Taking a Step Back with KCal: Multi-Class Kernel-Based Calibration for Deep Neural Networks
Authors: Zhen Lin
Shubhendu Trivedi
Jimeng Sun
Venue: ICLR 2023 poster (2023)
Keywords: Calibration
Kernel Density Estimation
Neural Networks
Healthcare
Classification

Abstract: Deep neural network (DNN) classifiers are often overconfident, producing miscalibrated class probabilities. In high-risk applications like healthcare, practitioners require fully calibrated probability predictions for decision-making. That is, conditioned on the prediction vector, every class’ probability should be close to the predicted value. Most existing calibration methods either lack theoretical guarantees for producing calibrated outputs, reduce classification accuracy in the process, or only calibrate the predicted class. This paper proposes a new Kernel-based calibration method called KCal. Unlike existing calibration procedures, KCal does not operate directly on the logits or softmax outputs of the DNN. Instead, KCal learns a metric space on the penultimate-layer latent embedding and generates predictions using kernel density estimates on a calibration set. We first analyze KCal theoretically, showing that it enjoys a provable full calibration guarantee. Then, through extensive experiments across a variety of datasets, we show that KCal consistently outperforms baselines as measured by the calibration error and by proper scoring rules like the Brier Score.

Published as a conference paper at ICLR 2023
TAKING A STEP BACK WITH KCAL: MULTI-CLASS -ARNEL-BASED CALIBRATION FOR DEEP NEURAL
NETWORKS
Zhen Lin"
Shubhendu Trivedi
Jimeng Sun
Department of Computer Science, University of Illinois Urbana-Champaign 2 Carle Illinois College of Medicine, University of Illinois at Urbana- Urbana-Champaii Champaign (zhenl in4, jimeng) 8illinol alinois..dd shubhendu@csai 1 mi edd
ABSTRACT
Deep neural network (DNN) classifiers are often overconfident, producing miscalibrated class probabilities. In high-risk applications like healthcare, practitioners require fully calibrated probability predictions for decision-making. That is, conditioned on the prediction vector, every class' probability should be close to the predicted value. Most existing calibration methods either lack theoretical guarantees for producing calibrated outputs, reduce classification accuracy in the process, or only calibrate the predicted class. This paper proposes new Kernel- Kernel-based calibration method called KCal. Unlike existing calibration procedures. KCal does not operate directly on the logits softmax outputs of the DNN. Instead, KCal learns a metric space on the penultimate-layer latent embedding and generates predictions using kernel density estimates on a calibration set. We first analyze KCal theoretically, showing that it enjoys a provable full calibration guarantee. Then, through extensive experiments across a variety of datasets, we show that KCal consistently outperforms baselines as measured by the calibration error and by proper scoring rules like the Brier Score. Our code is available at https::////.. /tttt::://gttthbi com/ z zlin7 com/2lnn//KCa /KCal
INTRODUCTION
The notable successes of Deep Neural Networks (DNNs) in complex classification tasks, such as object detection (Ouyang& & Wang. 2013), speech recognition (Deng et al. 2013), and medical diagno- sis (Qiao et al. 2020 Biswal et al. 2017), have made them essential ingredients within various critical decision-makir pipelines. In addition the classification accuracy, classifier should ideally also generate reliable uncertainty estimates represented in the predicted probability vector. An influential study (Guo al. 2017) reported that modern DNNs are often overconfident or misealibrated. which could lead to severe consequences in high-stakes applications such as healthcare (Jiang et al. (012)).
Calibration the process of closing the gap between the prediction and the ground truth distribution given this prediction. For a K K-classificatton problem, with covariates X E X and the label Y E y [K] denote our classifier + A as p (p1....,Pun with A K-1 being (K-I)--implexx Then, Definition 1. (Full Calibration /Vuicenavicius el al. 2019 P is fully-calibrated if E [K]:
pq [g.....gk] E A PPY kIp(X) 9 Qk-
worth noting that Def. ① implies nothing about accurucy. fact, ignoring and simply
predicting т, the class frequency vector, results in fully calibrated but inaccurate classifier. As result, our goal always to improve calibration while maintaining accuracy. Another important requirement is that
A K-1 Many binary calibration methods such Zadrozny & Elkan 2001 2002) result in vectors that are not interpretable as probabilities, and have to be normalized.
Many existing works only consider confidence calibration (Guo et al.) 2017 Zhang et al. 2020
Wenger et al.) 2020; Ma & Blaschko (021)), a much weaker notion than that encapsulated by Def. and only calibrates the predicted class (Kull et al. 2019] Vaicenavicius al. 2019).

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Definition 2. (Confidence Calibration) p is confidence-calibraeed if: Pu (X) max =q} = q.
Vg [0, PYY
arg maggax (x
k
However, confidence calibration is far from sufficient. Doctors need to perform differential diagnoses on patient, where mul- tiple possible diseases should be considered with proper probabil- ities for all of them, not only the most likely diagnosis. Figure shows an example where the confidence calibrated, but pre- diction for important classes like Seizure is poorly calibrated. A classifier can be confidence-calibrate but not useful for such tasks if the probabilities assigned to most diseases inaccurate.
CG-
Trequemcy
Recent research effort has started to focus on full calibration, for example, in Vaicenavicius et al. (2019); Kull et al. (2019); Wid mann et al. (219). We approach this problem by leveraging the la- tent neural network embedding in nonparametric manner. Non- parametric methods such as histogram binning (HB) (Zadrozny) & Elkan 2001) and isotonic regression (IR) (Zadrozny Elkan 2002). are natural for calibration and have become popular. Gupta & Ramdas (202T) recently showed a calibration guarantee for HB. However, HB usually leads to noticeable drops in accuracy (Patel et al., 2021), and IR prone to overfitting (Niculescu-Mizil Caruana (005). Unlike existing methods, we take one step back and train new low-dimensional metric space on the penultimate- layer embeddings of DNNs. Then, we use kernel density estimation- classifier to predict the class probabilities directly. We refer to our Kernel-based Calibration method as KCal. Unlike most calibration methods, KCal provides high probability error bounds for full calibration under standard assumptions. Empirically, we show that with little overhead, KCal outperforms all existing calibration methods in terms of calibration quality, across multiple tasks and DNN architectures, while maintaining and sometimes improving the classification accuracy.
