 2 and 3 in the proof of Lemma 41, we have that
Ao Po -aa1) jaa log (ū/()
(11) x Σ
log(n/n)
2 52-a 22 log(m/n)
E
(34)
where the last inequality follows from Lemma 44. Similarly, to control (1I)₂ large probability event, we have that
(11)2 S
Xaiidas Ad.) 2 Ka Kalli 1l2
log(ū/n) log(1/n)
Edas
(aas >a.))-222
n72
4a log log(1/m)
j2 log((m))
(35)
where the first inequality follows from Cauchy-Schwarz inequality, the third inequality follows from Lemma 44 and the last inequality follows from the assumption that _2a+2 S n. The lemma thus follows by substituting the results in (33), (34) and (35) into (32).
O
Lemma 26. Under Assumptions and 2a, for any E {0,1) and small constant E (0, 1/2). holds with probability least 1) that, for any j. k [J] such that 1 < 22a+2 log(1/n) $ n.
S J.
| (s) oas( - (t)x ds del S
jz- log 2og((n// (7/n)
52

---

Proof. By Lemma 46, we have that
|/ (o((st)(ouss())- (to - dan((s)} ds dt
/ (ma)(m)- - (). Pas (m)) dm
I (t)ss 0a,k(s)} ds dt
Σ Aas )a1)- 1 Ka(s, (0a - (an((s) ds dt b:
1 (Ra(m, (m. dm de.
(36)
and for any / N+.
/ Ko(sst)oo (0,((s) ds dt
/ Ka(s, ()0.))) ds dt f (Zas(()) - (an(m))) dm
Σ -aax-xor)) -1 rirg
[ Kals, t) (8) ds dt
/ [aa(m, (m,
dm de, (37)
Substituting the results in (37) into (36) and using the property of eigenfunctions, we have that (.(stt)( - (Qun(()) ()) (n,x(s) ds dt
f (.((s.t) f
= -1/ (ons(m) - (aa(m)))sss(m)) (m) dm Ara, - dai)- Aa.i
I (Ra(m,)) - Kalm, (m) dm de
+ +(asss 2o,2)= ' / (da.(m) (m)( 0u, (m) dm
1 (Ka(m,)) () Kalm,
Ka(m,)))00mm.. Fo. (m)0 (f) dm de
+ Σ Aas Xa.)) Arn -aa))
/ [Ka(m.)) Kalm, e) (m))))) dm de
/ (k.(m,)) Ka(m, (m)) dm de
=(I) + (IT) ( (1II).
(38)
Consider the following events,
& = (Ras - >a,el S <22/aa. Aa,el
for & E NNDb, j E [J]. a E 1),
E2 = = -Kllz S
log(1/n) for (0.1)).
53

---

Eg = { Ilaas Dasllee & / / for j E a [0.11].
and
E4 = 11/ (Ra(m, Km Q) ) (c)
dm de
a log(n/n) ñ
k,j E [7], a 10.11].
By Lemmas 32, 34, 35, 37 and similar argument the leads (51), it holds from union- bound argument that TP&& N& 2 The rest of the proof is constructed conditioning on these events happening, and we will control the three terms in (38) above separately in three large probability events below.
Step 1: Upper bound on (1). Note that by triangle inequality, we have that
(1) / (m))(m) P (m.))a.ss (m)) dm (Ro )ao)) - ''as
/ [Ra(m, Ka(m, })} @m.(()) - Dax )ax( (a)(m))a dm de
+ +1ʃ (ma)(m) 2a. Jaa dm Ora )o.s) -aas
| (Ra(m,)) E) Kalm, f) (m))as) dm de
(1)1 + (1)2.
Note that the control of (I), is similar the argument in (35) and will be dominated by the upper bound we give on (I)2. We omit the proof here. To control (I)2, we have that
(1) S An DO, - dasl (ma)(m) dns (m) (m) dm
-1/ [a( Ka e) & (m) dm de
j a log(n/n)
Ij Dajk daol
j2-3a 30 log (m/n)
< Da. Aays
n where the second inequality follows from Cauchy- Soachh--Shwwazziiii inequality. Therefore, we have that
22-30 30 - log" (7/n)
(T) S Daa dag9
dasl-
(39)
Step 2: Upper bound on (II). The argument to control (11) is similar to the one used in Step 1. We omit the proof here and we have that
22 3a j
log2(7/) (m/n)
(IT) S Daak
(40)
n²
Step 3: Upper bound on (III). To control (III), by
inequality, we have that
(11I)
54

---

E klijkk
AaN) SailDas -daid-s 3 (m
(m) (m) dm de}
Aad Prai - )oi) ) / (Ralm, (ma) dm de)
VIITT) (ITT))-
To control (II1)1, note that (IIT)
Σ higii
(ha - xa,1)
{ (Ra(m,)) Kalm, O)} ()(asmm)) - (m)(m)))anll dm de) }²
Σ Mai(Das--aall)) / [aamm C) Kalm, E) Dos (m)) dm de}
talsk
(A) + (B).
Similarly, the upper bound we provide on (A) will be masked off by the upper bound we provide on (B), and we focus on the upper bound on (B) below. We have, from a similar argument as the one used in Steps 2 and 3 in the proof of Lemma 41, that
(B) S Σ (ha Aa,)) 2. j a-a0- log log(m/n)
j log(ū/n)
(No - An.1) 21-20 20 j² log (7/1)
Σ
n
ñ
where the last inequality follows from Lemma 44. Similarly, we can show that
2 log(ū/n))
(IIT)2 S
Therefore, have that
log(n/n) 22-0 log (og((//)
j2 22 log2 (m/n)
(1I1) S
(41)
Step Combine results together. Substituting results (39), (40) and (41) into (38) and applying a union bound argument, we have that for any 1 S < k <
j2-3a 30k a log (og(mnn)
(41) S Dha Aas L
n22
22-3 -a log (T//n)
j2-a 2-0 log
log2 (7/1))
+ Daat - dasl-
n²
j2 32-30p
log 2 (n/n))
2 2- k2 log (n/n)
5 Daa - Ae.s -1.
log (8//))
22- 1og?( (7/1))
when jck {2j ^ JJ
n²
2A
k- Ing (m/1)
k2 log (7/y)
when k> {2j ^ J}
55

---

f2-ap2- (ogg(m/n)
where the second and the last inequality follows from the fact that j < k, the third inequality follows from the fact that
[ =
if j/2 - k $ 2j. if > 2j.
Das Aa,ul 2
The lemma thus follows.
Lemma 27. Under the same condition of Lemma 12, for any small constant E (0, 1/2), it holds with probability at least 1 1) that
Anai 0n, Oa.u.
0a,1, 0a.u.""
Q3
Xa.
Aas
j=1
Ja-2043 20+3
(J))
(ogg(()))) [ig((//)
a+1 when 2 < B S 2222,
24
J /in((m/m))
when B 2
at2
Proof. To control Q3, we have that
Ba1
Baus
Ba,1 Ou.u,"
Xas
Aadt
j=1
0a.15 0a.0.
0a,1. Oa0s
0a.15-0mm 0.,.
0a,1 0 0n.
Aa.
+
Aan
j=l
Nasi (Da,1.) -a,0s O0,1.5 + 0n,0s)2 + Σ (0a.1.) - 0a.0.)2 Paad
2
Na.
j=1
aaj
1=1
Aaj (0o,1.) 0o,0.3 fo,1, 0o,0,)² + (fa.1.5 - Ga,u.)?
Reos X2
dais?
Xa
Na,i
The lemma thus follows by applying a union bound argument to results in Lemmas 28 and 29.
Lemma 28. Under the same condition of Lemma 12, for any small constant E (0, 1/2), it holds with probability at least 1 ni that
Na. (Da,1.5 - Dn.o.s - Oa,1. + 0n,0.s)2
Xa.i
2040 log (og
1+1 when
2 <
3
<
a+2
log(i/n)
when
8 >
a+2
2
Proof. Consider the following events
1
2n log (n/n)
ε = /2a.3/2 < do. S 320.3/2 and
Ras
Aa.
Na.
7
j E e
(0, 10.
Eg = Paass - fa.1, + Da.o.s - Ba,0,1
j2 28 log 2
log (//n)
72
56

---

a+1
a+2
j [ [0,1],
3
2
2
and
E3 - 1a.1s-0aa. Da.1s Dapos Dapos x
j- log(m/n) "
[J], E {0, 10,
a a+2
2
By Lemma 39, similar argument used to control (44) and union bound argument, it holds that PEE 2 1 11-1, The rest of the proof is constructed conditioning on & &nngge happening
and we have that
Aa (Da.1.5 Da,0. f + oa,0,)2 S Σ
(o.1, - s,,os - 1 + 0a,0,)) (42)
Aaj
j=1
In the rest of the proof, we will present upper bounds on (42) in several cases depending on the relationship between a and B.
Case 1: When (ax + 12 < 3 < (a + 2)/2. In this case we have that
j 28 log 2 (j) log (7/n)
log(m/)) 228+2
(42)
j=l
j=1
Ja Ja-2 3 log" (J) log(m/n)
where the last inequality follows as a 23 + +222 0. Case 2: When B > (a + 2).2. 2. In this case we have that
(ng(n/n)
log (o/n)/n)
(42)
Lemma 29. Under Assumptions 1 and 2, for any small constant (0,1/2) and J N.. such that Jea+2 log2 log(n/n) S n. it holds with probability at least vj that
Σ (0..1 (aas))aa..s
J
1
log log(ūin))
dad
Aa.
n
Proof. Consider the following event,
E1 = (Na.s/2 5 Pass S 32a.3/2 and
1
log (n/n)
Xa.
Aa.5
2
a.j
for j E [J],a {0, 13.
By a similar argument used to control (44), we have that P(E1) 2 1 - 17. The rest of the proof is constructed conditioning happening. We thus have that
1 (a,1. oa,,,)² sas Tais
2
j j-2a log(ū/n)
(a.1,
Na.5
n
log log(ā/ m))
J (0a,1.5 0o.0,))
log(ū/n)
x
Xa. 7
The lemma thus follows.
57

---

Lemma 30. Under the same condition of Lemma 12, for any small constant (0, 1/2), it holds with probability at least
ti that
On,15 0,.
(0a.1.) 0n.0.)²
Aa.j
Aais
Log (J) log(i//))
when 1 < <3 3 - 502
log(i//)
when 3 a+2
Proof. By triangle inequality, we have that
(0a.1. - 00.0.)22
Σ (Da.13 - 0a,o.s) 19
Aaj
Aas
00.1.1 0,,0,)2- (00,1.) O 0,,0,²)
das
(l..1.) - 0a.0))2
(fa., - 0a..))² Aaii
Aas
(0.1. - 0a.0. + +o.1s
o ( (a, - 0a,0s Oa1i + 0a,0,5)
(o.,1.t 00,0,t)
(fa.1.) 04,0,)) Aaj
das
(0.1. - Ousil + 1-10,11 - Oa.,l)) 10.1. - Da.os 0a,1, + 0n.0,1
Aas
j=1
(0a.1.5 - 02,0,))
(0a.1.5 - 0a.0,)²
(43)
Dad
Aa
Consider the events
El { (aas/2 S Taas 32 and
1
log (//n)
Na.i
Na.i
22
a.j
7
for j [J], E 10, 13}.
(44)
Eg = [1a1. - 0 5--
for j E
a +
< B <
+ 2
2
2
and
7- log(n/n) 3, E== (1 - - S
3a + 2
for j E [J..a (0, 10,1).3>
ñ
2
Note that by (58), it holds with probability at least
1//6 that
Xa.i
2a
log(ū/n)
5 j-2a 20 log (ū/1)
3das
S Aa.j
< Ro.j S Aa.j
2
ñ
2
where the first and the fourth inequality follow from the fact that
log(m//)
Sj Nag
58

