COC curve: operating neural networks at high accuracy and low manual effort
Authors: Sara Sangalli
Ertunc Erdil
Ender Konukoglu
Venue: Submitted to ICLR 2023 (2023)

Abstract: In human-AI collaboration systems for critical applications based on neural networks, humans should set an operating point based on a model's confidence to determine when the decision should be delegated to experts.
The underlying assumption is that the network's confident predictions are also correct.
However, modern neural networks are notoriously overconfident in their predictions, thus they achieve lower accuracy even when operated at high confidence. Network calibration methods mitigate this problem by encouraging models to make predictions whose confidence is consistent with the accuracy, i.e., encourage confidence to reflect the number of mistakes the network is expected to make.  However, they do not consider that data need to be manually analysed by experts in critical applications if the confidence of the network is below a certain level. This can be crucial for applications where available expert time is limited and expensive, e.g., medical ones.
In this paper, we propose (1) Confidence Operating Characteristics (COC) curve that assesses a predictive model in terms of accuracy and manual analysis it requires for varying operating points on confidence, and (2) a new loss function for classification that takes into account both aspects and derived from the COC curve.
We perform extensive experiments on multiple computer vision and medical image datasets for classification and compare the proposed approach with the existing network calibration methods. Our results demonstrate that our method improves classification accuracy while delegating less number of decisions to human experts, achieves better out-of-distribution samples detection and on par calibration performance compared to existing methods.


Under review as a conference paper at ICLR 2023
TRAINING NEURAL NETWORKS TO OPERATE AT HIGH ACCURACY AND LOW MANUAL EFFORT
Anonymous authors Paper under double-blind review
ABSTRACT
In human-Al collaboration systems for critical applications based on neural net- works, humans should set an operating point based on a model's confidence to determine when the decision should be delegated experts. The underlying as- sumption is that the network's 's confident predictions are also correct. However, modern neural networks are notoriously overconfident in their predictions, thus they achieve lower accuracy even when operated at high confidence. Network calibration methods mitigate this problem by encouraging models to make predic- tions whose confidence is consistent with the accuracy, i.e., encourage confidence to reflect the number of mistakes the network is expected to make. However, they do not consider that data need be manually analysed by experts in critical ap- plications if the confidence of the network is below a certain level. This can be crucial for applications where available expert time is limited and expensive, e.g.. medical ones. The trade-off between the accuracy of the network and the amount of samples delegated expert every confidence threshold can be represented by a curve. In this paper we propose a new loss function for classification that takes into account both aspects by optimizing the area under this curve. We perform ex- tensive experiments on multiple computer vision and medical image datasets for classification and compare the proposed approach with the existing network cali- bration methods. Our results demonstrate that our method improves classification accuracy while delegating less number of decisions to human experts, achieves better
samples detection and par calibration performance compared to existing methods.
INTRODUCTION
Artificial intelligence (AI) systems based on deep neural networks have achieved state-of-the- results by reaching or even outperforming human-level performance in many predictive tasks Esteva jet al. y Rajpurkar et al. (2018); [Chnn et al. 2017); Szegedy et al. (2016). Despite the great potential of neural networks for automating various tasks, there pitfalls when they are used in fully automated setting, which makes them difficult to deploy in safety-critich applications such as healthcare Kelly et al. (2019); Quinonero-Candel et al. (2008): ; (Snngalli al. (2021). Human-Al collaboration aims at tackling these issues by keeping humans in the loop and building systems that take advantage of humans and AI by minimizing their shortcomings Patel et al. (2019).
A simple way of building collaboration between a network and a human expert would be delegat- ing the decisions the expert when the network's confidence score is lower than a predetermined threshold which we refer to as "operating point". For example, in healthcare, a neural network trained for predicting whether a lesion benign or malignant should leave the decision to the hu- man doctor if not very confident Jiang et al. (2012). In such cases the domain knowledge of the doctor could be exploited to assess more ambiguous cases, where for example education or previous experience can play crucial role in the evaluation. Another example of human-Al collaboration is hate speech detection for social media (Conneau & Lample| 2019), where neural networks highly reduce the load of manual analysis of contents required by humans. In industrial systems, curves are employed (Gorski et al. 2001) that assess a predictive model in terms of accuracy and num- ber of samples that requires manual assessment from human expert for varying operating points that loosely relate to varying confidence levels of the algorithm's 's prediction. We will refer to this performance curve Confidence Operating Characteristies (COC), it reminds of the classic Re-

