Title: Overcoming Common Flaws in the Evaluation of Selective Classification Systems
Venue: NeurIPS 2024
Accepted: yes
Keywords: Selective Classification

Method evaluation

Failure Detection
Abstract: Selective Classification, wherein models can reject low-confidence predictions, promises reliable translation of machine-learning based classification systems to real-world scenarios such as clinical diagnostics. While current evaluation of these systems typically assumes fixed working points based on pre-defined rejection thresholds, methodological progress requires benchmarking the general performance of systems akin to the $\mathrm{AUROC}$ in standard classification. In this work, we define 5 requirements for multi-threshold metrics in selective classification regarding task alignment, interpretability, and flexibility, and show how current approaches fail to meet them. We propose the Area under the Generalized Risk Coverage curve ($\mathrm{AUGRC}$), which meets all requirements and can be directly interpreted as the average risk of undetected failures. We empirically demonstrate the relevance of $\mathrm{AUGRC}$ on a comprehensive benchmark spanning 6 data sets and 13 confidence scoring functions. We find that the proposed metric substantially changes metric rankings on 5 out of the 6 data sets.

Overcoming Common Flaws in the Evaluation of Selective Classification Systems

Abstract

Selective Classification, wherein models can reject low-confidence predictions, promises reliable translation of machine-learning based classification systems to real-world scenarios such as clinical diagnostics. While current evaluation of these systems typically assumes fixed working points based on pre-defined rejection thresholds, methodological progress requires benchmarking the general performance of systems akin to the AUROC in standard classification. In this work, we define 5 requirements for multi-threshold metrics in selective classification regarding task alignment, interpretability, and flexibility, and show how current approaches fail to meet them. We propose the Area under the Generalized Risk Coverage curve ( AUGRC ), which meets all requirements and can be directly interpreted as the average risk of undetected failures. We empirically demonstrate the relevance of AUGRC on a comprehensive benchmark spanning 6 data sets and 13 confidence scoring functions. We find that the proposed metric substantially changes metric rankings on 5 out of the 6 data sets.

1 Introduction

Selective Classification (SC) is increasingly recognized as a crucial component for the reliable deployment of machine learning-based classification systems in real-world scenarios, such as clinical diagnostics [Dvijotham et al., 2023, Leibig et al., 2022, Dembrower et al., 2020, Yala et al., 2019]. The core idea of SC is to equip models with the ability to reject low-confidence predictions, thereby enhancing the overall reliability and safety of the system [Geifman and El-Yaniv, 2017, Chow, 1957, El-Yaniv et al., 2010]. This is typically achieved through three key components: a classifier that makes the predictions, a confidence scoring function (CSF) that assesses the reliability of these predictions, and a rejection threshold τ that determines when to reject a prediction based on its confidence score.

In evaluating SC systems, two primary concepts are essential: risk and coverage. Risk expresses the classifier's error potential and is typically measured by its misclassification rate. Coverage, on the other hand, indicates the proportion of instances where the model makes a prediction rather than rejecting it. An effective SC system aims to minimize risk while maintaining high coverage, ensuring that the model provides accurate predictions for as many instances as possible.

Current evaluation of SC systems often focuses on fixed working points defined by pre-set rejection thresholds[Geifman and El-Yaniv, 2017, Liu et al., 2019, Geifman and El-Yaniv, 2019]. For instance, a common evaluation metric might be the selective risk at a given coverage of 70% [Geifman and El-Yaniv, 2019], which communicates the potential risk associated with a specific confidence score c and selection threshold τ to a patient or end-user. While this method is useful for communicating risk in specific instances, it does not provide a comprehensive evaluation of the system's overall performance. In standard classification, this is analogous to the need for the Area Under the Receiver Operating Characteristic (AUROC) curve rather than evaluating sensitivity at a specific specificity, which can be highly misleading when assessing general classifier capabilities [MaierHein et al., 2022]. The AUROC provides a holistic view of the classifier's performance across all possible thresholds, thereby driving methodological progress. Similarly, SC requires a multithreshold metric that aggregates performance across all rejection thresholds to fully benchmark the system's capabilities.

Several current approaches attempt to address this need for multi-threshold metrics in SC. The Area Under the Risk Coverage curve ( AURC ) is the most prevalent of these Geifman et al. [2018]. However, we demonstrate in this work that AURC has significant limitations as it fails to adequately translate the risk from specific working points into a meaningful aggregated evaluation score. This shortcoming hampers the ability to benchmark and improve SC methodologies effectively.

In our work, we aim to bridge this gap by providing a robust evaluation framework for SC. Our contributions are as follows:

Refined Task Formulation: We are the first to provide a comprehensive SC evaluation framework, explicitly deriving meaningful task formulations for different evaluation and application scenarios such as working point versus multi-threshold evaluation.

Formulation of Requirements: We define five critical requirements for multi-threshold metrics in SC, focusing on task suitability, interpretability, and flexibility. We assess current multi-threshold metrics against our five requirements and demonstrate their shortcomings.

Proposal of AUGRC: We introduce the Area Under the Generalized Risk Coverage ( AUGRC ) curve, a new metric designed to overcome the flaws of current multi-threshold metrics for SC. AUGRC meets all five requirements, providing a comprehensive and interpretable measure of SC system performance.

Empirical Validation: We empirically demonstrate the relevance and effectiveness of AUGRC through a comprehensive benchmark spanning 6 datasets and 13 confidence scoring functions.

In summary, our work presents a significant advancement in the evaluation of SC systems, offering a more reliable and interpretable metric that can drive further methodological progress in the field.

2 Refined Task Formulation

Selective Classification (SC) systems consist of a classifier m : X → Y , which outputs a prediction and a confidence scoring function (CSF) g : X → R , which outputs a confidence score associated with the prediction. Assuming a supervised training setup, let { ( x i , y i ) } N i =1 be a dataset containing N independent samples from the source distribution P ( X,Y ) over X × Y . Given a rejection threshold τ , the model prediction is accepted only if the corresponding score is larger than τ :

̸

Given an error function ℓ : Y × Y → R + , the overall risk of the classifier m is given by R ( m ) := 1 N ∑ N i =1 ℓ ( m ( x ) , y ) . The error function thereby contains the measure of classification performance suitable for the task at hand. Commonly, SC literature assumes a 0/1 error corresponding to the failure indicator variable Y f with y f ,i ( x i , y i , m ) := I [ y i = m ( x i )] . In this case, the overall risk represents the probability of misclassification P ( Y f = 1) . By rejecting low-confidence predictions, the selective classifier can decrease the risk of misclassifications at the cost of letting the classifier make predictions on only a fraction of the dataset. This fraction is denoted as the coverage := P ( g ( x ) ≥ τ ) , with the respective empirical estimator 1 N ∑ N i =1 I ( g ( x i ) ≥ τ ) . Evaluation of SC

1. Scan over

rejection

thresholds

c

τ

c

<

τ

Method Development ('Multi-threshold eval.')

Patient

Accept pred.

Reject pred.

Incorrect prediction

('Silent' failure)

Correct prediction

Application ('Working point eval.')

Selective risk:

P(fail | accept)

Generalized Risk:

P(fail, accept)

Risk

Coverage

Classifier

CSF

Prediction

Conf.

score

Requirement

Description

R1

Task Alignment

sensitive to both factors: classification

performance and CSF ranking quality

✓

✓

✕

✓

R2

Monotonicity

improving one factor while keeping the other

one fixed leads to a better metric value

✓

✕

✕

✕

✕

R3

Ranking Interpretability

CSF ranking quality is based on permutation

distance to the optimum

✓

✕

✕

✓

✓

R4

CSF Flexibility

flexible w.r.t. CSF

✓

✓

✓

✓

✕

R5

Error Flexibility

flexible w.r.t. error function

(e.g. beyond accuracy)

✓

✕

✕

Current practice

(AURC)

Ours

(AUGRC)

AUROCf

AURC

AUGRC

NLL/

Brier

Score

e

-

AURC

(ours)

c

Generalized risk: 'The general risk of the system

to  produce a 'silent' failure for any prediction'

Selective risk: 'Given    and    , the risk of a

'silent' failure occuring for an accepted prediction'

τ

c

τ

0

. . .

