Assessing Uncertainty in Similarity Scoring: Performance & Fairness in Face Recognition
Authors: Jean-Rémy Conti
Stephan Clémençon
Venue: ICLR 2024 poster (2024)
Keywords: Uncertainty
Face
Recognition
Performance
ROC
Fairness
Bootstrap
Primary area: societal considerations including fairness, safety, privacy
TL;DR: This paper provides a valid bootstrap method for quantifying the uncertainty of ROC-based metrics of a similarity scoring function, such as performance and fairness metrics in Face Recognition.

Abstract: The ROC curve is the major tool for assessing not only the performance but also the fairness properties of a similarity scoring function. In order to draw reliable conclusions based on empirical ROC analysis, accurately evaluating the uncertainty level related to statistical versions of the ROC curves of interest is absolutely necessary, especially for applications with considerable societal impact such as Face Recognition.  In this article, we prove asymptotic guarantees for empirical ROC curves of similarity functions as well as for by-product metrics useful to assess fairness. We also explain that, because the false acceptance/rejection rates are of the form of U-statistics in the case of similarity scoring, the naive bootstrap approach may jeopardize the assessment procedure. A dedicated recentering technique must be used instead. Beyond the theoretical analysis carried out, various experiments using real face image datasets provide strong empirical evidence of the practical relevance of the methods promoted here, when applied to several ROC-based measures such as popular fairness metrics.

Published as a conference paper at ICLR 2024
ASSESSING UNCERTAINTY IN SIMILARITY SCORING: PERFORMANCE & FAIRNESS IN FACE RECOGNITION
Jean-Rémy Conti & Stéphan Clémençon
LTCI, Télécom Paris, Institut Polytechnique de Paris, Idemia (jean-remy conti, stephan. elemencon) @telecom-ppateee
ABSTRACT
The ROC curve is the major tool for assessing not only the performance but also the fairness properties of similarity scoring function. In order to draw reliable conclusions based on empirical ROC analysis, accurately evaluating the uncertainty level related to statistical versions of the ROC curves of interest is absolutely necessary, especially for applications with considerable societal impact such as Face Recognition. In this article, prove asymptotic guarantees for empirical ROC curves of similarity functions as well as for by- product metrics useful to assess fairness. We also explain that, because the false acceptance/rejectinn rates are of the form of U-statistics in the case of similarity scoring, the naive bootstrap approach may jeopardize the assessment procedure. dedicated recentering technique must be used instead. Beyond the theoretical analysis carried out, various experiments using real face image datasets provide strong empirical evidence of the practical relevance of the methods promoted here, when applied to several ROC-based measures such as popular fairness metries.
INTRODUCTION
The massive deployment of AI technologies brings with it a pressing demand for methodological tools to assess their trustworthiness. The reliability of Al systems concerns their estimated performance of course, but also their properties regarding fairness: ideally, the system should exhibit approximately the same performance, independently of the sensitive group (determined by e.g. gender, age group, race) to which it is applied. This particularly true for Face Recognition (FR) systems, the running exam- ple through this article, now under scrutiny by the general public and regulatory organizations, refer to
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e.g. Grother& & Ngan (2019) or Snow (2018).
Empirical ROC curves for three different models (ArcFace, CosFace, Ada- Cos) and for two distinct evaluation datasets (see 4). The ROC curves for the first dataset are depicted with solid lines while the ROC curves for the second dataset are displayed with dashed lines. A confidence band for the ROC computed with the ArcFace model on the first dataset is displayed in light blue.
Figure 1:
The task of designing a FR system is usually for- mulated as a similarity scoring/lcarning problem, e.g. Vogel et al. (2018). Assuming that the system processes pixellated face images X in Rd the goal is to build (symmetric) scoring function 8 Rd x ddd + such that the larger the simi- larity score sar, 2:) related pair of images, the larger (hopefully) the probability that both images correspond to the same individual. In this case, one assigns a positive label to the pair and a negative label otherwise. The gold standard to measure the performance of such a similarity s the Receiver Operating Characteristic (ROC) curve (D.e. & Swets 1966), namely the plot of the false rejection rate against the false acceptance rate, as the similarity scoring threshold varies.
Alternative correspondence: jeanremy. cont cont
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If, until now, the benchmark of FR systems has been essentially reduced to ad-hoc evaluation of the performance metrics, to the computation of empirical ROC curves based on FR evaluation datasets of reference (see Grother & Ngan (2019), the quantification of the uncertainty inherent in the randomness of the evaluation datasets (referred to as aleatoric uncertainty sometimes, see Hüllermeier & Waegeman (202T) is essential to compare and appreciate fully the merits of such systems regarding accuracy and fairness with confidence. The significance of the uncertainty quantification step is illustrated in Fig. I The ROC curves for three different FR models are computed on two distinct evaluation datasets (see for details on models and datasets). On the first dataset (solid lines), one would conclude that ArcFace is better model than CosFace, as the former has a lower empirical FRR than the latter, for any FAR value. However, one would draw the opposite conclusion, looking at the second dataset (dashed lines). Selecting models only depending on their empirical performance (ROC) has its flaws, as the uncertainty of those metrics is not taken into account. Note that the method for building confidence bands for the ROC, which we present in this paper, would have avoided such conclusions. Indeed, both models are indistinguishable in terms of performance, as their ROC curves are contained within the band. On the contrary, the AdaCos model performs worse than ArcFace/CosFace on both datasets and its empirical ROC curves are far from the confidence band. One could thus favor confidently ArcFace/CosFace over AdaCos regarding their performance.
