Improving Predictor Reliability with Selective Recalibration
Authors: Thomas P Zollo
Zhun Deng
Jake Snell
Toniann Pitassi
Richard Zemel
Venue: Accepted by TMLR ()

Abstract: A reliable deep learning system should be able to accurately express its confidence with respect to its predictions, a quality known as calibration. One of the most effective ways to produce reliable confidence estimates with a pre-trained model is by applying a post-hoc recalibration method. Popular recalibration methods like temperature scaling are typically fit on a small amount of data and work in the model’s output space, as opposed to the more expressive feature embedding space, and thus usually have only one or a handful of parameters. However, the target distribution to which they are applied is often complex and difficult to fit well with such a function. To this end we propose selective recalibration, where a selection model learns to reject some user-chosen proportion of the data in order to allow the recalibrator to focus on regions of the input space that can be well-captured by such a model. We provide theoretical analysis to motivate our algorithm, and test our method through comprehensive experiments on difficult medical imaging and zero-shot classification tasks. Our results show that selective recalibration consistently leads to significantly lower calibration error than a wide range of selection and recalibration baselines.

Toniann Pitassi Published in Transactions on Machine Learning Research (07/2024)
Improving Predictor Reliability with Selective Recalibration
Thomas P. Zollo
1p221055cokkmmbii.cc edu
Columbia University
Zhun Deng Columbia University
zhun. diccuumbba edu
Jake C. Snell Princeton University
jx2223@@pinneelon.cde eduu
toni@cs.
www.mbiaaaaaa.mmmomm teni@cs.columbia.co
edu
Columbia Universily
Richard Zemel
zemell@cs.chumbiit. edu
Columbia University
Reviewed on OpenReview: https: //
/ openreview. net/ forum? j9H6
Abstract
A reliable deep learning system should be able to accurately express its confidence with respect to its predictions, a quality known as calibration. One of the most effective ways to produce reliable confidence estimates with a pre-trained model is by applying a post-hoc recalibration method. Popular recalibration methods like temperature scaling are typically fit on small amount of data and work in the model's output space, as opposed to the more expressive feature embedding space, and thus usually have only one or a handful of parameters. However, the target distribution to which they are applied is often complex and difficult to fit well with such function. To this end propose selective recalibration, where a selection model learns to reject some user-chosen proportion of the data in order to allow the recalibrator to focus on regions of the input space that can be well-captured by such a model. We provide theoretical analysis to motivate our algorithm, and test our method through comprchensive experiments on difficult modical imaging and zero-shot classification tasks. Our results show that selective recalibration consistently leads significantly lower calibration error than wide range of selection and recalibration baselines.
1 Introduction
In order to build user trust in a machine learning system, is important that the system can accurately express its confidence with respect to its own predictions. Under the notion calibration common in deep learning (Guo al., 2017 Minderer et al. 2021), a confidence estimate output by a model should be as close as possible to the expected accuracy of the prediction. For instance, prediction assigned 30% confidence should be correct 30% of the time. This is especially important in risk-sensitive settings such medical diagnosis, where binary predictions alone not useful since 30% chance of disease must be treated differently than a 1% chance. While advancements in ncural network architecture and training have brought improvements in calibration as compared to previous methods (Minderer el. al.] 2021) ) neural networks still suffer from miscalibration, usually in the form overconfdence (Guo et al. 2017 Wang et al. 2021). addition, these models are often applicd complex data distributions, possibly including outliers, and may have different calibration error within and between different subsets the data (Ovadia et al. 2019 Perez- Lebel et al. 2023 We illustrate this setting in Figure Ta with Reliability Diagram, tool for visualizing

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(a) Pre-trained model
(b) Recalibration only
(c) Selection only
(d) Selective Recalibration
1.0
0.8
Accuracy
0.6
0.0
0.0
0.5
1.0
0.0
0.5
1.0
0.0
0.5
1.0
0.0
0.5 Confidence
1.0
Confidence
Confidence
Confidence
Figure 1: Reliability Diagrams for a model that has different calibration error (deviation from the diagonal) in different subsets the data (here shown in blue and green). The data per subset is binned based on confidence values; each marker represents a bin, and its size depicts the amount of data in the bin. The rod dashed diagonal represents perfect calibration, where confidence equals expected accuracy.
calibration by plotting average confidence against accuracy for bins of datapoints with similar confidence estimates.
To address this calibration error, the confidence estimates of pre-trained model be refined using post-hoc recalibration method like Platt scaling (Platt 1999). temperature scaling (Guo et al. 2017), histogram binning (Zadrozny & Elkan 2001). Given existing empirical evidence (Guo et al.) 2017] and the fact that they are typically fit on small validation sets (on the order of hundreds to a few thousand examples), these "recalibrators" usually reduce the imput space to the model's logits (e.g., temperature scaling) or predicted class scores (c.g., Platt scaling, histogram binning), as opposed the high-dimensionall and expressive feature embedding space. Accordingly, they are generally inexpressive models, having only or handful of parameters (Platt 1999 Guo et al. 2017 Zadrozny & Elkan 2001 Kumar et al. 2019 But the complex data distributions to which neural networks are often applied are difficult to fit well with such simple functions, and calibration error can even be exacerbated for some regions of the input space, especially when the model has only a single scaling parameter (see Figure 1bp.
Motivated by these observations, we contend that these popular recalibration methods are a natural for use with selection model. Selection models (El-Yaniv & Wiener 2010 Geifman & El- 2017) are used alongside a classifier, and may reject portion of the classifier's 's predictions in order to improve performance metric the subset of accepted (i.e., unrejected) examples. While selection models have typically been applied to improve classifier accuracy (Geifman & FJ- Yaniv 2017), they have also been used to improve calibration error by rejecting the confidence estimates of fixed model (Fisch et al.) 2022p. However, selection alone cannot fully address the underlying miscalibration because does not alter the confidence output the model (see Figure Ic), and the connection between selection and post-hoc recalibration remains largely unexplored.
In this work propose selective reculibration, where selection model and recalibration model are jointly optimized in order to produce predictions with low calibration error. By rejecting some portion of the data, the system can focus on region that can be well-captured by a simple recalibration model, lcading to a set of predictions with a lower calibration error than under recalibration or selection alone (see Figure Id] This approach is especially important when a machine learning model is deployed for decision-making in risk-sensitive domains such as healthcare, finance, and law, where a predictor must be well-calibrated human expert is use its output improve outcomes and avoid causing active harm. To summarize our contributions:
We formulate seloctive recalibration, and offer a new loss function for training such a system, Selective Top-Label Binary Cross Entropy (S-TLBCE), which aligns with the typical notion of loss under smooth recalibrators like Platt temperature scaling models.
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We test selective recalibration and S-TLBCE in real-world medical diagnosis and image classification experiments, and find that selective recalibration with S-TLBCE consistently leads to significantly lower calibration error than a wide range of selection and recalibration baselines.
We provide theoretical insight to support our motivations and algorithm.
2 Related Work
Making well-calibrated predictions is key feature of reliable statistical model (Guo et al. 2017 Hebert- Johnson et al.) 2018 Minderer al.) 2021 Fisch et al.
2022 Popular methods for improving calibration
error given a pretrained model and labeled validation dataset include Platt (Platt 1999 and temperature
scaling (Guo et al., 2017), histogram binning (Zadrozny & Elkan 2001 and Platt binning (Kumar et al.
2019) (as well as others like those in Nacini et al. (2015) Zhang et al. (2020)) ). Loss functions have also been
proposed to improve the calibration error of a neural network in training, including Maximum Mean Cali-
bration Error (MMCE) (Kumar et al. 2018): S-AvUC, and SB-ECE (Karandikar et al.) 2021). Calibration error is typically measured using quantities such as Expected Calibration Error (Naeini et al. 2015 Guo
ct al. 2017). Maximum Calibration Error (Nacini ct al.) 2015 Guo et al.) 2017), or Brier Score (Brier) 1950)
that measure whether prediction confidence matches expected oulcomes. Previous research (Roelols et al. 2022 has demonstrated that calibration measures calculated using binning have lower bias when computed using equal-mass bins.
Another technique for improving ML system reliability selective classification (Geilinan & El- Yaniv 2017
El-ianiv & Wiener) 2010), wherein a model is given the option to abstain from making a prediction on certain examples (often based on confidence or out-of-distribution measures). Selective classification has
been well-studied in the context of neural networks (Geifman & El-Yaniv 2017 2019 It has been shown to increase disparities in accuracy across groups (Jones et al. 2021 ), although work has
Madras et al. 2018).
been done to mitigate this effect in both classification (Jones et al. 2021) and regression (Shah et al. 2022 lasks by enforcing calibration across groups.