τ

n

Input Data

Selective Classification System

Reject if

2. Get area under

Risk-coverage curve

c

<

τ

a)

b)

systems often focuses on preventing "silent", i.e. undetected failures for which both y f ,i = 1 and g ( x i ) ≥ τ .

Deploying a SC system in practice requires the selection of a fixed rejection threshold τ . In the following refined evaluation protocol, we distinguish between the well-established application-specific task formulation where SC models are evaluated at individual working points (Section 2.1) and SC method development where evaluation is independent of individual working points (Section 2.2).

2.1 Evaluating SC systems in applied settings

Current Selective Classification evaluation commonly reports the selective risk of the system at a given coverage ('risk@coverage'), which implies a pre-determined cutoff on the risk or coverage (e.g. "maximum risk of 5%") leading to a working point of the system defined by a rejection threshold τ . The selective risk is defined as:

This formulation only considers accepted predictions with c > τ . For a binary failure error, this risk is an empirical estimator for the conditional probability P ( Y f = 1 | g ( x ) ≥ τ ) . Thus, this metric effectively communicates the risk of a "silent" failure for a specific prediction of classifier m that has been accepted ( c > τ ) at a pre-defined threshold τ . This information may be useful in applied medical settings, e.g. to communicate the risk of misclassification for a patient p whose associated

prediction has been accepted by a given SC system ( c p > τ ). Aside from the selective risk, other performance measurements for working point selection include for example Classification quality and Rejection quality [Condessa et al., 2017].

2.2 Evaluating SC systems for method development and benchmarking

Evaluating the general performance of a Selective Classification system requires a multi-threshold metric for a more comprehensive view compared to a working point analysis based on a fixed singular rejection threshold, analogous to binary classification tasks, where the Area Under the Receiver Operating Characteristic (AUROC) is used to assess the entire system performance across multiple classifier thresholds. However, the working point analysis can not be simply translated to multithreshold aggregation, as it is based on Selective Risk, which only considers the risk w.r.t accepted predictions, assuming that a specific selection has already occurred ( P ( Y f = 1 | g ( x ) ≥ τ ) ). This evaluation ignores rejected cases and thus contradicts the holistic assessment, where we are interested in the general risk for any prediction causing a silent failure independent of the future selection decision of individual predictions. This holistic assessment is reflected by the joint probability P ( Y f = 1 , g ( x ) ≥ τ ) , which reflects the risk of silent failure for any prediction processed by the system including cases that are potentially rejected in the future.

To address the discrepancy between the need for holistic assessment versus the selective risk's focus on already selected cases, we formulate the generalized risk:

This metric reflects the joint probability of misclassification and acceptance by the confidence score threshold. By aggregating this risk over multiple rejection thresholds, we can evaluate the overall performance of the SC system.

2.3 Requirements for Selective Classification multi-threshold metrics

Based on the refined task definition given above, we can formulate concrete five requirements R1-R5 for multi-threshold metrics in Selective Classification:

R1: Task Alignment. The general goal in SC is to prevent silent failures either by preventing failures in the first place (via classifier performance), or by detecting the remaining failures (via CSF ranking quality). As argued in Jäger et al. [2023], it is crucial to jointly evaluate both aspects of an SC system, since the choice of CSF generally affects the underlying classifier. R2: Monotonicity. The metric should be monotonic w.r.t both evaluated factors stated in R1, i.e. improving on one of the two factors while keeping the other one fixed results in an improved metric value. Note that R2 does not make assumptions about how the two factors are combined, but it represents a minimum requirement for meaningful comparison and optimization of the metric. R3: Ranking Interpretability. Interpretability is a crucial component of a metric [Maier-Hein et al., 2022]. AUROC is the de facto standard ranking metric for binary classification tasks, providing an intuitive assessment of ranking quality which is proportional to the number of permutations needed for establishing an optimal ranking. This can also be interpreted as the "probability of a positive sample having a higher score than a negative one". Ideally, the SC metric should follow this intuitive assessment of rankings. R4: CSF Flexibility. The metric should be applicable to arbitrary choices of CSFs. This includes external CSFs, i.e. scores that are not based directly on the classifier output. R5: Error Flexibility. Current SC literature largely focuses on 0/1 error (i.e. 1 -accuracy) in their risk computation. However, in many real-world scenarios, accuracy is not an adequate classification metric, such as in the presence of class imbalance and for pixel-level classification. Thus, the SC metric should be flexible w.r.t the choice of error function.

2.4 Current multi-threshold metrics in SC do not fulfill requirements R1-R5

AURC: Geifman et al. [2018] derive the AURC as the Area under the Selective Risk-Coverage curve. This metric is the most prevalent multi-threshold metric in SC [Geifman et al., 2018, Jäger et al., 2023, Bungert et al., 2023, Cheng et al., 2023, Zhu et al., 2023a, Varshney et al., 2020, Naushad and Voiculescu, 2024, Van Landeghem et al., 2024, 2023, Zhu et al., 2022, Xin et al.,

2021, Yoshikawa and Okazaki, 2023, Ding et al., 2020, Zhu et al., 2023b, Galil and El-Yaniv, 2021, Franc et al., 2023, Cen et al., 2023, Xia and Bouganis, 2022, Cattelan and Silva, 2023, Tran et al., 2022, Kim et al., 2023, Ashukha et al., 2020, Xia et al., 2024]. For the 0/1-error, AURC can be expressed through the following integral:

This integral effectively aggregates the Selective Risk (Equation 2) over all fractions of accepted predictions (i.e. coverage). However, as discussed in Section 2.2, the Selective Risk is not suitable for aggregation over rejection thresholds to holistically assess a SC system. This inadequacy effectively leads to an excessive over-weighting of high-confidence failures in AURC and in return to breaching the requirements of monotonicity (R2) and ranking interpretability (R3). We provide empirical evidence for these shortcomings on toy data (Section 3) and real-world data (Section 4.2). Further, current SC literature usually employs the 0/1 error function, even though the related Accuracy metric is well known to be an unsuitable classification performance measure for many applications. For example, Balanced Accuracy is used to address class imbalance and metrics such as the Dice Score are employed for segmentation. Moving beyond the 0/1 error yields a higher variance in distinct error values and may thus amplify the shortcoming of over-weighting high-confidence failures. As more sophisticated error functions being are an important direction of future SC research, it is crucial to ensure the compatibility of metrics in SC evaluation.

e-AURC: Geifman et al. [2018] further introduce the excessAURC as the difference to the AURC of an optimal confidence scoring function (denoted as AURC ∗ )

It relies on the intuition that subtracting the optimal AURC eliminates the contribution of the overall classification performance and collapses it to a pure ranking metric [Geifman et al., 2018, Galil et al., 2023, Jäger et al., 2023]. However, several works demonstrated that this intuition does not hold and that the e -AURC is still sensitive to the overall classification performance [Galil et al., 2023, Cattelan and Silva, 2023]. We attribute this deviation from the intended behavior to the shortcomings of the underlying AURC , as we find the desired isolation of classifier performance in our improved metric formulation in Section 3. e-AURC further inherits the shortcomings of AURC w.r.t monotonicity (R2), ranking interpretability (R3), and error flexibility (R5).