In order to make meaningful comparisons, the (possibly high) uncertainty inherent in the statistical nature of the estimation must be taken into account. Indeed, this evaluation is crucial to judge whether the similarity scoring function candidates meet the performance/farrees requirements in a trustworthy manner. The main purpose of this paper is precisely to explain how to quantify the uncertainty/variablliin of similarity ROC curves, by means of a dedicated bootstrap methodology in particular, in sound validity framework generalizing the one established by Bertail et al. (2008) in the non pairwise setup.
Related works. To our knowledge, the uncertainty issue inherent to ROC curve estimation and fairness metrics estimation is poorly documented in the literature, particularly for similarity scoring problems such as FR. It has been studied at length for scoring functions, but in the non pairwise setup, using bootstrap methods in (Bertail et al., 2008). The major difference between the analysis carried out therein and our framework lies in the fact that, in the similarity scoring context, false acceptance/rejection rates are not basic i.i.d. averages anymore but generalized U U-statistics (refer to Lee (999)) or Arcones & Gine (1992)). It has significant impact on the methodology that can be used to quantify the uncertainty of empirical performance/ffirnes measures. As shall be seen in this paper, naively applying the bootstrap of Bertail al. (2008) to similarity scoring problems (e.g. FR) strongly underestimates the ROC curve, resulting in confidence bands for the ROC curve which do not even contain the empirical ROC curve most of the time. In Vogel et al. (2018), non asymptotic confidence bounds for the estimation eттоr of empirical similarity ROC curves have been established by means of linearization techniques tailored to U-statisties in a slightly different probabilistic framework (stipulating random labels). It is the purpose of the present paper to investigate how to accurately approximate the distribution of the estimation error by means of dedicated resampling techniques and build bootstrap confidence bands with satisfactory probability coverage.
Contributions. As will be explained in the subsequent analysis, a naive application of the bootstrap procedure yields a systematic underestimation of the similarity ROC curve. We provide (i) recentering technique to counteract this, while still being accurate asymptotically. Resulting from this bootstrap variant, (ii) confidence bands for the ROC curve and FR fairness metrics are shown to be consistent, in addition to achieve nominal coverage on synthetic data. The recentered bootstrap also allows to define (iii) a scalar uncertainty measure for the ROC and fairness metrics, which can be employed to compare the robustness of several FR fairness metrics. Finally, in addition to the statistical analysis presented, the relevance of the approach is supported by (iv) illustrative numerical experiments, based on real data of face images, together with discussion about the practical use of the information produced, in order to make more reliable decisions concerning accuracy and fairness. These results pave the way for more valuable and trustworthy comparative analysis of the merits and drawbacks of FR systems.
Organization of the paper. The main concepts at work in similarity scoring/learning and in FR are briefly recalled in Section together with the notions pertaining to ROC analysis used in this article to evaluate predictive performance and build faimess criteria. The consistency of empirical similarity ROC curves, is stated section 33, It is also explained therein how to bootstrap empirical similarity ROC curves in valid manner, well as by-product summary statistics reflecting the
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accuracy fairness properties of the similarity scoring functions under study, Numerical experiments are presented and discussed in section for illustration purpose.
2 BACKGROUND AND PRELIMINARIES
We introduce here the main notations used throughout the article and briefly recall the standard similarity scoring/learning framework, involved in the design of FR systems in particular, and the key concepts pertaining to ROC analysis that are involved in the subsequent study. We next explain how to formulate fairness criteria based on similarity ROC curves in the FR context. Here and throughout, the indicator function of any event is denoted by [[E). the Dirac mass at any point x by 8z. and the useudo-inverss, of any cumulative distribution function (edf) k(t) on R by R (a) = infft E R: x(t)= 2 a).