---

By Lemma 36, we also have with probability at least 1 - n/66 that for all j E [J],
Paol
j 2xx log(n/n)
(og(n/n)
Xa.j
Aa
Aa.
0.
Aa.
77.
Additionally, by Lemma 39, have with probability least 1 7/3 that for all E [J].
0a,15- 0n0.s +
ja
29 log (og"())) log(R/n)
when a+d 2 < 3 < 242
0a,15 - 0a,0ol 2
2
0a.1. - Pa.0,l +
log(a/n)
when 3 a+2 2
when B S
3a+2
x
j V
log(A/n))
when
B
3x++2 2
where the second inequality follows from the fact that
When
a+1
2
<
a+2 2 it holds that
j2-20 20 log" (j) log(ū/n)
j² log" (j) log(ū/n)
50-
(45)
n
When a+2 2
5 32 it holds that
a log log(n/m)
22 a log T1
joat2 a log(n//n) Zj-
S Sj
7i
71
When 3 3
different term dominates depending on the value of j [J].
Therefore, by a union bound argument, we have that IP(E)) 2 n/3 for any k E (1,2,3). In addition, consider the following disjoint events,
E - 10als 0,.1. 00.0s 00,5s
j2 28 log logg(()) lo)
j E [J], {0.1). a a+
a a
< x
2
E, 1 0,.1 0a.0.s 00,0s $
j-a log(n/n) ñ
> a + 2
j E [J], E (0,
2
and by Lemma 39, we have that for each l E (4,5), PP&))) 1 11/3. In the rest of the proof, we will consider 3 different cases conditioning on various events happening based on the range of 3.
Case 1: When (a + 1)/2 < 8 < (a + 2)/2. In this case, the proof is constructed conditioning on E E & happening and by union bound argument we have that PE &g E,) 1 7). Consequently, it holds that
(43) S Σ j-8
j2 20 log 2 log(7/n)
(fa,15 - 0u.D,5) )
j=1 Xa.j
Ra.s
das
log 2 (J) log(n/n))
log(n//))
(fa. - 0a,0,5
23+1
+
Xa.j
1=1
J2 log" llgg(///))
59

---

where the last inequality follows from the fact that -1 -1 a 23
Case 2: When (a 2)/2 S (3a 2)/2. In this case, the proof is constructed conditioning
on & N& n& & happening and by union bound argument we have that IP(E, & n & 2 1 - 1/- Consequently, holds that
log(ū/n)
log(ū/n) n2
(0o,1.) - 00,0,)²
(43) S
Aaj
Xa.
==1
log ñ m n
log m//n) j 8
72
log i m)
71
where the last inequality follows from the fact that a/2 - < -1.
Case 3: When 3 > (3a + 2)/2. In this case, the proof is constructed conditioning on & & & happening and by a union bound argument we have that P(ε && &&) 2 1- Consequently, holds that
{s
j log(n/n) ñ
j- log(ū//))
log (n/n)
(43) 2 5=1
V
das
nt
j 23
log (i/1)
5 j- log(mm/))
log (7/
(V
Aa.
log(n/n)
log(ni/n) /n)
log (n/n)
f1
n
n
71
j=1
j=1
J log(n/n)
log(n//
log ( m)
log(n/n)
24
72
72
77
where the third inequality follows from the fact that a/2 B < - and the last inequality follows whenever 2 log" (J) log(n/n) S T.ehe lemma thus follows by combining results from three cases.
F Proofs for class probability estimation
In this section, we present auxiliary lemmas related to class probabilities. Results below holds for any o, y E (0, 1}. Lemma 31. Under Assumption for any small 6 > and a. E (0. 1), it holds that
P (ma.. Tayyl 2 5) S exp ( - --
Proof. Consider the sequence of bounded random variables {1(Y, U, o));-1 then it holds that
Σ = u,
Ta.y
71
Therefore, the lemma follows by applying Hoeffdings inequality for general bounded random vari- ables (e.g. Theorem 2.2.6 in Vershynin, 2018)
Note that in the lemma below, the constant 1/5 is arbitrary.
60

---

Lemma 32. With 0 < Cp. Cp < 1/5 being the absolute constants in Assumption 1, consider the following events,
& 8 { Cp (1- --))
n. nay Sū, for all a, E (0.1)).
2
and
& ( (p
(1-C))
2 ) n Ray S for all a.y E (0.1)].
2 ^
We have that P(E 0 6 2 1 for some small 7 e 10,) whenever { A nn 2 log(1/n).
Proof. For y E {0, 1] and a {0, 1}. consider the sequence of bounded random variables {1(Y=
y.A, A. م}}; By Hoeffdings inequality for general bounded random variables (e.g. Theorem 2.2.6 in Vershynin, 2018), we have that for any e1, 62 > 0,
1L, = 1, = 11 1 2 } S exp (-
E1
and
P
Σ
= -11--0---1--11----0
62 S exp (
(-
71
Therefore, we have that with probability at least 77/ that
m
Cp
Cp
11,1 Σ
1V= 2 - Volog(1/n) 2 Cyti
m
2
whenever 2 4log(1/n)) Similarly, we also have with probability at least 7//4 that
7
11,00
1(Y, = 0, 2 (1
11.1 V /lo((1/n)
1=1
(1 51-50h
(1-C)m
2 - (1-3C,,)
2
2
whenever 7 2 4 log(1 hn)((111111)// 3. The other cases can be justified similarly. The lemma thus follows from union bound argument.
G Proofs for functional data estimation
In this section, we present auxiliary lemmas related to mean, covariance, eigenvalue, eigenfunction and score estimation for the training data D. Denote n. Nao Na,1 and the group-wise mean and covariance function by
Ra.
Pa.y (t)
X 0,y (t),
Na,y
1=1
and
Ra(s,t)=
Σ 0E(0,13
no.y
(X) (s) (s){))) (t).
na.0 + Na.1 Nay
121 and (Ba.) Ja2n denote the eigenvalues and eigenfunctions of R₂ obtained by spectral expansion. The lemmas in the rest of the section holds for all E {0, 1} and y 1).
We further let Pay
61

---

G.1 Mean and covariance function
Lemma 33 (Lemma in Zapata et al., 2022), Assume Assumptions 1 and 2a hold. For any small 61 > O, it holds that
P PlIaay Payyle 2 61) S exp( ex and P(IRag - Kallaa 2 202) i exp( exp(-
where Ray 1/ 1) EA Ea (x) Pa. (s)) a -t))())))))
Proof. By standard properties of Gaussian processes and Assumption 2a, Assumption 2 in Zapata al. (2022) is automatically satisfied. Hence, the lemma follows.
Lemma 34. Assume Assumptions I and 2a hold. It holds for any small 0 < S that
IP (IIIR Kall 2 S (exp(-mae)
Proof. Note that
Ra(s, t) (s)
(Xay]()) - (aa)(s)} (aa - (ay(s)}
yEfo,1} 1=1
x (t) (t,y(t)} (P),((t) - (t))(t)] 18 (x(xx,,(() Hay (3)} (8))((aa))1)) (t,)
Ba.u
-
(Xān()) (oo (ay)) (t)
yE (0,1) i=1
B ((((aa(() Mo, { Haw (t)
Σ
May (Pa pag (8)) (Pa.) - (a,y
(46)
Tha
(E(0,1)
Therefore, triangle inequality implies that
ta(, - (s, E)1
na.
Σ
Σ
( (x)n(()) - (May(s){{ (1) (1,)
yE (0.1) 1=1
- -May ( Mayy (() (0)3))
| (Pa.o()) (oo()))))a((s))))))))))))))))))) (Pav(t) - pa,(t))² ds dt
Σ no. /
Na yE(0.1)
=
- Σ -
Pa.y
{ (s) - (a)(s) (t,
vE(0,1) 1=1
E{(x2an(s) (ap 18a (a)
Σ
na. - 14a,y
vE(0.13
62

---

= (II).
(47)
To control (I), note we have for any k E N+ that
[ [xxn(()) - (a,y (+) {
X (x) (1) (ayy(()
(an(()) (. ds dt Gagge da fl =;)),
E 1 - Ma,y (s) H{ [ (xa
Σ
yE(0,1) i=1
( (s) e ds. Therefore, by Lemma 45 and a similar argument as the one Lemma in Zapata et al. (2022) Lemma in Qiao al. (2019), we have that for some small 0 < 6 S
where for N4. Sa.y. = [(X2, ((6)-14m Mayy
a.y
P ((1) S exp(-mac?
(48)
To control (II), by Lemma 33, holds that for y E {0,1);
P (IPa. - Payyli
no
S exp(-
(49)
Na,y
Applying a union bound argument and substituting (48) and (49) into (47), we have that with prob- ability at least 1 - exp( cxp(--nc2). ?
RR, - Kall Set
Σ
+
S
The
Ra.y
na
(O
where the last inequality follows from the fact that S 1. Thus, the Lemma follows.
G.2 Eigenfunction
Lemma 35. Assume Assumptions 1 and 2a hold. For any j E N. such that j S Ja where Ja > function and any small 0 < < jJz (a+1)
have that
P(10a.3 - oa 122= 20) S exp ( "2
(-,p²)
Proof.
Note that by Lemma 46 and (5.16) in Hall and Horowitz (2007). it holds that
1l - 0a.ll12
< 2/ Σ
2a. ()(aa) Saxt)-1 - (R,ts.) (a( (s) ds dt
ki
< 2 Σ
pa. (o) - Aa,)) / (Ra(s,t)
ds dt
ki
2
Σ
Da (- daa))
(s, - (ns(s)) (oa
ds di
122
(50)
where the second inequality follows from the triangle inequality. Next, consider the event & = [ - r2 S e/j} and we have that P(E) > 1- cxp(-ñe²/ by Lemma 34, The rest of the proof is constructed conditioning on E, Construct another event
Eg (Bas -0.x)) 25 2()..5 Aa,2) < J 2(a+1)
for NYU), j [Jl]]. (51)
63

---

We want to show &g holds. By Weyl's inequality, it holds that Dās -ay--aash -a,5 I1 -Kll S sc/j for any j E N.. This then implies that
S Aa,e Aoo Aaii [7a.j Aoiol 2 Dhat A -Ejj e/j
2 Po,k - dail (1 - 2-1/2)/aa. - Aaysl 2 2 2-1//1 Aagle
(52)
where the first inequality follows from triangle inequality and the third inequality follows from Assumption 2a,
k
if < 3/2, j/2 S h S 2j. if 2j.
Mas - Aade 2
2 J (a+1)
and the fact that for 0 < 6 $ (1 - 2-11--2)j (a+1)
(a+1)
we have that
c ;
(a+1) (1-2 (1-2-113)) dagl-
Thus, from (52), we have that conditioning on ε &2 holds with probability 1. The rest of the proof constructed conditioning on both E₁ and Eg. To control (50), we can further upper bound it by
(50)
22
Σ k:
- Aart)
/
(a (t) ds dt )
Pass- Aa,x) -a,x) /Raal) - -a(x,K))) {
+ 2
Σ
k:b
(Tou(s) (a)(s) ds de }
(a / ds at d 29$
< 2 Σ
(Dau - )a,b)
(R.(s,t) -
+ 2
Σ
(Aour - sann Nonn) (Ra(s,)) - Ka(s,t)}
k:kØj
(5) (o.(()))aa ds dt } (a - Aa.t) / (Ra(s,)) - (a(s,t)) (aal: do dt 1
Σ
k=kofj
+ 2
Σ () Aaa)-1 (Ra(s,t) (s())t))
(oas(s) (a, ds dt
=2( -((I),
where the last inequality follows from & and the orthogonality of eigenfunctions. In the rest of the proof, we construct large probability events to control upper bounds on (T) and (II).
Step 1: upper bound on (7). Using the result in (46), we can further write (/)
1 S
(a Aa,k)
yc (0,0, 1=1 k
64