NLL / Brier Score: Importantly, proper scoring rules such as the Negative-Log-Likelihood and the Brier Score are technically not multi-threshold metrics. Yet, we include them here as they also aim for a holistic performance assessment, i.e. assessment beyond individual working points, by evaluating a general meaningfulness of scores based on the softmax-output of the classifier [Ovadia et al., 2019]. Thereby, they jointly assess ranking and calibration of scores, which dilutes the focus on ranking quality in the context of SC [Jäger et al., 2023]. In our formulation, the calibration aspect in these metrics breaks the required monotonicity w.r.t SC evaluation (R2). Further, they are not applicable to CSFs beyond those that are directly based on the classifier output (R4). AUROCf: The "failure version" of the standard AUROC assesses the correctness of predictions with a binary failure label [Jäger et al., 2023]. Based on the AUROC, it provides an intuitive ranking quality assessment. However, it ignores the underlying classifier performance (R1, R2), and is restricted to binary error functions (R5). OC-AUC: Kivlichan et al. [2021] introduce the Oracle-Model Collaborative AUC, where first a fixed threshold is applied on the confidence scores, above which error values are set to zero. Then, the AUROC f (or the Average Precision) are evaluated. This metric is also reported in [Dehghani et al., 2023, Tran et al., 2022], with a review fraction of 0.5%. OC-AUC is subject to the same pitfalls as AUROC f (R1, R2, R5). AUROC-AURC: Pugnana and Ruggieri [2023] propose the AUROC-AURC where the Selective Risk is defined as the classification (not failure) AUROC computed over the set of accepted predictions. However, it is only applicable to binary classification models with binary error functions (R5). Further, in employing Selective Risk it inherits the pitfalls described for AURC regarding monotonicity (R2) and ranking interpretability (R3). NAURC: Cattelan and Silva [2023] propose the NormalizedAURC , a min-max scaled version of the e -AURC , and claim that this modification eliminates its lack of interpretability. However, given the linear relationship with the AURC , it does not fulfill the requirements R2 and R3. F1-AUC: Malinin and Gales [2020], Malinin et al. [2021] introduce the notion of Error-Retention Curves, which corresponds to that of Generalized Risk Coverage Curves. However, in Malinin et al. [2021] the authors propose an F1-AUC metric which is only applicable to binary error functions (R5) and

breaks monotonicity (R2), as it decreases with increasing accuracy for accuracy values above ≈ 0 . 56 (see Appendix A.1.3 for a detailed explanation.) ARC: Accuracy-Rejection-Curves [Nadeem et al., 2009, Ashukha et al., 2020, Condessa et al., 2017] and the associated AUC directly correspond to the AURC and are therefore subject to the same pitfalls (R2, R3).

1

20

1

20

1

20

accuracy = 0.9

AUROCf = 0.5

AURC = 205

e-AURC = 200

AUGRC = 50

CSF-1

(a)

(b)

0.0

0.2

0.4

0.6

0.8

1.0

Coverage

0.0

0.5

1.0

Selective Risk

CSF-1

CSF-2

CSF-3

(d)

(e)

equal ranking quality

equal accuracy

worse ranking quality

worse accuracy

better AURC

better AURC

0.0

0.2

0.4

0.6

0.8

1.0

Coverage

0.0

0.1

0.2

Generalized Risk

CSF-1

CSF-2

CSF-3

(c)

correct

failure

1

200

400

600

800

1000

rank (descending confidence scores, N = 1000)

0

2

4

6

failure contribution

×

10 3

AURC-induced weighting

natural ranking

characteristic (AUGRC)

(f)

accuracy = 0.9

AUROCf = 0.5

AURC = 69

e-AURC = 64

AUGRC = 50

CSF-2

accuracy = 0.8

AUROCf = 0.44

AURC = 169

e-AURC = 175

AUGRC = 110

CSF-3

correct

failure

correct

failure

rank (conf. high     low)

rank (conf. high     low)

rank (conf. high     low)

R2

R2

3 Area under the Generalized Risk Coverage Curve

In Section 2.2, we illustrate that aggregating SC performance across working points requires to shift the perspective from the Selective Risk to the Generalized Risk (Equation 3) as an holistic assessment of the risk of silent failures for all predictions, before the rejection decision is made. We propose to evaluate SC methods via the Area under the Generalized Risk Coverage Curve ( AUGRC ). For the binary failure error it becomes an empirical estimator of the following expression:

This metric evaluates SC models in terms of the rate of silent failures averaged over working points and thus provides a practical measurement that is directly interpretable. It is bounded to the [0 , 1 / 2 ] interval, whereby lower AUGRC corresponds to better SC performance. The AUGRC is not subject to the shortcomings of the AURC , and we can derive a direct relationship to AUROC f and acc (the derivation and visualizations are shown in Equation 8 and Figure 5):

To illustrate, when drawing two random samples, it corresponds to the probability that either both are failure cases or that one is a failure and has a higher confidence score than the non-failure. This

is reflected by the second and first term of Equation 7, respectively. As an example, a CSF that outputs random scores yields AUROC f = 1 2 , and hence AUGRC = 1 2 (1 -acc) . The AUGRC for an optimal CSF ( AUROC f = 1 ) is given by the second term in Equation 7, hence subtracting the optimal SC performance yields a pure ranking measure re-scaled by the probability of finding a positive/negative pair. This overcomes the lack of interpretability of AURC and e -AURC (R3). Monotonicity (R2) is ensured since the partial gradients w.r.t. both AUROC f and acc are always negative. Further, it can accommodate arbitrary classification metrics via the error function ℓ (R4) as well as arbitrary CSF (R5).

Figure 2a depicts the metric contribution of individual failure cases depending on their ranking position. This shows empirically how the excessive over-weighting of high-confidence failures in AURC is resolved by AUGRC , and how the intuitive ranking assessment of AUROC f is established (see Section A.1.1 for a detailed derivation). We further showcase how AUGRC resolves the broken monotonicity requirement (R2) of AURC in Figures 2 b-d. Despite equal AUROC f and equal acc in CSF-1 and CSF-2, the AURC improves. And AURC even improves in CSF-3, which features lower AUROC f and lower acc compared to CSF-1. Figures 2 e-f depict the associated risk-coverage curves and reveal that the non-intuitive behavior of AURC is due to the excessive effect of the highconfidence failure of CSF-1 on the Selective Risk, which is resolved in the Generalized Risk. In Section 4.2 we demonstrate implications of AURC 's shortcomings on real-world data.

4 Empirical Study

We conduct an empirical study based on the existing FD-Shifts benchmarking framework [Jäger et al., 2023], which compares various CSFs across a broad range of datasets. The focus of our study is not on the performance of individual methods (CSFs) but rather on the ranking of methods based on AUGRC evaluation as compared to AURC . We utilize the same experimental setup as in Jäger et al. [2023] with the addition of CSF scores based on temperature-scaled classifier logits. Extending the benchmark by recent SC methods, e.g. Feng et al. [2022], is an interesting direction of future work. A detailed overview of the datasets and methods used can be found in Appendix A.2. The code for reproducing our results and a PyTorch implementation of the AUGRC are available at: https://github.com/IML-DKFZ/fd-shifts .

4.1 Comparing Method Rankings of AUGRC and AURC

To study the relevance of AUGRC , we investigate the changes induced by AUGRC on rankings of CSFs compared to AURC method rankings.