Recent work by Fisch et al. prodictions from a pro-trained model are rejected for the sake of improving sclective calibration crror. The authors propose a method for training a selective calibration model using an S-MMCE loss function derived from the work of [Kmaar et al. (2018). Our work differs from this and other previous work by considering the joint training and application of selection and recalibration models. While Fisch et al. (2022) apply selection directly to frozen model's outputs, we contend that the value in our algorithm lies in this joint optimization. Also, instead of using S-MMCF, we propose a new loss function, S-TLBCF,, which more closely aligns with the objective function for Platt and temperature scaling.
(002 introduces the setting of calibrated selective classification, in which
Besides calibration and selection, there are other approaches to quantifying and addressing the uncertainty in modern neural networks. One popular approach is the use of ensembles, where multiple models are trained and their joint outputs are used to estimate uncertainty. Ensembles have been shown to both improve accuracy and provide a means to estimate predictive uncertainty without the need for Bayesian model- ing (Lakshminarayanan et al.) 2017). Bayesian neural networks (BNNs) offer an alternative by explicitly modoling uncortainty through distributions over the weights of the network, thus providing a principled un- certainty estimation (Blundell et al.) 2015). Dropout can also be viewed as approximate Bayesian inference GGal 2016). Another technique which has received interest recently is conformal prediction, which uses past data to determine prediction interval set in which future points are predicted to fall with high probability (Shafer & Vovk 2008 Vovk et al. 2005). Such distribution-free guarantecs have been extended to cover wide set of risk measures (Deng el al. 2023 Snell et. al. 2023) and applications such as robot planning (Ren
et al. 2023) and prompting large language model (Zollo et al. 2024).
3 Background
Consider the multi-class classification setting with K classes and data instances (z. y) ~ D. where x is the input and y E (1, (1,2.... ..., K) is the ground truth class label. For a black box predictor f, fxx E RK is a
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vector where f()) is the predicted probability that imput has label k; we denote the confidence in the top predicted label f(z) = max; f(z).. Further, we may access the unnormalized class scores fa( I (which may be negative) and the feature embeddings fe(x) R The predicted class is ÿ arg max; fx) and the correctness of prediction is y = 1 5)
3.1 Selective Classification
In selective classification, a selection model g produces binary outputs, where indicates rejection and 1 indicates acceptance. A common goal is to decrease some error metric by rejecting more than 1 - B proportion of the data for given target coverage level B. One popular choice for input to g is the feature embedding [e(x), although other representations may be used. Often, a soft selection model ÿ IR - [0, 1] is trained and g is produced at inference time by choosing a threshold т on 9 to achieve coverage level B (i.e., (xx) 2 B).
3.2 Calibration
The model f is said to be top-label calibrated if Epls)f(<>) p To measure deviation from this condition, we calculate expected calibration error (ECE):
= pp for all E [0, 1] in the range f(x).
f(x) (EEE,-EE
ECE, = f(
(1)
where q is typically 1 or 2. A recalibrator model can be applied to f to produce outputs the interval [0, such that hx(()) E IR is the recalibrated prediction confidence for input a and h(((x))) = maxx h((()))
See Section 4.3 for details on some specific forms of h(.).
3.3 Selective Calibration
predictor is selec- tively calibrated if IEp [f(] P, 9(x) = p for all [0, 1] in the range of f(x) where g(x) = 1. To interpret this statement, for the subset of examples that are accepted (i.e., g(x) 1), for given confidence level the predicted label should be correct for proportion of instances. Selective expected calibration error is then calculated as:
Under the notion of calibrated selective classification introduced by Fisch et al. 2022
S-E.CE,- = (FF)co)) ((T00)1((0)),,)))) 19(0) 1000))))))))=
(2)
It should be noted that selective calibration a separate goal from selective accuracy, and enforcing it may in some cases decrease accuracy. For example, a system may reject datapoints with f(x) = 0.7 and p(y" = 1|x) = 0.99 (which will be accurate 99% of the time) in order to retain datapoints with f(x) = 0.7 and ply = 1|z) = 0.7 (which will be accurate 70% of the time). This will docrease accuracy, but the tradeoff would be acceptable in many applications where probabilistic estimates (as opposed discrete labels) are the key decision making tool. See Section 5.2.1 for a more thorough discussion and empirical results regarding this potential trade-off. Here we only concerned with calibration, and leave methods for exploring the Pareto front of selective calibration and accuracy to future work.
4 Selective Recalibration
In order to achieve lower calibration error than existing approaches, propose jointly optimizing selection model and recalibration model. Expected calibration error under both selection and recalibration is equal
to
SR-ECE,, =FFALIEDD (TD (TEp(V((((())))))) 1] (((())))) (o())))))>(()))))))))))))))))))) (o(o)=1)) 1
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Taking SR-ECE, as our loss quantity of interest, our goal in selective recalibration is to solve the optimization problem:
min SR-ECE, s.t. Eplg(x)) 2 3,
(4)
g.k
where 3 is our desired coverage level.
There are several different ways one could approach optimizing the quantity in Eqq 1 through selection and/or recalibration. One could apply only or g, first train à and then (or vice versa), or jointly train and h (i.e., selective recalibration). In Fisch et al. (2022), only is applied; however, as our experiments will show, much of the possible reduction in calibration error comes from h. While h can be effective alone, typical recalibrators are inexpressive, and thus may benefit from rejecting some difficult-to-fi portion of the data (as find to be the case in experiments models sequentially is also sub-optimal, the benefits of selection with regards to recalibration can only be fully realized if the two models can interact in training, since fixing the first model constrains the available solutions.
several real-world datasets in Section 5). Training the
Selective recalibration, where g and are trained together, admits any solution available to these approaches, and can produce combinations of g and h that are unlikely to be found via sequential optimization (we formalize this intuition theoretically via an example in Section 6 Since Eq. cannot be directly optimized, we instead follow Geifman & El- Yaniv (2019) and Fisch et al [0022] and define surrogate loss function L including both selective error quantity and term to enforce the coverage constraint (weighted by A):
min L = Liset + (
(5)
We describe choices for Lael (selection loss) and Loor (coverage loss) in Sections 4.1 and
4.2
along with and explain
recalibrator models in Section 4.3. Finally, we specify inference procedure in Section 4.4 how the model trained with the soft constraint in Eq. is used to satisfy Eq. 4
4.1 Selection Loss
In selective recalibration, the selection loss term measures the calibration of selected instances. Its general form for a batch of data D = {(T,, =)) with n examples is
(, 0(x,),
Lost
(6)
where measures the loss on selected examples and the denominator scales the loss according to the propor- tion preserved. We consider forms of l: S-MMCE of Fisch al. (2022), selective version of multi-class cross entropy, and our proposed selective top label cross entropy loss.
4.1.1 Maximum Mean Calibration Error
We apply the S-MMCE loss function proposed in Fisch et al. (2022) for training selective calibration system. For batch of training data, this loss function is defined as
1,, = n Σ M/
(7)
where & is some similarity kernel, like Laplacian. On a high level, this loss penalizes pairs of instances that have similar confidence and both are far from the true label y (which denotes prediction correctnoss 0 or 1). Further details and motivation for such an objective can be found in Fisch al. (2022).
4.1.2 Top Label Binary Cross Entropy
Consider a selective version of a typical multi-class cross entropy loss:
-1 Σ ((x;) log (x:()))
Is.CCE(9, f.
(8)
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In the case that the model is incorrect (y² = 0), this loss will penalize based under-confidence in the ground truth class. However, our goal is calibration according to the predicted class. Thus we propose a loss function for training a selective recalibration model based on the typical approach to optimizing a smooth recalibration model, Selective Top Label Binary Cross Entropy (S-TLBCE):
IS-TLBCE(C, h; f. =1 T Σg(z) of log h(f(x.)) + (1 y/) log(1 h(f(x.)))..
(9)
In contrast to S-MCE, in the case of an incorrect prediction S-TLBCE penalizes based on over-confidence in the predicted label. This aligns with the established notion of top-label calibration error, as well as the typical Platt or temperature scaling objectives, and makes this a natural loss function for training a selective recalibration model. We compare S-TLBCE and S-MCE empirically in our experiments, and note that the binary case these losses are the same.