2.1 SIMILARITY SCORING PROBABILISTIC AND STATISTICAL FRAMEWORK
The probabilistic framework considered here to formulate the similarity learning problem is the same as that of multi-class classification: Y is a discrete random label defined on a probability space (2, A. PP), valued in y = {1. .... }} with K 2 2, and X is a random vector defined on the same probability space and taking its values in a high dimensional space X' C R with d >> 1. For all E y. we denote the supposedly continuous conditional distribution of X given Y= k: by FR. the probability that Y equals & by pk: = P(Y = k). Equipped with these notations, the joint distribution P of the random pair (X, Y)) is fully characterized by {(Fk, (F): k == 1, .... K). In the rumning example considered through this paper, X is image depicting the face of an individual in population of K identities, the identity being indexed by Y. In similarity learning, the objective pursued is to find mapping X²2 + R U (=>0), called similarity scoring function, such that, given two independent pairs (X, Y) and (X,,)) Y), the larger the similarity score s(x, X'), the more likely the same label should be shared (i.e. one should observe the event Y Y') ideally. Before recalling performance/fairnee metrics in similarity scoring, we explain the usual methodology at work in FR.
Similarity scoring in Face Recognition. In FR, one learns, from a dataset of face images with identity labels, an encoder function f hx - Rp that embeds the images in a way that brings same identities closer together in certain sense. Each image is of size (h, w). while corresponds to the color channel dimension. It worth noticing that a pre- processocengg detection step (finding a face within an image) is required so that all face images have the same size (h, w). For an image r E Rixwxe its latent representation f(x)) E RP is referred to as the face embedding of x. Since the advent of deep learning, the encoder f is usually a deep Convolutional Neural Network (CNN) whose parameters are learned on a huge FR dataset, made of face images and identity labels. In brief,
(k)
the training consists in taking all images x; labelled with identity k. computing their embeddings fxx!*) ( and adjusting the parameters of f so that those embeddings are as close as possible (for a given similarity measure) and as far possible from the embeddings of identity I + k. The usual similarity measure is the cosine similarity, defined as
f(z)) x(I,, [If(x))) If(2))]
for two images x; zy, where ||+|| stands for the usual Euclidean norm. In some early works (Schroff al. 2015), the Euclidean metric [xf()))) - f(x;))] is also used.
Multi-sample statistical setup. In the subsequent analysis, the p& 's are supposed to be given and are part and parcel of the performance criterion (a possible choice consists in giving the same weight to all identities, i.e. P& = 1/K for k = 1, .... K), whereas the Fi 's are unknown in practice. In order evaluate empirically the properties of (trained) similarity scoring function s(z, a²), it assumed throughout the paper that K i.i.d. samples are available:
((k i.dd FR, where ny 2 for 1, K. m4
In the FR context, we thus suppose that ny images are available for identity k E {1, ..., K}. the size of the pooled sample being denoted by n = n, ....tng-- Those images belong to a test dataset, used to evaluate trained FR model f (or equivalently its associated similarity s in Eq. T. Based on
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these data, empirical versions of performance and fairness metries can be computed. Their statistical accuracy will be next investigated from an asymptotic perspective, as the n& 's simultaneously tend to infinity at the same rate as n: for all lc E y. there exists A > 0 such that ng/nn ) - +000
2.2 ROC ANALYSIS EVALUATION OF PERFORMANCE/FAIIREES IN SIMILARITY SCORING
As formulated in Vogel et al. (2018), similarity learning can be seen as a specific bipartite ranking problem, where the input space is of the form of a product space X' X:: given two independent observations (x, Y) and (X', Y") drawn from P. the input r.v. is formed by the pair (X, X'), while
Z = 211( Y - 1 is the binary label. The gold standard to evaluate bipartite ranking performance is ROC analysis: a statistical learning view can be found in e.g. Clémençon & Vayatis (209).. In the similarity learning context, the ROC curve of similarity scoring function s x') ) the plot of the False Rejection Rate (FRR) against the False Acceptance Rate (FAR) as the acceptance threshold varies, namely the mapping ROC: a E (0 - FRR o FAR (a), where, for all t E R.
k Y' ='= FAR(() = P(s(X,X'> > 2 = -13 = EAcIPAPP(s(Xxx)) > >t
EKSIPEPI
(RR(() P(s(X,X') St| 2 +11 EKey PEP(s(XXX)) Y
EkEy
note on the definition of the pseudo-inverse for the FAR quantity is available D.I
Remark 1. (ROC CONVENTIONS) In machine learning. the ROC curve usually refers to the PP-plot t E R + (FAR(t), FRR(t)). or equivalently a E (0, 1) + 1 FRR o FAR The FR community preferably plots FAR(() on the x-axis and FRR(() on the y-axis. Both components correspond to error rates that should be minimized, one possibly more than the other depending on the use case. Of course, these two curves provide exactly the same information as there is one-to-one correspondence between them. Note that we use the FR convention throughout the paper.
In practice, special attention is paid to certain points of the ROC curve. The FAR level a E (0, 1) determines the operational point of the FR system and corresponds to the security risk one is ready to take. According to the FR use case, it is typically set to 10 with E (I.....9..
