Minimum-Risk Recalibration of Classifiers

Recalibrating probabilistic classifiers is vital for enhancing the reliability and accuracy of predictive models. Despite the development of numerous recalibration algorithms, there is still a lack of a comprehensive theory that integrates calibration and sharpness (which is essential for maintaining predictive power). In this paper, we introduce the concept of minimum-risk recalibration within the framework of mean-squared-error (MSE) decomposition, offering a principled approach for evaluating and recalibrating probabilistic classifiers. Using this framework, we analyze the uniform-mass binning (UMB) recalibration method and establish a finite-sample risk upper bound of order $\tilde{O}(B/n + 1/B^2)$ where $B$ is the number of bins and $n$ is the sample size. By balancing calibration and sharpness, we further determine that the optimal number of bins for UMB scales with $n^{1/3}$, resulting in a risk bound of approximately $O(n^{-2/3})$. Additionally, we tackle the challenge of label shift by proposing a two-stage approach that adjusts the recalibration function using limited labeled data from the target domain. Our results show that transferring a calibrated classifier requires significantly fewer target samples compared to recalibrating from scratch. We validate our theoretical findings through numerical simulations, which confirm the tightness of the proposed bounds, the optimal number of bins, and the effectiveness of label shift adaptation.

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Peer review

Reviewer GPZN6/10 · confidence 4/52023-07-06

Summary

Background: Generating reliable probability estimates alongside accurate class labels is crucial in classification tasks. Calibration, which refers to the alignment between predicted probabilities and empirical frequencies of labels, is highly desirable in various applications. However, many machine learning algorithms lack inherent calibration. This paper aims to address these issues in two ways: 1. Development of a unified framework for recalibration that incorporates both calibration and sharpness in a principled manner. 2. Proposal of a composite estimator for recalibration under label shift, which converges to the optimal recalibration and enables sample-efficient adaptation of classifiers to label-shifted domains.

Strengths

1. This paper is exceptionally well-written, with a clear and logical structure, making it easily understandable to a wide audience. 2. The theoretical foundation of this paper is solid, the proof process is very detailed, making it easy for people to follow. 3. The algorithm proposed by the author is simple and efficient, making it easy to implement.

Weaknesses

1. Although the author provides solid theoretical analysis, this paper lacks enough experiments to demonstrate the effectiveness of their method. 2. The author employs three assumptions, and in my view, assumption 2 is a rather strong one, which is difficult to guarantee in practical applications. 3. The experiments in this paper were conducted primarily on toy datasets, and the author do not provide any experiments on how to apply Recalibration in real-world applications.

Questions

1. In Eq. (9), I understand that the function $\hat{h}(z)$ calculates the expected label within the bin that $z$ belongs to. So, why is $\hat{h}$ a monotonically increasing function of $z$ and its growth pattern similar to the cumulative distribution function (CDF) of $z$? Although you have made Assumption 2, can I understand that this assumption implies that $y$ must follow a specific distribution form in order to satisfy the monotonicity of $\hat{h}$? 2. In the experimental section, the author only presents a very simple toy scenario, and I am quite curious about the performance of the author's algorithm in more complex scenarios. For instance, in real-world classification datasets like cifar-10-long-tail or cifar-100-long-tail, how does the recalibration effect of more complex models, such as neural networks or decision trees? 3. In Fig. 1 (a), as the value of $n$ increases, regardless of the magnitude of $B$, $R^{cal}(\hat{h})$ exhibits a noticeable decrease. Does this imply that when the dataset size $n$ becomes sufficiently large, $R^{cal}(\hat{h})$ has already reached a level where recalibration is unnecessary? If the author can answer my question, I will change my score.

Rating

6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.

Confidence

4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.

Soundness

4 excellent

Presentation

3 good

Contribution

2 fair

Limitations

yes

Reviewer yKq48/10 · confidence 4/52023-07-07

Summary

The paper studies a very relevant problem of post-hoc calibration in probabilistic classifiers. There is a great body of work on calibrating probabilistic classifiers so that the predicted probabilities match the empirical label frequencies in the popular machine learning literature. However, calibrating probabilistic classifiers so that they retain their sharpness / refinement is not studied much. The paper focuses on the post-hoc calibration with retained sharpness properties from first principles. Its entry point is the classical decomposition of mean squared error of the classifier into sharpness and (mis)calibration. Traditionally, both the measures were important, but the works in machine learning literature has mostly focussed on (mis)calibration only. There are important contributions in this paper: a) proposing recalibration risk measure which attains its minimum value only when both sharpness and the calibration requirements are met. b) uniform mass binning algorithm to approximate the optimal recalibration map. c) Theoretical results on the binning scheme and binning scheme to bound both the calibration risk and the sharpness risk. d) Formal result on the trade-off between calibration and sharpness. e) recalibration studies in a simple (but important) distribution shift with label shift. All of these results are for binary classification setting.