4.2 Coverage Loss
When the goal of selection is improving accuracy, there exists an ordering under 9 that is optimal for any choice of B, namely that where I is greater for all correctly labeled examples than it is for any incorrectly labeled example. Accordingly, coverage losses used to train these systems will often only enforce that more than B proportion is to be rejected. Unlike selective accuracy, selective calibration is not monotonic with respect to individual examples and mismatch in coverage between training and deployment may hurt test performance. Thus in selective recalibration we assume the user aims to reject exactly 3 proportion of the data, and employ a coverage loss that targets a specific B,
Leov (3) (3 (3-- - ((x,)))
(10)
Such a loss will be an asymptotically consistent estimator of (B- =z)))) Alternatively, Fisch et al. (2022)
use a logarithmic regularization approach for enforcing the coverage constraint without specific target B computing cross entropy between the output of and a target vector of all ones. However, we found this approach to be unstable and sensitive to the choice of À in initial experiments, while our coverage loss enabled stable training at any sufficiently large choice of X, similar the findings of Geifman & El-Yaniv (2019)
4.3 Recalibration Models
We consider two differentiable and popular calibration models, Platt scaling and temperature scaling, both of which attempt to fit function between model confidence and output correctness. The main difference between the models is that Platt scaling works in the output probability space, whereas temperature scaling
is applied to model logits before softmax is taken. A Platt recalibrator (Platt 1999) produces output according to
h Platt (f(x)) + exp(w b)
(11)
where w are learnable scalar parameters. On the other hand, temperature scaling model Guoo et al. 2017 produces output according to
fa(x) T
h Temp (fo(z) softmax
(12)
where fa(x) is the vector of logits (unnormalized scores) produced by I and T' is the single learned (scalar) parameter. Both models are typically trained with a binary cross-entropy loss, where the labels and 1 denote whether an instance correctly classified.
4.4 Inference
Once we have trained 9 and h, we can flexibly account for 3 by selecting a threshold 5 on unlabeled test data (or some other representative tuning set) such that E[1(g(X) 7}] B. The model g is then simply g(x) (0(x)) r).
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4.4.1 High Probability Coverage Guarantees
Since 1(((x) 2 7 is a random variable with a Bernoulli distribution, we may also apply the Hoeflding bound [Hoeffding. 1963) to guarantee that with high probability empirical target coverage & (the proportion of the target distribution where ((x) 2 7) will be in some range. Given a set V of na ii.d. unlabeled examples from the target distribution, we denote empirical coverage on V as B. With probability at least 1 - -, 3 will
be in the range 13 e, 3 + c), where e c= / [oo[(2] For some critical coverage level 8, 7 be decreased until 3 2 B.
2nu.
5 Experiments
In this section we examine the performance of selective recalibration and baselines when applied to models pre-trained on real-world datasets and applied to a target distribution possibly shifted from the training distribution. In Section 5.1 we investigate whether, given a small validation set of labeled examples drawn ii.d. from the target distribution, joint optimization consistently leads to a lower empirical selective calibra- tion error than selection or recalibration alone or sequential optimization. Subsequently, in Section 5.2 we study multiple out-of-distribution prediction tasks and the ability of a single system provide decreasing selective calibration error across range of coverage levels when faced with a further distribution shift from validation data to test data. We also analyze the trade-off between selective calibration error and accuracy in this setting.
Since we are introducing the objective of selective recalibration here, we focus on high-level design decisions, in particular the choice of selection and recalibration method, loss function, and optimization procedure (joint vs. sequential). For selective recalibration models, the input to g is the feature embedding. Temperature scaling is used for multi-class examples and Platt scaling applied in the binary cases (following Guo et al.) (207)) and initial results on validation data). Calibration error is measured using ECE, and ECF22 with equal mass binning. For the selection loss, we use (s-TLBCE and Is-MMCE for binary tasks, and include a selective version of typical multi-class cross-cntropy (IsMCE) for multi-class tasks. Pre-training is performed where h is optimized first in order to reasonably initialize the calibrator parameters before beginning to train g. Models are trained both with h fixed after this pre-training (denoted as "sequential" in results) and when it is jointly optimized throughout training (denoted as "joint" " results).
Our selection baselines include confidence-based rejection ("Confidence") and multiple out-of-distribution (OOD) detection methods ("Iso. Forest", "One-class SVM"), common techniques when rejecting to improve accuracy. The confidence baseline rejects examples with the smallest f(z) (or h(f(z))), while the OOD methods are measured in the embedding space of the pre-trained model. All selection baselines are applied to the recalibrated model in order to make the strongest comparison. We make further comparisons to recalibration baselines, including the previously described temperature and Platt scaling as well as binning methods like histogram binning and Platt bimning. See Appendix A for more experiment details including calibration error measurement and bascline implementations.
5.1 Selective Recalibration with i.i.d. Data
First, we test whether selective recalibration consistently produces low ECE in a setting where there is a validation set of labeled training data available from the same distribution as test data using outputs of pretrained models on the Camelyon17 and ImageNet datasets. Camelyon17 (Bandi et al. 2018) is a task where the input is 96x96 patch of a whole-slide image of lymph node section from patient with potentially metastatic breast cancer and the label y is whether the patch contains tumor. Selection and rocalibration models are trained with 1000 samples, and we apply a Platt scaling h since the task is binary. ImageNet is a well-known large scale image classification dataset, where we use 2000 samples from a supervised ResNet34 model for training selection and recalibration models, and temperature scaling h since the task multi-class. Our soft selector 9 is a shallow fully-connected network (2 hidden layers with dimension 128), and we report selective calibration error for coverage level E {0.75, 0.8, 0.85, 0.9}. Full experiment details, including model specifications and training parameters, can be found in Appendix A
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ImageNet Selection Baselines
Re-Calibration 1..0)
Selective Re-Calibration
ECE)
0.04
ECE2
0.02
Uncalibrated
Temp Staling
Plan Staling
Plat Birner
Histograns Bioner
0.1
0.15
0.2
0.25
0.1
0.15
0.2
0.25
B
1-8
8
Confidence
Best
MCCC sequental
MMCE
Iso. Forest
Temp Scaling
S-MCE joint
S-TLBCE sequemia
sequential
joint
Camelyon17
Re-Caliratioon (8 1.0)
Selection Baselines
Selective Re- Ro-Calibration
ECE1
0.04
0.02
0.125
ECE2
0.075
0.050
0025
Uncalirraed
Temp Scaling
Platt Scaling
Plat Binner
Histogram Birmer
0.1
0.15
0.2
0.25
0.1
0.15
0.2
0.25
1- B
1-8
Confidence
Best recal.
SMMCE
S-TLBCE sequental
1so. Forest
Plant Scaling
M-MMCE joint
S-TLBCE joint
Figure 2: Selective calibration error ImageNet and Camelyon17 for coverage level B E (0.75, 0.8, 0.85, 0.9). Left: Various recalibration methods are trained using labeled validation data. Middle: Selection baselines including confidence-basel rejection and various OOD measures. Right: Selective recalibration with differ- ent loss functions.
Results displayed in Figure They show that by jointly optimizing the selector and recalibration models, able to achieve improved ECE at the given coverage level 3 compared to first training h and then We also find selective recalibration achieves the lowest ECE in almost every case in comparison to recalibration alone. While the confidence-basee selection stralegy performs well in these experiments, this is not a good approach to selective calibration in general, as this is a heuristic strategy and may fail in cases where a model's confident predictions are in fact poorly calibrated (soc Scction 5.2 for examples). In addition, the S- S-TLECE loss shows more consistent performance than S-MMCE, as reduces ECE in every case, whereas training with S-MMCE increase calibration error in some

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To be sure that the lower calibration error compared to recalibration is not because of the extra parameters in the selector model, we also produce results for a recalibration model with the same architecture as our selector. Once again the input is the feature embedding, and the model h has 2 hidden layers with dimension 128. Results for Camelyon17 are included in Figure 2 for clarity of presentation of ImageNet results, we omit the MLP recalibrator results from the plot as they were an order of magnitude worse than all other methods (ECE, 0.26, ECE2 0.30). In neither case does this baseline reach the performance of the best low-dimensional recalibration model.
5.2 Selective Re- De-CClibration under Distribution Shift
In this experiment, we study the various mets applied to genetic perturbation classification with RxRx1, as well as zero-shot image classification with CLIP and CIFAR-100-C. RxRx1 (Taylor et al. 2019) is a task where the input x is a 3-chamnel image of cells obtained by fluorescent microscopy, the label indicates which 1,139 genetic treatments (including no treatment) the cells received, and there is a domain shift across the batch in which the imaging experiment was run. CTFAR-100 dataset, and we perform zero-shot image classification with CLIP. We follow the setting of[Fisch al. (2022) where the test. data is drawn from shifted distribution with respect the validation set and the goal is not to target specific 3, but rather to train selector that works across a range of coverage levels. In the case of RxRx1 the strong batch processing effect leads to a 9% difference in pretrained model accuracy between validation (18%) and test (27%) output, and we also apply a selective recalibration model trained on validation output from zero-shot CLIP and CIFAR- to test examples drawn from the OOD CIFAR-100-C test set. Our validation sets have 1000 (RxRx1) or 2000 (CIFAR-100) examples, 9 is a small network with hidden layer of dimension 64, and we set 3 0.5 when training the models. For our results we report the under the curve (AUC) for the coverage vs. error curve, a typical metric in selective classification (Geifman & Yaniv 2017 Fisch et al. 2022) that reflects how model can reduce the error average at different levels of B. We mcasure AUC in the range 3 = [0.5, 1.0), with measurements taken at intervals of 0.05 (i.e., 8 E [0.5, 0.55, 0.6,... .... 0.95, 1.0)). Additionally, to induce robustness to the distribution shift we noise the selector/recalibrate input. See Appendix A for full specifications.
well-known image classification
Table 1: RxRx1 and CIFAR-100-C AUC the range 10.5, 1.0).