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ceiver Operating Characteristic (ROC) curve where analogous balance is sought after between Sensitivity and Specificity of a predictive model. COC curve can be used by domain experts, such as doctors, to identify the most suitable operating point that balances performance and amount of data to re-examine for the specific task. The underlying assumption in these applications is that the confidence level of networks indicates when the predictions are likely to be correct or incorrect. However, modern deep neural networks that achieve state-of-the-an results are known to be over- confident even their wrong predictions. This leads to networks that are not well-calibrated, i.e., the confidence scores do not properly indicate the likelihood of the correctness of the predictions Guo et al. (2017). Thus, neural networks suffer from lower accuracy than expected, when operated at high confidence thresholds.
Network calibration methods mitigate this problem by calibrating the output confidences of the model Guo et al. ())17):( Kummar et al. (2018); Karandikka et al. (2021): et al. (2021). However, they do not consider that data may need to be manually analyzed by experts in critical applications if the confidence of the network is below a certain level. This can be crucial for various applications where expert time limited and expensive. For example, in medical imaging, the interpretation of more complex data requires clinical expertise and the number of available experts is extremely limited, especially in low-income countries Kelly et al. (2019). This motivates us to take the expert load into account along with accuracy when assessing the performance of human-Al collaboration systems and training neural networks.
In this paper, we make the following contributions:
We propose a new loss function for multi-class classification that takes into account both of the aspects by maximizing the area under COC (AUCOC) curve.
We perform experiments on two computer vision and one medical image datasets for multi- class class classification. We compare the proposed AUCOC loss with the conventional loss functions for training neural networks as well as network calibration methods. The results demonstrate that our method improves other methods in terms of both accuracy and AUCOC.
We evaluate network calibration and out-o--distribution (OOD) samples detection perfor- mance of all methods. The results show that the proposed approach able to consistently achieve better OOD samples detection and on par network calibration performance.
1.1 RELATED WORK
In industrial applications, curves that plot network accuracy on accepted samples against manual workload of a human expert are used for performance analysis of the system (Gorski et al.) 2001 To the best of our knowledge this is the first work that explicitly takes into account during the optimization process the trade-off between neural network performance and amount of data to be analysed by a human expert, in a human-Al collaborative system. Therefore, there is no direct literature that we can compare with. We found the literature network calibration methods the closest to our setting because they also aim at improving the interaction between human and AI, by enabling the networks delegate the decision human when they are not very confident. Therefore, we compare our method with the existing network calibration methods in the literature.
In well-calibrated network, the probability associated with the predicted class label should reflect the likelihood of the correctness of the predictions. [uuo et al. (2017) defines the calibration error as the difference in expectation between accuracy and confidence in each confidence bin. One category of calibration methods augments or replaces the conventional training losses with another loss to explicitly encourage reducing the calibration error. Kumar et al. (0118) propose a method called MMCE loss by replacing the bins with continuous kernel to obtain a continuous distribution and a differentiable measure of calibration. Karandikar al. (202T) propose two loss functions for calibration, called Aoft-AvUC and Soft-ECE, by replacing the hard confidence thresholding in AvUC (Krishnan & Tickoo 2020) and binning in ECE Guo et al. (2017) with smooth functions, respectively. All these three functions are used secondary losses along with conventional losses such as cross-entropy. MMkhoti et al.] (2020) find that Focal Loss (FL) (Lin et al.) (017) provides inherently more calibrated models, even if it not originally designed to improve calibration, as it adds implicit weight regularisation. The second category methods are post-hoc calibration approaches, which rescale model predictions after training. Platt scaling (Plati 2000) and histogram
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binning Zadrozny & Elkan 2001 fall into this class. Temperature scaling (TS) Guo al. 2017 modern variant of Platt scaling, is the most popular approach of this group. The idea of TS is to scale the logits of a neural network by dividing by positive scalar such that they do not saturate after the subsequent softmax or sigmoid activation. TS can be used as a complementary method for all networks and it does not affect the accuracy of the models while significantly improving calibration.
METHODS
In this section, we illustrate more in detail the curve that assesses predictive model in terms of accuracy and the number of samples that are delegated to human expert for manual analysis, that has been used in industry, e.g., (Gorski et al. (001). We will refer it to Confidence Operating Characteristics (COC), as it reminds of the classic Receiver Operating Characteristic (ROC) curve where an analogous balance is sought after Sensitivity and Specificity of predictive model. Then, we describe novel cost function for training of neural networks: the area under COC (AUCOC) loss (AUCOCLosS).
2.1 NOTATION
Let D ((In, yn)) )) denote dataset composed of N samples from joint distribution (XX)) (") where In E X and In E y = {1.2. K} denote the imput data and the corresponding class label, respectively. Let fo(y'z) be the probability distribution predicted by neural network parametrized by for an input x. For each data point Ins 9 = argmaxyes/ folyla) denotes the predicted class label, associated to a correctness score Cn = 1 = Yn) and to a confidence [ri,...Fx]
score Tn fax (v)a=,), where In E [0. and 1(.) is indicator function. represents the vector containing all the predicted confidences for a set data points. p(r) denotes the probability distribution over values. In human-Al collaboration, samples with confidence lower than a threshold To would be delegated to a human expert for manual assessment.
2.2 CONFIDENCE OPERATING CHARACTERISTICS (COC) CURVE
Our first goal is find an appropriate evaluation method to assess the trade-off between neural network's performance and the number of samples that requires manual analysis from a domain expert a human-Al collaboration system. We decide to optimize the area under COC as it provides practictioners with flexibility in the choice of the operating point, similarly to classic ROC curve.
x-y axes of COC: order to construct the COC curve, first, we define a sliding threshold ro over the space of predicted confidences r and assume that the samples with confidence scores lower than Tu are given to a human expert for decision. Then, for each threshold To, we calculate te-axis) the percentage of samples that delegated human expert and axis) the accuracy of the network on the remaining samples corresponding to the threshold ro.
We formulate the axes of the COC curve as follows
axis: To p(r ro) J p(r)dr
y - E[c/r rol
For each threshold level ro. To represents the fraction of samples whose confidence lower than that threshold, i.e., the percentage of the samples that are delegated to expert for manual analysis. Eic/r 2 ro] corresponds to the expected value of the correctness score e for all the samples for which the network's confidence is equal or larger than the threshold ro, i.c., the samples for which network prediction will be used. This expected value can be computed as
Bicr 2 ro] [ (lclrlarrddr/111 -oo)
(2)
We provide the derivation of Eq. 2 in Appendix A.I
The area under COC curve (AUCOC), like the area under ROC curve, is global indicator of the performance of system. Higher AUCOC indicates lower number of samples that are delegated to human experts or/and higher accuracy for the samples that are analysed by the network. Lower AUCOC on the other hand, indicates higher number of delegations to human experts or/and lower
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accuracy. AUCOC can be computed by integrating the y y-axis expressed in Eq. range of To E [0, 1]:
D over the whole
AUCOC= / Bldr 2 roldTo J'' { Blcrrp(r)dr
} dro
TO
COC curves NN1 NN2
E[cr 2 ml
Predicted
(NN1)
Elcir re accursiy
0.66
Increasing accuracy
Predicted Confidences Neural Network 2 (NN2)
Decreasing number of delegated samples
1
sed AUCDE 0.64 MN2 AMCDC 0.45
0.5 0.66
samples to be analysed manually
ECEN
Acc NCc
AUCOC AUCOC
1.0
S
Bins
(b)
Figure I: (a) shows how one could improve AUCOC, increasing the accuracy of the network and/or decreasing the amount of data to be analysed by the domain expert. The dashed curve has higher AUCOC than the solid one. (b) illustrates toy example where two models have the same accuracy and ECE with 5 bins. However, they have different AUCOC values due to different ordering of correctly and incorrectly classified samples according the assigned confidence by the network.
2.3 AUCOCLOSS: MAXIMIZING AUCOC FOR TRAINING NEURAL NETWORKS
In the previous section, we described COC to assess the performance of a system in terms of accu- racy and number of samples delegated to human experts at various operating points and mentioned that higher AUCOC is desired. With this motivation, in this section, we introduce a new loss func- tion called AUCOCLoSS that maximizes AUCOC for training neural networks. AUCOC's explicit maximization would enforce the reduction of the number of samples delegated to human expert while maintaining the accuracy level assessed by the algorithm (i.c., keeping E[c/r 2 ro] constant) and/or the improvement in the prediction accuracy of the samples analysed by the algorithm while maintaining particular amount of data to be analysed by the human (i.e., keeping constant) as illustrated Figure la
We define our loss function to maximize AUCOC as
(4) We the negative logarithm AUCOC lies in the interval [0, 1]. which corresponds AU- COCLoss E [0, inf] which is suitable for minimizing cost functions. For computing AUCOC for training, we use kernel density estimation (KDE) with training samples to estimate p(r) in Eq.
AUCOCLoss log( (Og(ACCC).
p(r)
Σ K(r -n)
N
where K is a Gaussian kernel and we estimate its bandwidth using Scott's rule of thumb Scott 1979). Then, the other terms in Eq. E[crrp((r and are estimated as follows
N Σ K(r ra)
N K(r ra)dr.
B(clrip(r) ~
To
(6)
N ==1
N
n=1
2.4 TOY EXAMPLE: ADDED VALUE BY COC CURVE AND AUCOC
In this section, demonstrate the added value of assessing the performance of predictive model using COC curve and AUCOC compared to expected calibration error (ECE) Guo et al. (2017) and classification accuracy toy example.
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The toy example on the left side of Figure lb shows the confidence scores of 5 samples from two different models, NN1 and NN2, for a classification problem. The green circles denote the predicted confidences for correctly classified samples, while the red crosses the confidences of the misclassi- fied ones. ECE divides the confidence space into bins, computes the difference between the average accuracy and confidence scores for each bin, and returns the average of the differences the final measure of calibration eтrоr. Let us assume that we divide the confidence space into 5-bins as indi- cated with the gray dotted lines in the confidence spaces of NNI and NN2 for computing ECE. This results in equal ECE for both networks. Note that these models also have equal classification accu- racy since each classifies 2 out of 5 samples incorrectly. Looking at these two performance metrics, it is not possible to choose one model over the other since NN1 and NN2 perform identically.
On the contrary, the AUCOC larger for NN1 than for NN2, as shown in the rightmost plot in Figure [b] The difference in AUCOC is due to the different ordering of correctly and incorrectly classified samples, which COC and AUCOC are able to detect. By looking at the AUCOC results, one would prefer NNI compared to NN2. Indeed, NNI is better model than NN2 because it achieves either equal or better accuracy than NN2, for the same amount of data to be manually examined. Analogously, it delegates either equal or lower number of samples to expert for the same accuracy level.
2.5 IMPLEMENTATION DETAILS
COC: We implement the selection of the operating points in COC by thresholding on the predicted output confidences First, we arrange the confidences of the whole dataset (or batch) in ascending order. Each predicted confidence is then selected as threshold level for the vector r, and To and E(c/r ra] are computed. This allows us to save time threshold selections compared to exploring the confidence space with arbitrarily fine- ane-grained levels.
AUCOCLOSS: There are two important points about the implementation of AUCOCLOsS. First, instead of using B(clr)p(r)) Rs N EN Cn K(Ir -rn--rlll as given in Eq. 6 we approximated it as B[clr)p(r) x NV >n r "(Ilr - -nlll) where r, = fo (rr,) is the confidence of the correct class for a sample n. The main reason of this modification is that the gradient of the misclassified samples becomes zero because en is zero when sample x, is not classified correctly. Therefore, the network cannot learn how to classify samples correctly, once they are misclassified. To deal with this issue, replaced the correctness score Cn which can be either 0 1, with r. which can take continuous values between 0 and 1. With this new approximation, we can also compute and back- propp-pe the gradients for misclassified samples.
EXPERIMENTS
In this section, we present our experimental evaluations on multi-class image classification tasks. We performed experiments three datasets: CIFAR100 (Krizhevsky) 2009)) Tiny- ImageNet which is a subset of ImageNet (Deng et al., (209)) and on DermaMNIST (Yang et al. 2021). We com- pared AUCOCLOss with different loss functions, most of which are designed to improve calibration performance, while preserving accuracy: cross-entropy (CF), focal-loss (FL.) (Lin et al. 2017). adaptive focal-loss (Mukhoti et al. 2020), maximum mean calibration error loss (MMCE) Kumar let al. 2018), soft binning calibration objective (Soft-ECE) and soft accuracy versus uncertainty cal- ibration (Soft-AvUC) (Karandikar et al. 202T). We optimized MMCE, Soft-ECE, and UUC losses jointly with a primary loss which we used either CE or FL. consistently with the literature (Karandikar et al. (02I). We also performed experiments by optimizing the proposed AUCOCLoss as both primary and secondary loss with CE and FI.. KDE in AUCOCLoss is applied batch-wise during the training. In addition, for all the experiments we report results after applying temperature scaling (TS) (Guo et al. (017)) as post-training technique.
We use three metrics to evaluate the performance of the methods: classification accuracy, equal-mass expected calibration error (ECE) (Nixon et al., 2019) with 15 bins, and AUCOC. Classicification accuracy is simply the ratio between number of correct samples over total number of samples and AUCOC is computed using Eq. 3 Equal-mass ECE divides the confidence space into M bins such that each bin B,, contains equal number of samples and computes weighted average of the difference between the average accuracy and confidence of each bin. In addition, we report some
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examples of operating points of COC curve given a certain accuracy show the corresponding percentage of samples that need to be analyzed manually (To @acc) on the COC curves.
A reliable AI system should have lower confidence whenever it encounters out-of-distribution (OOD) data, deferring those samples to the human expert for further investigation. Evaluating the OOD samples detection performance is a common experiment in the network calibration literature. Therefore, we investigated the OOD detection performance of all the methods when CIFAR 100-C (Hendrycks & Dietterich] 2019) (with Gaussian noise) and SVHN (Netzer al. 201T) are OOD datasets, while the network is trained on CIFARIO0 100 (in-distribution). We evaluated the OOD detec- tion performance of all methods using Area Under the Receiver Operating Characteristics (AUROC) curve.
All the experiments were run times and we report the average results in Tables
2 and
3.1 SETUP DETAILS
For CIFAR (Krizhevsky) 2009) we used 45000/5000/10000 images respectively train- ing/validationn/est sets and Mide-RRsnnc-288 (Zagoruyko & Komodakis (016), consistently with Karandikar et al. (202T) We trained the model for 200 epochs, using Stochastic Gradient Descent (SGD), with batch of 512, momentum of 0.9 and an initial learning rate of 0.1, decreased after 60, 120, 160 epochs by a factor of 0.1.
Imay-lmageNet is subset of ImageNet (Deng et al. 2009) with 64 x 64 images and 200 classes. We employed 90000/100000 10000 images as trainingyvalidatiioneess set, respectively. As in Mukhoti eet al. (2020) we used ResNet-50 (He et al., 2016) as backbone architecture. SGD was used as optimiser with batch size of 512, momentum of 0.9 and base learning rate of 0.1, divided by 0.1 at 40th and 60th epochs.
We chose a medical imaging dataset, DermaMNIST YYagg al. 2021 as the third dataset since human-Al collaboration systems are particularly useful for medical domain. DermaMNIST is com- posed of dermatoscopic images with 7007/1003/2005 samples for training, validation and test set, respectively, categorised in 7 different diseases. We followed the training procedures of the original paper, employing a ResNet-50 He et al. (206)) Adam optimizer and batch size of 128. We found that more suitable initial learning rate is 0.001 and trained the model for 100 epochs, reducing the learning rate by 0.1 after 50 and 75 epochs.
We set the weighting factor for AUCOCLoss when it is used as secondary loss selected via cross- validation. For the baselines we used the hyperparameter combinations specified by the original papers if done If not, carried out cross- tross-validation and selected the settings that provided the best performance on the validation set. In order not to be biased towards a specific metric, for all the methods, we saved the models with respect the best ECE, accuracy, and AUCOC. We found empirically that models check-pointed using ECE provided very poor results and therefore we omit them while presenting results. Networks chenk-pointed using either accuracy or AUCOC provided comparable outcome. For Tiny-Imagenet, being the largest dataset, we report results for both check- pointing strategies in Sec. For CIFARIO0, DermaMNIST and OOD detection experiments, we only report the AUCOC checkpointing results in the main paper in Sec. 4 and the remaining results in Appendix E
RESULTS
In Tables 2 and 3 we report the results on CIFAR100, Tiny-Imagenet and DermaMNIST, respec- tively. For Tiny-Imagene,, the largest dataset, the results for both AUCOC-based and accuracy-based checkpoints are reported, while for the other datasets we reported results only for AUCOC check- points in the main paper and the rest in Appendix E Bold results indicate the methods that performed best for each metric. means the higher the better for a metric, while I the lower the better.
We empirically found that the optimization of AUCOC alone can be complex and AUCOCLOSS mainly provides the best results when it is used as a secondary loss, regularized by another cost function. We find that AUCOCLoss outperformed consistently all the baselines every dataset in terms of accuracy and AUCOC. In terms of ECE, the proposed loss function provided on par performance compared to the other loss functions particularly designed for network calibration. We