0.5 02
Figure I: Reliability diagrams for confidence calibration (top) and Seizure (bottom). The pop- ular temperature scaling (right) only calibrates the confidence, leaving Seizure poorly calibrated. See Figure 2 and the Appendix for complete reliability diagrams.
Summary of Contributions:
We propose KCal, principled method that calibrates DNNs using kernel density estimation on the latent embeddings.
We present efficient pipeline to train KCal, including dimension-reducci projection and stratified sampling method to facilitate efficient training.
We provide finite sample bounds for the calibration error of KCal-calibrated output under standard assumptions. To the best of our knowledge, this is the first method with a full calibration guarantee, especially for neural networks.
In extensive experiments on multiple datasets and state-of-the-are models, we found that KCal outperforms existing calibration methods in commonly used evaluation metrics. We also show that KCal provides more reliable predictions for important classes in the healthcare datasets.
The code to replicate all our experimental results is submitted along with supplementary materials.
RELATED WORK
Research on calibration originated in the context of meteorology and weather forecasting (see Murphy & Winkler 1984) for overview) and has a long history. much older than the field of machine learning (Brirr 1950; Murphy Winkler 1977 DDeroott & Fienberg 1983). We refer to Filho et al.) (202I) for a holistic overview and focus below on methods proposed in the context of modern neural networks. Based on underlying methodological similarities, we cluster them into distinct categorics.
Scaling: A popular family of calibration methods is based on scaling, in which mapping is learned from the predicted logits to probability vectors. Confidence calibration scaling methods include temperature scaling (TS) (Guo et al.) (017)) and its antecedent Platt scaling (Platt) 1199), ensemble of TS (Zhang et al. 2020), Couusian-PProeee scaling (Wenger et al.) (2020). combining a base calibrator (TS) with rejection option (Ma & Blaschko 2021). Matrix scaling with regularization
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was also used to perform full calibration (Kull et al. 2019). While some scaling- based methods can be data-efficient, there are no known theoretical guarantees for them to the best of our knowledge.
Binning: Another cluster of solutions relies on binning and its variants, and includes uniform- mass binning (Zadrozny & Elkan 200T). scaling before binning (Kumar et al.) 2019). and mutual- information-maximmzaaa based binning (Patel et al., (02T). Isotonic regression (Zadrozny & Elkan 2002) is also often interpreted as binning. Uniform mass binning (Zadrozny & Elkan 2001) has
distribution-free finite sample calibration guarantee (Gupta & Ramdas 2021] and asymptotic convergent ECE estimation (Vaicenavicius et al., 2019). However, in practice, binning tends to decrease accuracy (Patel et
Guo et al. 2017). Bimning can also be considered a member of the broader nonparametric calibration family of methods. Such methods also include Gaussian Process Calibration (Wenger et al. 2020), which however also only considers confidence calibration.
Loss regularization: There are also attempts to train calibrated DNN to begin with. Such methods typically add suitable regularizer to the loss function (Karandikar et al., 202T, Mukhoti et al.) 2020) Kumar et al. 2018), which can sometimes result in expensive optimization and reduction in accuracy.
Use Kernels: Although not directly used for calibration, kernels have also been used for uncertainty quantification for deep learning classification. In classification with rejection, the k-nearest-neighher algorithm (kNN), closely related to kernel-based methods, has been used to provide a "confidence measure" which is used to make binary decision (i.e., whether to reject to predict) (Papernot & McDaniel 2018 Diang et al. 2018). J. Recently, continuous kernels have also been used to measure calibration quality used as regularization during training (Widmann et al. 2019 Kumar al. 2018)) Zhang et al. (020) introduced a kernel density estimation (KDE) proxy estimator for estimating ECE. However, uses a un-optimized kernel over A A and shows the KDE-ECE estimator (but not the calibration map) is consistent. To the best of our knowledge, use of trained KDE to calibrate predictions hasn't been proposed before. Further, we also provide bound on the calibration error.
KCAL: KERNEL-BASED CALIBRATION
In this section, we formally introduce KCal, study its calibration properties theoretically, and present crucial implementation details and comparisons with other methods. Specifically, in Section 3.1 we discuss how to construct (automaticallyy calibrated predictions for test data using a calibration set Seal- Doing requires well-trained kernel and metric space, and we describe procedure to train such kernel in Section 3.2 In Section 3.3 we show that an appropriate shrinkage rate of the bandwidth ensures that the KCal prediction is automatically calibrated. Sections provides implementation details. Finally, in Section 3.5 we compare and contrast KCal with existing methods.
3.1 CLASSIFICATION WITH KERNEL DENSITY ESTIMATION
Following the calibration literature, we first require a holdout calibration set Scal (X,, Y) 1-1001110 In KCal, we fix a kernel function which is learned (the learning procedure is described in Section 3.2). For a new datum X the class probability (XN+1) takes the following form:
(x Sca) = Eiswessh (N41)
(3)
Eeewsss Ma, XN+1)'
where Shi ((x 8) Seally = k). The notation (x J. Seal) emphasizes the dependence on and Seat- However, we will use (XN+1) when the dependence is clear from context.
What we have described is essentially the classical nonparametric procedure of applying kernel density estimation for classification. For moment, suppose we know the true density function fa of Ps (the distribution of all the data in class k). and the proportion of class k. denoted m. (such that Eke(K] Th 1). Then, for any particular 2o. using the Bayes rule we get:
Remarks:
fa(2N)TT
P(Y = x|X mo} = EveRK Swr (2))
(4)
Now, replacing fa with the kernel density estimate fa(20) == (E (Einwess (o)(r, Sha and the class proportion mg with मे. |/ we get back Eq. (3)
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3.2 TRAINING
For good performance under the kernel density framework, it is crucial to employ an appropriate kernel function ø, which in turn relies the choice of the underlying metric. Therefore, train metric space on top of the penultimate layer embeddings deep learning models.
To begin, we assume deep neural network is already trained on Strain [Xtrain Y/ 1M We place no limitations on the form of loss function, optimizer, or the model architecture. However, we do require the neural net compute an embedding before final prediction layer, which is always modei the case in modern classification models. We denote the embedding function from + RA as f.
Given base "mother kernel" function c. such the Radial Basis Function (RBF) kernel, we denote the kernel with bandwidth as Øb == b ( We parameterize the learnable kernel as:
x'x,²') = (1, 2²) == 6e -I(((()))
(5)
where II IRAh + Rd is a dimension redducinn projection parameterized by shallow MLP (Sec- tion 3.4p. Since the inference time is linear in d, letting d < h also affords computational benefits.