RxRx1 ECF₂
CIFAR-100-C
Selection
Opt. of h
ECE,
Acc.
ECE,
FCF2
Acc.
Confidence
0.071
0.081
0.353
0.048
0.054
0.553
One-class SVM
0.058
0.077
0.227
0.044
0.051
0.388
Iso. Forest
0.048
0.061
0.221
0.044
0.051
0.379
S-MCE
sequential
0.059
0.075
0.250
0.033
0.041
0.499
joint
0.057
0.073
0.249
0.060
0.068
0.484
S-MMCE
sequential
0.036
0.045
0.218
0.030
0.037
0.503
joint
0.036
0.045
0.218
0.043
0.051
0.489
S-TLBCE
sequential
0.036
0.045
0.219
0.030
0.037
0.507
joint
0.039
0.049
0.218
0.026
0.032
0.500
Recalibration
Temp. Scale
0.055
0.070
0.274
0.041
0.047
0.438
(3 = 1.0)
None
0.308
0.331
0.274
0.071
0.079
0.438
Results are shown in Table I First, these results highlight that even in this OOD setting, the selection-only approach of Fisch et al. (2022) is not enough and recalibration is key ingredient in improving selective calibration error. Fixing h and then training performs better than joint optimization for RxRx1, likely because the distribution shift significantly changed the optimal Lemperature for the region of the feature space where g(x) = 1. Joint optimization performs best for CIFAR-100-C; and does still significantly improve ECE on RxRx1, although it's outperformed by fixing h first in that case. The confidence bascline performs quite poorly on both experiments and according to both metrics, significantly increasing selective calibration error in all cases.
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5.2.1 Trade-offs Between Calibration Error and Accuracy
While accurate probabilistic output is the only concern in some domains and should be at least some concern in most domains, discrete label accuracy can also be imporlant some circunstances. Table [ shows accuracy results under sclection, and Figure g shows the selective accuracy curve and confidence histogram for our selective recalibration model trained with S-TLBCE for RxRx1 and CIFAR-100 (and applied to shifted distributions). Together, these results illustrate that under different data and prediction distributions, selective recalibration may increase decrease accuracy. For RxRx1, the model tends to reject examples with higher confidence, which also tend to be more accurate. Thus, while ECE@B may improve with respect to the full dataset, selective accuracy al B is worse. On the other hand, for CIFAR-100-C, the model tends to reject examples with lower confidence, which also tend to be less accurate. Accordingly, both ECEO0 and selective accuracy at improve with respoct to the full datasct.
RxRx1
CIFAR-100-C
Confidencee Distribution (8 0.8)
Selective Accuracy
Confidence Distribution (8 0.8)
Selective Accuracy acy
No. 1000
1
Accunsyy
No. examples
Accuraoy
344
03 04
66 Confidencee
19
05
13
12
61
46
04
10
30
69
LG
Coverage
Confidence
Caverage
Figure 3: Plots illustrating 1) distribution of confidence among the full distribution and those examples
accepted for prediction (i.c., where g(x) = 1) at coverage level B 0.8 and 2) selective accuracy in the range B = [0.8, 1.0].
6 Theoretical Analysis
To build a deeper understanding of selective recalibration (and its alternatives), we consider a theoretical situation where a pre-trained model is applied target distribution different from the distribution on which it was trained, mirroring both our experimental setting and common challenge in real-world deployments. We show that with either selection rocalibration alone there will still be calibration error, while selective recalibration can achieve ECE = 0. We also show that joint optimization of g and h, as opposed to sequentially optimizing each model, is necessary to achieve zero calibration error.
6.1 Setup
We consider a setting with two classes, and without loss of generality we set y E [-1,1]. We are given a classifier pre-trained on a mixture model, a typical way view the distribution of objects in images (Zhu et a.. 2014 Thulasidasan et al., 2019). The pre-trained classifier is then applied to target distribution containing a portion of outliers from each class unseen during training. Our specific choices of classifier and training and target distributions are chosen for ease of interpretation and analysis; however, the intuitions built can be applied more generally, for example to neural network classifiers, which are too complex for such analysis but are often faced with outlier data on which calibration poor Ovadia et al. 2019).
Training Distribution (Mixture Model)
Target Distribution (with Outliers)
y=1
Figure 4: A classifier pro-trained on a mixture model applicd to target distribution with outliers.
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6.1.1 Data Generation Model
Definition 1 (Target Distribution) The target distribution is defined as (6", (aaa-)peruurbr misture model over (z, E Rº x {1,-1 -1): V zJ1 + (1 -1-13 (1-2)J2, where follows the Bernoulli distribution P(y 1) = P(y = - 2 follows a Bernoulli distribution P = 3, and is independent of y.
Our data model considers a mixture of two distributions with disjoint and bounded supports, J1 and J2, where J2 is considered to an outlier distribution. Specifically, for y É (-1,1}, Ji is supported in the balls with centers y0 and radius rı, J2 is supported in the balls with centers -yad" and radius r2: and both J1 and J2 have standard deviation a. See Figure 4 for illustration of our data models, and Appendix C. for full definition of the distribution.
6.1.2 Classifier Algorithm
Recall that in our setting f' is pre-trained model, where the training distribution is unknown and we only have samples from some different target distribution. We follow this setting in our theory by considering a (possibly biased) estimator i of Hr which the output of training algorithm (S*rr) that takes the i..d. training data set Sth {(*!", y )= as input. The distribution from which Str drawn is different from the target distribution from which have data to train the selection and recalibration models. We only impose one assumption data is abundant enough and 8g should be aligned with 6" with respect to direction (see Assumption 3. Appendix C.2 for formal statement). For ease of analysis and explanation, we consider a simple classifier defined by f = >i 1.
the model O: that a outputs i that will converge to 0g if training
t y/" /m when the training distribution is set to be an unperturbed Gaussian mixture
="'l"" Ni 8", o,o1) and yr follows a Bernoulli distribution P(y = 1) 1/2 This form of 0 is closely related to Fisher's rule in lincar discriminant analysis for Gaussian mixtures (scc Appendix C.2.1 for further discussion).
Having obtained 8, pretrained classifier aligns with the typical notion of softmax response in neural networks. We first obtain the confidence vector f(x) = (=)))) where
e00T
f-1(x) = caóT x +1
[x = ca0 x +1
(13)
and then output 9 = arg max (x). For E {-1,1}, the confidence score fu((r) represents an estimator of IP(y = k(a) and the final classifier is equivalent to 9 sgn(("x)) x).
6.2 Main Theoretical Results
Having established our data and classification modcls, we now analyze why selective recalibration (i.c., joint training of y and h) can outperform recalibration and selection performed alone or sequentially. To measure calibration error, we consider ECE, with g = 1 (and drop the subscript q for notational simplicity below). For the clarity of theorem statements and proofs, we will restate definitions of calibration error to make them explicitly dependent studying. We want to emphasize that are not introducing new comcepts, but instead offering different surface forms of the same quantities introduced earlier. First, we notice that under our data generating model and pretrained classifier, ECE can be expressed as
selection model g and temperature T and tailored for the binary case we are
ECE = Egr., x Ply 1 - 1 c 20
By studying such population quantities, analysis is not dependent on any binning methods that are commonly used in empirically calculating expected calibration errors.
The in-iistriuttion case also works under our data generation model.
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6.2.1 Selective Recalibration v.s. Recalibration or Selection
We the following ECE quantities according to our data model for recalibration alone (R-ECE), selection alone (S-ECE), and selective recalibration (SR-ECE). For recalibration, we focus on studying the popular temperature scaling model, although the analysis is nearly identical for Platt scaling.
0x = 2)
R-ECE(T) = EBg Ply 11
T
1 1+ e
1
gg == BØT= P'y = 10===== 1=0'2=0,(12 =
(g(z) =
1 e 20
0 T
SR- SR-ECE(G,T)) = = Ply 11
v,g(z) 1 2v
19(x) 1
Our first theorem proves that under our data generation model, S-ECE and R-ECE can never reach 0, but SR-ECE can reach 0 by choosing appropriate g and T.