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Test results on CIFARI00 for accuracy, AUCOC and ECE. For each loss function, we report the results of the model check-pointed on the best AUCOC validation set, and post TS. The last two columns report To corresponding to 90% and 95% accuracy (pre TS). bold the best result for each metric. The average results over 3 runs reported.
Table 1:
Pre TS
Post TS
To 4
acc.
Loss funct.
AUCOC
Accuracy t
ECE
AUCOC 1
ECE
90%
95%
CE
91,43
75,71
11,56
92,98
2,41
29,03
44,61
FL (7=3)
93,91
77,83
4,05
93,82
1,92
24,71
40,48
AdaFLS3
93,89
77,64
4.15
93,79
1,47
25,08
40,74
CE+MMCE
92,42
75,35
9.99
92,7
2,32
30,01
44,71
FL+MMCE
93,9
77,78
1,87
93,92
1,92
25,62
40,9
CE+SoftAvVCC
93,99
77,65
6,81
93,87
3,23
24,62
45.0
FL.+ Soft-AvUC
93,97
77,64
1,89
93,93
1,83
25,97
40,82
CE+SoftEECR
93,88
77,57
7,09
93,75
3,85
24,76
40,14
FL4Soft-ECE
93,41
76,69
6,43
93,56
1,74
28.13
43,29
CEAAUCOCLoss
94,49
78,94
6,7
94,35
1,65
21,6
36,73
FL+AUCOCLOss 94,18
78,31
2,85
94,24
1,34
23,5
36,68
AUCOCLoSs Prim.
94,24
78,32
5,92
94,32
2,16
23,65
38,3
also observed that the largest improvement in ECE is obtained after temperature scaling (TS) in all methods.
order to have an idea what a certain AUCOC corresponds to in practice, Figure 2 show some examples of COC curves (more results available in Appendix B Overall, the plots AUCOCLoss lie above all the baselines, which a desirable behavior as corresponds to better operating points. Moreover, in Tables
report some operating points of the COC curves for each model. In particular, we measured the percentage of samples delegated to expert (7p) at 90% and 95% accuracy for TIFARI00 and DermaMNIST, and at 65% and 75% accuracy for Tiny-Imagenet (as the initial accuracy is also much lower in this dataset). In all the experiments, to varying degrees, AUCOCLoss able provide lower delegated samples than the baselines.
In OOD experiments reported in Table 4 we used the model trained on CIFAR 100 and evaluated the OOD detection performance on both CIFAR100-C and SVHN dataset. We used the confidence scores of the models with and without applying TS when determining where sample is OOD or in- distribution. The bold results highlight the best results in terms of AUROC. On both OOD datasets, AUCOCLoSS provided the highest AUROC, both before and after applying temperature scaling..
Finally, we investigated how the performances of the models vary when the batch size varies, which may be crucial for KDE-based methods like AUCOCLoss as it may affect the accuracy of density estimation. We investigated this by reducing the batch size from 128 (used the main experiments) to 64 and 32 in DermaMNIST dataset. We report the results in Table 9 in Appendix E for space reasons. We observe that reducing the batch does not significantly affect the performance of our method.
CONCLUSION
In this paper we propose a new cost function for multi-class classification that takes into account the trade-off between neural network's performance and the amount data that requires manual analysis from domain expert when the network is not confident enough, by maximizing the area under COC (AUCOC) curve. Extensive experiments on various computer vision and medical image datasets presented, comparing the new loss with various baselines. The results demonstrate that our approach improves the other methods terms of both accuracy and AUCOC and provides com- parable ECE. Additionally, evaluate the performance of different losses for OOD samples detection and show that our method outperforms the baselines.
While presented COC and the loss based AUCOC for multi-class classification, extensions to other tasks is possible and we will explore this direction in future work. Another possible direction would be investigating other performance metrics embed in the y-axis of COC other than accuracy.