Fairness metrics. In order to inspect the faimess properties of FR system based on a similar- ity scoring function s, one generally looks at differentials in performance amongst several sub- groups//egmment of the population, sensitive attribute (e.g. gender, race, age class, ...) making them distinguishable. For a given (discrete) sensitive attribute that can take M > 1 different val- ues, in A 10 M - 1}} say, we enrich the probability space and now consider random vector (X, Y, A) where A E A indicates the subgroup to which the individual indexed by Y be- longs to. For every fixed value E A. can further define the FAR/FRR related to subgroup (A = P(s(X,X)) > t Y Y A == A' a} and (t) = (X, < <tlY= Y', A A' = a} for all f R. where by (X', Y", A') is meant an independent copy of the random triplet (X, Y.Y.). Ideally, fair scoring function would exhibit nearly constant F. values when varies, for all & (and the same property for the FRR, (t) values). A FR fairness metric quantifies how much a model is far from this property. The FR faimess metrics considered in this paper are those used by the U.S. National Institute of Standards and Technology (NIST) in their FRVT report (Grother 2002). They attempt quantify the differentials in (FAR, (t) JasA and (FRR,(t))ae.) (t)) Each fairness metric has two versions (one for the differentials in terms of FAR, the other in terms of FRR). A typical fairness metric is the max-min fairness below:
max.e. A FAR (t)
(1) max.e. (t)
FARMAX (1)
FRR
max min
min.EA FAR.(
m FRR,(1) (t)
In practice, the threshold t is set as for the ROC curve, i.e. it achieves a level FAR(t) = a E (0. for the global/total population, and not for some specific subgroup. Three other popular FR fairness metrics are the max -eo-geemman metric, the log-geomean metric and the Gini coefficient. Their definition is postponed to A. for conciseness.
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3 SIMILARITY SCORING METRICS ASSESSING UNCERTAINTY
Motivated by the need to make trustworthy decisions taking into account the uncertainty inherent in the evaluation data,
investigate the statistical accuracy of empirical counterparts of the ROC curve of given similarity scoring function s(x, ) in the statistical multi-sample framework described in 2.1 We next explain how use the bootstrap methodology to estimate the related uncertainty level and build accurate confidence bands for the ROC and fairness metrics. For notational simplicity, the results are stated and proved in the case where Pk 1/K for all k V, extension the general case being straightforward.
3.1 STATISTICAL INFERENCE CONSISTENCY RESULT
An estimator of the ROC curve naturally obtained by replacing the quantities FAR and FRR with their natural statistical counterparts in the definition of the ROC curve. In the multi-sample statistical setup defined in 2.1 the empirical versions of (IR(()) and FRR(() can be expressed as follows, using the symmetry property of similarity scoring functions: for all E IR
(1)
2
1x (1) .x,)) 1.
K(K
Tig
k<l
K FRR,(t) (t) = K
E I5KJSm
[s((!
xx, (K) xx(x) $ t).
(3)
(T&k 1)
Notice that the terms involved in the two averages above are not independent, as each X x!()) is involved in many terms of both averages, in contrast to the standard bipartite ranking framework (Bertail et al. 2008) where one deals with i.i.d. mean statistics. As detailed in A.2] these averages are actually generalized U-statistics, the simplest extensions of standard i.idd. mean statistics. Properties and asymptotic theory of U-statistics can be found in Lee (199)) while concentration properties are investigated in Clémençon et al.) (2016). The quantíties (2) and (3) can then be used compute the empirical similarity ROC curve based on the available evaluation datasets:
ROC,: a E (0, 1) + FRR, (aA)
Empirical versions of fairness metrics are naturally obtained in a similar plug-in fashion (see D.3
The result stated below reveals the uniform consistency of the curve (4) in the multi-sample asymptotic framework considered here.
Proposition (STRONG CONSISTENCY) With probability one, we have:
sup (ROC,, ROC(a)) 0, as +00. (E(0.1)
Refer to D.2 for the technical proof. While the true ROC curve is unknown, this result gives confidence in the quantity ROC,, computes based on data. A similar consistency result is stated in[Hsieh & Turnbull (1996), when the negative and positive cdf's are estimated by basic i.i.d. averages, and proved by means of classic results for empirical processes. The case of empirical similarity ROC curves cannot be handled in the same way because, as previously emphasized, (2) and (3) are generalized U-statisties and the terms involved in these averages exhibit a complex dependence structure. Linearization tricks (i.e. Hoeffding decomposition), such as those used in Vogel et al. (2018). would be required to establish addition the asymptotic Gaussianity of rescaled version of) the fluctuation process
(a) = Vn ROC(a)), a E (0.
As underlined in Bertail et al. (2008), where the (much simpler) statistical framework considered is the same as in Hsich & Turnbull ( 1996), the identification of the Gaussian limit of the process of Eq, 6 is of very poor interest regarding the construction of (asymptotic) confidence bands for the similarity ROC curve: beyond the computational difficulties inherent in simulating Brownian bridges, the presence of the unknown quantity ROC(t) in the complex limit law makes its impracticable to build confidence bands. Appropriate bootstrap techniques, whose asymptotic validity can be proved, should be preferably used instead.