Theorem Under Assumption for any E (0,1) and â output by at there erist thresholds M E N+ and t > 0 such that if max T2,0, []0 < T and m > M, there exists a positive lower bound L, with probability at least 6 over Str
min
min S-ECE(g), min R-ECE(()) L. g3
TEB
However, there erists To and go satisfying Elgo(x)) 2 B, such that SR-ECE, To) 0.
Intuition and interpretation. Here give some intuition for understanding results. Under perturbed mixture model, R-ECE is calculated
1fe E A}
1fU E B}
R-ECE(T) = 0
-29T8- g- o3 1811
2n0 0-
E 2-20/
+ exp
1 exp
v)
for disjoint sets A and B, which correspond to points on the support of J1 and J2 respectively. In order to achieve zero R-FCE, when v E A, we need T = 0 0- (002)000))) However, for v E B we need T (101).) These clearly cannot be achieved simultancously. Thus the presence of the outlier data makes it impossible for the recalibration model to properly calibrate the confidence for the whole population. A similar expression can be obtained for S-ECE. As long as @T 0r (a²||0] and -aÔT fr /()2)((1))) are far from 1 (i.e., miscalibration exists), no choice of g can reach zero S-ECE. other words, no selection rule alone can lead to calibrated predictions, since no subset of the data calibrated under the pre-trained classifier, However, by setting 9o (x) for all B and go(x:) 1 otherwise, and choosing To = 0 (00((()())))) SR-ECE = 0. Thus can conclude that achieving ECE 0 on eligible predictions is only possible under selective recalibration, while selection or recalibration alone induce positive ECE. See Appendix C for more details and analysis.
-aô 6t ///////10000) (
6.2.2 Joint Learning versus Sequential Learning
We can further demonstrate that jointly learning selection model and temperature scaling parameter T can outperform sequential learning of and T. Let us first denote ğ == arg S-ECE(())) such that E[ğ(+)] 2 3 and f = arg min, R-ECE(T). We denote two types of expected calibration error under sequential learning of g and T, depending on which is optimized first.
ECE
min S-ECE(g, T);
g:
SCE == min R-ECE T). TER
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Our second theorem shows these two types of expected calibration error for sequential learning are lower bounded, while jointly learning g. T can reach zero calibration error.
Theorem Under Assumption 3 3 > 2(1- 8), for any E (0, 1) and 0 output by a, there exist thresholds M E N+ and T2 Ti > if max r2, o < T2, Ti < 0, and m M, then there exists a positive lower bound L, with probability at least over St
min {ECER-S, ECEE-RR L.
However, there exists To and go salisfying Elgo(x)) 2 B, such that SR-ECE To) = 0.
Intuition and interpretation. If we first optimize the temperature scaling model to obtain T, T will not be equal to 0 0* /00(0000)))) Then, when applying selection, there exists g that can reach 0 calibration error since the temperature is not optimal for data in A or B. On the other hand, if we first optimize the selection model and obtain D. I will reject points in A instead of those in B because points in A incur higher calibration error, and thus data from both A and will be selected (i.e., not rejected). In that case, temperature scaling not will be able to push calibration error to zero because, similar the case in the earlier R-ECE analysis, the calibration error in and B cannot reach 0 simultaneously using a single temperature scaling model. On the other hand, the optimal jointly-learned solution yields set of predictions with zero expected calibration error.
7
Conclusion
We have shown both empirically and theoretically that combining selection and recalibration a potent strategy for producing a set of well-calibrated predictions. Eight pairs distribution and were tested when i.idd validation data is available; selective recalibration with our proposed S-TLBCE loss function outperforms every single recalibrator tested in 7 cases, and always reduces S-ECE with respect to the calibrator employed by the selective recalibration model itself. Taken together, these results show that while many popular recalibration functions are quite effective at reducing calibration error, they can often be better fit to the data when given the opportunity to ignore a small portion of difficult examples. Thus, in domains where calibrated confidence is critical to decision making, selective recalibration is a practical and lightweight strategy for improving outcomes downstream of deep learning model predictions.
Broader Impact Statement
While the goal of our method is to foster better outcomes in settings of societal importance like medical diagnosis, mentioned in Section sclective classification may create disparitics among protected groups. Future work selective recalibration could focus on analyzing and mitigating any unequal effects of the algorithm.
Acknowledgments
We thank Mike Mozer and Jasper Snoek for their very helpful feedback on this work. This research obtained support by the funds provided by the National Science Foundation and by DoD OUSD (R&E) under Co- operative Agreement PHY-2229929 (ARNI: The NSF AT Institute for Artificial and Natural Intelligence). JCS gratefully acknowledges financial support from the Schmidt DataX Fund at Princcton University made possible through major gift from the Schunidt Futures Foundation. TP gratefully ackuowledges support by NSF AF:Medium 2212136 and by the Simons Collaboration on the Theory of Algorithm Fairness grant.
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Appendix
A Additional Experiment Details
In training we follow Fisch al. (2022) and drop the denominator Laet, as the coverage loss suffices
keep 9 from collapsing to O. Recalibration model code taken from the accompanying code releases from
Guo et al. 22077 (Temperature Scaling) and Kumar al. 2001 (Platt Scaling, Histogram Binning, Platt
Binning).
A.1 Calibration Measures
We calculate ECE, for q [1,2} using the python library released by Kumar al. (009 ECE, is caleulated as:
1
BBI Σ
EieB,
1 0i} EieB, f(ara)
ECE3
(14)
Byl
[By]
where B = BI..., Bm are a set of m equal-mass prediction bins, and predictions are sorted and binned based on their maximum confidence f(r). We set m 15.
A.2 Baselines
Next we describe how baseline methods are implemented. Our descriptions are based on creating an ordering of the test set such that at a given coverage level B, 3 proportion of examples from the end of the ordering are rejected.
A.2.1 Confidence-Barte Based Rejection
Confidence based rejection is performed by ordering instances in a decreasing order based on f(x), the maximum confidence the model has any class for that example.
A.2.2 Out of Distribution Scores
The sklearn python library Pedreeoss et al. 2011 is used to produce the One- Class SVM and Isolation Forest models. Anomaly scores are oriented such that more typical datapoints are given higher scores; instances are ranked in a decreasing order.
A.3
In-distribution Experiments
Our selector is shallow fully-connected network with 2 hidden layers of dimension 128 and ReLU activa- tions.
A.3.1
Camelyon17
Camelyon17 (Bandi et al. 2018) is task where the input is a 96x96 patch of whole-slide image of lymph node section from a patient with potentially metastatic breast cancer, the label is whether the patch contains a tumor, and the domain d specifies which of 5 hospitals the patch was from. We pre-train a DenseNet- 121 model on the Camelyon17 train set using the code from Koh et al. 2021 The validation set has 34,904 examples and accuracy of 91%, while the test has 84,054 examples, and accuracy of 83%. Our selector y trained with learning rate of 0.0005, the coverage loss weight À is set. to 32 (following Geifman & El-Yaniv 2019)); and the model is trained with 1000 samples for 1000 epochs with batch size of 100.
httpe://gttubb.cmm//-aambbb cal.aratiiooaaaiiiiiooooooo
http:///ghhub..cmmmlaaad///////
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A.3.2 ImageNet
ImageNet a large scale image classification dataset. We extract the features, scores, and labels from the 50,000 ImageNet validation samples using a pre-trained ResNet34 model from the torchvision library. Our selector g is trained with learning rate of 0.00001, the coverage loss weight À is set to 32 (following (Geifman & El- Yaniv 20191); and the model is trained with 2000 samples for 1000 epochs with batch size of 200.
A.4 Out- diitrr-diit Experiments
Our selector g is a shallow fully-connected network (1 hidden layer with dimension and ReLu activation) trained with a learning rate of 0.0001, the coverage loss weight, À is set 8, and the model is trained for 50 epochs (to avoid overfitting since this is OOD setting) with batch size of 256.