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Table 2: Test results on Tiny-ImageNet for accuracy, AUCOC and ECE. For each loss function, the first row reports the results of the model check-pointed on the best AUCOC (in gray) and the second row on the best accuracy on validation set, pre and post TS. The last two columns report to corresponding to 65% and 75% accuracy (pre TS). In bold the best result for each metric. The average results over runs are reported.
Pre TS Accuracy
Post TS
TD
acc.
Loss funct.
AUCOCT
ECE
AUCOCT
T
ECE
65%
75%
72,56
47,39
21,39
72,99
1,54
39,88
56,29
72,68
47,62
22,04
73,07
1,97
39,35
56,03
FL (9=3)
73,12
47,71
10,33
73,09
1,46
38,4
55,08
73,05
47,61
10,72
73,02
1,61
38.71
55,15
AdaFL553
73,19
47,81
10,98
73,19
1,35
38,56
55,2
73,21
47,78
10,47
73,18
1,52
38,33
55,38
CEAMMCE
72,47
47,11
21,12
72,73
1,79
39,94
56,7
72,5
47,04
20,76
72,69
1,65
39,99
56,79
FL+MMCE
73,51
48,03
2,82
73,47
1,94
37,83
55,15
73,44
47,89
3,62
73,41
1,62
37,78
55,35
CE+Soft-AwUC
72,92
47,89
21,77
73,47
2,13
38,28
54,91
72,89
47,89
21,7
73,44
1,87
38,41
55,08
FL.+ Soft-AvUC
74,3
48,69
7.18
74,34
1,51
35,83
53,21
73,23
48,15
22,01
73,81
1,86
37,93
54,19
CE+Soft-ECE
72,94
47,65
17.56
72,95
1,66
38,81
55,84
72,94
47,65
17,56
72,95
1,66
38,81
55,84
FL+Soft-ECE
72,61
47,4
1,964
72,6
2,6
39,94
56,71
72,61
47,39
2,99
72,59
2,1
39,71
56,69
CE+AUCOCLosS
74,56
49.1
12.33
74,66
1,65
34.78
52,01
74,74
49,21
12,48
74,7
1,67
34,62
52,28
FL+AUCOCLoss
74.3
49,19
3,23
74,25
1,86
34,85
53,15
74,3
49,12
3,17
74,27
2,06
34,86
53,01
AUCOCLoSS Prim.
73.03
47.7
8.18
73,22
2.5
37,93
55,64
72,89
47,5
8,13
73,08
2,58
38,49
55,76
Table 3: Test results DermaMNIST for accuracy, AUCOC and ECF. For each loss function,
report the results of the model check-pointed on the best AUCOC on validation set, pre and post TS. The last two columns report To corresponding to 90% and 95% accuracy (pre TS). bold the best result for each metric. The average results over 3 runs are reported.
Pre TS
Post TS
Tp
acc.
Loss funct.
AUCOC
Accuracy
ECE 4
AUCOCT 1
ECE
90%
95%
CE
89,84
71,59
8,6
90,17
3,07
43,51
56,21
FL (1=3)
90.5
72,64
11,97
90,94
4.24
40.63
53,9
AdaFL53
90.11
73.1
12,54
90,73
3,88
40,78
55,86
CE+MMCE
89,71
70,99
10,87
89,77
3.59
45,82
58,07
FL+MMCE
89,34
71,72
4.35
89,57
3,65
46,81
59,1
CE+Soft-AwUC
89,67
71,51
5,58
90,15
3,79
43,04
57,09
FL+ Soft-AvUC
89,42
71,04
4,09
89,62
3,48
45,47
58,69
CE+Soft-ECR
89,54
71,46
9,35
89,91
3,23
43,48
57,82
FL+Soft-ECE
90,22
72,62
10,31
90,53
4,5
40,95
57,46
CE+AUCOCLoss
90.77
74.3
12,98
91,01
5,7
39,01
52,7
FL+AUCOCLoss 91,35
74,8
7,69
91,44
5.1
37,3
53,9
AUCOCL Prim.
90,87
73,42
9,11
90,93
8,25
40.4
55,44
We believe that this new direction of considering expert load in human-Al collaborated system is important, and COC curve and AUCOCLoss will serve baseline for future work.
8