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3.2 BOOTSTRAPPING THE PERFORMANCE/FAIRNESS METRICS CONFIDENCE REGIONS
Provided that representative datasets of the target populations are available, the empirical ROC curve (4) (and its related scalar summaries) of similarity scoring function =') is the main tool to assess performance and fairness in various applications such as FR. However, in order to make meaningful comparisons, the (possibly high) uncertainty inherent in the statistical nature of the estimation must be taken into account. Indeed, this evaluation is crucial to judge whether the similarity scoring function candidates meet the performance/fairnes, requirements in a trustworthy manner, as will be discussed on real examples in the next section. We now explain how to use specific bootstrap resampling methodology to quantify the variability of the fluctuation process (6) in an asympiotically valid manner and why a naive bootstrap technique fails in the present situation.
Objective. When computing ROC,, (a) to estimate the true ROC curve, one makes the error
(a) = (O) - ROC(a),
(7)
which is unknown, just like ROC(a). The variability of the random variable é, (a) fully characterizes the uncertainty of the empirical ROC curve. The objective is to approximate én (a) so that its variability can be estimated. This variability (iee the uncertainty of ROC,, will be used to build confidence bands around the empirical ROC curve and to define a scalar uncertainty metric. In order to approximate én (a). the bootstrap approach makes it possible to sample an estimate of é, (a). With many samples, one can retrieve the variability of the error é (a).
Naive bootstrap. The bootstrap paradigm, introduced by Efron (199)) and developped at length in Bertail et al. (2008), suggests to recompute the empirical similarity ROC curve from K independent sequences of i.i.d. variables x=***,
x!A)= i.dd Fx
1 Eisisn xx("), con- 24
.... x! k= ..... KJ. In other words, for each identity & within the dataset, one simply randomly samples with replacement ny images x(()) ..... X xk))* among the ny face images x ...., X available for identity k. The boot- strap sample, which should be viewed as sort of replicate of the evaluation dataset, is obtained by concatenating all the images thus sampled. Notice that the bootstrap sample may contain the same images several times due to the use of the sampling with replacement scheme, in contrast to the original dataset D. In practice, the resampling scheme is replicated B > 1 times in order to compute a Monte-Carlo approximation of the distribution of the bootstrap ROC, i.e. the curve -1
ditioned upon the evaluation dataset D == {X (k)
+ ROC, FRR., (FAR,)) (a)) with
2 (t) = K(K- - 1)
12g
> t}.
(8)
k<l
i=1 j=1
K
FRR (t)
2
[s((x x(())) S st).
(9)
TER (nig - 1)
The curve ROC, n. is nothing but the empirical ROC curve computed with a bootstrap sample, instead of the original dataset. It turns out that, conditionally to the dataset D. the quantity en (a) = (a) - ROC, (a) approximates (a)'s law. The approximation procedure asymptotically valid, as proved in [..] so that it satisfies our objective. In practice, one would use B bootstrap samples, get B realizations of (a), and thus B realizations of ((1) (a) which allow to compute the variability of (a)(,, in order to estimate the variability of é (ax), i.e. the uncertainty of the ROC curve. The bootstrap percentile method enables to build accurate confidence regions for the quantity ROC, of interest. Indeed, for a fixed a E (0, 1). a confidence interval, at level 1 - aci E [0, 1]. around ROC,.(a) can be obtained by considering the "g acz-th and the 1-ac1-th quantiles of B realizations of ROC,, (a) + 2 (a)). It is worth noticing that, in spite of its asymptotic validity, the method can be seriously compromised in the non asymptotic regime when the distribution of ROC, (a:) is not centered at ROC,, (a). This is typically the case in the present situation, as depicted in Fig. 2 the B = 200 realizations of ROC, (a) (light blue) are not at all centered around the empirical ROC curve (dark blue), the confidence band formed by the naive
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Figure 2: Bootstrap versions of the ROC curve (ROC,, in light blue) and the empirical ROC curve (ROC, in dark blue). The V-statistic counterpart ROC,, is depicted in red.
Figure 3: Confidence bands at 95% confidence level for the empirical ROC curve (dark blue), using two methods: the naive bootstrap (light red) and the recentered bootstrap (light blue).
bootstrap, at level 1 - acr 95%, being displayed in Fig. 3 (light red). Note that the naive bootstrap Bertai et al. (2008) yields strongly inaccurate confidence bands (in the non asymptotic regime) as the bands do not even contain the empirical ROC curve most of the time. For both Figures, the pretrained encoder f defining the similarity function s (ef Eq, [ is ArcFace, while MORPH is the evaluation dataset (see for details models and datasets). We now explain why the distribution of
ROC, (a)) is not centered around ROC,, (a) i.e. why the naive bootstrap of Bertail et al. 2008 fails.
--statistic version of the ROC and recentering. Firstly, one can easily check that E. (*(FAR | DJ FAR,, (f), where by E- D] and P {-1 D} are meant the conditional expectation and probability given the dataset D used to compute the empirical criterion ROC,.