CSF Rankings with AUGRC substantially deviate from AURC rankings . Figure 3 illustrates the method ranking differences for all i.i.d. test datasets, showing changes in ranks across all CSFs. Notably, AUGRC induces changes in the top-3 methods (out of 13) on 5 out of 6 datasets. To ensure that these ranking changes are not due to variability in the test data, we evaluate the metrics on 500 bootstrap samples from the test dataset and derive the compared method rankings based on the average rank across these samples ("rank-then-aggregate"). The size of each bootstrap sample corresponds to the size of the test dataset. Metric values for each bootstrap sample are averaged over 5 training initializations (2 for BREEDS and 10 for CAMELYON-17-Wilds). We analyze the robustness of the resulting method rankings for AURC and AUGRC separately, based on statistical significance. To that end, we perform pairwise CSF comparisons in form of a one-sided Wilcoxon signed-rank test [Wilcoxon, 1992] on the bootstrap samples. To control the family-wise error rate at a 5% significance level, we apply the Holm correction for multiple testing per metric and dataset, following [Wiesenfarth et al., 2021] (see Figure 7 for the results without correction for multiple testing). The resulting significance maps displayed in Figure 3 indicate stable method rankings for both AURC and AUGRC . This suggests that the observed ranking differences are induced by the conceptual difference of AUGRC compared to AURC and, as a consequence, that the shortcomings of current metrics and respective solutions discussed in Section 2, are not only conceptually sound, but also highly relevant in practice. For the results in Figure 3, we select the DeepGamblers reward parameter and whether to use dropout for both metrics separately on the validation dataset (details in Table 1 and Table 2).

The shortcomings of AURC affect CSF comparison across datasets and distribution shifts . The method rankings for all datasets and distribution shifts, based on the original test datasets, are shown

in Table 4, which also includes results for equal hyperparameters for AURC and AUGRC . Results for the bootstrap-based method ranking analysis for distribution shifts are displayed in Figure 6. The substantial differences in method rankings across all datasets and distribution shifts underline the relevance of AUGRC for SC evaluation. The complete AUGRC results on the FD-Shifts benchmark are shown in Table 3.

In few cases, as shown in Figure 3, we observe that a CSF A is ranked worse than a neighboring CSF B even though CSF A's ranks are significantly superior to those of CSF B based on the Wilcoxon statistic over the bootstrap samples. This discrepancy can occur if the rank variability in CSF A is larger than in CSF B. While this does not affect our conclusions regarding the relevance of the AUGRC , it indicates that selecting a single best CSF for application based solely on method rankings should be approached with caution. We recommend closer inspection of the individual CSF performance in such cases, although this is not the focus of our study.

BREEDS-Entity-13

CIFAR-10

CAMELYON-17-Wilds

AUGRC

AURC

iWildCam

CIFAR-100

SVHN

CSF rankings

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AURC

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CSF rankings

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CSF rankings

CSF rankings

CSFs:

MSR

PE

MLS

MCD-MSR

MCD-PE

MCD-EE

TEMP-MLS

DG-TEMP-MLS

DG-PE

ConfidNet

Devries et al.

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*

tied ranks

4.2 Showcasing Implications of AURC Shortcomings on Real-world Data

Figure 4 gives an in-depth look at how the conceptual shortcomings of AURC affect method assessment on real-world data. The example uses the confidence like reservation score of the DeepGamblers method (DG-Res) [Liu et al., 2019] and the Monte Carlo Dropout-based predictive entropy (MCD-PE) [Gal and Ghahramani, 2016] as CSFs on the CIFAR-10 test dataset. Despite DG-Res having higher classification performance and ranking quality than MCD-PE, the AURC erroneously favors MCD-PE over DG-Res. This violates the monotonicity requirement (R2). This can be attributed to the excessive contributions of only few high-confidence failures, which aligns with the theoretical findings on failure contributions shown in Figure 2 (R3). In this example, the highconfidence failures are associated with high label ambiguity or incorrect labeling, suggesting that the AURC may exacerbate the influence of label noise in practice.

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0.00

0.01

0.02

0.03

0.04

0.05

Selective Risk

MCD-PE

GT: truck

Pred: automobile

Confid: 1.0000

GT: truck

Pred: automobile

Confid: 0.9999

GT: truck

Pred: automobile

Confid: 0.9998

GT: cat

Pred: frog

Confid: 0.9998

GT: cat

Pred: frog

Confid: 0.9896

GT: cat

Pred: dog

Confid: 0.9892

GT: truck

Pred: automobile

Confid: 0.9891

GT: truck

Pred: automobile

Confid: 0.9889

0.0

0.2

0.4

0.6

0.8

1.0

Coverage

0.00

0.02

0.04

Generalized Risk

MCD-PE

MCD-PE

AUROCf = 0.9340

Accuracy = 0.950

AURC = 5.53

AURC = 5.35

Accuracy = 0.946

AUROCf = 0.9338

>

>

>

AUGRC = 4.34

AUGRC = 4.82

<

(a)

(b)

5 Conclusion

Despite the increasing relevance of SC for reliable translation of machine learning systems to realworld application, we find that the current metrics have limitations in providing the comprehensive assessment needed to guide the methodological progress of SC systems.

In this work, we establish a systematic SC evaluation framework, thereby promoting the adoption of more comprehensive, interpretable, and task-aligned metrics for comparative benchmarking of SC systems.

We find that none of the existing multi-threshold metrics, particularly the AURC , meet the key requirements we identified for comprehensive SC evaluation, leading to deviations from intended and intuitive performance assessment behaviors. To address this, we introduce the AUGRC as a suitable metric for comprehensive SC method evaluation. Substantial differences in method rankings

between AURC and AUGRC , demonstrated through extensive empirical studies, highlight the importance of selecting the right SC metric. Thus, we propose the adoption of AUGRC for meaningful SC performance evaluation.

When evaluating specific applications for which certain risk or coverage intervals are known to be irrelevant, adaptations such as a partial AUGRC (analogous to the partial AUROC) may be considered. Further, investigating the relation between confidence ranking and calibration and on evaluating SC in settings where calibrated scores are of interest is an interesting direction of future work. Overall, our proposed evaluation framework provides a solid basis for future work in Selective Classification, including developing novel methods as well as analyzing the properties of individual methods.

Acknowledgments and Disclosure of Funding

This work was partly funded by Helmholtz Imaging (HI), a platform of the Helmholtz Incubator on Information and Data Science. We thank Lukas Klein, Maximilian Zenk and Fabian Isensee for insightful discussions and feedback.

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A Appendix

A.1 Technical Details

A.1.1 AURC Failure Contribution

Let { τ t } N t =1 provide a total order on the predictions (relaxing the assumption to a partial order leads to a more complicated but very similar formulation). Then, for a rejection threshold τ t , a failure case at rank t ∗ contributes to the Selective Risk (Equation 2) by 1 t if t ∗ ≤ t . Averaging over all thresholds yields a contribution through the failure at rank t ∗ to the AURC of 1 N ∑ N t = t ∗ 1 t . For the Generalized Risk, the contribution at t ∗ ≤ t is always 1 N , resulting in an overall contribution of N -t ∗ -1 N 2 to the AUGRC . The failure contribution curves are displayed in Figure 2.

A.1.2 Relationship between AUGRC and failure AUROC

̸

Let Y f = I ( m ( x ) = y ) be a binary indicator variable for classification failures. Then, the Generalized Risk corresponds to the joint probability that a sample is accepted and wrongly classified. We can write the AUGRC as follows:

At ( ∗ ), we use that P ( g ( x ) ≥ τ ) = P ( g ( x ) ≥ τ | Y f = 0) · acc + P ( g ( x ) ≥ τ | Y f = 1) · (1 -acc ) . On choosing two random samples, the second term in the final equation represents the probability that both are failures and the first term represents the probability that one is a failure, the other is not, and that the failure has higher confidence score than the other.

The AUGRC is monotonic in acc and AUROC f as both partial derivatives are negative ∀ acc ∈ (0 , 1) , ∀ AUROC f ∈ [0 , 1] :

0.0

0.2

0.4

0.6

0.8

1.0

Coverage

0.0

0.2

0.4

0.6

0.8

1.0

Risk

Selective Risk

Generalized Risk

0.0

0.2

0.4

0.6

0.8

1.0

AUROC

0.0

0.2

0.4

0.6

0.8

1.0

0.000

0.075

0.150

0.225

0.300

0.375

0.450

AUGRC

A.1.3 F1-AUC does not fulfill R2

For the optimal CSF, all failure cases are attributed to lower confidence scores than correct predictions. Hence, the precision P ( Y f = 0 | g ( x ) ≥ τ ) is one for coverages up to P ( Y f = 0) and decreases as P ( Y f = 0) /P ( g ( x ) ≥ τ ) above. Following the task formulation as defined in Malinin et al. [2021], we can derive the following analytical expression for the optimal F1-AUC ∗ :

F1-AUC ∗ increases with increasing accuracy up to P ( Y f = 0) ≈ 0 . 56 , then it decreases, favoring models with lower classification performance. Thus, the F1-AUC does not fulfill the monotonicity requirement (R2).