Given that the embedding function fz) from the neural network fixed, the only learnable entities are b and II. In the training phase. we fix
1, and train II using (stochastic) gradient descent and los-loss. The specific value of b does not matter since it can be folded into II. Let us denote St.in = {(z, y) E Suain y = k.. In each iteration, we randomly sample two batches data from Swain - the prediction data, denoted as S to evaluate II, and "background" data for each k, denoted as B², from Stain \ S to construct the KDE classifier. Then, the prediction for any 2i is given by
P& (Jr; Stmin \. ( =- ( Σ
1
S ,(o, /
Σ
1 \ S o(2,2,) (6)
|B*]
|B41
(r.8))8k
where Ô is shorthand for δIL. defined in Eq.
The log-loss is given formally by
Algorithm 1 Overview of KCal
Input:
Σ
log (, d Sman S
Suain: (Xtrrain Ytrrin)) )= used to train the NN
B
Seat {(X,, Y Ji= calibration set
(AN))s
f: Embedding function X' - A (trained NN)
Finally, we pick bb= 6' on the calibration set Seal using cross-validation. This because should be chosen contingent
XN+1: Unseen datum for prediction
Training (of the projection II):
the sample size (Section 3.3 Choosing b can be done efficiently
Denote Ste
((x y) Srain k).
(Section 3.4 Algorithm [T summarizes the steps we explicated upon so far.
Denote Φ as a base kernel function (e.g. RBF)
with bandwidth
repeat
Sample Sta. f 9;)] - from Stain
3.3 THEORETICAL
Compute p(z,) via Eq. (6)
ANALYSIS: CALIBRATION COMES FREE
Loss / + B ソ LogLoss(p(z,).ou
In the previous section, we have only described procedure to improve the prediction accuracy for P on Swain- This section will show that cali- bration comes free with the P obtained using Algorithm [ In particular, we show that as the sample-size size for each class in Scal increases. converges to the true frequency vector of Y given the prediction. For smoother presentation, we only state the relevant claims in what follows. Detailed proofs are presented the Appendix.
Update II with (stochastic) gradient descent.
until the loss / does not improve.
Set do Ô...E for inference.
Inference:
Denote 5h := ((x. v) E Seally = k). Tune 6" on Scal by minimizing log loss. (eaneest &e- (1)
(x((N+1) Eennee oor (x.Xx+1)*
To begin, we make few standard assumptions, such as in Chacón & Duong (2018), 8), including:
(vk) The density the embedded space, [f k)), denoted fiiof,k is square integrable and twice differentiable, with all second order partials bounded, continuous, and square integrable. o is spherically symmetric, with finite second moment.
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arbitrary class i and ignore the subseript k to the density f for readability, We denote the size S = m. Intuitively, due to the bias- variance trade-off, a suitable bandwidth b will depend on m: A small b reduces bias, but with the finite m, a smaller b also leads to increased variance. Thus, b should go to O "slowly", which is formally stated below: Lemma 3.1. For almost all x, if mm + 00 and + 0 as m + oc, then we have
Lemma 3.1 and 3.2 focus
llfucen(z) f L as m oc.
Here STOOP, & is the estimated STEER using Seal- Recall that d is the dimension of II((())). We will call such bandwidth b admissible, and we sometimes write b(m) to emphasize the dependence on m. The following lemma gives the optimal admissible bandwidth:
Lemma 3.2. The optimal bandwidth is b = O(m )..), which leads to the fastest decreasing MSE (i.e. Blll f() - friof & (z) (22) of O(m a+*).
Now we are in a position to present the main theoretical results. In the following, m denotes the rarest class's 's count (m = ming [|St. }. Theorem 3.3 provides a bound between P and the true conditional probability vector on the embedded space P (XX)))
Theorem 3.3. Fixing I such that the density of [I((())) is positive, with b(m) = Om 2ोअ 2tz),, any X (0, P()p(()) (II((((())) > (3K + 1)Cm s Ke-Bm 444
for
(8)
where P& (II()()))) = P(Y = k (I(I(((X)) Iffx)))))
(9)
for some constant B and C. As a corollary, p(xx) almost almoss qurely (II((()))) as m xc.
Next, we bound the full calibration error with additional standard assumptions. More specifically, use and build upon the main uniform convergence result for classical KDE presented in Jiang (2017)
obtain Theorem 3.4
Theorem 3.4. Assume SHot, is a-Hölder continuous and bounded away from 0 for any k. For an Øース for some constants B and C we have:
admissible b(m) with shrinkage rale O(( log
m
[ssup (X) - P(Y k|p(x))) > (3K + 1)C( X.
log m ) S K(m (1m-11 m z2za 4+20
(10)
m
We now proceed to present details pertaining to the efficient implementation of KCal.
3.4 IMPLEMENTATION TECHNIQUES
Efficient Training: As might be immediately apparent, utilizing algorithm Darritthmoorr prediction using
full Syain S can be an expensive exercise. In order afford a training speedup, we consider random subset from Steai S using a modified stratified sampling. Specifically, we take m random samples from each Str denoted as train and replace the right-hand side of Eq. 6 with:
Sk.m.
|Staaa H(ا (o)r.
| ~(2, (0)).
(11)
(R.u)e Ah.m
ME(K), (t st
The re-scaling term |stt is crucial get an unbiased estimate of faFTR. The stratification employed makes the training more stable, while also reducing the estimation variance for rarer classes (more details in Appendix B). The overall complexity is now ((KmdhB) per batch. In all experiments, we used m 20 and B = 64.
mh
Form of II: While there considerable freedom in choosing a suitable form for II, we parameterize II with two layer MI. with a skip connection. Consequently, II can reduce to linear projection when sufficient, and be more expressive when necessary. We also experimented with using only a linear projection, the results for which are included in the appendix. We fix the output dimension to d min(dim 32). except for ImageNet (d 128).
Bandwidth Selection: Finally, to find the optimal bandwidth using Seal, use Golden-Section search (Kiefer 1953) to find the sag-loss-miniminnn
b*. This takes Oloog ub-lb, steps where [bb, ub] tol is the search space, and tol is the tolerance. Essentially, we assume that the loss is convex function with respect to b, permitting an efficient search (see Appendix H which presents empirical evidence that the convexity assumption is valid across datasets).