A.4.1 RxRx1
RxRx1 Taylor al. 2019 is task where the input 3-channel image of cells obtained by fluorescent microscopy, the label y indicates which of the 1,139 genetic treatments (including no treatment) the cells received, and the domain d specifies the batch in which the imaging experiment set has 9,854 examples and accuracy of 18%, while the test set has 34,432 examples, and accuracy of 27%. 1000 samples are drawn for model training. Gaussian noise with mean 0 and standard deviation is added to training examples in order to promote robustness.
run. The validation
A.4.2 CIFAR-100
CIFAR-100 is well-known image classification dataset, and perform zero-shot image classification with CLIP. We draw 2000 samples for model training, and test on 50,000 examples drawn from the 750,000 examples in CIFAR-100-c. Data augmentation in training is performed using AugMix (Hendryeks et al. 2020 with a severity level of 3 and a mixture width of
B Additional Experiment Results
B.1 Brier Score Results
While our focus in this work is Expected Calibration Error, for completeness we also report results with respect to Brier Score. Figure 5 shows Brier score results for the experiments in Section 5.1 Selective recalibration reduces Brier score in both experiments and outperforms recalibration. The OOD selection baselines perform well, although they show increasing error more data is rejected, illustrating their poor fit for the task. Further, Brier score results for the experiments in Scction 5.2 are included in Table 2 Selective recalibration reduces error, and confidence-basee rejection increases error, which is surprising since Brier score favors predictions with confidence near 1.
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ImageNet Selection Baselines
Re-Calibratin (ß 19=1. 1.0)
Selective Re- Calibration
0.15
Brier Score
030
0.5
0.00
Unsalibrated
Tamp Skaling
Plat Skaling
Platt Binner
Hetegran Birner
0.1
0.15
0.2
0.25
0.1
0.15
0.2
0.25
1-8
Confidence
Best recal.
SIMCE sequential
S-MMCE joir.
Temp Scaling
One-dlass SVM
S-MMCE sequential
S-TLBCE joint
Camelyon17
Re-Caibrratio (8 = 1.0)
Selection Baselines
Selective Re- Calibration
0.5
Brier Score
0.10
0.8
0.00
Unsalibrated
Temp Staling
Plat Sealing
Plat Binner
Histgran Binner
0.1
0.15
0.2
0.25
0.1
0.15
0.2
0.25
B
1-8
Confidence 150. Forest
recal.
S-MMCE sequenial MMCE joint
S-TLCCE sequenial TLBCE joint
Platt Scaling
Figure 5: Selective calibration error ImageNet and Camelyon17 for coverage level B E (0.75, 0.85, 0.9). Left: Various re-calibration methods are trained using labeled validation data. Middle: Selection baselines including confidence-basee rejection and various OOD measures. Right: Selective re-calibration with dif- ferent loss functions.
Table 2: RxRx1 and CIFAR-100-C AUC in the range 3 00.5, 1.0].
Selection
Opt. of h, RxRx1
CIFAR-100-
Confidence
0.169
0.180
One-class SVM
0.077
0.051
Iso. Forest.
0.061
0.051
S-MCE
sequential
0.138
0.166
S-MCE
joint
0.138
0.166
S-MMCE
sequential
0.126
0.165
S-MMCE
joint
0.126
0.164
S-TLBCE
sequential
0.126
0.166
S-TLBCE
joint
0.126
0.165
Recalibration
Temp. Scale
0.140
0.164
(B 1.0)
None
0.252
0.170
C Theory: Additional Details and Proofs
C.1 Details on Data Generation Model
Definition 2 (Formal version of definition 1 For H* E RP, (0" or,,,,,r,,73 re)--prruubbbdd truncated-
Gaussian model defined as the following distribution over (a,y) R2 x -1:
y zJ1 + 2)J2.
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Here, J1 and J2 are two truncated Guassian distributions, i..,,
J1 ~ PIN(y 8 , f B(0" (o(("""))BBBB"".. J2 ~ P2N(-y-- att of E(aa- . r2) (B(
where P1,22 are normalization coeffcients to make J1 and J2 properly defined; y follows the Bernoull distribution 1) P(y
-1) - 1/2; and follows a Bernoulli distribution P(z P(== 3.
For simplicity, throughout this paper, set P1 P2 and this is always achievable by setting r1/r2 appropri- ately. We also set E (0, 1/2).
C.2 Details on /
Recall that we consider the 0 that the output of a training algorithm (S that takes the i.i.d. training data set Str = {(o]", v." 12) as input. We imposed the following assumption on 0.
Assumption 3 For any given 8 E (0, 1), there exists 8g E R with | = ((),, that with probability at least
110-0 0oll < m))
and Φ60, m) goes to 0 as m goes to infenity. Also, there exist threshold M E N+ such that if m > M, (66.m) is decrensing function of & and n. Moreover,
0 0
min
|00 1 0 0r,
|10b1| > 0.
We will prove the following lemma as cornerstone for our future proofs.
Lemma 1 Under Assumption
S for any 5 E (0, 1), there erists a threshold M E N+, and constants < < Is < Is, such that if m > M, with probability least over the
I I2, 0 I3 14 < al3, randomness of Str,
σT T |101
E [2,,
σT E Idl. ||o|| E [os
Proof 1 Under Assumplion 3 we know m + 0o lends to ô + 0o- In addition, for any 5 E (0, 1) there exists a threshold M E N+ such that if m > M, Φ65, m) a decreasing function of 5 and m, which leads to
θT
0
0g
E
+ +e), σTO- E 10.00 -,00 +el, ||ōl| E [[|0o]] - -, 1,110 + +
10|12
10012
10
for some small > 0 that makes the left end of the above intervals larger than 0 and 0g 0 -5 al0 0- -E) hold for all T2, m as long as m > M. Then, we set I, 's accordingly to each value above.
C.2.1 Example Training /
In this section, provide one example to justify Assumption 3 i.e., θ = Ei= 2. y!"" /m, where the training set is drawn from an unperturbed Gaussian mixture, i.e., xf yy Nyy -0" 00,0== and y" follows Bernoulli distribution " 1) = 1/2. Directly following the analysis of Zhang et al. (2022),
have
σT - -0( |||0|| + 110-112
For |1012, notice that
0 = 0" + Em
20

---

Published in Transactions on Machine Learning Research (07 (224))
where cm ~ ~O( ). Then, we have
110+112 P m |10|2 = 110"1| 26,00 0" + |||ml|
Or( Or( vP Orl 110011 Vm
Given p/m O(), combined with the form of classic concentration inequalities, can verify this example satisfies Assumption 3
C.3
Background: ECE Calculation
Recall denote f(x) E.E is:
(a)) and denote the predicition result. 1 C(r). The definition of ECE = E ((x) IPly - - ff()) = pl - -pl.
Notice that there are two cases.
Case 1: f(x)= = (1(),, by reparameterization, we have
22v
22
Ply = 9 f(x)) = p] -pl-- = Ply = 1 11((x) =
1 c
1 1 c2v
200
Ply = 1 110 x= = v]
1 620
Case 2: f(x) (x)) by reparameterization, we have
1
Ply = fxx p] pl
Fly - f(z)
1 c2vl
1 e22
Fly 1 0 == r]
1 220
2r
1 Ply = -1 0T 0) (1
+ 2v
20
Ply = 1 0T x ==
1 620
To summarize,
ECE E (oo = (xx p] pl Egg Ply 1 ØT v)
1 1 20
Temperature scaling.
2 x/T
p (x) c2 +1
o (x)
(15)
c2.6T x/T
Thus,
R-ECE Eggz Ply BT x vT
+ e 20
Platt scaling.
(=))(x)=
1
p (x)
1
(16)
e2w.0T Tw
e - 2626 +1
E.F = -ETT Ply =
| w 0T x. b= -]]
1te
21

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Published in Transactions on Machine Learning Research (07/2024)
ECE calculation.
The distribution of HT x has the following properties.
xTyy =1 10 0r 00 x B(OTor, -
+ + 2)p2\( ~T (0, B(adTTr,ra100)) ( 8", 1)}:
Jxly - 2p. of (0 B(T .r. rr:l@11) U B "0r |r))
+ (1 - =a)=>(aaaaaa- 0 0 00 x e B(od r² rr2110 (ôll u UB(-aul at 0,, r2 []]-
Now, we are ready to calculate ECE. Specifically, given 8, For notation simplicity, we denote 0",
A = B( rr,r) U B rr and B = B(a θ 0r, r2 r2 U (B-(a rrll@)))) Mean- while, for simplicity, we choose 71,72 such that P1 = P2 p. This is always manageable and there exists infinitely many choices, we only require Si S2 (B(OTO", rr,rr((01),,_. (( (-0 -0000 0rrrrr((()), B(a r2ll@) B(-aT 0*, r2) In able to achieve #i P2 p. such that if r r2 are both smaller than
for any S * S2, S1 E
it only depends on r1/12- Apparently, there exists threshold & (one can choose Ti, as functions of with appropriate choosen σ), then An B 0 can be achieved.