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Under review as a conference paper at ICLR 2023
Table 4: Test A on OOD detection for models trained on CIFARI00 and tested on
CIFAR100-C (Gaussian noise) and SVHN, pre and post TS, for models check-pointed for AUCOC.
Loss funct.
CIFAR100-C, AUROC f
SVHN, AUROC f
Pre TS
Post TS
Pre TS
Post TS
CE
74,37
75,22
77,42
79,42
FL (y=3)
75,12
75,35
76,61
77,08
AdaFL53
74,53
74,68
80.3
81,28
CE+MMCE
74,67
74,53
75,79
77,39
FL+MMCE
74,42
74,39
77.57
77,41
CE+Soft-AvUC
73,63
73,62
78,04
78,9
FL+ Soft-AvUC
72,78
76,72
79.68
79,68
CE+Soft-ECH
73,29
73,18
77,02
77,69
FL+Soft-ECE
74,29
73,97
79,68
81,17
CE+AUCOCLosE
76,03
76,8
82,03
83,5
FL+AUCOCLOS
76,31
76,02
80,11
79,35
AUCOCLoss Prim.
79,02
78,86
81,12
79,82
Bler- no accurary of network OS!
90
Eldr> fo accuracy of the
90
80
80
70
70
20
60
100
20
a0
100
Tel samples be anslysed manually (%)
13: samples be aralysed manually (%)
(a) Checkpoints for AUCOC, TS
(b) Checkpoints for AUCOC, TS
Bddr accuracy oftre network 196)
Eldr- ral accuracy of the network
90
90
80
80
AUCOC 55.02
93.00
70
40 60
100
20
40
100
Tel: samples be analysed manually (%)
Tsl samplos be aralysed manusily (56)
(c) Checkpoints for accuracy, TS
(d) Checkpoints for accuracy, TS
Figure 2: COC curves on CIFARION 100, run for one seed. In the first row the plots for models check- pointed with AUCOC, the second with accuracy. In the first column models without TS, in the second with TS.

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Under review as a conference paper at ICLR 2023
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Under review as a conference paper at ICLR 2023
A APPENDIX
A.1 DERIVATION OF EQUATION
The y-axis in COC curve is expressed mathematically by:
plar 2 20) plr 2 ro)
Sc' ple, r)dr Sro p(r)dr
Blcr 2 ro] = p(drr ro)c=
Σ
Σ
Sr p(dr))(())dr
Sh Ep(dr)ep(r)dr Ir p(r)dr
St Bldrlp(()ddr Sr p(r)dr
Sra p(r)dr
So Blclrlp(r)dr 1
The x-axis in COC curve is expressed mathematically by:
To = p(r < ro) - - n
drr)dr
Using the Ty formulation, we can rewrite the y-axis as
J ((c))
Elc/r 2 rol
Maximizing the area under this curve, over To E [0, 1] corresponds to dro
max A = max /'' ( B(cr)p(()dr
Let us assume use Gaussian kernel with this expression:
(r rnn
K[Ilr r1 V2Ta exp
2n²
Developing the equation, the area calculation becomes:
dro
A
E(clrp(())dr
11-70
To
N
dro
(c.)) - - -lll)drr
11-70 7D
N
1
(r - rx
dro
((c))
exp
dr
V2Ta
20x2
TO
(8)
(r )2
dTo
1(cx)
eex
dr
V2TTa
2a
To
(((x)
Indrr
(1-
(1-th)
Voou
In
ndtr
("o0 Veoo
ro In
dru
To
#thresh
[TTON-1, S(TO, ro.k)
=
(To.k - To.k-1)
2
1=1
Where:
( To
To
N Σ
T0.k
p(r)dr
N
K(Ilr - rn 11)dr
N
To, -Tn In
-Tn Voor
ndtr
ndtr
N
Voou
n=
12

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Under review as a conference paper at ICLR 2023
Where ndtr expresses the Gaussian cumulative distribution function and the last row in Equation exploits the trapezoidal rule for integrals computation.
A.2 DERIVATIONS OF THE GRADIENTS OF AUCOC
d
d d0
{ Blclrip(r)dr
d7o
A=
d0
Here, we use the assumption discussed in Section 2 that To does not depend on any parameter, thus allowing us apply Leibnitz's integration rule, obtaining:
d
f. 1 de
d
B(cr))(()dr)) } dro TO
A
(10)
d8
d
dro db
dio
(11)
db
To can be expressed as:
To = p(r S = w p(r)dr
(12)
Consequently:
dTu
dp(r)
dru
dr + p(ro)
0
d0
d0
d0
(13)
dp(r) dr de
dro
d0
p(rp)
Plugging this expression back into Equation 10 we obtain:
dA
d0 / U
d0 B(drlp(()dr + E[c|ro] /-' dp(r)
dro
d0 dr
(14)
TO
Assuming the use of a Gaussian kernel:
(_ (r- rnn
K(Irr - rxll)
exp
(15)
V2Ta
2a²
And re-writing:
B(clrolp(ro) p(ro)
ΣN ((,,) rnll)
(16)
| K(1roo -ryll)
N
The gradient of the area becomes:
dA
/ {1" dr.
pro dp(r)
dto
Widrll(()dr+ Elclrol
dr
dr
drn
1 1- TO
(r r2
J/ dra 2 N Σ 1(c,)
exp
dr+
2012
Bccro h- dr, V2TTe n=1
1 exp
(r-rm)
dio
dr
20
To
= - 1
(r -
(17)
V2ma3N 1(nn) . (r (r-ra) ra) exp
dr+
2a²
E(c[ro] V2Ta2N (n)) --)) exp (- (r - r2))
(T)
dTu
dr
2a²
To
= =/0²
rn )
(ro r2n
{-
V2TaN 1(cp):
exp
exp
2a²
2a²
1
(ro - r2
(-rn)²
dro
E|c/ro]
exp
exp
dr)
V2TaN
2a
2a
T0
13

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Under review as a conference paper at ICLR 2023
Also for the gradients in the code implementation exploited the trapezoidal rule for the compu- tation of the external integral between [O.1].
B ADDITIONAL CALIBRATION METRICS
In this section
adding results on additional calibration metrics: Kolmogorov- -miinn (KS)
Gupta et al, 2021), Brier score (Brier 1950) and class-wise ECE (cw-ECE) (Kull et al. 2019).
Results reported in Tables respectiveeyl for CIFAR100, Tiny-ImageNe and DermaMNiST. For all the metrics and datasets, AUCOCLoss provides either the best or comparable results with respect the baselines.
Table 5:
Test results on CIFARI00 for Brier score, class-wise ECE (cw-EC3) and Kolmogorov-
Smirnov (KS), both for checkpoint AUCOC (gray line)and accuracy (white line).
Pre TS
Post TS
Loss funct.
Brier 4
KS
cw-ECE +
Brier +
KS
ew-ECE +
CE
36,26
11,55
0,293
33,91
1,1
0,211
34,94
13,56
0,315
31,58
1,28
0,206
FL (y=3)
31,35
4,05
0,19
31,16
1,46
0,184
31,36
4,00
0,189
31,19
1,42
0,183
AdaFL53
31,49
4.17
0,191
31,3
1.22
0,183
31,54
3,95
0,191
31,35
1,29
0,184
CE+MMCE
36,15
9,99
0,271
34,6
1,23
0,209
35,07
13,25
0,311
31,87
1,46
0,199
FL+MMCE
31,24
1,62
0,182
31,24
1,82
0,181
31,25
1,29
0,181
31,24
1,52
0,180
CE+Soft-AVUC
32,29
6,82
0,215
31,71
0,53
0,199
32,42
7.14
0,219
31,76
0,62
0,197
FL+ Aoft-AVUC
33,34
10,05
0,258
31,74
1,58
0,177
33,43
10,23
0,179
31,78
1,8
0,176
CE+Soft-ECE
32,44
7,06
0,22
31,85
1.3
0,197
32,44
7,08
0,219
31,83
0.83
0,197
FL+SoRt-ECE
32,87
6,43
0,235
32,19
1,56
0,180
32,88
6,92
0,245
32,1
1,25
0,179
CE+AUCOCLoss
30,54
6.72
0,21
29,78
0,75
0,184
30,62
6.3
0,204
30,0
0.93
0,188
FL+AUCOCLOss
30,33
2,32
0,188
30,24
0,95
0,175
30.35
2,28
0,187
30,25
0,93
0,177
AUCOCLoss Prim.
31.01
5,93
0,242
30,49
1,52
0,185
31,14
5,95
0,245
30,62
1,59
0,188
14