Strengths

1. It is clear that the paper makes theoretical contributions to an important problem. It has been observed that major recalibration algorithms in machine learning literature not just ignore the sharpness measure, but they degrade it [1]. There is a trade-off between these quantities, and this paper formally state this with some actionable insights to balance this trade-off. 2. Theorem 1 generalises a previous result in the literature with an additional insight on the sharpness risk. Generalisations and connections to previous results is a sign of great work. 3. While the results on label shift are easy consequences of the general recalibration results presented earlier in the paper, it is still great to see this extension. 4. In my opinion, the paper is significant and relevant to the machine learning community. I believe the paper will spur more work on post-hoc calibration methods. 5. The paper is excellently written, easy to read and understand. [1] Aditya Singh et al. On The Dark Side Of Calibration For Modern Neural Networks. (2021) (http://www.gatsby.ucl.ac.uk/~balaji/udl2021/accepted-papers/UDL2021-paper-074.pdf)

Weaknesses

I do not have major concerns with this paper. I have some questions though (please see below):

Questions

1. The optimal recalibration function is same as the canonical calibration map defined in [2] (Section 3.1). While it is clearly interesting to see that a canonical calibration map is the optimal recalibration map, could authors comment on this connection? The optimal recalibration map is certainly a calibrated function (as the Proposition 1 in [2] states), I am curious about some insights on why it would also be a map to minimise the sharpness risk. 2. Additionally, [2] also propose a sort of general methodology to estimate this canonical calibration map (Section 4.1). While they do not comment on the partitioning scheme, it would be interesting to draw insights on general partitioning schemes and the bound on the recalibration risk. 3. While the authors mention that extending the results presented in the paper to multi-class setting is a future research direction, could authors see that the estimator provided in [2] could be useful for multi class case (as [2] do not restrict itself to binary classification)? [2] Juozas Vaicenavicius et al. Evaluating model calibration in classification. (2019)

Rating

8: Strong Accept: Technically strong paper, with novel ideas, excellent impact on at least one area, or high-to-excellent impact on multiple areas, with excellent evaluation, resources, and reproducibility, and no unaddressed ethical considerations.

Confidence

4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.

Soundness

4 excellent

Presentation

4 excellent

Contribution

3 good

Limitations

Obviously, there are limitations to this work, as there are with any works. However, the authors have been very clear in stating them.

Reviewer USpM7/10 · confidence 3/52023-07-09

Summary

This paper introduces the concept of minimum risk recalibration, utilizing Mean Squared Error (MSE) decomposition. The authors provide justification for their approach by demonstrating that minimizing the proposed risk yields simultaneous minimization of the calibration risk while preserving the sharpness of the probability forecaster. Furthermore, the authors employ the MSE decomposition to analyze the recalibration method known as UMB. Theoretical analysis reveals that selecting an appropriate number of bins allows for achieving a balance between the calibration risk and sharpness. Expanding on their findings, the authors apply their methodology to address the problem of recalibration in label shift. They provide theoretical and experimental validation for their approach in the context of label shift, further supporting the effectiveness of their proposed method.

Strengths

1. The authors provide a rigorous introduction to key statistical measures essential for the recalibration task, such as recalibration/calibration risk and sharpness risk. They demonstrate that minimizing these measures can result in a well-calibrated forecaster. 2. Building upon the introduced statistics and their properties, the authors enhance the analysis of UMB. They unveil a tradeoff between calibration risk and the preservation of sharpness by establishing a high-probability error bound. Furthermore, they leverage this upper bound to guide the selection of the number of bins in the UMB method, highlighting the practical significance of the proposed bound and decomposition. 3. The authors extend the application of their proposed method to address the challenge of label shift, showcasing the validity of their work in practical downstream tasks. 4. The paper thoroughly discusses the mildness of the assumptions made and experimentally validates the non-trivial nature of the proposed bounds and schemes.

Weaknesses

While the outcome of this study is fruitful, it is encouraged that the authors delve into a further discussion regarding the potential extension of their framework to the multi-class scenario. This extension would provide additional intuitive information and enhance the applicability of their proposed methodology.