---

(xag(t) Mayy (5) ya
(8){((a)()) - [o,w(t))
F (x))(()
Ma, (8)) (8)) (ta (1) ds di
12
+
Σ
Σ
- -aaa)
ye(0,1) kecky
(Ta. (8) (ay(()) (Ta.s(!) - May (1) Do. (s))))) ds di /
Σ
Da (-)a( -a Sa..nsSa.uull
na (0,1)
i=1 kick(t))
1 Σ
2a. (-) - )a,t) " Mannkassoaaa
1E(0,1) keb
-(), (
where for k. M+. Saya [(xáv()) (a)(()))ann()) ds N(O,)au)) Eaw.k
0. Eia ei
So.y.k.
ne Step 1-1: upper bound on (T),. Denote {f2,g] Pimn Kanieeeee Jyen, and { f= PREN, two collections of independent standard Gaussian random variables. With the above notation, by Lemma 45, to control (I)1. it suffices to control
Σ
E
Σ ki
(-) - daxz)= v da Zin
Note that for any E [R we have that
{
Σ
- Aa,e) Vajdaa 2i.
12
= E Dris - )o,k)) Aaa 2,
k
(Nas \a,k)) 27 =1
(Ma - )o,x) -2k-o 1 2;B
Σ Exskee )a,1t) 2k -
2
j
Enire (Nas )a,k) -2 -k
E A (Dai )a.ts) - E
Sj
(Nau -
Eickes (Na..) daxe) kk-
5J 3 (1 + +-++222 8-1
(Nout -aah)) -2 - (2hzz(h))
- *(1 + f0t2)
(Nou \,, -2 E 20 B(
45 i (1+ $-20"11--166111 + gat2) = 20b1
65

---

where the first equality follows from the orthogonality of { the first inequality follows from Assumption 2a, the second inequality follows from Jensen's inequality, the third inequality follows from Lemma 44, the fourth inequality follows from Cauchy-Schwarte inequality and the fifth inequality follows from Lemma 44 and the fact that
1/22" 26 + 1
E (2
bb.
Therefore, pick 1, = j² and L2 x j. we have that
Σ E
v,, (- )a,k) v Aa 2 _=
k=
3 46))) x UINL
Hence, it holds from Lemma 45 that
P((): 2 { s 2xpp (-
iaa 2
mate
S exp
(53)
2j² + 2ÿe
whenever S.J.
Step 1-2: upper bound on (1)2. control (1)2, it holds that
(1) S
Σ
Σ
(anx()) - >a,x)) (a,n)
(E
Σ
Tayy
(Dai (a,k) (o,k)~2e0 262 Sa.y.3 "a.y.k
(54)
(0,1)
where the first inequality follows from triangle inequality and the last equality follows from the orthonormality of (0a. (keN, Also, since for any E N4 and a, E {0, 1J, we have that Sayy. (0,, Aa.x) by the independence property. Then, by standard property of independent Gaussian random variables, this implies that Ea.y.k ~ (0, Aa.x/No.y).
iid.
Moreover, by standard properties of sub-Gaussian variables and Lemma 47, we have that Extree (Aa.j - )a,k)) -262 E2 upper bound its sub-Wes SebbWll norm. Note that
Sa.g.J5a.y. follows sub- Weibull distribution with parameter 1/2. We next
Σ (Maus )a,k)-
Sa.
Σ (Daus 21o.p Niskogte
41
(haus \a.k)) 11
Σ k:
(has Aa,1))
Aa, 712
kilefj
0.9
j 202 Tay kzkfj
82
Σ (Das - Ao,k) -2
where the first incquality follows from triangle incquality, the second inequality follows from stan- dard property sub-Gaussian random variables (c.g. Lemma 2.7.7 in Vershynin, 2018), the third inequality follows from the fact that for any equality follows from Assumption 2a and the last inequality follows from Lemma 44. Therefore, by Lemma 47.1, it holds for any small 8 > 0 that
No. ||Ea,, pz S /Na.u/ aa. the fourth in-
P
Σ
(Na.j
Na.k) 2
222 Sa 2 20) S exp {
Ro.y $12 j2
Pick 8 7 (ma 2),, we have that
n To
P
Σ
Dhas )a,k))
222
Sa.a
S exp
(55)
j2
66

---

Substituting the result in (55) into (54), we have that with probability at least exp( (2))) that
Thayy
2
(1) S
se
(56)
222 y
yE (0,1)
where the last inequality follows from the fact that j 2 1 and < 1.
Step 2: upper bound on (II). To control (II), we have that
(1T)² Σ (Da.s - )a,x)=
(Das
ki
[/ [Raalo,)) -Bab6:))-Kl650)000
(a)(())aa (a ds dt]
S 2((+1])
Σ
20
[ { ( -all - pe)(s)}
ds ] fa at]
km
2(a+1))
(((l))))) day do} dt
1 [ / ( ( {s)( ds) // / (Ka(s,t) ka(s,t)}² ds dt)
1 -2/nt1)00ns Da, 1l2211111 11ia||Rn - -Kalli
2 j2 1ina - Day 112> S (1 - 2-1211/3 Paglliz,
where the first inequality follows from &g in (51), the first equality follows from Parseval's identity, the second inequality follows from Sauchy-Schward inequality, the third inequality follows from E and the last inequality follows for all such that < S jJ jo Therefore, we have that
- @agil12 exp (- E
(57)
Step 3: Combine results together. Substituting the results in (53), (56) and (57) into (50) and applying union bound argument, it holds with probability at least 1 exp( -ia 22 2/2)) that
Iba. da.ille S e + (1 1/3) - 20.1le2.
which implies that 1 daollee S and the lemma thus follows.
G.3 Eigenvalue
Lemma 36. Under Assumptions I and 2a, for any small constant 1) (0,1/2), it holds with proba- bility at least 1 that, for any j E [J] such that f2a+2 log(ū/n) /1) S Na.
log(na/n)
Buss Aent S
na
Proof. By Lemma 46 and the triangle inequality, for any j E [J], it holds that
(1 I1 oa Oassle=)) Aa.a
s // (Ra(s,t) - Ka(s, ( ds dt
67

---

+ +IRaas )aalles
|| (Ba(s,t) - (x, Das ( ds
$1 (Ra(stt)) Ka(s, (o ds dt 1ōa5 P Ra -Ka L2²
where the last inequality follows from the fact that
11 [Ra(0,
1) Ka (),a, do S Ro - Ka 12
sup JE
Therefore, by Lemmas 34, 35 and 37 and union bound argument, have that with probability least 1 37,
j-20 log(na/n)
Folog (ña/n)
j 20 log(Ta/n)
.aais Aassl S
for fll j E [J],
nla
n
Na
(58)
where the last inequality follows since 2 log(na/ /n) S Na.
G.4 Projection of difference between covariance function and its estimator
Lemma 37. Assume Assumptions and 2a hold. Then for e, k. E N+ and 0 <
S
/ma.y ((k) -a (Ta, holds that
p R/ (Ra(s,1) Ka(s, () ()) ds di 2 e) S exp ( (-
(ek))-a)
Proof. Note that
1
Ra.y
(x( Ka(s,))=
X (8)
nia
VE (0.1)
E B (x))()) - (n (+) - Ma, ma,y(1)] (113)
We have that for any k l E.
/ (Ra(s,t)- (..o))))((()))))alss))))....
(Saggt Ean,)) - Eay.k) - do,ellfe= k}
yE(0,1) 1=1
Bu,5
Na.y
No
vE(0,1}
ne
Na.y
Sanye
Say.k
(59)
The
Pa.yy
Tayy
0E(0.1)
j=1
where for & N+. S..y. ( pa,y (a,y (s) ds N(0,>a,e) Ea,,, Mayy Diny ei Sa.g.e- By standard properties of sub-Gaussian random variables (e.g. Lemma 2.7.7 in Vershynin, 2018) and Assumption 2a, we have that ddd.p.efd...elle I VAa,eda, x (2k) for any 10 and i E Mña.y-- Using the standard property of the covariance operator, it holds that
((Et,y.SSa..hk = EE (//((Ka)()) Ma. (a, (s) (t) - (t.( ds dt
68

---

- Ka(s, e(t) ds dt Aa, kj.
Hence, for E 1) holds from Bernsteins inequality (e.g. Theorem 2.8.1 in Vershynin, 2018), have that for any ō1,y 0,
P
Tla.w
E (aag fe -b)) 51.0) x exp (-((e)))
(a na.yô..v
}} (0k)-a/2)
Na
Moreover, it holds from General Hoeffding inequality (e.g. Theorem 2.6.2 in Vershynin, 2018) that for any 82,y: 63.y > O,
R
nayy 22
2 62 2 exp (-
P
Sa
Payy
8-a
1=1
and
Beyy
2 ]3 S exp (
Mo. 03.0
PP
Sa.y.
Payy
k-a
Pick
81.y =
VV
Na
82. = V/ Ma.kk-a'
na
and
63,y = eV Nayl
Tha
Payy
and by union-boume argument, we have that
R(1
/ (Rals,t) (a(s,)) ((9))) ds dt <
Σ
Naw (OL.) + 62,00au)
ME (0.1)
noc
2 exp
(Ek)-a/
Note that
Σ
(f1, + 62.00.u) Nayy
Σ
nayy {
Na (9
Da
+2 Tha
+
na,y V (ek)
na
Nay
(0,1
a.3E/0,13
$ Sett 3/2
Se,
(ak)-a
where the last inequality follows from the fact that E S V (Mayy(ek Maa hence
Nayy 21
S
(Ek)
-a
nh
and the lemma follows.
Lemma 38. Assume Assumptions I and 2a hold. For j N+ and O < <c$j-af we have that
P(ll / (Ka(s, - Ra(s, ()) (1) dell 2 20] Sexp exp (-
j-a
Proof. The proof follows from a similar and even simpler argument as the one used Step 1-1 in the proof of Lemma 35. We only include the difference in the following. With the same notation as the one used the proof of Lemma 35, it holds that
E
x N ()) - (., - ) 12
(SaneeSann (a_l(=))))]]
69

---

Aa.j (Ze - [[(= }})
t=
00
00
Aa,t
Σ
B ZitZis 1(e = =)}
2 blj (j (j- 8)0-2
Therefore, by picking L1 = ja and L2 j in Lemma 45, we have that
R(ll (Ka(s, (s, ou (t) dt 12
2 2 2
cxpp exp (- jja++ja --a+--ze
ng
The lemma thus holds by picking satisfying j- zj
G.S Projection score
Lemma 39. Under Assumptions and 2a, we have with probability at least 1 - 77 that, for any j E [J] such that J2a+2 log (J) log(ū//)) S n,
20 log (i) log(ū/n)
when 2+1l < 3 < a+2 2
0m,15 0an (0,,1--0,,0.
- 0u,Dsl) S
2
log(ū/n)
when a+2 2
Proof. Applying union bound argument to the arguments in Lemmas 40, 41 and 42, we have with probability least 1 17 that
0a,15 0..u.- (fu,1.5 - fa,u,j)
/ (1a,(1)-Ho,11
- Ma,, - (1) + (o.(1)) (1) di
+ *1/ (/mmi - pm -ma (0))0)0))00 - (1) dt
Baii
/ (8 (1a,(()) - (1) (1) di
Σ
Day i=1
yE (0.1}
(o (ta dt
Phy
vE(U,1}
+ / (1)
- (1)) (1) - Dn.(1)) di
j2 20 log log(m/n) (m/n)
12 loggoo/// (7/11) log(1/n)
{1 j= 1 & log(s)}
j log(n/n) +
[1 (1v 9+1 log(j)}
ñ
whenever log" (J) log(n/n) $ n. Thus, the result follows.
Lemma 40. Under Assumptions and 2a, for any small e > O, we have that
Tea
/ (1 - Ma.y (1) (1) (1) del 2 { I exp (
(-myacc
Na.y i=1
j-a
70