A.2 Experiment Setup

A.2.1 Datasets and Methods

We compare the following CSFs: From the classifier's logits we compute the Maximum Softmax Response (MSR), Maximum Logit Score (MLS), Predictive Entropy (PE), and the MLS based on temperature-scaled logits, for which a scalar temperature parameter is tuned based on the validation set [Guo et al., 2017]. Three predictive uncertainty measures are based on Monte Carlo Dropout [Gal and Ghahramani]: mean softmax (MCD-MSR), predictive entropy (MCD-PE), and mutual information (MCD-MI). We additionally include the DeepGamblers method [Liu et al.], which learns

a confidence like reservation score (DG-Res), ConfidNet Corbière et al. [2019], which is trained as an extension to the classifier, and the work of DeVries and Taylor [2018].

We evaluate SC methods on the FD-Shifts benchmark [Jäger et al., 2023], which considers a broad range of datasets and failure sources through various distribution shifts: SVHN [Netzer et al., 2011], CIFAR-10, and CIFAR-100 [Krizhevsky] are evaluated on semantic and non-semantic new-class shifts in a rotating fashion including Tiny ImageNet [Le and Yang, 2015]. Additionally, we consider sub-class shifts on iWildCam [Koh et al., 2021], BREEDS-Enity-13 [Santurkar et al., 2020], CAMELYON-17-Wilds [Koh et al., 2021], and on CIFAR-100 (based on super-classes) as well as corruption shifts based on Hendrycks and Dietterich [2019] on CIFAR-100. The data preparation and splitting are done as described in Jäger et al. [2023]. For the corruption shifts, we reduce the test split size to 75000 (subsampled within each corruption type and intensity level).

The following classifier architecture are used in the benchmark: small convolutional network for SVHN, VGG-13 [Simonyan and Zisserman, 2014] on CIFAR-10/100, ResNet-50 [He et al., 2016] on the other datasets.

If the distribution shift is not mentioned explicitly, we evaluate on the respective i.i.d. test datasets.

Our method ranking study focuses on the evaluation of CSF performance based on the existing FD-Shifts benchmark, hence we required no GPU's for the analysis in Section. 4. As both AURC and AUGRC can be computed efficiently, (CPU) evaluation time for a single test set is less than a minute; evaluation on 500 bootstrap samples on a single CPU core take around 3 hours.

A.2.2 Hyperparameters and Model Selection

The experiments are based on the same hyperparameters as reported in Table 4 in Jäger et al. [2023]. Based on the performance on the validation set, we choose the DeepGambler reward hyperparameter and whether to use dropout (for the non-MCD-based CSFs). For the former, we select from [2 . 2 , 3 , 6 , 10] on Wilds-Camelyon-17, CIFAR-10, and SVHN, from [2 . 2 , 3 , 6 , 10 , 15] on iWildCam and BREEDS-Entity-13, and from [2 . 2 , 3 , 6 , 10 , 12 , 15 , 20] on CIFAR-100. When performing model selection based on the AURC metric, we obtain the same configurations as reported in Jäger et al. [2023]. When performing model selection based on the AUGRC metric, we obtain the parameters as reported in Tab. 1 and Tab. 2. For temperature scaling, we optimize the NLL on the validation set using the L-BFGS algorithm with lr = 0 . 01 .

A.3 AUGRC Computation Time

The main bottleneck of the AUGRC computation is the sort operation on the confidence scores. This is the same as for the AURC and there is no computational overhead compared to the AURC . For a small benchmark of the computational time, we evaluate the AURC , AUGRC , and AUROC f on 1000 random scores and failure labels. We obtain the following results (averaged over 5000 cases): 564 µs ± 2 µs ( AURC ), 562 µs ± 2 µs ( AUGRC ), 730 µs ± 13 µs ( AUROC f ).

A.4 Additional Results

Table 3 shows the AUGRC results for the 13 compared CSFs across all datasets and distribution shifts. While the MSR baseline is not consistently outperformed across all experiments by any of the compared CSFs, MCD improves MSR scores in most settings. Temperature scaling does not exhibit consistent improvement of MLS scores.

Table 3: FD-Shifts benchmark results measured as AUGRC × 10 3 (score range: [0, 500], lower is better ↓ ). The color heatmap is normalized per column, whereby whiter colors depict better scores. "cor" is the average over 5 intensity levels of image corruption shifts. AUGRC scores are averaged over 10 runs on CAMELYON-17-Wilds, over 2 runs on BREEDS, and over 5 runs on all other datasets. Abbreviations: ncs: new-class shift (s for semantic, ns for non-semantic), iid: independent and identically distributed, sub: sub-class shift, cor: image corruptions, c10/100: CIFAR-10/100, ti: TinyImagenet.

iWildCam

BREEDS

CAMELYON

CIFAR-100

CIFAR-10

SVHN

study

iid

sub

iid

sub

iid

sub

iid

sub

cor

s-ncs

ns-ncs

iid

cor

s-ncs

ns-ncs

iid

ns-ncs

ncs-data set

c10

svhn

ti

c100

svhn

ti

c10

c100

ti

MSR

48.0

55.0

6.86

117

8.13

91.3

54.6

150

187

210

334

203

4.93

67.9

166

291

154

3.44

48.5

48.2

48.6

MLS

60.3

62.4

9.28

121

8.13

91.3

65.6

155

200

213

337

195

5.56

68.4

163

288

150

3.97

46.5

46.1

46.6

PE

48.2

55.2

6.87

116

8.13

91.3

56.4

150

188

210

333

202

4.91

67.2

165

290

152

3.44

47.9

47.5

48.0

MCD-MSR

42.8

54.3

6.89

116

5.64

95.1

53.1

146

176

210

336

202

4.56

60.4

166

294

156

3.40

47.3

47.4

47.9

MCD-PE

43.4

55.1

6.98

116

5.64

95.1

56.6

146

179

210

335

198

4.73

60.6

164

293

153

3.47

46.3

46.4

46.9

MCD-MI

43.7

58.1

7.28

118

6.13

101

59.4

148

182

211

333

197

4.99

64.9

167

302

161

3.55

47.0

47.7

48.2

ConfidNet

82.0

91.2

6.89

117

4.36

86.1

57.5

154

192

214

340

200

4.70

65.4

165

291

153

3.43

48.6

48.4

48.8

43.9

59.6

6.31

112

3.47

149

52.5

145

175

210

336

203

4.75

60.8

168

296

159

3.38

47.6

47.7

48.0

66.5

63.8

9.39

122

3.46

124

64.3

230

194

215

330

197

4.40

63.8

167

287

151

4.09

46.8

46.2

46.6

Devries et al.