Whereas the quantity FAR,, (4) is well centered around the empirical FAR,(() given D, this not the case for the FRR metric in general. Indeed, we have:
E FRR (t) D = K
Σ Hs((x"),,) 5) == (t). ISt.imm
(10)
The quantity above is an average of K independent V-statistics, that may slightly differ from FRR,, (t), the difference being of order ((1/n) This is due to the presence of the diagonal terms (s((! _x("*) < l). which results from the fact that perfect similarities (i.e. similarities equal to 1 in the cosine similarity ( can be observed in the bootstrap samples with non zero probability, while this cannot occur when computing FRR,,t) Hence, the empirical FRR tends to be underestimated by its naive boostrap version in general. This is reflected in the bootstrap ROC curves ROC, being centered around their V-statistic version ROC,, (a) == FRR., (FAR,) ) (ax), and not around the empirical ROC curve ROC,, as depicted by Fig. 2 Notice incidentally that this phenomenon is specific to similarity scoring, because of the pairwise nature of the statistic (3). and does not occur in the classic bipartite ranking framework (Bertail et al. 2008). In fact, the V-statistic version of the
ROC curve is of great interest since the recentered error en (a): = ROC,,() - ROC,, (a) also approximates é, (a)'s 's law, as proved in D.5 In the same way than for (a), we are able to build confidence intervals, at confidence level 1 act E [0, 1], for the quantity ROC,(a). Considering B bootstrap samples, one would compute 1fm. (a) (resp. (n.B) (a)) the 2e1-th (resp. the L-gei-th)
quantile of the B realizations of ROC,, + e (a). ((n.B) (a) and uo (a) are respectively the lower and upper bounds of the confidence interval, obtained with a variant of the naive bootstrap
(n.B)
which we call recentered bootstrap. The difference with <(1) (a) is that, using Eq. 10 one finds that E* (aC,((a) + a( (a) D) = ROC,, (a), i.e. the confidence intervals are well centered around the empirical ROC curve, as depicted in Fig. 3
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The theoretical considerations within this section can be summarized by the result below. It states that the confidence interval, at confidence level 1 - aci, for the ROC curve, using the recentered bootstrap, has a probability of containing the true ROC curve which is truly equal to acs when n and B tend to +00. Refer to D.6 for the technical proof. The result holds when applying the recentered bootstrap to the considered fairness metrics (see (D7].
Theorem 1. Let (0,1) and aci (0,1). Under the (mild) assumptions in D.4 we have:
P(tCP()) S ROC(a) S whaB) (a)} + 1--a
as n. and B both tend to +00.
This is an asymptotic result. In the non asymptotic regime, i.e. with a finite evaluation dataset, an interesting question is find what happens this probability of containing the true ROC curve. For that purpose, it is common practice to use synthetic datasets allowing for an approximation of the true ROC curve. On each dataset, one can compute confidence intervals and check whether the frequency, over all datasets, of containing the true ROC curve is truly equal to - acT-- In B. we generate 200 synthetic datasets and conclude that it is very close to 1 1 OCT, underlining the soundness of the recentered bootstrap, while it is not the case for the naive bootstrap of Bertail et al. (2008).
Uncertainty metric. To quantify the uncertainty about the ROC curve, one might be interested in a scalar quantity which summarizes its uncertainty. For instance, v /Var [e, [ax] seems appropriate for such quantification. However, in order to compare this scalar uncertainty at several FAR levels it
seems reasonable to consider a relative quantity such as (a)r (RO,, (a)) To estimate this quantity, we use the recentered bootstrap and define the normalized uncertainty of the ROC curve as:
U]R] V Varle! D
(11)
(OC,,(a)
(2) (a) ROC, (a) (a) approximates (a)'s law. In practice, one would use B as 6n
bootstrap samples, get B realizations of (a), thus of (2) (a). From those data, one would compute their standard deviation, normalized by the empirical ROC curve. The definition of the normalized uncertainty is naturally extended to fairness metrics (see (see D.7)
The pseudo-codes for the naive/recentered bootstrap methods, the computation of confidence intervals, as well as of the normalized uncertainty for the ROC curve and fairness metrics available in C
4 NUMERICAL EXPERIMENTS APPLICATIONS
Models and datasets. We take as encoder several pre- tre-trained models (AdaCos of Zhang et al.
(2019) ArcFace of Deng et al. (2019)), CosFace of Wang et al. (0188), CurricularFace of Huang et al.
(0020 whose backbone is MobileFaceNet (Chen et 2018)). trained on the MS-Celeb- 1:06--1M-a M--
datase This dataset is a cleaned version of the MS-CelebIM dataset (Guo et al. 2016) and
contains 3. 3.28M images of 73k identities. We choose the MORPH dataset (Ricanek & Tesafaye 2006
as evaluation dataset. It is composed of 55k face images from 13k distinct identities. This dataset is widely used for fairness evaluation since it is provided with ground-truth age and gender labels (the available labels for the latter are female and malc). All images are pre-processed by the Retina-Face detector (Deng et al., 2019c) and are of size 112 x 112 pixels. Unless specified, all experiments use B = 200 bootstrap samples.