---

Proof. By the standard property of Gaussian processes, we have that for each E [Ma.g]
/ ((xzy()) - (Ka
(H)))))))))))))) ltd N(0, Aa.j).
Therefore, the lemma follows by Assumption and General Hoeffding inequality (e.g. Theorem 2.6.2 in Vershynin, 2018).
Lemma 41. Under Assumptions I and 2a, for any small constant E (0,1/2), it holds with proba- bility at least that
/ (xay(() Pay (t)) - (t,((t) dt S f= log 2 (i) log loggoo(( (i/n)
Rayy
for any j E [J], a, E (0, with 2 (og(1/1) S iz.
Proof. Consider the following events:
E [1oa Aa,'o S v22/aa. da, for {0,1), &e NMD. [e)
log(1/n)
Eg = {1RKs - Kall $
a (o.11)
n
j2 (og((i/))
E3 { |1oa. 0asll1e E V
jorjeM, e
n
na
di $ &V5 log log(m/n)
E4
(Xo, (0))))())00))mmmm.mm.
(1)
Paay
i=1
for j E [.] a, E (0,1)}.
and
Es nav x n. for any a,y (0.1)
Applying a union bound argument, it holds from Lemmas 32, 34, 35 and 40 and a similar argument as the one leads to (51) that P&& n& && N 2 1 1- 7/4, for some small 17 E (0. 1/2). The rest of the proof is constructed conditioning on & N & n & & happening.
Note that by Lemma 46, have that
(t) (Far(t)) (t) dt X (t) Ma. (8)
na.y
i=1
Raiy
(Xa,(() (t) (t.o(()))) dt (das(s) ( (s)(s)) (s) ds
Na.y
Ra.y
/ (a dt) dt)
+ Σ Aass \a,k)
Na.y
kikak
J
- (o ds de
Ba.y
(1)g(1) - M4 ( (1) di (0as(s) -0 (m)( (s) ds
Ta.y
71

---

Ra.y
+ Σ (aas )a,x)
(x),p() Mo.y (a,((0))0 de}
nayy
i=1
| (Rols,t) (a(s, - (s)(s)) ds de
Ra.
+ Σ Cas Aa,h)
(X Ho.((t)) dt}
Rayy i=1
k:
| (Ra(s,)) Oals, ds de
(T) (TT) + (ITT).
(60)
In the rest of the proof, we construct seperate large probability events which give control upper bounds on terms (I), (II) and (IIT).
Step 1: upper bound on (I). To control (I), we have that with probability at least 1 - n/4 that, for any j E [J)
/ {8 (8xm(() May((1) a (1) del Is {ons(s) )a.( (a ds
1(01 S
Nay i=1
Ravy
(x)n(t) - (a,(()) dt
Ra.y
/ ( doy/ (s)
log ( n1)
ja (og(n/n)
j2-a log (7/1)
(61)
ñ
n2
where the second inequality follows from Cauchy-Schwarz inequality and the last inequality follows from &g and E
Step 2: upper bound on (1T). To control (IT), we have that
((I)) < (1) 2 lez Kha Kallis
Σ
Das Aa,i
Nayy
Rawy
{8i (t) Ma,y (t) (t)) dt
S (0) Pay( Lez 1ka Kalles
Royy
/ (K ot. de). (62)
Σ |as Na,
Na.y
k:kej
where the first inequality follows from Schwarz inequality and the second incquality fol- lows from E,. Note that by the standard property of Gaussian process, have that
(Kay(t) (t) (t) ou iid. N (o. Ha.
Aa,k
Therefore, it holds from standard properties of sub-Gaussian norms that
Σ kickpt
Don' Aadk
(Xa. Ma, (t) delllys
Na. 1=1
72

---

na.y
1
Aa,kl
(X (aay (1)(()) dl
ma.y
k:be
i=1
da.k mayy
j log (j)
Ao.j
Aa, A
S
Dag - da.el - k0/2 S
Mayy k=
Naay
where the last inequality follows from Lemma 44. Thus, by standard properties of sub- Gaussian random variables (e.g. Proposition 2.5.2 in Vershynin, 2018), we have that for any 61 >> 0,
P
Σ k:kyy
Da.
Aa,'l
(Xo (1) (1) de 2 5
Pay 1=1
(-
oftiny
S exp
j2t. log²(1)) (1)
Pick ō1 = V2zta 1o2talog²000 log (j) log(f//) / x jzzta log (i) (og((//n)/ / we then have that with
probability least 1 - 7/4 that
Ra.v
PMu - dadel
(Xa (Xa,(() (1) 14 May 0o, di
nay
2 log (j) (j) log(finn)
(63)
71
Substituting (63) into (62), it holds from union-bound argument that with probability at least 1 - n/4 that
j log(m/m))
log(1/n)
jzta log log2(i) log (ñ/n)
(ID) $
jat log" (j) log" (7/n) /n) log(1/n)
(64)
3
for any j E [J].
Step 3: Upper bound on (111). To control (111), firstly, note that under & holds that
( Σ
Das Aa,al
(Xxx(t) Ha. -Ha. (t) (u, dt
Thyy
i=1
/ (Ra(s,)) (R((s,)))))
ds de
Rayyy
1
Maj
Sa
Ma,p - (o. dt
Na.
k
(Ra(s(())))
((0))))((()))))),))))))...... ds del
Next, control its -rccliz norm and we have that
ne
f (Xaan(t) (t, dell
(111)) 2 Edos
(Naus )o,k)
Na.y d=
ko
73

---

J
(Ra( Kals, ( (() ds dell
h-a
Σ (Nais )a, K:
Das Ma, ka
n
log(s)
π
where the first inequality follows from the triangle inequality and Lemma 2.7.7 in Vershynin (2018), the second inequality follows from Lemmas 37 and 40 and the last inequality follows from Lemma 44. Consequently, by standard properties of sub-Exxe sub-Exponential random variables (e.g. Propo-
sition 2.7.1 1 in Vershynin, 201 8), it holds for any 82 >> 0 that
P P((ann 52) I exp { 1 log(i)
82n
}
By union bound argument and by picking ō2 j²-a/2 log(i) log log(fi/n)/ū, we have that with probability least 1 - 7/4 that
1 31-3 log(z) log(ū/n) I(IIN) x
(65)
for any j E [J].
Step 4: Combine results. Substituting the results in (61), (64) and (65) into (60) and applying union bound argument, we have with probability at least 77 that
na.
18 - Ma.y (()) (0) Dn.i (1)} dt
Na.y
1=1
12 2- log (7/n)
juta jotalog² (i) lo//)
+ f² log(() log log(ū/n)
712
313
j2 log" (o)
π²2
for any j E [J] such that 2 log(1/ (2/11/) 1/1) $ n.
Lemma 42. Under Assumptions and 2a, for any small constant 17 E (0, 1/2), it holds with proba- bility at least 17 that
/ (t)) -a.a(ta) (oan(t)) (a.(()) dt
j2-23 239 log(ū/n)
Log ( n1) log(1/n)
{1 + (ott-1
72
a log(n/n)
+
{1 + -8 88 8+1 log(j)}
n
for any j E [J] with J20+2 Slog n.
Proof. The proof follows using a similar and simpler argument as the one used in the proof of Lemma 41. We only include the difference here.
By Lemma 46, we have that
f (Ha,)()) (H..((t) (a.(())))))))))) (t)} dt
74

---

= {ua.i(t) -o,o((t) de f (daa(s) (ay(()) ds
Σ (has )c,x)=1 {Ma,(() Mu,((t) dt
k:
/ (Ra(s,)) Ka(s, (a(s,))]0nu ( (o) (f) ds de - (Man10)) Ma,,a()) (()(()))))))))))))))))))))) de
f f (Das(t) (sn( (a)(s)] Day( ds
Σ (Pas Sc,x)=1 {Ma,(() (a,(tt dt
k:kg
(o((s (s)(s))0a ds de S (Ra(s,)) Ka 6)} (0)
+ Σ (aas- -a,,) f {Ma,(()) pa,u(t)) dt
k:kyJ
[ (Ra(s,)) Ka(s, 0ns (f) (f) ds de
+I)+ + (III).
(66)
Step 1: Upper bound on (1) Using a similar argument as the one used in Step in the proof of Lemma 41, we have with probability at least n/3 that
(()) $ // (ta,((t) Ot. de) / (ōar(s) ou)(5))0u ds
< /1 (ta,(t)- Ha.o(t))oasst aely -panss)-Ou((s)} )s)(s)}² dog (og((s) ds
55 8 je2 (olg((i/))
j 1 log(ū/n)
(67)
where the third inequality follows from Assumption 2c
Step 2: Upper bound on (II) To control (II), then using a similar argument as the one used in
Step 2 in the proof of Lemma 41, we have with probability at least 1 - n/3 that
((II) S (s) day (s)1lea111 Kall
Σ
Pais - Aa,i / (Hai(t) (a) dt
ki
1 log(ū/n) log(1/n)
Daj daye -a -
k:
1 log( (ï 1)) log(1/n)
(1 + f log(i))
(68)
n?
Step 3: Upper bound on (1II) To control (III), then using a similar argument as the one used in
Step 3 in the proof of Lemma 41, have with probability least 1 n/3 that
(1II)) S Σ Pas Aa / (Ho, (t)) (t) dt
ko
75

---

[ Ka(s, ((l((s)))-KK(s)))))m, ds dē
log (1/1)
{1 -821-++ log(ū)}-
(69)
72
Step 4: Combine results. Substituting the results in (67), (68) and (69) into (66) and applying union bound argument, it holds with probability at least 1y that
/
{ ( (t) (le,o(t)))aa,,.) (a, de
j2 299 Log(n/n)
j2 Plog n/ /n) (1/n)/
{1 + j log(1)}
ñ
ñ²
a log (/n))
+1+ j 1 8-1 log(i)}
7
H Technical lemmas
For completeness, provide all technical lemmas in this section.
Lemma 43 (Generalized Neyman Pearson lemma, e.g. Lemma 3.1 in Zeng et al., 2024a). Let ou, .1...., &m be m + real-valued functions defined on a Euclidean space X. Assume they are v-integrable for a o-finite measure V. Let f E F be any function of the form
Deyman-PParson
1,
do(x) > Em ( 6o(x) = Σ 1= GO)(x))
(x) = T(x),
0,
(o(x) < EI (O)(T),
where 0 < (xx) for all E X. For given constants f1, 11,..., bm E R. let Fig be the class of measurable functions f X + R satisfying
J for fó. St,, ie (L.2.....m))
(70)
and let F= be the set of functions in Fg satisfying (70) with all inequalities replaced by equalities.
(1) If E F_, then
f E arg max fET=
f dv.
Moreover, if v(tx 66(x) = Emh (o)()))) f' === almost everywhere with respect to v.
0, for all f E arg arg Jx f du,
(2) Moreover, if 2 for all I,..,,m, then
arg max / f dv. JEF x
Moreover. if v(v(xx do(x)= E. (x)) = 0. for all f E max fx f du, we have f'(x) f( almost everywhere with respect to v.
Lemma 44 (Lemma 7 in Dou et al., 2012). Under Assumption 2a, for each r 2 1, there exists constant C, > 0 depending on such that
h Doss Aaul I
[C, (((1+ ifr 1,
Livi)))
(1 12t0-+-++-ooog ir -r=1 1,
KEN
for all E {0, 1) and j N+.
76