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Under review as a conference paper at ICLR 2023
Table Test results on Tiny-ImageNet for Brier score, class-wise ECE (cw-ECE) and Kolmogorov- Smirnov (KS), both for checkpoint on AUCOC (gray line)and accuracy (white line).
Pre TS
Post TS
I Loss funct.
Brier +
KS 4
cw-ECI: 4
Brier
KS 4
ew-FCE +
CE
72,56
21.38
0,287
66,25
0,74
0,163
72,89
22,05
0,295
66,14
1,05
0,160
FL (y=3)
67,29
10,34
0,198
65,91
1,14
0,152
67,42
10,73
0,198
65,93
0,74
0,151
AdaFL53
67,36
10,98
0,207
65,71
0,85
0,159
67,19
10,47
0,197
65,72
0,97
0,157
CE+MMCE
72,5
21,12
0,286
66,42
0,93
0,157
72,25
20,76
0,283
66,43
0,82
0,161
FL+MMCE
65,53
2,7
0,158
65,49
1,63
0,151
65,67
3,38
0,158
65,53
1,05
0,151
CE+SOft-AvUC
72,23
21,77
0,287
65,2
1,15
0,146
72,25
21,67
0,286
65,3
0,98
0,146
FL+ Soft-AvUC
65,3
7.17
0,182
64,62
1,13
0,15
65,54
8,01
0,292
64,65
1,02
0,145
CE+Soft-ECE
70,34
17,56
0,251
66,
0,81
0.15
70,34
17,56
0,251
66,14
0,82
0,150
FL+Soft-ECE
66,35
1,77
0,156
66,4
2,37
0,161
66,5
2,81
0,159
66,43
1,91
0,152
CE+AUCOCI
66.3
12,23
0,209
64,23
1,29
0,157
66,25
12.33
0,210
64,23
1,29
0,157
FL+AUCOCLoss
64,54
3,13
0,165
64,45
1,29
0,155
63,65
3,11
0,164
64,47
1,24
0,158
AUCOCLoss Prim.
66,81
8,18
0,183
65,88
2,5
0,168
66,91
8,15
0,184
65,98
2,51
0,174
C CHOICE OF THRESHOLDS тP
The vector of thresholds ro [ro. Ko is chosen to equal to the predicted confidences vector [r1, [r.,)..', This choice has few noticeable consequences. First, it saves time and energy consumption, as the thresholds are inherently learnt from the predicted confidences for each model. Moreover, ro is automatically "calibrated" to the output of each model, the interval of TO E 1] is spanned uniformly, providing full flexibility to choose the most suitable operating point. Finally, and most importantly for feasible implementation of the proposed method as reported in Appendix A.2 this choice makes not dependent on the network parameters. In fact, what relevant for the computation To is the proportion of samples whose confidence is smaller than the corresponding To and this adaptive choice of thresholds guarantees consistency irrespective of the specific values r. This can be explained with the aid of Figure 3 The first two axes at the top are the output confidence space for two neural networks (the same conclusion would hold we considered single model at two different epochs). The arrows indicate smallest output confidence of the dataset for each model, which have two different values. Equally, the green arrows point to the second smallest samples. Given our threshold selection strategy, these samples correspond to the two smallest thresholds for each model, n:NN1 r.V and NN ro. for NN1 and NN2 respectively. Irrespective of the different threshold levels between the models, the corresponding values To the same for both NNI and NN2, respectively 0% and 25% for the first two thresholds To.1, To,2, in this example with four samples.
15

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Under review as a conference paper at ICLR 2023
Table Test results on DermaMNIST for Brier score, class-wise ECE (cw-ECE) and Kolmogorov- Smirnov (KS), both for checkpoint on AUCOC (gray line)and accuracy (white line).
Pre TS
Post TS
I Loss funct.
Brier +
KS
cw-ECI: 4
Brier
KS 4
ew-FCE +
CE
39.71
8,06
3,22
38,11
2.15
2,22
45,48
19,83
6,01
37,01
1,45
2,53
FL (y=3)
40,00
11,14
4,06
36,55
1,77
2,03
41,68
17,02
5.12
35,69
1,81
1,91
AdaFL53
39,94
12,32
3,93
36,19
1,95
1,66
39,42
10,68
3,67
36,7
1,43
1,90
CE+MMCE
41,89
10,88
3,69
38,96
2,73
2,2
44,05
18,67
5,59
36,82
2,44
2,48
FL+MMCE
38,96
3,10
2,55
38,91
2,29
2,34
40,71
13,74
4,72
35,73
1,06
1,89
CE+SOft-AvUC
38,64
5,18
2,55
38,28
2.69
2,21
45,07
18,29
5,67
38,12
3,62
2,7
FL+ Soft-AvUC
39,01
6,75
3,09
37.98
2.13
1,82
39,08
12,00
3,89
36,38
2,04
1,88
CE+Soft-ECE
40,37
9,39
3,31
38,39
2,73
1.92
44,67
19,53
5,87
36,7
1,55
2,15
FL+Soft-ECE
38,87
10.13
3,4
36,91
2.67
2,08
39,23
11.74
3,78
36,05
1,35
1,92
CE+AUCOCI
38,3
7,16
4.11
36,12
1,56
1.78
38,15
12,43
3,8
35,33
1,67
1,98
FL+AUCOCLoss
36,07
6,55
2,55
35,46
2.04
2,04
36,02
4,06
2,27
35,71
2,11
1,93
AUCOCLoss Prim.
37,18
5,22
2,68
38,06
1,17
2,37
36,98
4,7
2,53
36,74
3,00
2,8
NN1
MN1
Predicted Confidences
Neural Network (NN1)
0
0.1
0.2
Predicted Confidences
Neural Network (NN2)
0
1
T022
X
X
0%
25%
100%
Figure 3: Illustrative example that shows how To does not depend on any parameter nor on any specific threshold level ro. Even if the models have different threshold levels, the points on To axis are the same.
D RESULTS ON A NEW MEDICAL DATASET: RETINAMNIST
In this Section we report the results on RetinaMNIST Yang 2021 using ResNet-50 as back- bone and batch size 128. The RetinaMNIST is based on the DeepDRiD24 challenge, which provides dataset of 1,600 retina fundus images. The task is ordinal regression for 5-level grading of diabetic retinopathy severity. Consistently with the original paper, we split the source training set with ratio of into training and validation set, and use the source validation set as the test set. For the selection of the common hyperparameters, we followed the suggestions of Yang et al. (202 for all baselines and AUCOCLosS.
16