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3.5 COMPARISONS WITH EXISTING CALIBRATION METHODS
Most existing calibration methods discussed in Section 2 and KCal all utilize a holdout calibration set. However, unlike KCal, existing works usually fix the last neural network layer. KCal, on the other hand, "takes a step back", and replaces the last prediction layer with a kernel density estimation based classifier. Since the DNN f fixed regardless of whether we use the original last layer or not, are really comparing KDE classifier (KCal) with linear models trained in various ways, after mapping all the data with f. Note that this characterization is true for most existing methods, with a few exceptions (e.g., those summarized under "loss regularization" in Section 2
Employing a KDE classifier affords some clear advantages such as a straightforward convergence guarantee and some interpretability" Furthermore, KCal can also be improved in an online fashion, benefit especially desirable in certain high- applications such as in healthcare. For example, a hospital can calibrate trained model prior deployment using its own patient data (which is usually not available to train the original model) as it becomes available.
Another important advantage of KCal is concerning normalization. In fact, simultaneously calibrating all classes while satisfying the constraint that P E A K K- is a distinguishing challenge for multi-class calibration. Many calibration methods perform one-vs-rest calibration for each class, and require separate normalization step at test time (Zadrozny & Elkan [001] [002 Patel et al.) 2021] Gupta et al.) 2021). This creates gap between training and testing and could lead to drastic drop in performance (Section 4). On the other hand, KCal automatically satisfies p A K-1 and the normalization is consistent during training and testing.
A disadvantage of KCal is the need to remember the (Saa)) used to generate the KDE prediction. This is however mitigated to a large extent by the dimension reduction step, which already reduces the computational overhead significantly F there are 160K (5K images, d 32) scalars to remember, which is only 0.2% of the parameters (85M+) of the original DNN (ViT-base-parl base-patch11). Moreover, KDE inference is trivial to parallelize on GPUs. There is also a rich, under- explreex literature to further speed up the inference. Examples include, KDE merging (Sodkomkham al.. 2016). Dual-Tree (Gray & Moore 2003), and Kernel Herding (Chen et al. 2010). These methods can easily be used in conjunction with KCal.
For example, in one of our experiments on CIFAR-I00,
4 EXPERIMENTS
4.1 DATA AND NEURAL NETWORKS
We utilize two sets of data: computer vision benchmarks on which previous calibration methods were tested, and health monitoring datasets where full calibration is crucial for diagnostic applications. Table summarizes the datasets and their splits.
Table 1: Dataset summary: Splits and number of classes (K).
Dataset
In
IICCppt))
ISRUC
ISRUC(pat)
PN2017
C10
C100
SVHN
ImageNet
Train
103.818
1,936
61,841
69
15,087
45,000.
45,000
65,931
1,281.167
Calibration
1.787
77
1,372
6
253
5,000
5,000
7,326
25,000
Test
33,953
26,070
24
10,000
10,000
26.032
25,000
K
5
5
10
100
10
1,000
Benchmark data Following Kull et al. (2019). we use multiple image benchmark datasets, including
CIFAR-10. CIFAR-100, and SVHN (Krizhevsky 2009 Netzer et al. (201T). We reserve 10% of the training data as the calibration set. We fine-tune pretrained ViT (Dosovitskiy et al. (02T) and
MLP-Mixer (Mixer) (Tolstikhin al. (02T) from the t imm library [Wightman| 2019). We chose ViT and Mixer because they are the state-of-the-ae neural architectures in computer vision, and accuracy
should come before calibration quality. We also included the ImageNet dataset (Deng et al. 2009) Patell
and use the pretrained Inception ResNet V2 (Szegedy et al. 2017) following Patel et al. (2021
Health monitoring data We also use three health monitoring datasets for diagnostic tasks: III (Jing sinterrittalijjuu--cooooi
et al. 2021), an
(IIIC patterns classification dataset; ISRUC (Kha lighi et al. 2016). a sleep staging (classification) dataset using polysomnographic (PSG) recordings;
'That is, one could understand how the prediction is made by examining similar samples. Experiments about the effect of performance and overhead are provided in the Appendix.

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PN2017 (2017 PhysioNet Challenge) (Clifford et al. 2017 Goldberger et al. 2000 a public electro- cardiogram (ECG) dataset for rhythm (particularly Atrial Fibrillation) classification. For the training set, we follow Hong et al. (2019); Jing et al. (202I) for PN2017 and IIIC, and used 69 patients' data for ISRUC. For the remaining data, 5% is used as the calibration set and 95% for testing. We perform additional experiments after splitting into training/calibratiiontess sets by patients for III and I marked as the "pat" version in tables. The calibration/test split is 20/80 in "IIIC (pat)" and "ISRUC (pat)" because the number of patients is small. For IIIC and ISRUC, we follow the standard practice and train CNN (ResNet) on the spectrogram (Biswal et al. 2017 Ruffini et al., 2019 Yuan et al. 2019 Yang et al. 2022). For PN2017, we used top-performing model from the 2017 PhysioNet Challenge, MINA (Hong et al. 2019).
Table 2: Accuracy in % (f means higher=better). Accuracy numbers lower than the uncalibrated predictions are in dark red and the best are in bold (both at p=0.01). KCal typically improves or maintains the accuracy.