---

Lemma 45 (Theorem 2.5 in Bosq, 2000). Let {X; be independent random variables in sep- arable Hilbert space with norm - 1 for all i e [n] and
b! 2-2 for b = 2, ...,
Σ B(IIx+1))
2 2nLLL3
with L1, L2 being two constants, then for all > 0, it holds that
P (1
nE² S 2exp (- 21 + 2L
Lemma 46 (Lemma 5.1 Hall and Horowitz, 2007). If we can write
((8,()= Σ (3 and ((,,1) = Σ (0),0,
then it holds for any j 14 that
F5 -x-- >/- / (((s))))-K(ss))o ds dt
$ 110; 2zا1ce (1īj 2/1+ (R(s,t) K(s ds del]
Moreover, if ktj 12,- - >ul > 0, it holds for E [0, that
= (,( / (0,(s) (3) ds
+ Σ (x - Ax) / / (R(s,t))
ds dt.
ki
Lemma 47 (Wong et al., 2020, Sub- Weibull properties). Let X be random variable. Then the following statements are equivalent for every a O. The constants C₁ C2 C3 > differ at most by a constant depending only
1. The tail of satisfies
2exp (t/Ci)) for all t t2 0.
2. The moments of X satisfy
| = (E(xP])) 1/p < Copp for all 2 aaa.
3. The moment generating functioo of | is finite at some point; namely
exp((x1/C2)) 22
We further call random variable X which satisfies any of the properties above a sub- Weibull random variable with parameter
77