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Under review as a conference paper at ICLR 2023
Table 8: Test results on RetinaMNIST for accuracy, AUCOC and ECE. For each loss function, we report the results of the model check-pointed on the best AUCOC validation set, and post TS. The last two columns report To corresponding to 65% and 75% accuracy (pre TS). bold the best result for each metric. The average results over 3 runs reported.
Pre TS
Post TS
TD 4
acc.
Loss funct.
AUCOC
Accuracy
ECE
AUCOC
ECE
65%
75%
CE
70,86
51,08
13,27
72,05
11,42
49,4
63,25
FL (x=3)
69,22
50,83
17,08
69,27
11,59
51,5
76,63
AdaFL53
68,84
50,75
17,38
69,21
11,58
52,88
67,00
CE+MMCE
71,46
51,00
10,96
70,64
11,75
55,79
70,92
FL+MMCE
70,52
51,25
10,94
70,83
10,23
56,92
83,38
CE+Soft-AvUC
70,48
51,08
7,71
63,33
10,21
70,08
61,54
FL+ Soft-AvUC
68,71
49,75
11,59
58,43
12,67
60,88
67,67
CE+Soft-ECR
72,15
52,00
9,27
70,91
11,05
49,04
62,5
FL4Soft-ECE
67,99
51,83
13,44
67,9
11,05
48,33
58,96
CEAAUCOCLOss
73,54
53,33
7,49
72,11
10,83
48,00
53,79
FL+AUCOCLOss 71,71
53,33
11,19
70,2
9,02
48,3
69,83
AUCOCLoSs Prim.
67,94
48,08
22,63
68,09
13,24
52,5
66,54
Table Test results on DermaMNIST for accuracy, AUCOC and ECE for batch size 64 and 32 for models check-pointed on the best AUCOC on validation set.
Batch size 64
Batch size = 32
Loss funct. CE+AUCOCLoss
AUCOC1
Accuracy
ECE
AUCOCT
Accuracy +
ECE +
91,16
75.18
11,00
90,14
73,8 73,47
15.5
FL+AUCOCLosS
91,13
74,71
9,35
91.1
6,69
AUCOCLoss Prim.
88,85
70,31
8,9
88,68
71,19
8,36
E ADDITIONAL RESULTS
In this section we report additional results and plots from our analysis. Moreover Table reports an ablation study on the impact of batch size of both the baselines as well as AUCOCLosS.
17

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Under review as a conference paper at ICLR 2023
Table 10: Test results CIFARI00 100 for accuracy, AUCOC and ECE. For each loss function, report the results of the model check check-pointed on the best accuracy on validation set, pre and post TS. The average results over runs reported. The best results highlighted in bold.
Pre TS
Post TS
@ acc.
Loss funct.
AUCOC 1
Accuracy t
ECE +
AUCOCT
ECE
90%
95%
CE
82,43
78,02
13,66
93,89
4,11
23,59
38,13
FL (x=3)
93,88
77,83
3,93
93,79
1.76
24,98
40,7
AdaFL53
93.87
77.77
3,95
93,78
1.74
25,14
40,54
CE+MMCE
85,34
77,81
13,33
93,77
4,21
24.00
38,61
FL+MMCE
93.87
77.74
1,55
93,89
1,79
25,58
40,98
CE+Soft-AwUC
93,94
77,53
7,13
93,82
3,17
24,88
44,67
FL+ Soft-AvUC
93.1
77.76
10,23
93,77
3,7
24.68
39,8
CE+Soft-ECE
93,87
77,57
7,09
93,73
3,42
24,81
40.12
FL+Soft-ECE
93,42
76,8
6,92
93,59
1,43
28,08
43,38
CE+AUCOCLOSS
94,38
78,74
6,2
94,26
1,55
22,21
36,96
FL+AUCOCLoss 94,16
78,22
2,47
94,22
1,24
23,58
38,7
AUCOCLoSS Prim.
94,17
78,18
5,95
94,24
2,09
23,83
38,67
Table 11: Test results on DermaMNIST for accuracy, AUCOC and ECE. For each loss function, we report the results of the model check-pointed on the best accuracy on validation set, pre and post TS. The average results over rums are reported.
Pre TS
Post TS
To
acc.
Loss funct.
AUCOC
Accuracy
ECE
AUCOC 1
ECE
90%
95%
CE
65,41
73,41
26,42
90,07
2,58
41,51
62,05
FL (y=3)
89,85
73,83
17,08
91.1
3.67
40,78
55,84
AdaFL53
89,44
73,52
10,71
90,65
2,77
42.39
58,07
CE+MMCE
74.89
73,25
19,13
90,43
3,84
40.4
56,74
FL+MMCE
90,53
74,21
15,59
90,92
3,52
39,88
56,39
CE+Soft-AvUC
73,27
73,64
19,04
89,98
3,9
42,54
60,47
FL+ Soft-AvUC
86,78
73,18
12,02
90,84
2,77
39,42
55,59
CE+Soft-ECE
73,74
73,63
20,4
90,31
3,05
41,6
58,12
FL+Soft-ECE
89,68
73,35
11,78
90,8
3,88
41,95
56,76
CE+AUCOCLoss
90,59
74,78
12,47
90,9
5,8
36,63
53,15
FL+AUCOCLoss 91.18
74,85
7,24
91,24
5,8
36,56
51,17
AUCOCLosE Prim. 90,55
73,8
8.3
90,57
7,9
41,7
60,77
Table 12: Test AUROC(%) on OOD detection models trained on CIFAR100 and tested on CIFAR100-C (Gaussian noise) and SVHN, pre and post TS. Results correspond to model check- pointed for accuracy.
Loss funct.
CIFAR100-C, AUROC
SVHN, AUROC
Pre TS
Post TS
Pre TS
Post TS
CE
75,75
75,91
78,9
80,41
FL (y=3)
74,94
75,14
77.37
77,88
AdaFL53
74,44
74,57
79.7
80,68
CE+MMCE
75,91
75,88
75.51
76,94
FL+MMCE
74.17
74,11
78,28
78,11
CE+Soft-AvUC
73,78
73.7
77,46
78.34
FL+ Soft-AvUC
75.92
75.91
79,22
80,52
CE+Soft-ECE
73,32
73,19
77.17
77,93
FL+Soft-ECE
75,92
74,49
83,41
81,63
CE+AUCOCLoGG
75,69
76,38
81,37
82,97
FL+AUCOCLOSS
76,08
75,83
79,03
78,24
AUCOCLOSS Prim.
78,8
78,5
79.4
78.13
18