Accuracy
UpCa1
Diycell
I-Raax
Focal
Spline
102
KCall
(pa))
53.68:1444 1.22
58 68-10110 442
1.42
57,
58.51411335
58.68-1.44
58.05-1.37
61.67:2.22
mnC
58.53110.06
58 38.53.10.05
63.50L.0.10
10.01
58.36.10:20
58.5210.06
58.06.10.04
ISRUC (144)
75.11=0.77
75. 75.11=00:77
75.57=0.91
78.54-0.68
73.79+0072
75.11
70.11
75.13=0.76
76.26=0.59
76.13=0.899
ISRUC
74.66:0008
74.00 74.66:00.00
76:08:00116
75. 05:10:00:07
73.36:1000
74.00:0000
7..66100008
74,6620.09
75.05
77,45
PN2017
54.67:0.114
54.67-00.14
6.200-9..23
57.55:20.39
13.78:10:13
55.1 55.11:2:.844
15.15
34,0920.15
51.90
60. 0..61
CIO (VI))
98.96-0.05
98.94-0.05
98.94-0.01
98.94-0.015
96. 96.76++0.06
98.94+11.85
98.94-0.05
98.94-0.06
98.93-
98. 98.98-11.09
CIO (Miser)
98.1710.00
96. 96.17_.0.00
98 98.033.0.00 0.00
98.133.0.08
96.98.100006
98.17.10.08
98.17.10.06
98.16..0005
96.15
95. 14 0.06
CIDO
92.1920010 16
12.09:11.16
92.08=00:14
91. 91.45:00.17
91.21 11.21:00122
92.09:2:.366
92.09200166
92.09::0.16
92.41=00117
92.3 01.35
C100 (Miser)
67,55++0.20 20
0..53000.20
87.24 22-24-022 0.22
83. 10:00.11
06.40-00333
87:53+00.30
07.53+0.330
87.51+0.20
13-13--0.25
87 55+05- 16
SVII (VIT)
95.93110.05
95.93100005
95,9 0.09
98 98.85 10006
95. 05.70.10.00
95 05.93.10.05
95 35.9310.05
95.931.0.05
96, 48 96,04
96.42.1..0
SVHN (Miser)
24.85-20.04
95 05:004
05.85500.05
05.2440004
05.55000004
05.85=00005
11:
ImageNet
80.44::0.24
09.55:00:4
00.34:0.288
80.22:10.27
80.4410.34
80.44:0.24
64:20.22
Table 3: Class-wise ECE in 10-2 (L means lower=better). The best eccuracy-preservvnng method is in bold (p=0.01). The lowest but not prreeerriiii number is underscored. KCal almost always achieves the lowest class-wise ECE, while maintaining accuracy.
CECE
DaCal
55
DirCal
I-Max
Pocal
Spline
I0P
GB
XBCC
KCal
IIC (jut)
807:00227
8.97:085
5.13::1.48
9.231
539920.53
8.561:0.2
8.3310.50 50
7.95:0064
7.12:00.43
4.68:
IIC
7.96+0.02
8.6--052
2.24-0.13
8.76+0026
8.78+0022
84340221
8.1++0.25
7.5240023
6.70-00255
2.03-00.26
ISRUC (pat)
4.48:0024
0.76
4.18:0..0
8.56+1.00
9,23+0.21
4.68
4.60:00.0
4.64=0.43
0.08+0.36
3.82=1.24 1.24
ISRUC
4.49:0.02
5,17:0.77
2.71:0040
9.22:0885
9.05:0033
4.7310.15
4.67:0.36
4.67:00.27
4.10:0.22
1.90:000 0.28
PN2017
12.17-0107
12,51+0.225
4.30=00.7
9.92+1.16
17:31+00.9
8.140073
1209+00.4
12.17-40.07
12.35-00.0
4.25 1.26
C10 (NTT)
3.19+0.01
0.76+0.04
0.83=0.06
0.68:00.5
4.820007
0.904:0:04
0.81+0006
0.74:00.06
1.11+0.27
0.74-0007
C10 (Miser)
3.11:0.02
1.45:0.12
1,23+0.10
1.24:0.17
6.7011.03
1.28:00.09
1.30+0.07
1.21::0.07
1.43-0.119
1.17:00.10
C100 (NIT)
5.9010.05
5.27100.00
4.64110.13
4.9610.17
5.53110.06
4.4110.14
4.7210.11
4.6510.166
4.2710.23
4.32
C10D (Miser)
5.3910.04
5.8210.17
5.2510014
5.7910.24
5.7210.05
4,9210.18
5.34100.33
5.0910.15
5.2610.19
4.621:0.10
SVHN
3.37.10.01
2.31:0.56
1.22::0.06
2.6410.30
5.8910.03
1.3410.05
1.39:10.06
1.40110.05
1,4700.11
0.23::0.10
SVHN (Mixer)
3.20+0001
3.06:0661
1,21:0.12
2.64:0.17
5,59:0022
1.45+0.09
1,42+00.66
1.46:00.66
1.64-00133
1.40=00.55
ImageNet
2.96+00.22
3.25=0.07
5.60-+0.3
2.82+0.19
2.17+00.66
2.30++0.14
2,42+0.06
1.94-004
Table 4: ECE in 10 (L means lower=better). The best (p=0.01). The lowest but not accuracy- preerecvigpee number is underscored. KCal is usually on par or better than the best baseline.
method is in bold
ECE
UnCsl
DirCsl
I-Nex
Focal
Spline
10F
SNCC
KCal
IIC (pat)
9.3211001
2.75
10.52.14.05
7.53.10.55
4.5812.04
4.57.12.14
6.331.3.28
4.34
IIC
9:28-0.03
1.52
1.39-0.19
10.1640.81
7:2=0005
3.20400664
3.50+0044
1.00-0.49
4.78+-2.24
2.6240.59
ISKUC (pat)
3.59:032
1.53
2.9720977
8.86:11.99
44.88:0.44
19881035
2.45
0.03
2.1210.93
2.78
ISROC
3,4600066
3,53 160
2.2740.69
9.58++1.3
14.00-0.006
1,50+1551
2.71-4996
2.000074
2.12+1.03
1.36 +-.41
PN2017
16.70=0.22
16.9920.73
5.64:0075
10.40+135
24.63-0.13
6.4+2.099
16.07=203
16:66=0:21
13.29+1.00
4,78+1 1.48
CVIT)
9.15
0.7520.11
0.40:20.04
0.514:0077
7.17=0.07
0.39:00.88
0.39::004
0.21-0006
0.42:00.29
0.40:+00.05
CIO (Miser)
9.4400006
106::0112
0.61:0.07
0.91-00.14
12.53:0006
0.36-10866
0.66:10.00
0.34:0.10
0.91+:0.44
0.59-+0..9
C1OD IVIT)
11.64=0.14
2.77+0.46
0.74-0.16
3.28++0.22
9.7-0009
1.08++0118
1,07-+0.19
0.88-0.11
1.05++0.50
1.50++0.32
C100 (Miser)
13.7110..5
3.0310.34
106.10.26
4.75.10.27
14.3510221
0.25
1.70.10.66
0,06.0.26
1,9310..49
3.07.10.49
SVHN (VIT)
10 10:10::000
2.43:2272
0.60:19.07
2.05:10.18
12.17:00088
0.74+-0.10
0.62
0.64:9.07
0.72:10.21
0.64:20.12
SVHN (Miser)
10.2910.04
3.191:2555
0.66
2.13.10.10
11.09100006
0.78.10.11
0.60-10.08
0.72110.06
0.72.10.28
0.7310.110
ImageNet
3.2110.155
3.52
4.3010.6
7.97.0035
L10.10.00
1.31110.47
0.87..0.12
L.4310.34
4.2 BASELINES METHODS
We compare KCal with the multiple state-of-the-art calibration methods, including Temperature Scaling (IS) (Guo et al.) 2017). Dirichlet Calibration (DirCal) (Kull et al.) 2019). Mutual- maximization based Binning (I-Max) (Patel et al. 2021). Gaussian Process Calibration
PPN211 did not provide patient IDs, so cannot split by patient.