61.7

74.7

8.24

120

24.3

145

69.6

150

214

222

342

211

4.57

70.8

166

287

152

5.56

48.3

48.5

49.9

TEMP-MLS

48.2

55.1

6.86

116

8.13

91.3

54.8

150

187

210

333

203

4.92

67.6

166

291

153

3.44

48.3

47.9

48.4

DG-PE

52.0

69.7

6.33

114

3.46

123

55.2

151

186

210

329

197

4.09

62.9

167

288

154

3.41

48.9

48.4

48.7

DG-TEMP-MLS

52.2

69.9

6.35

114

3.46

123

55.0

222

184

211

331

199

4.09

63.1

167

288

155

3.41

49.0

48.5

48.8

iWildCam

BREEDS

CAMELYON

CIFAR-100

CIFAR-10

SVHN

study

iid

sub

iid

sub

iid

sub

iid

sub

cor

s-ncs

ns-ncs

iid

cor

s-ncs

ns-ncs

iid

ns-ncs

ncs-data set

c10

svhn

ti

c100

svhn

ti

c10

c100

ti

metric

α

β

α

β

α

β

α

β

α

β

α

β

α

β

α

β

α

β

α

β

α

β

α

β

α

β

α

β

α

β

α

β

α

β

α

β

α

β

α

β

α

β

optimized both (w.r.t. dropout and DG reward parameter)

MSR

5

5

6

2

6

4

7

9

9

9

2

2

3

3

5

5

7

7

3

1

3

6

11

10

12

12

11

11

5

6

9

7

8

9

8

6

10

10

9

9

9

9

MLS

10

10

10

8

13

12

13

12

9

9

2

2

11

12

7

11

12

12

9

10

7

11

1

1

13

13

12

12

1

1

2

3

1

1

11

11

3

3

1

1

1

1

PE

6

6

7

5

8

6

6

4

9

9

2

2

10

7

9

5

11

9

10

1

12

4

6

8

10

10

9

9

4

4

6

6

2

3

8

6

7

7

6

6

6

6

MCD-MSR

1

1

1

1

3

7

3

4

6

6

6

7

2

2

2

2

2

2

1

1

9

8

9

8

3

4

2

2

5

6

12

12

10

12

4

2

5

5

5

5

5

5

MCD-PE

2

2

2

3

4

9

3

4

6

6

6

7

7

8

2

2

3

3

1

1

6

7

3

4

5

7

3

3

3

2

11

11

8

6

6

9

1

1

4

4

4

4

MCD-EE

3

3

3

6

5

10

3

4

8

8

6

6

8

9

2

2

4

4

6

8

7

8

6

6

7

9

1

1

2

2

9

7

2

3

8

10

2

2

3

3

3

3

ConfidNet

13

13

13

13

10

7

9

9

5

5

1

1

9

10

7

10

9

10

11

11

11

12

4

6

9

6

8

8

9

4

6

7

6

6

5

5

11

11

10

10

10

11

4

4

9

7

1

1

1

1

4

4

12

13

1

1

1

1

1

1

3

1

10

8

11

10

6

8

4

4

10

13

13

13

13

13

1

1

6

6

7

7

7

6

12

12

11

9

12

13

12

13

1

1

9

11

12

11

13

13

10

11

12

12

1

2

2

2

8

3

7

7

13

10

3

1

5

2

12

12

4

4

2

2

2

1

Devries et al.

11

11

12

12

11

11

11

11

13

13

13

12

13

13

10

5

13

13

13

13

13

13

13

13

4

5

13

13

5

6

1

1

2

3

13

13

8

8

13

12

13

13

TEMP-MLS

7

6

8

3

8

4

7

4

9

9

2

2

5

4

5

5

8

7

3

1

3

4

10

10

11

11

10

10

5

6

8

7

6

6

7

6

8

8

8

8

8

8

DG-PE

8

8

4

10

7

2

10

2

1

1

10

9

6

6

11

9

6

6

6

1

2

1

5

2

1

1

5

5

11

10

4

3

10

9

2

3

12

12

10

10

10

10

DG-TEMP-MLS

8

9

4

11

2

3

2

2

1

1

10

9

4

5

12

12

5

5

8

8

3

3

6

5

1

1

6

6

11

10

5

3

12

11

3

3

13

13

12

12

12

11

optimized AUGRC (w.r.t. dropout and DG reward parameter)

MSR

5

5

4

2

7

4

8

9

9

9

2

2

3

3

7

5

7

7

4

1

5

6

11

10

12

12

11

11

5

6

9

7

8

9

8

6

10

10

9

9

9

9

MLS

10

10

8

8

13

12

13

12

9

9

2

2

11

12

10

11

12

12

10

10

8

11

1

1

13

13

12

12

1

1

2

3

1

1

11

11

3

3

1

1

1

1

PE

6

6

5

5

8

6

7

4

9

9

2

2

9

7

6

5

9

9

1

1

4

4

8

8

10

10

9

9

4

4

6

6

2

3

8

6

7

7

6

6

6

6

MCD-MSR

1

1

1

1

4

7

4

4

6

6

6

7

2

2

2

2

2

2

1

1

10

8

8

8

3

4

2

2

5

6

12

12

10

12

4

2

5

5

5

5

5

5

MCD-PE

2

2

2

3

5

9

4

4

6

6

6

7

7

8

2

2

3

3

1

1

7

7

4

4

5

7

3

3

3

2

11

11

8

6

6

9

1

1

4

4

4

4

MCD-EE

3

3

3

6

6

10

4

4

8

8

6

6

8

9

2

2

4

4

7

8

8

8

7

6

7

9

1

1

2

2

9

7

2

3

8

10

2

2

3

3

3

3

ConfidNet

13

13

13

13

10

7

10

9

5

5

1

1

10

10

10

10

10

10

11

11

12

12

5

6

9

6

8

8

9

4

6

7

6

6

5

5

11

11

10

10

10

11

4

4

7

7

1

1

1

1

4

4

12

13

1

1

1

1

1

1

4

1

11

8

11

10

6

8

4

4

10

13

13

13

13

13

1

1

6

6

7

7

7

6

12

12

9

9

12

13

12

13

1

1

9

11

12

11

13

13

11

11

12

12

1

2

2

2

8

3

7

7

13

10

3

1

5

2

12

12

4

4

2

2

2

1

Devries et al.

11

11

12

12

11

11

11

11

13

13

13

12

13

13

5

5

13

13

13

13

13

13

13

13

4

5

13

13

5

6

1

1

2

3

13

13

8

8

13

12

13

13

TEMP-MLS

7

6

6

3

8

4

8

4

9

9

2

2

4

4

7

5

8

7

4

1

5

4

10

10

11

11

10

10

5

6

8

7

6

6

7

6

8

8

8

8

8

8

DG-PE

8

8

10

10

1

2

2

2

1

1

10

9

5

6

9

9

6

6

7

1

1

1

3

2

1

1

5

5

11

10

4

3

10

9

2

3

12

12

10

10

10

10

DG-TEMP-MLS

9

9

11

11

3

3

2

2

1

1

10

9

6

5

12

12

5

5

9

8

3

3

5

5

1

1

6

6

11

10

5

3

12

11

3

3

13

13

12

12

12

11

CIFAR-100

→

CIFAR-10 (s-ncs)

AUGRC

AURC

CSF rankings

1

2

3

4

5

6

7

8

9

10

11

12

13

1

2

3

4

5

6

7

8

9

10

11

12

13

1

13

2

3

4

5

6

7

8

9 10 11 12

1

13

2

3

4

5

6

7

8

12 11 10 9

CIFAR-100

→

SVHN (ns-ncs)

AUGRC

AURC

1

2

3

4

5

6

7

8

9

10

11

12

13

1

2

3

4

5

6

7

8

9

10

11

12

13

CSF rankings

1

13

2

3

4

5

6

7

8

9 10 11 12

1

13

2

3

4

5

6

7

8

12 11 10 9

CIFAR-100

→

TinyImagenet (ns-ncs)