P x y= 1)
Ply = 0 =o]
P(O x = 1 1) PiO x= 1= 11)
1
1fve A} +
1( E B}
-20TT-- 11 021|0
200TT- 0 1)
+ exp
1 1 exp
C.4 Proof of Theorem
C.4.1 Temperature Scaling Only
A simple reparametcrization lcads to:
1 1fv E A}
1fv E B)
1
R-ECE E T
1 1 exp
-20
0)
1 + exp
220TT 0 3 v)
2-20/I
The lower bound contains two parts. We choose the threshold mentioned previously small enough such that I > max(ri,r2). This can be achieved because Ta is independent of 12,
Part I. When v B rı [[]]] C A, and we know that AnB 0. Let us choose threshold
min os)/o² > 0. Then for any T > 0, it must fall into one of the following three cases.
22

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Published in Transactions on Machine Learning Research (07/2024)
Case 1: T-1 and 0T ((o2)0@1)) are far: T-1 - rT /00(((0)))))))) t, recall 0Tx,, then
1
E
REB(TTT 0-,ri
-- or 02111122 v)
1 1 e-20T 20/T
1 exp
1
1
P(x B(0", ,n)))
+ exp
10T-
or -ri) (ri))
1+ exp
-29T0" (ŌTO- + ri))
02116112
8
1
-2(0T 8+ /0/(021011)) + (1( -n))
-20 0r (Ø A: + +r1) 021101=
+ exp
1 + exp
8
1
1
2
1 exp
-2 8+
-ri))
+ exp
-20 0- 02110
(ÔT g- + tmm))
8
min da]]n_/a,,dE((aaa4
: 81
e +) r1))
+ exp (-2c/ (dl
Case 2: T-1 and 0T 1021(1))) are far: / T >
B BEEB(OTO (r.n.ld0))
1 exp
o=1101| 1)
0- 0+
1 201fT
1
B
+ exp
-28 0- (ÔT 8- -i)) @allē
exp
-2T 0- + r1)
1
B
+ exp
-0 0r (Ō gr -i)) 02110
1 + exp
/2(0T0/(0021000 - 7) 0r +m1)
2V
min
82
2 cE (e).Fa
1 expl (- 2c -i)) 1 exp exp (-2(c/02 (dd
Case When T-1 and ØT ( are close: TT (08' I T, then when v e
B(-af 8", rr c. B. For small enough 7 satisfying T S 0.2(1 - 2
1
= zEB(-ad rx(((1)
2000 ?>liا]|
1 e 20/T
+ exp
exp(2va)
2(1 TT
2
8E| "0p p +r| vE 0 ralld1))
uo To aTe
min
min
(1 + exp(2au))²
02110112
-7
2
20 ex
1- = 8a
min sals
min
1.8(1 - a) Φ²
RE|
tr|
(1 exp(2ar))2
2
Part III. Combining together, we have
R-ECE 2 min(31, (3,33,,85) 83).
Finally, we take ri S min TO2 To2/T) Ts which ensures Br for all i= 1,2,3.
23

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Published in Transactions on Machine Learning Research (07 (024))
C.4.2 Selective Calibration Only
We hope B[g(z) = 1] 2 B. Let us first define g = {8T x g(z) = 1}. Then, for any 9, we have
Ply = 1 x U, E
, y 1)
9 y + 0
110 E A} +
1fc B} 1 9}
00 02101p2
Ta 00100
exp
+ exp
E Ang)+ +
1f2 E Bng)
-20T0- 63110r12
220TT. +
1 exp
1 + exp
Then, by choosing small enough o, such that 1/0² > 1, the corresponding ECE
1
ECEs OT zlOT rEG
E
-2
1 + exp
0210
1
1ivE Bng] 2 2v
2a0T o2
+1
1 + exp
2 E
em ØT alpTTEG T
-2 0611611
AnEANg
c- 2n +1
1 exp
1
+ E v= BT Te0
1 fo E Bng)
2 T
2" +1
exp
Aiu P Angle g + (v Bng
Since P( Angg () + 9) = 1, it is not hard to verify that
S-ECE 2 min X2]
where
2 exvx)(2v
I1
X1 min min
ael1, 3) VEA (1 + exp (app)ar))))
02
20 exp()va)
ah1
A min
min aE(- alg .j BEB
(1 + exp 22
0²
C.4.3 Selective Re-calibration
We choose g = B and set T-1 0 then SR-ECE = 0. Thus, there exists appropriate choice of g and T such that
o² =
SR-ECE(g, 7) = 0.
C.5 Proof of Theorem 2
Usually, is much larger than 1 B, for example, 3 90%. In this section, we impose the following assumption.
24

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Published in Transactions on Machine Learning Research (07/2024)
Assumption 4 The selector g will retain most of the probabilty mass in the sense that
B 2(1 8).
Let us denote Ç = 3/2 (1- (3) and Ç is a positive constant. First, we have the following claim.
Claim 5 Under Assumption if we further have
min "ER( ar m (10|))
-2TT0-
e 2n +
+ exp
0711011
oC -0 rrlr@(@
1 + exp
Ta
2v
+21101122
then for 91 = arg min 9:Bigg:BBggiai.... g(1))1122 S-ECE, we have that BrE (o000rrr))000 .T)l (x)) )- = 1] = =F(( B(-a0", (r2)))
Proof 2 The proof is straighlforward. We denole O = xx: E BB--0000,,. ,22), g(x) 0}. We will prove that P( E 00 =0
If not, let us denote P P(x E 0) > O. Since we know 3 > 2(1 B), which means even if we "throw away" by only setting points in B( r1) with value equals to 0, there will still be other remaining probability mass retained TB TB(0", r1) with g value equals to Then, there erists 92 such that 92(x) for all B(-a0", , ,22 and leads to P(r B(-a0", g2()) 1) Piz E B(-a0" .2)) 1) + (enabled by the fact B > 2(1 B) ) and P( (z) 1) P(g2(z) 1) for B(0 r1) U( r2). Since
all the probability mass
min vE R" m (101)) + exp
--
2
02110112
max vEB(-o8 r211@1)
2a0 o- +
2v
1 exp
which means "throwing away' points in B(a0", r2) more effectively lower the calibration error and must have
S-ECE(g2) < S-ECE(n).
Next, we state how to set the parameters such that the condition in Claim holds. As long as we choose 0, (T, 12 small enough, such that
1
1 1
+ 16)
1 exp( exp((22 (T,
r1)))
1 + +xpp -24 + r216))
1 1 cxp(2all//22444 r2
then,
min BEB( 0" ((01))
-2AT0-
e 2u +
+ exp
max wEB(-ade r(1
Ta0T 6- d1le
2 +00+1
1 exp
1)
25

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Published in Transactions on Machine Learning Research (07/2024)
Then, following similar derivation in Section C.4.1 ECES-+ > 0.
prove with suitably chosen parameters 71, 12,72,0, 0,
Lastly, let us further prove ECET-S > 0. We choose TI and small enough such that v > 0 for all v E B(θ 8" ,r,r1 ,r11001) B(ad 8" r2 (00l) and < for all E B(- (B
"00rr,rl100 rr1l001) UB(-af 0", r2]101]).
For 1/T E -=00" 0" we can calculate the derivative for R-ECE as the following: = I'llo
1
1
R-ECE- T (r.111001)
220
e 20/7
+1
+ exp
0113
+ E -EB((-TTO.
-20TT-- a-
c 2--20/T +1
1 + exp
a=|101= 10)
+ R2 VE
+
200TT.
2-2u/ +1
1 + exp
0=110
2)
1
+ E BUEB(-oBTo
200T0. 02110
e 2v/T +1
+ exp
v
Next, we take a derivative over x = 1/T for x E a- 0" which leads to
dR-ECE
2ve
2ue 20x (e2va + 1)
dx
2E
(22v0 + 1)2
+ 2
UEB(aBT or All)
Consider the two values
2ve 2ux
2ave
(e2vx + 1)21
222vr 1)2
the ratio
2ave 2
2n
(eZavz + 1)2 (
/ (e2ve + 1)2
tv40 20.
That means if we take suitably small r1,12 and let [er, 02) with appropriately chosen C1, 02
dR-ECE dr
0.
Thus, we know the best choice of 1/T should not be equal to 8 oلlle
Then, notice 3 > 2(1 B), which means the probability mass in (B 0r [0]] cannot be all be "thrown away"; following similar derivation in Section C.4.1. we can prove with suitably chosen parameters Ti. 0, ECET-S 0.