Reviews:
[{"review_id": "xGkSCXs1Wa", "reviewer": "Reviewer_tKM6", "summary": "This paper addresses fairness-aware binary classification for functional data by introducing a post-processing algorithm called Fair Functional Linear Discriminant Analysis (Fair-FLDA). The main contributions include theoretical guarantees on fairness and excess risk control under homoscedastic Gaussian processes, and a novel analysis of eigenspace estimation effects on classification performance. The work provides both theoretical foundations with finite samples and algorithmic implementation, validated through synthetic and real data experiment.", "questions": "1. Why choose this sophisticated approach over simpler alternatives (e.g., dimension reduction + standard fair classification)?\n2. How does the method scale computationally and handle practical challenges (eigenspace estimation, noisy/missing sensitive attributes)?\n3. Can the framework extend to multiple sensitive attributes or multi-class classification?\n4. Can we compare the proposed method with standard fairness methods adapted to functional data?", "limitations": "1. Limited empirical evaluation (multiple datasets, baselines, adapted fairness methods) and lack of justification on this\n2. Justification against simpler alternatives\n3. Analysis of practical limitations and computational efficiency\n4. Broader applicability (multiple attributes, missing data scenarios)", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 2, "strengths_and_weaknesses": "**Strengths:**\n1. Novel theoretical contribution combining fairness and functional data analysis, with comprehensive guarantees under weak assumptions\n2. Clear theoretical connections to both FDA and fairness literature and strong motivation of the topic\n\n**Weaknesses:**\n\n1. Limited empirical validation (single synthetic/real dataset) and missing comparisons with adapted existing fairness methods\n2. Lack of justification against simpler alternatives (e.g., dimensionality reduction + standard fair classification)\n3. Practical limitations: In many cases, sensitive attributes may not be avilalbe during both training and testing. Also, missing discussion on multi-class sensitive attribute (e.g., race, age category, etc.)\n4. Concerns around eigenspace estimation reliability and computational efficiency", "was_revised": true, "final_justification": "I appreciate the authors for their detailed response to my concerns and questions. Few remaining questions includes 1) how such comparing methods for additional experiments selected (as some are not SOTA methods); 2) concern on sensitive information not available in training time. However, as most of my questions have been answered in rebuttal, I am leaning towards raising the final score.", "rebuttal": "We thank the reviewer for the appreciation of our theoretical contributions and the strong motivation for studying fair functional classification. Below we address the weaknesses (W), questions (Q) and limitations (L) raised by the reviewer. We respectfully hope the reviewer could consider raising the score given the additional comparisons to adapted methods, the clarified advantages of our method and extensions to scenarios with missing or multiple sensitive attributes. \n\n**1. Additional empirical evaluation (W1, Q4, L1)**\n\n**Comparisons with standard fairness methods adapted to functional data.** To the best of our knowledge, our work is the first to derive the Bayes optimal fair classifier and to establish explicit convergence rates in the context of functional data. There are no existing fair functional classifiers available for direct comparison. \n\nAs suggested, we have incorporated additional baselines following a \"dimension reduction + standard fair classification\" strategy. Specifically, we first apply functional principal component analysis to extract features, and then employ fair classification methods designed for multivariate data. These include three post-processing methods FPIR [1], PPF [2] and PPOT [3], and one pre-processing approach FUDS [1], with default parameters as in the open source code of [1].\n\nResults in Tables 1 and 2 show that our proposed fair classifier, Fair-FLDA, consistently achieves the lowest classification errors while effectively controlling disparity under the pre-specified levels. In contrast, the other four baseline methods exhibit higher classification errors. In particular, PPF shows poor disparity control when $\\delta=0$, and FUDS fails to adequately control disparity on the NHANES dataset. These extensive numerical results highlight the superiority and practical necessity of our method for fair functional classification.   \n\nTable 1: Median classification error and DD over 500 runs under Gaussian with mean difference $\\beta=1.5$ and sample size $n=1000$.\n\n|$\\delta$|0.00||0.05||0.10||0.15||0.20||\n|--|--|--|--|--|--|--|--|--|--|--|\n| |Err|$U_{DD,50}$|Err|$U_{DD,50}$|Err|$U_{DD,50}$|Err|$U_{DD,50}$|Err|$U_{DD,50}$|\n|Fair-FLDA|0.234|0.022|0.227|0.048|0.219|0.098|0.213|0.149|0.208|0.197|\n|FPIR|0.276|0.024|0.269|0.039|0.264|0.080|0.258|0.122|0.253|0.168|\n|PPF|0.240|0.323|0.269|0.039|0.264|0.080|0.258|0.121|0.253|0.167|\n|PPOT|0.275|0.023|0.269|0.040|0.264|0.080|0.258|0.122|0.253|0.168|\n|FUDS|0.276|0.030|0.269|0.040|0.263|0.087|0.256|0.142|0.251|0.195|\n\nTable 2: Median classification error and DD over 500 runs under NHANES.\n\n|$\\delta$|0.00||0.05||0.10||0.15||0.20||\n|--|--|--|--|--|--|--|--|--|--|--|\n| |Err|$U_{DD,50}$|Err|$U_{DD,50}$|Err|$U_{DD,50}$|Err|$U_{DD,50}$|Err|$U_{DD,50}$|\n|Fair-FLDA|0.314|0.021|0.305|0.048|0.297|0.099|0.289|0.149|0.285|0.200|\n|FPIR|0.385|0.016|0.377|0.050|0.369|0.101|0.360|0.151|0.354|0.200|\n|PPF|0.343|0.722|0.378|0.047|0.369|0.097|0.361|0.144|0.355|0.193|\n|PPOT|0.385|0.016|0.377|0.050|0.369|0.101|0.360|0.150|0.353|0.200|\n|FUDS|0.403|0.157|0.371|0.189|0.358|0.254|0.351|0.317|0.346|0.349|\n\n**Synthetic/real datasets.** In the original submission, we conducted extensive simulations under various model configurations to illustrate the effects of sample sizes, alignment of mean difference, model misspecification and perfect classification. These synthetic experiments provide insights into the practical implications of our theoretical convergence rates. Detailed results are deferred to the Supplementary Material due to space constraints. \n\nSince no prior work specifically targets fair functional classification, there are no widely established benchmark datasets in this context. We therefore use the NHANES dataset, a common dataset in functional data studies [4, 5], to demonstrate the practical significance of our fair functional classifier. It is important to note that our method is broadly applicable; for instance, it can be used on the Siena Scalp EEG dataset in the PhysioNet database to classify seizures while ensuring gender fairness.\n\n**2. Advantages of our method over simpler alternatives (W2, Q1, L2)** \n\nWe clarify our advantages from both theoretical and numerical aspects. Theoretically, our method fully respects the functional nature of the data by deriving the explicit form of the Bayes optimal fair classifier in Eq(4), which naturally motivates its practical approximation via truncation in Algorithm 1. We establish that the excess risk converges to zero and provide an explicit convergence rate in Theorem 5. In contrast, simpler alternatives such as \"dimensional reduction + standard fair classifiers\" lack any theoretical guarantees. Numerically, through extensive comparisons with FPIR, PPF PPOT and FUDS on both synthetic and real data, our method consistently achieves the lowest classification error while effectively controlling disparity, significantly outperforming these simpler alternatives. \n\n**3. Extensions to scenarios with missing or multiple sensitive attributes (W3, Q3, L3, L4)**\n\nOur framework in Section 2.2 can be naturally extended to settings where the sensitive feature is unavailable at testing, as well as to multi-class classification problems. We provide a sketch of methods in this response, and we will incorporate this addition in the revision.\n\n**Missing sensitive attributes.** The extension consists of three steps.  \n\n**Step 1.** A key ingredient in our framework, when sensitive features are available, is that both the misclassification error $R$ and disparity measure $D$ are linear in classifiers $f:\\mathcal{X}\\times \\mathcal{A}\\rightarrow [0,1]$. To extend the framework to settings where sensitive attributes are unavailable at testing, it is necessary to show that $R$ and $D$ remain linear to classifiers $f:\\mathcal{X} \\rightarrow [0,1]$, which is solely defined on $\\mathcal{X}$. Following a similar idea as in the proof of Prop.1, it can be verified that DO, PD and DD are still linear and of the form $D(f) = \\int_{\\mathcal{X}}f(x)w_{X, D}(x)dP_{X|Y=0}(x)$, where $w_{X,D}:\\mathcal{X} \\rightarrow \\mathbb{R}$ is a weight function.\n    \n**Step 2.** By the generalised Neyman--Pearson lemma and a similar argument to the one used in the proof of Theorem 2, we can derive an explicit formula for $\\delta$-fair Bayes optimal classifier of the form $f_{D}^\\star(x) =\\mathbf{1}[\\pi_{1} \\frac{dP_{X|Y=1}}{dP_{X|Y=0}}(x) -\\pi_{0}  \\geq \\tau^\\star w_{x, D}(x)].$ \n\n**Step 3.** Construct a plug-in classifier. Further to the estimation in Algorithm 1, an additional non-parametric estimator can be used to estimate the probability of A given X and Y.\n\n**Multi-class classification.**  Consider predictive disparity as an example. For a multi-class sensitive attribute $a \\in \\mathcal{A} = [1, \\ldots, |\\mathcal{A}|]$, motivated by the proof of Theorem 2 in our paper and Theorem 4.8 in [1], we conjecture that the generalised Neyman-Pearson lemma leads to $f_{PD}^\\star(x,a) = \\mathbf{1} [\\frac{dP_{a,1}}{dP_{a,0}}(x) \\geq  \\frac{\\pi_{a,0}+\\tau^\\star_a}{\\pi_{a,1}}]$, where the threshold $\\tau^\\star_a$ is selected in a way similar to Eq(3) in the paper.\n\n**4. Eigenspace estimation and computational efficiency (W4, Q2, L3)**\n\nEigenspace estimation plays a fundamental role in FDA. We justify the performance of eigenspace estimation from both theoretical and numerical perspectives. In our paper, the theoretical guarantees for eigenfunction and eigenvalue estimation have been established in Lemmas 35 and 36, respectively. These results are optimal, matching the minimax rate established in [6] up to poly-logarithmic factors.\n\nTo support our theory, we provide further simulation results in Table 3, where Fair-FLDA refers to the proposed classifier in Section 4; Truth uses true eigenfunctions and eigenvalues; Fourier replaces estimated eigenfunctions with Fourier basis and eigenvalues with covariance projection scores. Overall, disparity control is comparable across methods, with all meeting their fairness criteria. Comparing the results of Fair-FLDA with Truth, we see that the misclassification errors of the proposed classifiers are even smaller than those obtained without eigenspace estimation. This improvement is attributed to the data-adaptive nature of the estimated eigenfunctions, which captures more variance than the fixed true basis. Substituting the estimated eigenfunctions with the pre-specified Fourier basis leads to a noticeable increase in misclassification errors.\n\nRegarding computational complexity, the eigensystem is estimated using a discretisation-based approximation in implementation, with an overall complexity of $O(K^2J)$, where $K$ is the number of evaluated points and $J$ is the truncation level. Due to the inherent smoothness of functional data, choosing $J$ in 50–100 will provide a sufficiently accurate approximation. \n\n\n\nTable 3: Median classification error and DO over 500 runs under Gaussian with mean diff $\\beta=1.5$ and sample size $n=1000$.\n\n|$\\delta$|Fair-FLDA||Truth||Fourier||\n|--|--|--|--|--|--|--|\n| |Err|$U_{DO,50}$|Err|$U_{DO,50}$|Err|$U_{DO,50}$|\n|0.00|0.221|0.029|0.235|0.028|0.274|0.027|\n|0.05|0.213|0.049|0.228|0.055|0.266|0.053|\n|0.10|0.207|0.099|0.223|0.101|0.261|0.101|\n|0.15|0.203|0.150|0.220|0.140|0.256|0.151|\n|0.20|0.200|0.191|0.218|0.172|0.253|0.200|\n\n[1] Zeng et al., Bayes-optimal fair classification with linear disparity constraints via pre-, in-, and post-processing. arXiv:2402.02817.  \n\n[2] Chen et al., Post-hoc bias scoring is optimal for fair classification. ICLR 2024.  \n\n[3] Xian et al., Fair and optimal classification via post-processing. ICML 2023.\n\n[4] Chang and McKeague, Empirical likelihood-based inference for functional means with application to wearable device data. JRSSB, 2022.\n\n[5] Lin et al., Causal inference on distribution functions. JRSSB, 2023.\n\n[6] Wahl. Lower bounds for invariant statistical models with applications to principal component analysis. Ann. Inst. H. Poincaré Probab. Statist, 2022."}, {"review_id": "udQ7vrwfhW", "reviewer": "Reviewer_4ZfG", "summary": "This paper introduces the first framework for fairness-aware classification in the context of functional data analysis (FDA). The authors provide a theoretical characterization of the Bayes-optimal fair classifier under disparity constraints (Theorem 2), propose a practical post-processing algorithm (Fair-FLDA), and establish high-probability guarantees for both fairness satisfaction (Theorem 3) and excess classification risk (Theorem 5). This work bridges the gap between algorithmic fairness and FDA, a domain where fairness considerations have not been previously explored. The mathematical development is rigorous, and the results generalize known finite-dimensional theories to the functional setting.", "questions": "Please see weaknesses.", "limitations": "N/A", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths: \n1. The idea of fairness-aware functional classification is new and important for the FDA community, and the paper makes the first steps to address it in a principled way. \n2. The theoretical results are sound – they appear to be derived correctly and with appropriate conditions, and they generalize known results.\n\nWeaknesses:\n1. While the paper is carefully crafted and mathematically rigorous, the core methodology—such as group-wise threshold adjustment grounded in the Neyman–Pearson principle and use of Radon–Nikodym derivatives—builds upon existing techniques in fairness-aware classification and functional data analysis. The novelty primarily lies in thoughtfully adapting and unifying these tools in the functional setting, rather than proposing fundamentally new statistical models or algorithmic paradigms.\n2. Limited diversity in real-world functional datasets; generalizability to broader FDA applications is not demonstrated.", "was_revised": true, "final_justification": "The paper takes the first steps toward fairness-aware functional classification, and the theoretical results are sound. The authors have addressed the concerns through the rebuttal, and I keep the positive score.", "rebuttal": "We thank the reviewer for recognising the importance of fairness-aware functional classification and our first contribution in this area. Below, we address the weaknesses (W) raised by the reviewer.\n\n**1. Methodological and theoretical novelty (W1)**\n\nThank you for your appreciation of our work. To the best of our knowledge, this paper is the first to study fairness with functional data.  While there is a rich body of literature on fairness-aware classification for multivariate data, most existing methods do not extend naturally to functional data due to the infinite dimensionality of the function space. Rather than the posterior probabilities $\\mathbb{P}(Y=1|A=a, X=x)$ considered in most of the previous papers, our methodology fully respects the functional nature of data by constructing the classifier based on the Radon--Nikodym derivative $\\mathrm{d}P_{a,1}(x)/\\mathrm{d}P_{a,0}$, which is a more natural functional used for functional classification. Theoretically, we provide not only a fairness guarantee of the designed classifier Fair-FLDA in Theorem 3, but also an explicit convergence rate of excess risk in Theorem 5.\n\nTo further demonstrate the advantages of our approach, we have included additional numerical results in which dimension reduction using functional principal component analysis (FPCA) is first performed, followed by the application of existing post- and pre-processing fairness methods. Specifically, we first apply FPCA to extract features, and then employ fair classification methods designed for multivariate data. These include three post-processing methods FPIR [1], PPF [2] and PPOT [3], and one pre-processing approach FUDS [1]. \n\nResults in Tables 1 and 2 show that our proposed fair classifier, Fair-FLDA, consistently achieves the lowest classification errors while effectively controlling disparity under the pre-specified levels. In contrast, the other four baseline methods exhibit higher classification errors. In particular, PPF shows poor disparity control when $\\delta=0$, and FUDS fails to adequately control disparity on the NHANES dataset. These extensive numerical results highlight the superiority and practical necessity of our method for fair functional classification.   \n\n|   | $\\delta=0.00$ |  | 0.05 |  | 0.10 |  | 0.15 | | 0.20 | |\n|----|---|---|---|---|---|---|---|---|---|---|\n|            | Error   | $U_{DD,50}$ | Error   | $U_{DD,50}$ | Error   | $U_{DD,50}$ | Error   | $U_{DD,50}$ | Error   | $U_{DD,50}$ |\n| Fair-FLDA  | 0.234   | 0.022     | 0.227   | 0.048     | 0.219   | 0.098     | 0.213   | 0.149     | 0.208   | 0.197     |\n| FPIR       | 0.276   | 0.024     | 0.269   | 0.039     | 0.264   | 0.080     | 0.258   | 0.122     | 0.253   | 0.168     |\n| PPF        | 0.240   | 0.323     | 0.269   | 0.039     | 0.264   | 0.080     | 0.258   | 0.121     | 0.253   | 0.167     |\n| PPOT       | 0.275   | 0.023     | 0.269   | 0.040     | 0.264   | 0.080     | 0.258   | 0.122     | 0.253   | 0.168     |\n| FUDS       | 0.276   | 0.030     | 0.269   | 0.040     | 0.263   | 0.087     | 0.256   | 0.142     | 0.251   | 0.195     |\n\nTable 1: The median classification error and median DD measure over 500 repetitions under the Gaussian setting with mean difference $\\beta=1.5$ and sample size $n=1000$.\n\n|   | $\\delta=0.00$ |  | 0.05 |  | 0.10 |  | 0.15 | | 0.20 | |\n|---|---|---|---|---|---|---|---|---|---|---|\n|            | Error   | $U_{DD,50}$ | Error   | $U_{DD,50}$ | Error   | $U_{DD,50}$ | Error   | $U_{DD,50}$ | Error   | $U_{DD,50}$ |\n| Fair-FLDA | 0.314        | 0.021            | 0.305        | 0.048            | 0.297        | 0.099            | 0.289        | 0.149            | 0.285        | 0.200            |\n| FPIR      | 0.385        | 0.016            | 0.377        | 0.050            | 0.369        | 0.101            | 0.360        | 0.151            | 0.354        | 0.200            |\n| PPF       | 0.343        | 0.722            | 0.378        | 0.047            | 0.369        | 0.097            | 0.361        | 0.144            | 0.355        | 0.193            |\n| PPOT      | 0.385        | 0.016            | 0.377        | 0.050            | 0.369        | 0.101            | 0.360        | 0.150            | 0.353        | 0.200            |\n| FUDS      | 0.403        | 0.157            | 0.371        | 0.189            | 0.358        | 0.254            | 0.351        | 0.317            | 0.346        | 0.349            |\n\nTable 2: The median classification error and median DD measure over 500 repetitions under NHANES data.\n    \n**2. Diversity and generalizability (W2)**\n\nThank you for the comment. In our work, we use the publicly available NHANES dataset as a representative data example to demonstrate the effectiveness of our method. It is worth noting that our approach is broadly applicable to a wide range of functional data classification problems where fairness is a concern. For example, the Siena Scalp EEG dataset in the PhysioNet database contains EEG recordings from male and female subjects and can serve as a natural application of our fair functional classifier, where the response variable can be defined as the occurrence of a seizure. Overall, our method offers a principled and broadly applicable tool for fair classification in functional data analysis. We will include a relevant discussion on this point in the revised version. \n\n[1] Zeng et al., Bayes-optimal fair classification with linear disparity constraints via pre-, in-, and post-processing. arXiv:2402.02817.  \n\n[2] Chen et al., Post-hoc bias scoring is optimal for fair classification. ICLR 2024.  \n\n[3] Xian et al., Fair and optimal classification via post-processing. ICML 2023."}, {"review_id": "pNWMZ5Ad4a", "reviewer": "Reviewer_uzpK", "summary": "In this work the authors consider Bayes-Optimal classification of functional data. While the notions of fairness and Bayes-optimal classifiers does not change in any meaningful way, standard analysis of these results does not readily extend to functional data. For this the authors systematically develop the necessary tools and propose an algorithm Fair Functional Linear Discriminant Analysis classifier (Fair-FLDA) which is a post-processing algorithm to enforce fairness constraints in functional data. This requires a held-out calibration dataset of i.i.d. samples. The authors validate briefly on synthetic and real datasets.", "questions": "See questions in the weaknesses sections.", "limitations": "The empirical validation of this approach is substantially below what we should expect.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:  \n\nThe problem of fair classification is a substantial one and in the context of medical applications (neuroscience or genetics) and legal decisions (perhaps based on hand-writing recognition) unfairness or discrimination can cause significant harm. The authors do a good job of motivating the project, though I would remove the White House line given the recent 180 shift from the White House along the lines the authors highlight. \n\nThe problem is comprehensively explored and clearly enumerated. The method the authors develop to improve fairness classification also seems to be reasonable (I was unable to check all details of their proofs). \n\nWeaknesses:\n\n  \nWhile the high-level motivation is clear and the transition between each of the derivations is clear, there are several key aspects that I am unclear on. Additionally, I find the experimental validation to be below standard. I enumerate my questions about some areas that lack clarity below:   Are there any works that establish unfairness of classifiers on functional data in particular? The problem of fairness is an important one, but practically the actual problem solved by this paper seems minimal and the experimental validation is lacking.   The authors to not label their axes, provide any legend, or properly outline their experimental set up. Additionally, there are many post- and pre- processing fairness method that would clearly work in this setting and the authors make no attempt (as far as I can see) to discuss or benchmark against them. Without this the evaluation of the paper is incomplete. Why are no pre-processing e.g., reweighing considered as baselines?   In real-world settings how much does the need for calibration data affect the results? As it seems most functional datasets are of limited size, and if you have to split into test, train, validation, and calibration datasets does this not substantially reduce the utility of the proposed approach? This also may need to be explored.   Perhaps this is too harsh of a phrasing, but the communication both in the figure and description of the experimental section is significantly inadequate to understand (1) the scope of fairness issues in the context, (2) the performance of the proposed method versus baselines, (3) the robustness of the proposal to various assumptions. I am happy to hear the counter arguments by the authors on these points and change my mind if they are compelling.", "was_revised": true, "final_justification": "The authors have convincingly updated their results to address my concerns and I have increased my score in line with my clarified understanding of the work.", "rebuttal": "We thank the reviewer for appreciating the strong motivation of our work, the importance of the problem and the comprehensive exploration we conducted. Below, we address the weaknesses and limitations raised by the reviewer. We respectfully hope the reviewer considers raising the score given additional comparisons with alternative methods, and more clarifications and interpretation regarding the advantages of our approach.\n\n**Fairness in functional data classification**\n\nWe appreciate your recognition of the importance of fairness. Our data application demonstrates that unfairness can indeed arise in functional classification settings, highlighting the practical relevance of this problem.  This can be seen from Figures 4-6 in the appendix of our submission. Despite its significance, to the best of our knowledge, there is no previous work addressing unfairness in functional data classification. Our work bridges the gap by introducing a principled approach with theoretical guarantees.\n\nRegarding the experimental studies, we provide additional comparisons with multivariate methods adapted to functional data, please see the response to your following point. Moreover, we emphasise that our approach is broadly applicable to a wide range of functional data classification problems where fairness is a concern. For example, the Siena Scalp EEG dataset in the PhysioNet database contains EEG recordings from male and female subjects and can serve as a natural application of our fair functional classifier, where the response variable can be defined as the occurrence of a seizure.\n\n**Comparisons with methods adapted to functional data**\n\nThanks for the comment on the numerical evaluation. Since no existing fair classifiers can be directly applied to functional data, we adopt a ``dimension reduction + standard fair classifiers\" strategy for comparison. Specifically, we first apply functional principal component analysis (FPCA) to extract features, and then employ fair classification methods designed for multivariate data. These include three post-processing methods FPIR [1], PPF [2] and PPOT [3], and one pre-processing approach FUDS [1]. Since the open-source code provided by [1] only supports evaluation using the DD measure, we currently report results under DD with the default parameters. In the revised version, we will include the complete numerical results under the other measures PD and DO.\n\nBased on the numerical results, our proposed fair classifier, Fair-FLDA, consistently achieves the lowest classification errors while effectively controlling disparity under the pre-specified levels. In contrast, the other four baseline methods exhibit higher classification errors. In particular, PPF shows poor disparity control when $\\delta=0$, and FUDS fails to adequately control disparity on the NHANES dataset. These extensive numerical results highlight the superiority and practical necessity of our method for fair functional classification. \n\n|   | $\\delta=0.00$ |  | 0.05 |  | 0.10 |  | 0.15 | | 0.20 | |\n|----|---|---|---|---|---|---|---|---|---|---|\n|            | Error   | $U_{DD,50}$ | Error   | $U_{DD,50}$ | Error   | $U_{DD,50}$ | Error   | $U_{DD,50}$ | Error   | $U_{DD,50}$ |\n| Fair-FLDA  | 0.234   | 0.022     | 0.227   | 0.048     | 0.219   | 0.098     | 0.213   | 0.149     | 0.208   | 0.197     |\n| FPIR       | 0.276   | 0.024     | 0.269   | 0.039     | 0.264   | 0.080     | 0.258   | 0.122     | 0.253   | 0.168     |\n| PPF        | 0.240   | 0.323     | 0.269   | 0.039     | 0.264   | 0.080     | 0.258   | 0.121     | 0.253   | 0.167     |\n| PPOT       | 0.275   | 0.023     | 0.269   | 0.040     | 0.264   | 0.080     | 0.258   | 0.122     | 0.253   | 0.168     |\n| FUDS       | 0.276   | 0.030     | 0.269   | 0.040     | 0.263   | 0.087     | 0.256   | 0.142     | 0.251   | 0.195     |\n\nTable 1: The median classification error and median DD measure over 500 repetitions under the Gaussian setting with mean difference $\\beta=1.5$ and sample size $n=1000$.\n\n|   | $\\delta=0.00$ |  | 0.05 |  | 0.10 |  | 0.15 | | 0.20 | |\n|---|---|---|---|---|---|---|---|---|---|---|\n|            | Error   | $U_{DD,50}$ | Error   | $U_{DD,50}$ | Error   | $U_{DD,50}$ | Error   | $U_{DD,50}$ | Error   | $U_{DD,50}$ |\n| Fair-FLDA | 0.314        | 0.021            | 0.305        | 0.048            | 0.297        | 0.099            | 0.289        | 0.149            | 0.285        | 0.200            |\n| FPIR      | 0.385        | 0.016            | 0.377        | 0.050            | 0.369        | 0.101            | 0.360        | 0.151            | 0.354        | 0.200            |\n| PPF       | 0.343        | 0.722            | 0.378        | 0.047            | 0.369        | 0.097            | 0.361        | 0.144            | 0.355        | 0.193            |\n| PPOT      | 0.385        | 0.016            | 0.377        | 0.050            | 0.369        | 0.101            | 0.360        | 0.150            | 0.353        | 0.200            |\n| FUDS      | 0.403        | 0.157            | 0.371        | 0.189            | 0.358        | 0.254            | 0.351        | 0.317            | 0.346        | 0.349            |\n\nTable 2: The median classification error and median DD measure over 500 repetitions under NHANES data.\n\n**Presentation of numerical experiments**\n\nThank you for your suggestions on our numerical presentations. In the revised version, we will rewrite Section 4 to update figures to include axis labels and legends, and clarify the experimental setup to provide more insights into our results. \n\nDue to the restrictions of the rebuttal format, we will talk through Figure 2 in the original submission here as an example for clarification.  In Figure 2 of the original paper, we compare the performance of Fair-FLDA in terms of both misclassification error and fairness control as the disparity threshold $\\delta$ varies. Focusing on the left panel of Figure 2, the orange dots represent the FLDA classifier, which ignores fairness constraints and serves as the baseline in the comparison. The blue stars and pink triangles are the results for Fair-FLDA and Fair-FLDA$_c$, respectively. Focusing on the misclassification error plots in the first column, we observe that in the fair-impacted regime (i.e. $\\delta$ is small), the error decreases as $\\delta$ increases, and eventually approaches the baseline as the fairness constraint is relaxed. To evaluate fairness performance, the second and third columns report disparity measures under various values of $\\delta$. Specifically, points below the grey dashed line indicate that the fairness constraint is satisfied. We show that FLDA consistently fails to meet these fairness requirements, while Fair-FLDA maintains the desired median disparity level, and Fair-FLDA$_c$ ensures that the disparity remains below $\\delta$ with at least 95\\% probability. We have also included results on model misspecification in Appendix A.3 to illustrate the robustness of Fair-FLDA.\n\n**Usage of calibration data**\n\nThe calibration data are primarily introduced here for technical convenience to bring independence among samples in our theoretical studies. In practice, our Algorithm 1 can be implemented by executing both **S1** and **S2** on the whole dataset. In all numerical experiments in Section 4, we mimic the effect of sample splitting via a cross-fitting approach detailed in Appendix A.1, where two classifiers are trained by alternating the roles of data used for model estimation and threshold calibration, and then averaged. To further illustrate the effect of sample splitting, we include additional numerical results on both simulated and real datasets in Table 3 below. Although the reduction in sample size from data splitting slightly increases the misclassification error, the unfairness measure under NoSplit is usually higher than the threshold $\\delta$ due to the dependence of the data used in training and calibration. \n\n|           |     Fair-FLDA      |              |   NoSplit     |              |\n|----|---|---|---|---|\n| $\\delta$    | Error          | $U_{DO,50}$  | Error          | $U_{DO,50}$  |\n| **Simulated data:**  |               |               |               |              |\n| 0.00        | 0.221         | 0.029        | 0.211         | 0.029        |\n| 0.05        | 0.213         | 0.049        | 0.203         | 0.059        |\n| 0.10        | 0.207         | 0.099        | 0.198         | 0.110        |\n| 0.15        | 0.203         | 0.150        | 0.195         | 0.164        |\n| 0.20        | 0.200         | 0.191        | 0.193         | 0.208        |\n| **Real data:**            |        |              |      |              |\n| 0.00        | 0.291         | 0.034        | 0.284         | 0.039        |\n| 0.05        | 0.286         | 0.054        | 0.281         | 0.056        |\n| 0.10        | 0.284         | 0.099        | 0.278         | 0.101        |\n| 0.15        | 0.282         | 0.148        | 0.277         | 0.152        |\n| 0.20        | 0.281         | 0.196        | 0.276         | 0.205        |\n\nTable 3: Comparison of data splitting for Fair-FLDA on the real and simulated datasets. Results are reported as the median over 500 iterations. NoSplit: the results of Fair-FLDA applied without data splitting.