AUGRC

AURC

1

2

3

4

5

6

7

8

9

10

11

12

13

1

2

3

4

5

6

7

8

9

10

11

12

13

CSF rankings

1

13

2

3

4

5

6

7

8

9 10 11 12

1

13

2

3

4

5

6

7

8

12 11 10 9

CIFAR-100

→

sub-class shift

AUGRC

AURC

1

2

3

4

5

6

7

8

9

10

11

12

13

1

2

3

4

5

6

7

8

9

10

11

12

13

CSF rankings

1

13

2

3

4

5

6

7

8

9 10 11 12

1

13

2

3

4

5

6

7

8

12 11 10 9

CIFAR-100

→

corruption shift

AUGRC

AURC

1

2

3

4

5

6

7

8

9

10

11

12

13

1

2

3

4

5

6

7

8

9

10

11

12

13

CSF rankings

1

13

2

3

4

5

6

7

8

9 10 11 12

1

13

2

3

4

5

6

7

8

12 11 10 9

SVHN

→

CIFAR-10 (ns-ncs)

AUGRC

AURC

1

2

3

4

5

6

7

8

9

10

11

12

13

1

2

3

4

5

6

7

8

9

10

11

12

13

CSF rankings

1

13

2

3

4

5

6

7

8

9 10 11 12

1

13

2

3

4

5

6

7

8

12 11 10 9

SVHN

→

CIFAR-100 (ns-ncs)

AUGRC

AURC

1

2

3

4

5

6

7

8

9

10

11

12

13

1

2

3

4

5

6

7

8

9

10

11

12

13

CSF rankings

1

13

2

3

4

5

6

7

8

9 10 11 12

1

13

2

3

4

5

6

7

8

12 11 10 9

SVHN

→

TinyImagenet (ns-ncs)

AUGRC

AURC

1

2

3

4

5

6

7

8

9

10

11

12

13

1

2

3

4

5

6

7

8

9

10

11

12

13

CSF rankings

1

13

2

3

4

5

6

7

8

9 10 11 12

1

13

2

3

4

5

6

7

8

12 11 10 9

iWildCam

→

sub-class shift

AUGRC

AURC

CSF rankings

1

2

3

4

5

6

7

8

9

10

11

12

13

1

2

3

4

5

6

7

8

9

10

11

12

13

1

13

2

3

4

5

6

7

8

9 10 11 12

1

13

2

3

4

5

6

7

8

12 11 10 9

CSFs:

MSR

PE

MLS

MCD-MSR

MCD-PE

MCD-EE

TEMP-MLS

DG-TEMP-MLS

DG-PE

ConfidNet

Devries et al.

*

tied ranks

BREEDS-Entity-13

→

sub-class shift

AUGRC

AURC

1

2

3

4

5

6

7

8

9

10

11

12

13

1

2

3

4

5

6

7

8

9

10

11

12

13

CSF rankings

1

13

2

3

4

5

6

7

8

9 10 11 12

1

13

2

3

4

5

6

7

8

12 11 10 9

CAMELYON-17-Wilds

→

sub-class shift

AUGRC

AURC

1

2

3

4

5

6

7

8

9

10

11

12

13

1

2

3

4

5

6

7

8

9

10

11

12

13

CSF rankings

1

13

2

3

4

5

6

7

8

9 10 11 12

1

13

2

3

4

5

6

7

8

12 11 10 9

*

*

*

*

*

*

*

*

*

*

*

*

CIFAR-10

→

corruption shift

AUGRC

AURC

1

2

3

4

5

6

7

8

9

10

11

12

13

1

2

3

4

5

6

7

8

9

10

11

12

13

CSF rankings

1

13

2

3

4

5

6

7

8

9 10 11 12

1

13

2

3

4

5

6

7

8

12 11 10 9

CIFAR-10

→

CIFAR-100 (s-ncs)

AUGRC

AURC

1

2

3

4

5

6

7

8

9

10

11

12

13

1

2

3

4

5

6

7

8

9

10

11

12

13

CSF rankings

1

13

2

3

4

5

6

7

8

9 10 11 12

1

13

2

3

4

5

6

7

8

12 11 10 9

CIFAR-10

→

TinyImagenet (ns-ncs)

AUGRC

AURC

1

2

3

4

5

6

7

8

9

10

11

12

13

1

2

3

4

5

6

7

8

9

10

11

12

13

CSF rankings

1

13

2

3

4

5

6

7

8

9 10 11 12

1

13

2

3

4

5

6

7

8

12 11 10 9

CIFAR-10

→

SVHN (ns-ncs)

AUGRC

AURC

1

2

3

4

5

6

7

8

9

10

11

12

13

1

2

3

4

5

6

7

8

9

10

11

12

13

CSF rankings

1

13

2

3

4

5

6

7

8

12 11 10 9

*

*

1

13

2

3

4

5

6

7

8

9 10 11 12

BREEDS-Entity-13

CIFAR-10

CAMELYON-17-Wilds

AUGRC

AURC

iWildCam

CIFAR-100

SVHN

CSF rankings

1

2

3

4

5

6

7

8

9

10

11

12

13

1

2

3

4

5

6

7

8

9

10

11

12

13

AUGRC

AURC

1

2

3

4

5

6

7

8

9

10

11

12

13

1

2

3

4

5

6

7

8

9

10

11

12

13

AUGRC

AURC

1

2

3

4

5

6

7

8

9

10

11

12

13

1

2

3

4

5

6

7

8

9

10

11

12

13

AUGRC

AURC

1

2

3

4

5

6

7

8

9

10

11

12

13

1

2

3

4

5

6

7

8

9

10

11

12

13

AUGRC

AURC

1

2

3

4

5

6

7

8

9

10

11

12

13

1

2

3

4

5

6

7

8

9

10

11

12

13

AUGRC

AURC

1

2

3

4

5

6

7

8

9

10

11

12

13

1

2

3

4

5

6

7

8

9

10

11

12

13

CSF rankings

CSF rankings

CSF rankings

CSF rankings

CSF rankings

CSFs:

MSR

PE

MLS

MCD-MSR

MCD-PE

MCD-EE

TEMP-MLS

DG-TEMP-MLS

DG-PE

ConfidNet

Devries et al.

1

13

2

3

4

5

6

7

8

9 10 11 12

1

13

2

3

4

5

6

7

8

12 11 10 9

1

13

2

3

4

5

6

7

8

9 10 11 12

1

13

2

3

4

5

6

7

8

12 11 10 9

1

13

2

3

4

5

6

7

8

9 10 11 12

1

13

2

3

4

5

6

7

8

12 11 10 9

1

13

2

3

4

5

6

7

8

9 10 11 12

1

13

2

3

4

5

6

7

8

12 11 10 9

1

13

2

3

4

5

6

7

8

9 10 11 12

1

13

2

3

4

5

6

7

8

12 11 10 9

1

13

2

3

4

5

6

7

8

9 10 11 12

1

13

2

3

4

5

6

7

8

12 11 10 9

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

tied ranks

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: The described pitfalls of existing metrics affect Selective Classification evaluation in general. Hence, we thoroughly discuss all common metrics in the field and demonstrate the relevance on a state-of-the-art benchmark.

Guidelines:

The answer NA means that the abstract and introduction do not include the claims made in the paper.

The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.

The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.

It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.

2. Limitations

Question: Does the paper discuss the limitations of the work performed by the authors? Answer: [Yes]

Justification: The empirical study is based on an extensive, state-of-the-art, yet naturally limited benchmark. However, the scope is clearly defined in the paper.

Guidelines:

The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.

The authors are encouraged to create a separate "Limitations" section in their paper.

The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.

The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.

The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.

The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.

If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.

While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.

3. Theory Assumptions and Proofs

Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?

Answer: [Yes]

Justification: For the analytical relationships, we provide the assumptions and derivations. Guidelines:

The answer NA means that the paper does not include theoretical results.

All the theorems, formulas, and proofs in the paper should be numbered and crossreferenced.

All assumptions should be clearly stated or referenced in the statement of any theorems.

The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition.

Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.

Theorems and Lemmas that the proof relies upon should be properly referenced.

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 conclusions of the paper (regardless of whether the code and data are provided or not)?

Answer: [Yes]

Justification: The evaluation protocol as well as all experiment setups are described. Additionally, we provide the code for reproducing the results.