26


Reviews:
[{"review_id": "HIaYmVacdW", "reviewer": "Reviewer_XaU7", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "### Pros\n\n1. **Clarity**. Overall, the writing is clear and easy to follow. In addition, the organization of the main draft is well-established.\n2. **Well-motivated problem and principled method**. Improving the reliability of pre-trained classifiers is an important problem for many practical scenarios. In addition, while the proposed method is simple and can be viewed as a naive combination of two existing approaches, the motivation is clear and demonstrated with the theoretical justification. \n\n### Cons\n\n1. **Limited empirical improvement**. The improvement from the proposed method is not significant compared to a simple baseline (”Selection with Confidence”), even though it requires an additional training process. For example, this baseline outperforms all the variants of the proposed method on the ImageNet dataset and shows comparable performances on the Camelyon 17 dataset. While the proposed method outperforms this baseline under distribution shift (Section 5.2), this baseline still significantly improves the calibration of the pre-trained model (e.g., ECE-1 on RxRx1: 0.308 (None) > Confidence (0.071) > Proposed (0.036)). The advantage of this simple baseline is more noticeable as the effectiveness of the proposed framework varies depending on the configurations and tested datasets. \n2. **Limited justification on current design choices**. Currently, there are some fixed design choices such as the choice of feature extractor and the architecture of selector g. However, such a choice is not demonstrated yet; for example, it is unclear whether the proposed framework can be generalizable across different feature extractors on the same dataset.\n\n### Editorial comments\n\n1. Typo\n    - ECE_q in Eq.1, while ECE-q in Section 5 and Tables.\n2. To help the understanding of the reader, it would be better to provide more details about each dataset in the main text, including the statistics, used backbone, and the figures of examples.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes a new framework, called “selective recalibration”, to improve the calibration of a pre-trained classifier. Post-hoc recalibration and selection approaches have been developed separately for this purpose, but the authors show that combining these two approaches could give a unique advantage and make further improvements. The authors demonstrated this advantage theoretically, under simple assumptions. Then, they also provide empirical evidence through extensive experiments on popular benchmarks.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "More empirical demonstrations for the current design choices and various feature extractors.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "tqDZSBhbfC", "reviewer": "Reviewer_bH9Y", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The paper and setting is well presented. I am not fully up to date on the latest research in the area, but from a cursory search and reading it seems the authors have covered the important related papers. The methodological improvement is straight-forward and easy to understand and the results that are generated are solid. On the basis of these strengths I am inclined to vote for accept on this work. \n\nOne critical missing piece of this work is the final accuracy of the models reported in Table 1 and Table 2. It does not seem necessary that the accuracy of the model would drop when the proposed method is applied, but having the concrete accuracy numbers would allow readers to better understand what trade-off exists, if any. I feel this is my strongest suggestion. \n\nThere are some minor presentation issues that ought to be fixed as well. Before equation (1) it is not made clear what $\\hat{f}$ is precisely. In Equation (6) the authors are missing indices which make the equation improperly defined.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper studies the problem of calibrating the predictions of a pre-trained model. To do so, the authors employ simultaneous selection and recalibration. While this combination on its own is not novel, the authors propose to jointly optimize the parameters of the selection and recalibration models rather than sequentially optimize them as has been studied in prior works. Methodologically, this modification is made quite simply by pushing the recalibrator model inside of the cross-entropy term. \n\nThe authors validate this intuitive modification with a selection of experiments on RxRx1 and CIFAR-100-C and show strong results indicating that their method is strictly better than prior approaches with respect to ECE. However, it is a little less clear in the current presentation of the paper exactly how the accuracy compares between the methods. This will be a necessary addition to fully recommend acceptance (more on this below). \n\nThe authors finish the paper by providing theoretical justification for their method in a toy setting. While I feel this gives a good intuition for why the approach works and may be useful to readers I do not see it as a strong contribution on its own, more of an aspect that strengthens the papers presentation.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Addition of accuracy statistics in Table 1 and Table 2. Adjustments to preamble of equation (1) and re-write of equation (6).", "broader_impact_concerns": "I do not have broader impact concerns about this work.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "ioywU5Tn77", "reviewer": "Reviewer_K1dr", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths**\n1. The paper is well written and easy to follow.\n2. The problem addressed in the paper is relevant to the community.\n3. The proposed approach is simple, effective, works well in practice, and is convincingly motivated.\n\n**Weaknesses**\n\n4. Although the paper is mostly clear, I did find section 4.1 to be a bit ambiguous. For starters, eq 6 is stated without defining that $(x, y)$ is the batch. Then, the losses $l(f, \\hat{g}, h, x, y)$ are not written in such a way that makes it clear for the reader what is fixed an what is being optimized, e.g. I think $l(\\hat{g}, h; f, x, y)$ would be better notation. More importantly, I do not believe that $L_{set}$ as defined in eq 7, along with the choice of $l$ in eq 9 recover the loss used by Fisch et al., as claimed in the current manuscript; both because the scaling changes to $\\sum_{i,j}\\hat{g}(x_i)\\hat{g}(x_j)$ and because $q$ is not used in the numerator in eq 6 (compare to eq 13 in Fisch et al.). Also, should eq be multiplied by $1/n^2$ instead of $1/n$? This makes it ambiguous if, in the experiments, the authors compare against the method as described in Fisch et al., or the slightly different version they describe in sec 4.1.2.\n5. I do not believe that eq 10 provides an unbiased estimate of $(\\beta - \\mathbb{E}[\\hat{g}(x)])^2$. I am aware that at test time you set the threshold so as to ensure coverage, but I still believe this should be discussed.\n6. While the analysis in sec 6 is very interesting and convincingly motivates the need for joint training, the used classifier remains very simple. It would be good to discuss what might change in the setting where a neural network classifier is used instead. To be clear I'm not asking for theoretical results in the setting, just a discussion.\n7. This paper mostly focuses on calibration, which can be understood as attempting to have proper quantification of aleatoric uncertainty. I think it would be appropriate to discuss methods aiming to also quantify epistemic uncertainty in the related work section, e.g. Bayesian neural networks, deep ensembles, and conformal prediction.\n8. The organization of the paper, where the theoretical analysis in section 6 comes after the experiments in section 5, is bizarre.\n9. Finally, a list of typos:\n- First paragraph: \"1% chance\" -> \"a 1% chance\"\n- Last paragraph of sec 2: cite Kumar et al. 2018 using \\citet\n- Equations are sometimes missing punctuation, e.g. missing period at the end of eq 2, and missing comma at the end of eq 4\n- The sentence \"Intuitively, to optimize...\" after eq 4 is rather unclear, please rephrase\n- eq 12: missing s subindex for f\n- Missing indicator in $\\hat{g}(x) \\geq \\tau$ in sec 4.4.1.\n- Top of page 7: \"ECE-1 and ECE-2\" -> ECE_1 and ECE_2 for notational consistency\n- Definition 1, while understandable, should use more formal notation, $x|y$ as written, is $x|y,z$, and the dependence on $y$ should be made clear in the notation, rather than in the paragraph following the definition.\n- Please double check you cite the correct version of papers. I noticed you cite Fisch et al. (2022) as an arxiv preprint, but I believe tha's a TMLR paper. I did not go over all the citations, but please check them.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper uses several concepts which I will begin by briefly summarizing.\n\nRecalibration refers to techniques which modify the output of a classifier to ensure it is properly calibrated. Selective classification aims to only classify points where the classifier is confident enough (while obeying some coverage constraints) so as to increase classification accuracy, which is achieved by training a selection model which, given an input, determines if it should be classified or not. The recently proposed selective calibration uses a selection model for calibration purposes, i.e. to ensure that the classifier is properly calibrated on the selected points.\n\nThis paper proposes selective recalibration, which combines recalibration with selective calibration. In other words, the authors simultaneously use a selection model along with a small head which changes the output of a classifier to improve its calibration on selected datapoints. The authors propose a sensible objective to jointly train the recalibration and the selection networks given a pretrained classifier. The authors also argue that this simultaneous optimization is desirable by constructing an example where perfect calibration is unachievable when using only recalibration, only selective calibration, or even when using selective recalibration where the two networks are trained sequentially; yet perfect calibration is achievable when doing simultaneous optimization in selective recalibration.\n\nOverall this is a good submission which is clearly written, well motivated, and which has good empirical results.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Please see weaknesses section of my review.", "broader_impact_concerns": "I have no broader impact concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper addresses the problem of calibrating the predictions of a pre-trained classifier. The authors introduce a new technique, selective recalibration, which combines recalibration with selective calibration by jointly optimizing the parameters of both approaches. Several methods for this joint optimization are proposed, and they are easy to implement. The authors demonstrate its effectiveness through theoretical justification in a toy setting and empirical evidence on two benchmarks.

All three reviewers acknowledged the clarity of this paper, the significance of the problem, and the soundness of the proposed method. During the revision process, the authors successfully addressed the reviewers’ comments; consequently, all of the reviewers now agree to the acceptance. Therefore, AE recommends a clear acceptance of this paper.
Decision: Accept (poster)