Questions

Please refer to the Weakness part.

Rating

7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

4 excellent

Presentation

3 good

Contribution

4 excellent

Limitations

The authors have analyzed the limitation of the used method in the appendix.

Reviewer kAN56/10 · confidence 1/52023-07-25

Summary

This work proposes a calibration method for probabilistic classifiers. It is known that most machine learning models produce predictions with high confidence that yield a distribution different than the underlying true label distribution. The proposed method focuses on calibration without a loss on the prediction performance and is claimed to adjust for label shift.

Strengths

- This paper is written clearly with good and easy-to-follow notations. The different definitions are well-placed and help the reader to follow the flow of the paper. Overall good structure of the paper. - The equal focus during the calibration on both calibration and sharpness goals is well-motivated and importantly this setting. The theory is developed rigorously and the different choices and assumptions are justified. - This paper extends the proposed work to label-shfit setting and provides a good discussion on "recalibration under label shift" . - This paper provides experimental results that show the validity of the theoretical work proposed.

Weaknesses

I think this paper lacks the following two points: - A discussion in section 1.1 highlighting to the reader how this paper approaches the recalibration problem differently than existing work. Only the work in [26] has been addressed. - Section 6 does not include a discussion about the empirical results of previous works.

Questions

N/A

Rating

6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.

Confidence

1: Your assessment is an educated guess. The submission is not in your area or the submission was difficult to understand. Math/other details were not carefully checked.

Soundness

3 good

Presentation

4 excellent

Contribution

3 good

Limitations

N/A

Reviewer a48G6/10 · confidence 3/52023-07-27

Summary

This work looks at methods for calibrating probabilistic classifiers for Mean Squared Error by decomposing it into calibration error and sharpness errors. It gives finite sample error guarantees on both of these for the Uniform Mass Binning method. By balancing sharpness and calibration error, they also propose the optimal number of bins to use. Additionally, they look at the problem of calibrating classifiers in the case of label distribution shift between train and test. Their results show that transferring a calibrated classifier requires significantly fewer target samples compared to recalibrating from scratch. They validate our theoretical findings through numerical simulations.

Strengths

- Existing works have looked at finite sample bounds for the calibration error but not the sharpness (as mentioned in this paper). This is the first method to look at both sharpness and calibration together which is an important criterion. - The paper is generally well written.

Weaknesses

- The main contribution is to give risk bounds for sharpness along with the calibration error. In my opinion, this is not a significant contribution. There is not any particular strategy proposed using this analysis but only choosing the number of bins. - The second problem of using this method for label distribution shift also seems straightforward and I have also asked a related question below.

Questions

- Why did the authors choose to analyze only Uniform Mass Binning and not other methods like Uniform width binning? - For the label distribution shift problem, is the proposed method equivalent to just scaling all the predicted probablities according to the target distribution and then using the uniform binning technique? - In definition 4, the sharpness risk of h over f, is this the sharpness risk of h.f - sharpness risk of f? - Why is Assumption 3 needed for the label distribution case? - Just to confirm, is this the first sharpness risk bound? - In figure 2a, why is the gap increasing between the theoretical and empirical bound as n increases? - In table 1, the different methods compared are not clear to me. Can the authors please explain?

Rating

6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

3 good

Presentation

3 good

Contribution

2 fair

Limitations

Yes

Reviewer yKq42023-08-17

Post rebuttal comment

Thanks to authors for the detailed response. I'm certainly glad that authors found my comments useful, and have been able to draw further connections. Unfortunately, I haven't been able to engage in the discussion to the capacity that I'd have hoped due to some personal issues, but I find the response very insightful. I remain confident that the current paper should appear at NeurIPS, and hence I'm also increasing my score.

Reviewer kAN52023-08-18

Thanks to the authors for addressing my concerns. I will keep my score as I am unfamiliar with the overall related work as shown in my confidence score.

Reviewer a48G2023-08-19

Response

I would like to thank the authors for providing the detailed response. After reading the rebuttal and other reviewers' comments, it seems like providing guarantees for sharpness and a strategy to choose optimal bins is an important contribution to a fundamental problem. I have increased my score.

Reviewer USpM2023-08-20

Thanks to the authors for providing related works in the multi-class scenario. I will keep my score since the difficulty of the extension to the multi-class scenario is inherent in the task of probability calibration and does not weaken the contribution of this work.

Program Chairsdecision2023-09-21

Decision

Accept (spotlight)

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