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Table 5: Brier Score in 10-2 (1 means lower=better). The best accuracy-preserving methods are in
bold (p=0.01). The lowest but not accuracy-preservinn number underscored.
UnCal
25
DisCal
I-30ax
Spline
132
203CE
KCal
HIC (pat)
21.30-025
20.70=0069
18,94=0.55
11.09=1..29
28.
20.4++0.50
20552+0.55
20.35+0042
21.11=0.71
19.33--0.78
InC
21.35-0.01
20.62+0.227
18.33=0.00
20.83=0.119
21 21.46:0:01
20.1100.099
20.99:0.09
2000=00.08
20.60=0.25 0.26
17.54:-0.10
ISRUC
15226022 25
15.3010.31
15.3740.38
16.2540.49
18.5540..18
15.1110.226
15.1640.31
15.161:0.99
14.09-0.22
14.97.10.29
ISRUC
15.16-0..3
15.5010..19
15.07.0.099
16.62.0..3
18.77.00.01
15.3110:05
1533910.10
15355.0.06
14.9110.005
14.28.00.08
PN2017
2661-005
26.74:10.22
22.44:0.155
24.58:00.59
17,29:0.03
23.28:10.37
26.39:1..00
26.61:10.05
26.41:0044
22.56.10.26
C10 (NIT)
1. 1.76=00.03
0.89=0006
0.78
0.84+0004
1,7550003
0.79+:0.84
0.70::0.04
0.78=00.04
0.85=+0.10
0.75=00005
C10 (Mixer)
2.29+0.03
1.48-0.07
1.2:00.05
1.4++0.08
4.16++0.04
1,39400044
1.40+00055
1.37-+0.04
1.45=00.16
1.34-00.04
C100 (NT))
6.3420008
5.35
5.17=0.10
5,48:0.14
6,93+0.07
5.19:0.09
5.18+0.10
5.14:0.09
4.81=0.10
5.015:0.0
(Mixer)
10.15=0.11
7,9410.17
7,8220.12
8.2310.17
10.9110.08
7.7610.12
7.8240.115
7.72:0.13
7.36
7.61::0.09
SVHN (VIT)
3.99#0.03
3.08=0.34
2.78=0004
2.99=#.07
5,03+003
2.80++0.03
2.79+00.04
2.79:0.04
2,43+00.22
2.49==0.03
SVHN (Mixer)
4.03-+0.03
3.21
2:77=00.03
3.04
5.06=00004
2,84
2.81:+0.04
2.81==0.04
3.03=00.02
2.68::0003
ImageNet
11.15=0.14
11.2040.15
12,03:00.21
11.9320.18
10.6810.13
10.69.0.113
10.67.00.12
11.1410.10
Table 6: Ranks for different evaluation metrics. The best rank is underscored. In general, KCal consistently outperforms baselines on Accuracy, CECE and Brier, and the difference between most methods on ECE is small.
Ranking
UnCa1
TS
Dir
I-Max
Rocal
Sp1ine
MNCE
BCB
8.42+1.43
6.68+1111
3.38#1.80
1.73+1.55
9.39=0.95
3,51=1.06
4,25+1135
1.91
4.52=0.98
3.84=:1.35
Accuracy
5.0311.30
5.031.300
4.5312..9
6.4112.366
9.99:0.03
5.56100.3
5.0111227
5.6411.16
4.7413.30
2.701:2.01
CECE
6.99+1195
741+1.60
3.31+2008
6.2242667
9.46-0.61
4.59+-2.06
5,1241.13
4.37-11.27
4.69=1.99
183+-0.76
Brier
8.18+1.52
6.91
3.86+2.08
7.42+1.06
8.98=2.67
1.23+1.05
4.88+1.24
3.89
4.11#22.9
2.08
Average
7.16
6.51
3.76
7.09
9:46
4.81
4.51
2.61
(GP) Wenger et al. 2020). Intra Order-preserving Calibration (TOP) (Rahimi et al. 2020 Splines-
based Calibration (Sp1 in)e (Gupta et al. (2021) Focal-loss-based calibration (Feca Mukkott 2020), MMCE-based calibration (MMCE) Kumar et al. 2018)
et al.
4.3
EVALUATION METRICS
We report standard evaluation metrics: Accuracy, class-wise expected calibration error (CECE) (uull
al. 2019 Patel et al., 202T Nixon et al. 2019). expected calibration error (ECE) (Guo al. 2017 and Brier score (Brier 1950). CECE is typically used as proxy to evaluate full calibration quality, because directly binning basing on the entire vector p requires exponentially (in K) many bins. Similar to Patel et al. (2221); ; Nixon al. (219)), we ignore all predictions with very small probabilities (less than max 0.01, K D. ECE, on the other hand, only measures confidence calibration (Def For both ECE and CECE, we use the "adaptive" version with equal number of samples in each bin (with 20 bins), because this is shown to measure the calibration quality better than the equal-width version (Nixon et al. 2019). Brier score can be viewed the sum of "calibration"
term, and a "refinement" term measuring how discriminative a model is (Kull & Flach 2015). Here
we focus the brier score of the top class. We refer to (Guo et al. 2017: Kull et al.)
2019 Nixon
let al. 2019) for further discussion on these metrics.
4.4 RESULTS
The results are presented in Tables 3 calibration and test sets, and the standard deviations are reported. For ImageNet, skipped Focal and MMCE because the base NN is given and these methods require training from scratch. Due to space constraints, we include ablation studies in the Appendix.
4 and 5 All experiments are repeated 10 times by reshuffling
In general, KCal has the best CECE, accuracy and Brier score, and is highly competitive in terms of ECE as well. Note that KCal is also the only method with provable calibration guarantee. TS effective in controlling overall ECE but shows little improvement on CECE over UnCal. DirCal often ranks high for the calibration quality but tends to decrease accuracy as K increases. DirCal's performance also has a higher cost: Every experiment requires training over hundreds of models with SGD and taking the best ensemble, accounting for most of the experiment computation cost in this paper. The amount of tuning suggested for good performance indicates sensitivity to the choice of hhheeh-paamettee parameters, which we have indeed observed to be the case. Spline, IOP and GP similar D rCal on vision datasets, but generally perform worse on the healthcare datasets.