Guidelines:

The answer NA means that the paper does not include experiments.

If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.

If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.

Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.

While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example

(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.

(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.

(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).

(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.

5. Open access to data and code

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Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?

Answer: [Yes]

Justification: We don't see any potential negative societal impacts. We address enhanced reliability of real-world decision systems through the studied methods as a potential positive impact. However, we propose no new Selective Classification method in our work.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors introduce a new metric AUGRC for evaluating classifiers under the Selective Classification framework, whereas the classifier\nhas an option to reject low-confidence predictions. The authors introduce desirable properties for evaluating the Selection Classifiers, and show that the new metric has all those properties, unlike any of the currently used metrics. Experiments show that AUGRC produces significanly different rankings of Confidence Scoring Functions compared with currently used metrics.\n\nSTRENGTHS:\nThe authors address relevant question and make useful contribution. The paper is well written, clear and easy to read. The experimental analyses are, to my judgement, sound. The code is provided.\n\nWEAKNESSES:\nFig. 4: the risk vs. coverage curves for AURC vs. AUGRC do not look that different. Sure, there is non-monotonicity in AURC, but that is\nin the low-coverage region, which presumably is not of practical interest. The rest looks monotonic and fairly similar. It is also true\nthat the two metrics suggest different CSFs, but again, they aren't that different. To be clear, I still think AUGRC is favorable, just\nsaying that the difference appears modest.\n\nI think the key conclusion, which is the recommendation to adopt AUGRC, is valid, however the language in the section is a bit too strong (there is \"substantial\" twice and \"significant limitations\"). I recommend to tone-it-down.\n\nQUESTIONS:\nThe authors claim that Selective Classification is a crucial components for clinical diagnostics ML systems (among other application domains). Can you provide examples of classifiers in clinical use that incorporate Selective Classification?\n\nWhat is AUROC_f? It is introduced on line 172, however the description is unclear and there are no references.\n\nLIMITATIONS:\nThere is no limitations section. I don't have a good idea regarding limitations of this manuscript.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-06-26"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper presents 5 requirements for multi-threshold metric for Selective Classification and a novel metric to evaluate selective classifiers called AUGRC. The proposed metric satisfies 5 requirements that are not met by current approaches. The proposed metric changes the rankings on 5 out of 6 datasets considered by the authors.\n\nSTRENGTHS:\nThe paper's main strengths are:\n\n\n1. the paper aims to tackle a long-standing concern in abstaining classifiers literature, i.e. how to evaluate with a single measure these classifiers\n2. the theoretical derivations seem sound\n3. the contribution is well framed within current literature\n\nWEAKNESSES:\nThe main concerns of the paper are:\n\n1. The empirical evaluation can be improved\n2. The interpretability of the proposed measure is not that straightforward\n3. The paper presentation can be improved. \n\nRegarding the empirical evaluation, I have a few remarks.\n* there are some contradictory lines, which should be double-checked: for instance, in lines 268-269, it is not clear to me why you claim that AURC erroneously favors DG-RES over MCD-PE, as DG-RES's performance seems to be better than MCD-PE (accuracy-wise and ranking-wise). Similarly, it is not clear to me why in Figure 4 the authors state DG-RES is favored \"despite a lower classification performance and ranking quality\", while in Figure 4.a is reported a higher accuracy and higher $AUROC_f$ for DG-RES;\n* second, the authors never specify whether they correct for multiple outcomes testing (e.g. using Bonferroni correction). I think since the authors are performing multiple pairwise ranking tests, they also have to account for this. \n\nRegarding the interpretability requirement, the authors claim (correctly) that the AUROC can be interpreted as \"the probability of a positive sample having a higher score than a negative one\". I did not fully grasp what is the straigthforward interpretation of the AUGRC score in this context (lines 206-208).\n\nRegarding paper presentation, the paper heavily relies on acronyms. I would personally reconsider this choice to improve the overall readability.\n\nQUESTIONS:\nI have a few questions for the authors:\n\n* [Q1] can the authors clarify my concerns regarding the empirical evaluation?\n* [Q2] can the authors discuss further my concern regarding interpretability? Is there a straightforward interpretation of the AUGRC score?\n* [Q3] when considering a degenerate abstaining classifier that always abstains, i.e., whose confidence score is always zero, is the AUGRC 0 (as the one obtained by a perfect classifier)? If so, is AUGRC favoring classifiers that tend to be underconfident and asbtain too much? Is there a way to avoid such a behaviour? \n* [Q4] if I correctly understand how AUGRC works, why do not the authors consider the _classification quality_ by [Condessa et al, 2017] instead of the Generalized Risk Score? Such a metric also considers the failures that are correctly rejected and could avoid favoring abstaining classifiers that over reject.\n\n\nTypos:\n\nline 128: schore  $\\to$ score\n\nLIMITATIONS:\nThe authors do not directly discuss limitations of their proposed approach in the main paper. For instance, I think a brief discussion regarding the computational time required to compute AURGC should be included in the main paper.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-06-27"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors propose 5 requirements that should be satisfied by selective classification (SC) metrics such that they can be successfully used to rank SC models for a task. They then propose a new metric, called AUGRC, which is shown to satisfy all 5 requirements. Finally, the authors show empirically that their method performs better rankings thank previous metrics.\n\nSTRENGTHS:\n1. The paper is well-motivated, clearly written, and has an excellent coverage of previous works.\n2. The 5 requirements were well-chosen and a metric that satisfies them is likely to be a good choice for SC ranking.\n3. The metric is simple, yet effective, and the maths seems sound.\n4. Results are convincing and statistically analysed; the toy dataset was didactic.\n5. There's enough information in the main body of the paper to fully understand the experiments and corresponding results, i.e. it's not necessary to read the appendix, even if further details are available there.\n\nWEAKNESSES:\n1. Limited discussion of future works.\n2. NLL and Brier Score are listed among the multi-threshold metrics, but there's no thresholding involved in their evaluation. In fact, it should be possible to use them as risk measures, which would cover more for than just 0/1 loss, especially where probabilities are important for decision making.\n\nQUESTIONS:\n1. A similar task to SC is classification with rejection of OOD samples (sometimes called open-set recognition). In such cases, there isn't a ground truth to evaluate the risk on (in practice). Can the authors envision a way to adapt their metric to these scenarios?\n2. The metrics were scaled by ×1000 in Figure 2. Were they also rescaled for all the other results?\n3. Regarding interpretability, the authors say that \"The optimal AUGRC is given by the second term in Equation 7\". What does the optimal AUGRC mean? What would be the AUGRC of the Bayes-optimal model for a task? Is there an AUGRC value associated to a random classifier, as in AUROC?\n5. Any intuitions about why the rankings of AURC and AUGRC where the same for iWildCam?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 8, "REVIEWER_CONFIDENCE": 5, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2024-07-06"}

comments: SUMMARY:
The paper tackles the problem of Selective Classification (SC). The authors show problem with the existing metrics used in evaluating SC and propose to use the Area under the Generalized Risk Coverage curve (AUGRC). Empirical results provide useful insights about the effect of using this metric and shows how this changes  the relative ordering of state-of-the art methods.

STRENGTHS:
- The problem tacked is very important, especially for the safe deployment of ML models which requires the knowledge of when not to trust the model.
- The proposed metric is well-motivated.
- The experiments are extensive.
- It is important for the community to know about this work as this changes the perception of SC and the best method to use.

WEAKNESSES:
- It is important to add the details of the methods in the main paper. For example, DeepGamblers (DG) is referred to as DG in the main paper and this abbreviation is only explained in the appendix. Similarly for other baselines.
- Although the experiments are extensive, it is good to include the recent state-of the art SC methods such as Feng et al (2023) to understand which method is the best to use.

References:
Feng et al: Towards Better Selective Classification

QUESTIONS:
See the previous section

LIMITATIONS:
NA
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2024-07-16