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Under review as a conference paper at ICLR 2023
network (S5)
accurary of the
Elir>> fol accuracy of the
RL AUCOC 94.33
FL. AUCOC 93.
ADDEL AICOG 6E.5
AdaEL AUEDE 93.02
RNCE BUCOE 95.00
MIEE, AUCOE $1.00
Son ECE, AUCOC 93.51
FL Soft ECE, AUCOC 93.70
ScfAAIIC AUCOC 98.31
FL. Soft-ayKC AUCGE 93.01
AUCOC.oss AUCOC 94.2
TL. AUCOCLoes, AUDOC 94.34
70
manually
(a) Checkpoints for AUCOC, TS 0
(b) Checkpoints for AUCOC, TS
network 196)
Eldra>- par>-aal -al: accurscy the natwork 180
Bler m:)): accurscy the
FL AUCOC 24.03
FL. AUCOC 93.20
AdOYL AUCOC 53.20
AdDPL AUEDE 25.
RL NHCE AU.0C
FL. MICE, AUCOC 93.97
SoftLEE, AUCDE 93.40
Sulaace, AUDOC stati
SoRAAUC, AUDOC 91.35
FL. Sott-ArJ AUCDE 23.07
AUCOCLass AUCOC 94.25
FL. AUCDCLoSS, AUCOC 94.31
70
samples analysad manually
samples analysed manually
(c) Checkpoints for accuracy, TS 0
(d) Checkpoints for accuracy, TS
Figure 4: Additional COC curves CIFAR100. In the first the plots for models check-pointed
with AUCOC, the second with accuracy. In the first column models without TS, in the second with TS.
19


Reviews:
[{"review_id": "d9x5b_nnb6", "reviewer": "Reviewer_4BfS", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "", "summary_of_the_paper": "The paper presents a COC curve and area under the COC as a loss to obtain a better trade-off between network confidence and accuracy. The proposed loss together with the standard cross-entropy loss yields a better trade-off as measured by AUCOC.", "summary_of_the_review": "There are concerns about the value addition of the proposed method and an important reference missing. Due to this I'm recommending rejection but looking forward to the response from the authors.\n\n## Post rebuttal\nI thank the authors for the detailed response. I'm convinced that the proposed AUCOC loss adds value beyond calibration and I appreciate the effort to include KS and other calibration metrics. The discussion about the relation to calibration should be included in the paper to improve clarity. I'm increasing the score to marginal-accept.", "strength_and_weaknesses": "## Strengths\n1. AUCOC is interesting as a measure of a trade-off between confidence and accuracy and also as an auxiliary loss.\n2. The experiments show marginal improvements in this trade-off as measured by AUCOC.\n\n## Weaknesses\n1. An important paper is not cited [a]. For instance, the COC curve introduced in this paper has similarities to the ones used in [a] and the Kolmogorov-Smirnov (KS) metric has similar properties to AUCOC. Specifically, the counter-example provided in Fig.1 for ECE is not applicable to the KS metric. This raises questions about the value addition of the COC/AUCOC. Please address this.\n2. The value of AUCOC as a loss is not very clear to me. Is it improving calibration if so it should be shown in the evaluation. I think it would be better to show the KS metric in table 1 to show calibration. As we know, ECE is susceptible to the binning scheme.\n3. Section 2.3 makes multiple approximations and it is not clear about the effect of them in practice.\n\n\n[a] Gupta K, Rahimi A, Ajanthan T, Mensink T, Sminchisescu C, Hartley R. Calibration of neural networks using splines. ICLR 2021.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "ah2v16Pi4O", "reviewer": "Reviewer_DLK2", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "", "summary_of_the_paper": "This paper focuses on the trade-off between the error rate of the classifier and the number of examples that need to be processed manually because the system was not confident enough to process them automatically. This trade-off is at the heart of all uses of automatic classifiers for industrial tasks with maximum error rate constraints. The authors propose a curve (Confidence Operating Characteristics (COC)) and a measure on this curve (AUCOC) to compare classifiers by taking into account the rate of examples rejected to a treatment because of a lack of confidence of the classifier. The authors compare the use of a loss function derived from this curve for training neural networks with standard calibration methods. The experiments are conducted on 3 datasets (CIFAR100, TinyImageNet and DermaMNIST)\n\n", "summary_of_the_review": "The method proposed in this article is not totally new and its experimental evaluation does not show a major gain compared to other calibration methods. The experiments remain somewhat limited, on data sets that do not really correspond to industrial use cases", "strength_and_weaknesses": "The paper addresses an important but often neglected problem in the field: the search for an operating point in the implementation of an automatic prediction system coupled with a manual validation. This problem is closely related to the search for a good calibration for machine learning models and in particular neural networks.\n\nHowever, the proposed curve is not new and has been used for a long time in industrial systems.  See for example the read/substitution curves, used to determine the number of rejected bank checks: \n\n N. Gorski, V. Anisimov, E. Augustin, O. Baret, and S. Maximov, “Industrial bank check processing: the A2iA CheckReader,” Int. J. Doc. Anal. Recognit., pp. 196–206, 2001.\n\nExperiments seem to show an advantage to the proposed method, but this advantage remains limited. The datasets chosen are perhaps not the most adequate to highlight the contribution of the proposed method because, apart from DemaMNIST, they do not correspond to use cases where a minimum performance is fixed and implies manual processing to achieve it. An illustration on real cases of the contribution of the proposed method in terms of reduction of the percentage of manual processing would be interesting. Figure 4 in the appendix gives an indication in this sense and shows that the difference between the different methods is very small.\n\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "91btbqbH98e", "reviewer": "Reviewer_FvY2", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "", "summary_of_the_paper": "This paper considers the fact that there should be an operating point based on the confidence of a model (deep neural network) in which an expert should examine the decisions in human-AI collaboration settings. The authors propose a confidence operating characteristics (COC) curve to summarize a model's performance. In COC, the x-axis is the proportion of samples with confidence scores less than a threshold; that is the proportion of samples that should be delivered to an expert. The y-axis is the model's expected accuracy for samples with higher confidence scores than the threshold. The curve is constructed by varying the threshold value. Then, the authors propose the area under the COC curve (AUCOC) that summarizes the curve as a single metric. They use the metric as an objective (usually combined with other metrics such as cross-entropy) to train deep nets. Kernel density estimation is utilized to compute the (extra) terms in the objective. They claim that the trained networks with this extra objective are more accurate and require fewer samples to be delegated to the experts.", "summary_of_the_review": "I think the paper is well-written and well motivated. The results look good and the code is also provided. A bit more details on the experiments and analysis of the introduced loss function might be required.", "strength_and_weaknesses": "The paper is, in general, well-written.\n\nThe authors consider an important problem where the expert load versus the confidence/accuracy of a model is examined.\n\nTaking expert load into account might be helpful in some applications.\n\nTheir method seems to have good results both for in and out of distribution cases.\n\nWhile the toy example is intuitive, the problem is raised because of the binning issue in the ECE metric. Binning-free metrics like the Brier score may not suffer from this issue. Evaluating on other metrics such as Brier, NLL, KS[1], and classwise calibration (calibration on other classes rather than the top one) can further strengthen the paper.\n\nIt would be better to discuss the computational time and the extra overhead the proposed method may have in the main paper. \nSome details are missing: e.g., the number of bins (M) used for calculating the ECE metric, and the kernel density is computed using batches or the whole dataset.\n\nThe ECE errors are sometimes higher than other methods (especially on the DermaMNIST dataset). This brings a few questions: Is AUCOC loss a proper scoring rule? Is this going to have some effect on the results of AUCOC?\n\nTwo checkpoints are considered: one by the accuracy of the model and the other by the AUCOC value. Which one do the authors suggest using?\n\n[1] Gupta et al., \"Calibration of neural networks using splines,\" ICLR 2022.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "Not applicable"}, "extra_text": {}}]

Author rebuttal:

Decision: Reject