\n\n[1] Zeng et al., Bayes-optimal fair classification with linear disparity constraints via pre-, in-, and post-processing. arXiv:2402.02817.  \n\n[2] Chen et al., Post-hoc bias scoring is optimal for fair classification. ICLR 2024.  \n\n[3] Xian et al., Fair and optimal classification via post-processing. ICML 2023."}, {"review_id": "DgaUqEAdLI", "reviewer": "Reviewer_V7nb", "summary": "This paper introduces a novel framework for fairness-aware functional classification, a rapidly growing area within machine learning that addresses disparities across subpopulations. The authors propose a unified approach to classify functional data while controlling the disparity level below a pre-specified threshold, overcoming challenges like the absence of density ratios and intractable posterior probabilities in infinite-dimensional functional spaces. They also develop a post-processing algorithm called Fair Functional Linear Discriminant Analysis (Fair-FLDA), designed for homoscedastic Gaussian processes, which achieves fairness through group-wise thresholding. The paper provides theoretical guarantees for both fairness and excess risk control for Fair-FLDA, including excess risk control for standard FLDA as a byproduct. Some numerical experiments on synthetic and real datasets validate the practicality of their proposed algorithm.", "questions": "How does the fairness problem specifically manifest within functional classification, and what motivated this particular area of investigation?\n\nWhat is the justification for assuming Gaussian processes for non-sensitive features in the proposed algorithm, and what literature supports this choice in the functional fairness setting?\n\nWill the numerical experiments using less common datasets have a broad impact on the fairness community, and what are the authors' plans to address this?", "limitations": "N/A", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 1, "significance": 2, "originality": 3, "strengths_and_weaknesses": "### Strengths\n\nThe theoretical analysis here is incredibly robust, spanning over 50 pages of proofs – I truly appreciate the authors' dedication to that! It's clear they've put a lot of effort into deriving numerous theoretical results, even tackling fairness guarantees. This feels like a significant theoretical step forward, as, to my knowledge, not many papers in this field offer theoretical guarantees for fairness-aware algorithms.\n\n---\n\n### Weaknesses\n\nFirst off, I have to point out that this paper leans heavily on theory (with proofs taking up pages 18 to 70 in the appendix!). Compared to all that intricate theory, the fairness background feels relatively straightforward. This makes me wonder why \"Social and economic aspects of machine learning\" was chosen as the Primary Area instead of \"learning theory,\" which seems like a much better fit for the paper's core.\n\nSetting aside the scope for \"Social and economic aspects of machine learning,\" I also found the paper's motivation a bit unclear and, at times, a little far-fetched.\n\n* For example, the logical connection between the second and third paragraphs in the introduction is weak. Paragraph 2 dives into functional classification, but then paragraph 3 jumps straight to the fairness issues of existing algorithms without explicitly linking them to \"functional classification.\" Why investigate fairness specifically within functional classification? That connection needs to be clearer.\n* Another point: the proposed post-processing algorithm assumes non-sensitive features are Gaussian processes, but there's no explanation for this choice. What's the rationale behind it? Is there existing literature that supports using Gaussian processes in this specific context?\n* Finally, the numerical experiments feel a bit limited, using datasets that aren't widely established in prior literature. I'm not entirely convinced that the setup in this paper will have a broad impact on the fairness community.\n\n**I want to emphasize that, if I've misunderstood anything, please correct me – I'm happy to adjust my rating accordingly! Thanks!**\n\n---\n\nOverall, this strikes me as more of a pure theoretical paper (specifically, a learning theory paper, rather than one focused on \"Social and economic aspects of machine learning\"), analyzing scenarios somewhat related to fairness. It's a bit outside my usual area, and I'm not familiar with some of the theoretical tools, so I can't really judge the novelty of the theoretical contributions. However, given that \"Social and economic aspects of machine learning\" is the chosen primary area, I think these concerns need to be properly addressed for the paper to truly resonate with the fairness community.", "was_revised": true, "final_justification": "I'm really not quite sure how much theoretical contribution there is (since theory takes up a very large proportion in the paper, it's necessary to consider whether theoretical innovation is really important in this article). Due to the author's detailed explanation and supplementary experiments, I tend to increase my score.", "rebuttal": "We thank the reviewer for acknowledging our theoretical contributions and recognising the importance of fairness in functional classification. Below, we address the weakness (W) point by point and further elaborate on the motivation behind our work. We respectfully hope the reviewer could consider raising the score in light of the additional motivation provided, comparisons to adapted methods, and clarifications of our model assumptions.\n\n**Practical motivation** \n\nIn our paper, we focus on developing a novel and practically effective methodology for fairness-aware classification in functional data settings. This is an important and underexplored problem with significant real-world relevance. To the best of our knowledge, our proposed framework, Fair-FLDA, is the first to explicitly address fairness in the context of functional classification. Functional data pose unique challenges due to their infinite-dimensional nature, which makes it difficult or ineffective to directly apply fairness techniques developed for multivariate data.\n\nOur motivation is primarily practical: in many applications involving functional data, such as biometric signals, longitudinal health records, or time-varying sensor data, ensuring fair and accurate classification is crucial. As demonstrated in Section 4, applying standard classification algorithms to such data can lead to substantial fairness violations (see the second and third columns in the right panel of Figure 2). In this response, we also examine a simple ``dimension reduction + standard fair classification\" strategy: applying dimension reduction followed by fair multivariate classification. Additional baseline results using this approach are included in Tables 1 and 2 at the end of this response, along with detailed comparisons between Fair-FLDA and three post-processing and one pre-processing fairness methods originally designed for multivariate data. Across all experiments, Fair-FLDA consistently achieves lower misclassification rates while effectively mitigating disparity, highlighting its practical merits. In the revised version, we will provide a clearer explanation of the methodological and practical motivations that guide our study.\n\n**Theoretical contribution** \n\nWhile the main thrust of our contribution lies in methodology and empirical performance, we include theoretical analysis to support and validate the proposed approach. This analysis serves to reinforce the practical methodology rather than being the central focus of the paper. We appreciate your recognition of this theoretical component. Notably, theoretical guarantees for functional classification remain largely open in general settings, particularly when the underlying eigenspace is unknown. Our results address this challenging scenario while incorporating fairness constraints.\n\n**Writing for the second and third paragraphs (W1)** \n\nWe appreciate your suggestions on our writing in the introduction. In the revised version, we will rewrite these two paragraphs to improve the logical flow and strengthen their connection. Specifically, in the third paragraph, we will refer to the real data example in Section 4 to demonstrate that existing classifiers exhibit substantial unfairness, thereby highlighting the practical importance of addressing fairness in functional data classification. We will also emphasise the lack of existing work in the literature for fair functional data classification to further motivate our contribution.\n\n**Assumptions on Gaussianity (W2)**\n\nThank you for your question regarding our model assumptions. The classification problem for Gaussian processes has been extensively studied in the literature (e.g., [1], [2], [3]). In functional data analysis, it is well known that linear methods can often achieve optimal performance, largely due to the infinite dimensionality of the functional space (e.g., [1]). In some scenarios, linear classifiers can even lead to perfect classification (see Remark 2). When the data is generated from a Gaussian process, this optimality can be further understood through its connection to the structure of reproducing kernel Hilbert spaces (RKHS), as discussed in [2] and [3].\n\nIn our paper, the Gaussianity assumption offers us a theoretically tractable foundation for studying fairness in functional classification. Under this assumption, explicit expressions for the Radon–Nikodym derivative can be derived, which in turn facilitate the construction of the plug-in classifier, Fair-FLDA. To demonstrate the robustness of our method, we also include simulation results under non-Gaussian models in Appendix A.3, highlighting its practical performance beyond the Gaussian setting.\n\nAs our work is, to the best of our knowledge, the first to study fairness in functional data classification, we consider the Gaussian process model a natural and principled starting point. It provides a clean and interpretable platform for exploring fairness constraints in infinite-dimensional spaces. In future work, we plan to extend our methodology to more general, non-Gaussian settings.\n\n**Additional empirical evaluation (W3)**\n\nOur primary goal in this work is to introduce the first principled framework for fair functional classification with theoretical guarantees, providing the fairness community with a trustworthy algorithm.  Our theoretical contributions are numerically justified through both synthetic and real datasets, along with additional experiments done in the rebuttal stage.\n\nSince no prior work specifically targets fair functional classification, there are no widely established benchmark datasets in this context. We therefore use the publicly available dataset NHANES, which is frequently used in the functional data literature [4,5], to demonstrate the practical significance of our fair functional classifier. It is important to note that our method is broadly applicable to a wide range of functional data classification problems where fairness is a concern. For example, the Siena Scalp EEG dataset in the PhysioNet database contains EEG recordings from male and female subjects and can serve as a natural application of our fair functional classifier, where the response variable can be defined as the occurrence of a seizure.\n\nMoreover, we have included more comparisons with other baseline methods to further demonstrate the superiority and practical relevance of our method for fair functional classification. We have incorporated additional baselines following a ``dimension reduction + standard fair classification\" strategy. Specifically, we first apply functional principal component analysis (FPCA) to extract features, and then employ fair classification methods designed for multivariate data. These include three post-processing methods FPIR [6], PPF [7] and PPOT [8], and one pre-processing approach FUDS [6].\n\nResults in Tables 1 and 2 show that our proposed fair classifier, Fair-FLDA, consistently achieves the lowest classification errors while effectively controlling disparity under the pre-specified levels. In contrast, the other four baseline methods exhibit higher classification errors. In particular, PPF shows poor disparity control when $\\delta=0$, and FUDS fails to adequately control disparity on the NHANES dataset. These extensive numerical results highlight the superiority and practical necessity of our method for fair functional classification.   \n\nOverall, the proposed fair functional classifier, equipped with theoretical guarantees, fills an important gap in the literature on fair functional classification and contributes to advancing research in the fairness community. We hope these clarifications adequately address your concerns.\n\nTable 1: Median classification error and DD over 500 runs under Gaussian with mean difference $\\beta=1.5$ and sample size $n=1000$.\n\n|$\\delta$|0.00||0.05||0.10||0.15||0.20||\n|--|--|--|--|--|--|--|--|--|--|--|\n| |Err|$U_{DD,50}$|Err|$U_{DD,50}$|Err|$U_{DD,50}$|Err|$U_{DD,50}$|Err|$U_{DD,50}$|\n|Fair-FLDA|0.234|0.022|0.227|0.048|0.219|0.098|0.213|0.149|0.208|0.197|\n|FPIR|0.276|0.024|0.269|0.039|0.264|0.080|0.258|0.122|0.253|0.168|\n|PPF|0.240|0.323|0.269|0.039|0.264|0.080|0.258|0.121|0.253|0.167|\n|PPOT|0.275|0.023|0.269|0.040|0.264|0.080|0.258|0.122|0.253|0.168|\n|FUDS|0.276|0.030|0.269|0.040|0.263|0.087|0.256|0.142|0.251|0.195|\n\nTable 2: Median classification error and DD over 500 runs under NHANES.\n\n|$\\delta$|0.00||0.05||0.10||0.15||0.20||\n|--|--|--|--|--|--|--|--|--|--|--|\n| |Err|$U_{DD,50}$|Err|$U_{DD,50}$|Err|$U_{DD,50}$|Err|$U_{DD,50}$|Err|$U_{DD,50}$|\n|Fair-FLDA|0.314|0.021|0.305|0.048|0.297|0.099|0.289|0.149|0.285|0.200|\n|FPIR|0.385|0.016|0.377|0.050|0.369|0.101|0.360|0.151|0.354|0.200|\n|PPF|0.343|0.722|0.378|0.047|0.369|0.097|0.361|0.144|0.355|0.193|\n|PPOT|0.385|0.016|0.377|0.050|0.369|0.101|0.360|0.150|0.353|0.200|\n|FUDS|0.403|0.157|0.371|0.189|0.358|0.254|0.351|0.317|0.346|0.349|\n\n[1] Delaigle and Hall. Achieving near-perfect classification for functional data. JRSSB, 2012.\n\n[2] Berrendero et al. On the use of reproducing kernel Hilbert spaces in functional classification. JASA, 2018.\n\n[3] Torrecilla et al. Optimal classification of Gaussian processes in homo-and heteroscedastic settings. Statistics and Computing, 2020.\n\n[4] Chang and McKeague, Empirical likelihood-based inference for functional means with application to wearable device data. JRSSB, 2022.\n\n[5] Lin et al., Causal inference on distribution functions. JRSSB, 2023.\n\n[6] Zeng et al., Bayes-optimal fair classification with linear disparity constraints via pre-, in-, and post-processing. arXiv:2402.02817.  \n\n[7] Chen et al., Post-hoc bias scoring is optimal for fair classification. ICLR 2024.  \n\n[8] Xian et al., Fair and optimal classification via post-processing. ICML 2023."}]

Author rebuttal:

Meta-review: The manuscript introduces the first framework for fairness-aware classification in the context of functional data analysis (FDA), providing a theoretical characterization of the Bayes-optimal fair classifier under disparity constraints, proposing a practical post-processing algorithm (Fair-FLDA), and establishing high-probability guarantees for both fairness satisfaction and excess classification risk. Reviewers acknowledged that this work bridges the gap between algorithmic fairness and FDA, a domain where fairness considerations have not previously been explored. Nonetheless, concerns were raised regarding the novelty, as the core methodology draws on existing techniques in fairness-aware classification and FDA, such as group-wise threshold adjustment grounded in the Neyman-Pearson principle and the use of Radon-Nikodym derivatives, and about the absence of simple baselines involving multivariate fairness methods applied to functional data after dimension reduction. Most of these concerns were addressed in the rebuttal and subsequent author–reviewer discussion, including through additional experiments in extended settings where the sensitive feature is unavailable at test time and for multi-class classification problems. In the end, all reviewers agreed that the manuscript is suitable for publication, though they emphasized the need for a careful revision of the motivation and a more thorough discussion of scenarios where sensitive information is not available during training.
Decision: Accept (poster)