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In Patel et al. (2021). I-Max lowers ECE and CECE significantly. However, it has critical issue it does not produce a valid probability vector Once normalized, as reported in our experiments, the performance worsens. Since calibrating all the classes simultaneously is the distinguishing challenge in multiclass classification, we interpret the observation as: If this normalization constraint is removed, the "optimization problem" (to lower calibration error) is much simpler, but the results are invalid hence unusable probability vectors. Spline also requires a re-normalization step, but its performance stays consistent. Focal is worse than the UnCal in many experiments. While calibration performance may improve by combing Focal with other methods, the drop in accuracy harder to overcome We also observed that for healtheare datasets, being able tune on a different set of patients boosts the performance significantly. This is reflected in the accuracy gain for Di rr and KCal, and suggests that the embeddings/logits are quite transferable, but the prediction criteria itself can vary from patient to patient.
Finally, we summarize the rankings of all datasets in Table It is clear that KCal consistently improves calibration quality for all classes and maintains improves accuracy. And if we look at only the confidence prediction (Brier or ECE), KCal is still highly competitive.
USGaI
Spire
MNCK
RCW
D,55 0.50 905
9.50
0.25 0.00
5.50 610
028
2.15
5.58 6.18
5.25 0.00 0.52 0.00
50/
82:
Figure 2: Reliability diagrams for the predicted class (top) and Seizure (bottom) in IIIC. All methods calibrate confidence well, but only KCal achieves reasonable calibration quality for Seizure
4.5 CASE STUDY FOR SEIZURE PREDICTION
We show the reliability diagrams Kull et al. 2019: Guo et al. 2017) on the IIIC dataset to illustrate the importance of full calibration in Figure We include both the the predicted class (confidence calibration) and Seizure. More reliability diagrams can be found in the Appendix, and the results are consistent for all classes. The un un-calibrated predictions have large gaps for both confidence and Seizure. Most baselines provide calibrated confidence calibration, but fail to calibrated the output for the rare class Seizure. KCal, on the other hand, achieves the most consistent results. We note again that since all competing classes must be considered together for any clinical decision, full calibration is indispensable in medical applications.
CONCLUSION
This paper proposed KCal, Gaamed-kermel-baseee calibration method for deep learning models. KCal consists of a supervised dimensionality reduction step on the penultimate layer neural network embedding to improve efficiency. A KDE classifier using the calibration set is employed in this new metric space. As a natural consequence of the construction, KCal provides a calibrated probability vector prediction for all classes. Unlike most existing calibration methods, KCal is also provably asymptotically fully calibrated with finite sample error bounds. We also showed that empirically, it outperforms existing kate-of-the-an calibration methods terms of accuracy and calibration quality. Moreover, KCal is more robust to distributional shift, which is common in high-risk applications such as healthcare, where calibration is far more crucial. The major limitation of KCal is the need to store the entire calibration set, which is small overhead with the dimension reduction step and potential improvements.
IIt generates vector whose sum ranges from 0.4 to 2.0 in our experiments. The range is wider for a larger K. IIn PN2017, rare classes are oversampled during training (Hong et al. (2019). While this did not cause issues for other calibration methods, the distributional shift at test time seems catastrophic for Focal.
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ACKNOWLEDGMENTS
This work was supported by NSF award SCH-2205289, SCH-2014438, IS-1183842, NIH award ROI IRO1NS107291-01.
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14


Reviews:
[{"review_id": "NNJa1DyLSf", "reviewer": "Reviewer_W8Nj", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 5, "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 proposes a new Kernel-based calibration method called KCal which  learns a metric space on the penultimate-layer\nlatent embedding and generates predictions using kernel density estimates on a calibration set. Theoretical and empirical results shows the effeciveness of KCal. ", "summary_of_the_review": "This paper proposes KCal which is claimed as the first kde-based calibration method that brings theoretical analysis. The method itself is interesting and promising but some claims and empirical results needs improving.", "strength_and_weaknesses": "Strengths: \n1. A well-established method that builds a kernel-based calibration method. \n2. Theoretical proof of KCal.\n3. Experimental result analysis.\n\nWeakness:\n1. Somewhat bold claims of first theoretical guaratee/kde calibration method: The conformal predictions is well-studied with theoretical guarantee[Shafer 2008, Angelopoulos 2020].  The kernel mean embedding for uncertainty calibration [Kumar 2008, Cui 2020, Bhatt 2022]. \n2. Calibration performance not better in all cases \n3. Computation time not fully explained/demostrated. As said in the paper, computation overhead of KCal is generally larger than ordinary methods and this needs to be empirically analyzed in different scales and acceleration settings. \n\n\nReferences\n1. Shafer G, Vovk V. A Tutorial on Conformal Prediction[J]. Journal of Machine Learning Research, 2008, 9(3).\n2. Angelopoulos A, Bates S, Malik J, et al. Uncertainty sets for image classifiers using conformal prediction[J]. arXiv preprint arXiv:2009.14193, 2020.\n3. Kumar A, Sarawagi S, Jain U. Trainable calibration measures for neural networks from kernel mean embeddings[C]//International Conference on Machine Learning. PMLR, 2018: 2805-2814.\n4. Cui P, Hu W, Zhu J. Calibrated reliable regression using maximum mean discrepancy[J]. Advances in Neural Information Processing Systems, 2020, 33: 17164-17175.\n5. Bhatt D, Mani K, Bansal D, et al. f-Cal: Aleatoric uncertainty quantification for robot perception via calibrated neural regression[C]//2022 International Conference on Robotics and Automation (ICRA). IEEE, 2022: 6533-6539.\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": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "Dha0YrnBAYJ", "reviewer": "Reviewer_2JZP", "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 proposes a method called Kcal for the (full) calibration of deep neural networks.\nKcal consists in replacing the last layer (usually a softmax) of a neural network with a learned projection layer for dimensionality reduction, followed by a Kernel Density (KDE)-based estimator to produce output probabilities. The learnable parameters introduced by Kcal are the parameters of the dimension-reducing projection layer $\\Pi$, as well the kernel bandwidth $b$. The parameters of $\\Pi$ are learned by (stochastic) gradient descent with an appropriate training strategy: contrary to classical neural networks, the KDE “layer” involves all training points and not only the point which is flowed through the network. The kernel bandwidth $b$ is chosen on a calibration set to minimise the log-loss.\n\nA theoretical analysis is provided in the form of high probability error bound, which shows that as the per-class sample size increases, the estimated probabilities converge to the true probability of the class $Y$ given the input score, i.e, the estimated probabilities are calibrated.