To highlight the significance of the tackled problem in this paper, we show in Fig. the empirical ROC curves of three models (ArcFace, CosFace and AdaCos), computed on two distinct datasets.
Those datasets are obtained by splitting MORPH two parts, with the same number of images, each identity being present in both splits (see A.3). A confidence band at 95% confidence level, computed for AreFace on one split of data, suggests that ArcFace and CosFace are indistinguishable in terms of performance, for the FAR levels displayed on the z-axis. This insight is interesting there is no model between ArcFace and CosFace that performs better than the other on both datasets.
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Figure 4: Confidence bands 95% confidence level for the FRRmin models (ArcFace, AdaCos). The empirical faimess metrics are depicted as solid lines.
Figure 5: Normalized uncertainty of several fairness metrics (FAR faimesss in solid lines, FRR fairness in dashed lines). The gender label is chosen as the sensitive attribute.
max fairness metric, for two
Then, we investigate the uncertainty related to the fairness metric FRRMAX The gender label is used here as the sensitive attribute. We display in Figure confidence bands at 95% confidence level for the FRR faimess metric (see B.3 for other fairness metrics), for two models (AdaCos and ArcFace). Three zones (A, B, C) are delimited by dashed lines. For the zone A (resp. C), the empirical fairness is better for AdaCos (resp. ArcFace), while the upper-bound of the confidence band is lower for AdaCos (resp. ArcFace). One would conclude that, for each zone, one model is better than the other in terms of FRR fairness (AdaCos for zone A, ArcFace for zone C). The case of zone B is more complex. Only using the empirical fairness metrics, one would choose ArcFace as the fair model. However, the uncertainty for ArcFace is high, and one may choose AdaCos for its robustness, especially in the case where there would be a strict fairness constraint to deploy the technology (e.g. legislation requiring FRR < 4 at FAR 6 x 10 for any evaluation dataset).
Finally, we compute the normalized uncertainty of Eq. TT for all fairness metrics. illustrated in Figure 5] the max- max-geomean metric displays (almost always) the lowest uncertainty, both in terms of FAR and FRR, which makes it particularly suitable for fairness evaluation. This finding is supported by similar experiments in B.2 where the trained model, the evaluation dataset and the used sensitive attribute change. In particular, we employ ArcFace model with a ResNet backbone, evaluated on RFW (Wang al. 2011)). In addition be more robust than other fairness metrics, the max-geomean metric has the significant advantage to be interpretable.
CONCLUSION
In this paper, we consider the problem of assessing the uncertainty inherent in estimating ROC curves, using evaluation datasets, in the context of similarity scoring. Quantifying this uncertainty of great interest since the ROC curve is the gold standard evaluate performance and fairness in various applications, such as Face Recognition. We show the consistency of empirical similarity ROC curves and propose a variant of the bootstrap approach to build confidence bands around performanceffainees metrics, in order to quantify their variability. The procedure is proved to be asymptotically valid and its relevance is illustrated through applications in Face Recognition. Finally, some popular Face Recognition fairness metrics are compared in terms of their uncertainty, revealing that the max- max-geomean metric the more robust to assess fairnsss. While the gold standard by which fairness will be evaluated in the future is not fixed yet, we believe that it should definitely incorporate uncertainty measures, since it could lead to wrong conclusions otherwise. The bootstrap approach is simple, fast and could greatly improve the reliability of accuracy and fairness metrics, especially within the Face Recognition community.
Reproducibility. Pseudo-codes for the experiments of the paper are available in the Supplementary Material The open-source pre-trained models used for the experiments are available with download links.

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Published as a conference paper at ICLR 2024
ACKNOWLEDGMENTS
This research was partially supported by the French National Research Ageney (ANR), under grant ANR-20-CE23-0022 (LIMPID project).