\n\nExperiments are provided for 4 vision datasets (CIFAR10, CIFAR100, SVHN, ImageNet) each time with a choice of pre-trained architecture, and 3 health monitoring datasets. Kcal is then compared to 8 calibration baselines in terms of ECE (top-label and class-wise), accuracy and Brier score of the top class, and shown to provide favourable performance across all measures (or on par for ECE).", "summary_of_the_review": "I think that the clarity of the paper as well as the favourable empirical results outweight the weaknesses.", "strength_and_weaknesses": "**Strength**.  \nKcal targets **full calibration**, i.e., a calibration conditioned on the full probability vector rather than just conditioned on the probability of predicted class (confidence calibration). This is an important problem since as stated by the authors, small probabilities should also be reliable in high stakes applications such as healthcare.\n\nKcal provides improvements in terms of Accuracy, Class-wise ECE and Brier score across most datasets.\n\n**Weaknessess**.   \nUsing a KDE-based estimator requires an adapted training scheme, and comes with a presumably high computational cost. What is the training time of Kcal across the different datasets?  \nIt also introduces multiple hyper parameters: the kernel bandwidth, the projection architecture and output size, the batch size used to learn the projection, the number of samples used to build the KDE.  \nTherefore, it comes with both a certain computational and implementation complexity.\n\nNo insights are provided to explain the performance of Kcal compared to the other baselines. I am wondering wether it could be due to the fact that it re-uses the whole train set to fit the projection map, while other methods such as Temperature Scaling only calibrate based on the smaller calibration set.\n\n**Question**.  \nThis is a question regarding the choice of kernel bandwidth $b$. Since it can be folded in the projection map, it is implicitly learned while training the projection map. We see in appendix that across all datasets, the best kernel bandwidth equals 1. it would make sense if it has already been learned with the projection map. Hence the question: it is useful to have a bandwidth selection step on the calibration set?", "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": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "54kNwDvs5W", "reviewer": "Reviewer_VY8v", "summary": "", "questions": "", "limitations": "", "rating": 8, "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 proposes a kernel-based calibration KCal where class proabilities are determined by per-class kernel density estimates in an embedding space. The embedding space is obtained by a multi-layer perceptron applied on top of the penultimate layer of the neural network. Extensive experiments demonstrate that KCal outperforms the methods on most relevant measures and on many datasets. KCal has asymptotic guarantees and finite sample bounds.\n", "summary_of_the_review": "The paper is well-written and has very few minor weaknesses. The proposed method is performing better than existing methods for most evaluation measures and datasets. Training of the projection MLP is an extra step that has to be done on the validation data.", "strength_and_weaknesses": "Strengths:\n* The paper is well-written and clear;\n* The method is novel, theoretically justified and improves the state-of-the-art;\n* The experiments have been carried out with great care in optimizing the hyperparameters for the existing methods;\n\nWeaknesses:\n* I didn't find any major weaknesses.\n\nMinor weaknesses:\n* The list of measures includes ECE, Accuracy, CECE, and Brier score. However, cross-entropy (NLL) could have been considered as well, because all proper losses decompose into refinement and calibration losses, not just the Brier score. NLL has different properties than the Brier score, and pays more attention to the probabilities near the extremes.\n* The number of instances per bin could have been shown for Figure 2 (and for similar figures in the appendix), for example as a separate figure. This would help one to visually assess roughly how high ECE would be.\n", "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": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works."}, "extra_text": {}}, {"review_id": "-R7zYfyN3Y", "reviewer": "Reviewer_LaNX", "summary": "", "questions": "", "limitations": "", "rating": 8, "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 tackles the question of how to generate “fully calibrated” predictions in multi-class classification problems. It proposes a method, “KCal”, which learns a low-dimensional projection of a DNN’s final-hidden-layer embeddings, and then uses kernel density estimation (KDE) in this projected space to generate the fully-calibrated probability vectors.\n\nTheoretically, this paper shows that as the size of the calibration set increases (where size here is measured in terms of the number of occurrences of the rarest class), the probability vectors generated by KCal get closer to being fully calibrated, and in the limit m -> infinity, they are perfectly calibrated.\n\nEmpirically, this paper compares KCal to a large number of other calibration methods, on a variety of computer vision and health monitoring datasets. It shows that KCal generally maintains or improves the accuracy of the uncalibrated model, while generally attaining better calibration scores relative to the baseline methods.\n", "summary_of_the_review": "Overall, I believe the proposed KCal method is a novel and promising approach for generating more calibrated predictions for multi-class classification problems, and thus that this paper should be accepted.  However, my review is only medium confidence, given that this is not an area of research I have worked directly in.", "strength_and_weaknesses": "Strengths\n- The method is relatively simple, well-motivated, and easy to understand. Using KDE for calibration seems like a natural idea, given the connection between KDE and the Bayes classifier mentioned at the end of section 3.1 (on that note, it’s surprising to me that KDE has not been proposed elsewhere to generate calibrated predictions, as is claimed at the end of section 2).\n- The method performs well empirically, across a wide range of experiments, and compared to a wide range of baselines.\n- The method is theoretically tractable, and provably leads to calibrated predictions under certain conditions/assumptions.\n\nWeaknesses\n- Although I didn’t study the proofs in detail, I was curious whether there was anything in the proofs that was specific to the proposed learning algorithm. In particular, would the results hold for any feature space? Or is there something specific about the learned low-dimensional feature space that allows these theorems to hold? Are these standard results based on using KDE on a fixed feature space?\n- As far as I can tell, no ablations are performed to better understand what aspects of the algorithm are crucial for attaining good calibrated predictions. For example, what happens if KDE is applied directly on the final-hidden-layer outputs of the DNN?\n- I was also curious whether theorems 3.3 and 3.4 gave tight or loose bounds, given the real values of m, B, C, K, d, and alpha on the real tasks (can all of these be estimated?).\n", "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": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}]

Author rebuttal:

Decision: Accept: poster