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Reviews:
[{"review_id": "OLTfstmauV", "reviewer": "Reviewer_DZ4X", "summary": "This paper introduces introduces a method to quantify the uncertainty of similarity ROC curves in order to make meaningful comparison between different FR models. A naive bootstrap procedure tends to underestimate the ROC curve. Thus, a recentering technique is proposed so that a scalar uncertainty measure for ROC and fairness metrics is defined. With the statistical analysis and numerical experiments, a discussion about the practical use of the uncertainty value is done to achieve more reliable decisions related to accuracy and fairness.", "questions": "* In Figure 4, the $FRR^{max}_{min}$ is used as the fairness metric, why don't the authors use the best fairness metrics in Figure 5 (max-geomean)? It could lead to more consistent analysis.\n* Why do the authors not evaluate the method in the fair face recognition dataset?\n* Does the training dataset affect the uncertainty of the face recognition methods?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "+ The problem to measure the uncertainty in FR similarity function is well motivated. While ROC curve has become the standard to compare face recognition models, inconsistent comparison might occur due to the differences in evaluation dataset. The proposed method is able to build the confidence bands around the performance/fairness metrics by incorporating the uncertainty measures.\n\n+ The recentering bootstrap technique refines the naive bootstrap that tends to underestimate the empirical FRR as shown in Figure 3. \n\n+ The numerical experiments in face recognition problem is done to discuss how the proposed uncertainty metric and confidence band around ROC curve can be used in real-application. If the upper bound of the confidence band is lower, then the empirical fairness is better.  The differences between upper bound and lower bound of the confidence means that the uncertainty is high, thus it is better to choose a method that has small difference especially in the case where a strict fairness constraint is needed.\n\n+ Multiple fairness metric has been evaluated using multiple FR methods. It is shown the max geomean metric has the lowest uncertainty in terms of FAR and FRR.", "weaknesses": "- There is a lack of justification of why Adacos has lower uncertainty than Arcrface in fairness metric. To fairly compare the performance of Adacos and Arcface, training those methods on Fair Face Recognition dataset [A] might help to justify the performance better. Note that there are multiple FR methods [B,C] that focus on solving fairness problem in face recognition. Comparison with those methods might be useful to justify the proposed metric. \n\n- Instead of the background and preliminaries, it is more important to include more comparison of fairness metrics with various FR methods in the supplementary material in the main manuscript. As the paper focuses on the application of fair recognition technology, it is important to add more justification on the corresponding problem.\n\nAdditional references\n- [A] FairFace Challenge at ECCV 2020: Analyzing Bias in Face Recognition, ECCVW 2020\n- [B] Consistent Instance False Positive Improves Fairness in Face Recognition, CVPR 2021\n- [C] RamFace: Race Adaptive Margin Based Face Recognition for Racial Bias Mitigation, IJCB 2021", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "PsvB5Nrje9", "reviewer": "Reviewer_EFt7", "summary": "This paper focuses on the uncertainty of ROC curve in Face Recognition applications based on similarity scoring. It first proves asymptotic consistency guarantees for empirical ROC curves of similarity functions, which is the theoretical basis of the methodology of building confidence bands for ROC. Then, it provides recentering technique to counteract the underestimation of the similarity ROC curve caused by naïve bootstrap. Resulting from this bootstrap variant, the confidence bands and uncertainty measures for the ROC are defined and shown to be consistent. Finally, the results of various experiments using real face image datasets are illustrated to discuss the practical relevance of the proposed methods.", "questions": "1. This paper is not the first to apply the bootstrap method to ROC curves and there isn’t enough difference between it and Bertail’s work proposed in 2008. The concept of uncertainty proposed in this paper in relatively new in the field of ROC curve estimation, but it is too simplistic in the field if uncertainty estimation, which needs deeper elaboration.\n2. The experiments demonstrates the scenarios and ways in which the method proposed in this paper can be applied, but it is not sufficient to illustrate its effectiveness, and there is a lack of experimental evidence that the proposed uncertainty is a better evaluation metric to compare the performance and fairness of Face Recognition models. Besides, there is no comparison of the proposed method with other methodologies.", "limitations": "", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1. In some cases, the simple ROC curve is not enough to compare the performance of different FC models, and the idea of introducing uncertainty into the evaluation metric fills this gap and is relatively practical. \n2. The proposed method for estimating the confidence intervals and uncertainty of ROC is intuitive and simple, and can be easily applied directly to various Face Recognition models.", "weaknesses": "1. This paper is not the first to apply the bootstrap method to ROC curves and there isn’t enough difference between it and Bertail’s work proposed in 2008. The concept of uncertainty proposed in this paper in relatively new in the field of ROC curve estimation, but it is too simplistic in the field if uncertainty estimation, which needs deeper elaboration.\n2. The experiments demonstrates the scenarios and ways in which the method proposed in this paper can be applied, but it is not sufficient to illustrate its effectiveness, and there is a lack of experimental evidence that the proposed uncertainty is a better evaluation metric to compare the performance and fairness of Face Recognition models. Besides, there is no comparison of the proposed method with other methodologies.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "BWggangGXb", "reviewer": "Reviewer_XZHc", "summary": "This paper proposes a method for evaluating uncertainty on the ROC curve in face recognition problem.", "questions": "None", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper provides a theoretical analysis of the properties and uncertainties associated with the ROC curve in face recognition evaluations and introduces a novel method for quantifying this uncertainty.", "weaknesses": "1. This paper dedicates excessive length and involves many complex formulas to introduce some basic knowledge of face recognition evaluation, which seems unnecessary.\n2. While fairness in face recognition is a pressing issue, the contributions of this paper appear limited. As the authors themselves note, typically, global ROC curves and their comparisons with specific attribute-driven ROC curves would suffice for the uncertainty analysis in most face recognition scenarios.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (poster)