Confidence Calibration Using Logits Transformations

Deep neural networks have proven useful in various fields such as image classification [1, 2, 3], object detection [4, 5], and speaker verification [6]. Motivated by these successes, deep neural networks are now being integrated into safety-critical systems such as medical systems [7, 8]. However, as observed by Guo et al. [9], neural networks are often over-confident on their predictions. This over-confidence can be a critical problem in the safety-critical applications where over-confident neural networks can be wrong with high confidence.

Various calibration techniques have been proposed to alleviate the miscalibration issue. A common approach is to map uncalibrated logits or confidences to calibrated ones  [9, 10, 11, 12, 13, 14, 15, 16]. Another option is to train a neural network with a modified loss function to produce calibrated confidence [17, 18, 19]. Although existing methods successfully improve the confidence in terms of calibration metrics such as the expected calibration error (ECE) [20], there is further space for improvement, especially in high-dimensional settings where vast amounts of data are required for good calibration.

In this work, we propose Hoki, a calibration algorithm that achieves strong empirical performance as compared with state-of-the-art techniques. The intuition behind Hoki is illustrated in Figure 1. Suppose we are given two images of Persian cats that are initially correctly classified and suppose that we randomly perturb each image a number of times.

If all perturbed versions of the first image are still correctly classified whereas the second image transformations lead to label switches (e.g., to Siamese cat), then intuitively we would have higher confidence in the first image’s label. More generally, the idea, as shown in Figure 2, is to use logit transformations to test the network’s sensitivity and group examples according to the proportion of observed label switches.

Based on this intuition, we first present a sufficient condition for perfect calibration in terms of ECE through leveraging the information from changes after transformations. An added benefit of this theoretical result is that it also leads to a natural implementation, through minimizing the empirical calibration error (including transformations). The proposed algorithm is also efficient in terms of runtime, especially in the case of logit transformations due to their reduced dimensionality (as compared to the input space dimensionality).

To evaluate Hoki, we perform experiments on MNIST, CIFAR10/100, and ImageNet, and we use several standard models per dataset, including LeNet5, DesneNet, ResNet, and ResNet SD. On these datasets, we compare Hoki with multiple state-of-the-art calibration algorithms, namely temperature scaling [9], MS-ODIR, Dir-ODIR [14], ETS, IRM, IROvA-TS [21], and ReCal [11]. In terms of ECE [20], Hoki outperforms other calibration algorithms on 8 out of 15 benchmarks; we emphasize that Hoki achieves lower ECE than other methods on all ImageNet models. Finally, in terms of learning time, Hoki achieves similar performance to temperature scaling, the fastest algorithm we compared with.

The main contributions of this paper are as follows:

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  we propose Hoki, an iterative calibration algorithm using logit transformations;

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  we provide a sufficient condition for perfect calibration;

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  we show that Hoki outperforms other calibration algorithms on 8 out of 15 benchmarks, and achieves the lowest ECE on all ImageNet models;

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  we demonstrate that Hoki is also efficient in terms of learning time, achieving performance comparable to temperature scaling.

This paper is structured as follows. In Section 2, we summarize the related works, and in Section 3, we describe the problem statement. Then, we present the theory of calibration through transformations in Section 4. In Section 5, we demonstrate the transformation selection process, and we illustrate the calibration algorithm using transformations in Section 6. In Section 7, we show the experimental results, and we conclude this paper in Section 8.

Various approaches have been tried for obtaining accurate confidences, and in this paper, we consider the papers most related to our approach. We review post-hoc calibration techniques, transformation based calibration, and also calibration with a theoretical guarantee.

Post-hoc Calibration. Many researchers have proposed calibration methods for a neural network classifier so that the predicted probabilities match the empirical probabilities using a validation set [10, 12, 13, 9, 14, 15, 17, 21]. These post-hoc approaches learn a (simpler) mapping function for the calibration without re-training the given (complex) classifier, and different functions for the different types of inputs have been proposed for the mapping. For example, temperature scaling [9] uses a linear function on logits, Platt scaling [13] employs an affine function on logits, and Dirichlet calibration [14] trains an affine function on log logits.

Transformation-based calibration. Approaches using input transformations for calibration have also been proposed. Bahat and Shakhnarovich [22] apply semantic preserving transformations such as contrast change, rotation and zoom to augment given inputs and compute confidence using the augmented set of inputs. Jang et al. [11] introduce a lossy label-invariant transformation for calibration. They define a lossy label-invariant transformation and use it to group inputs and apply group-wise temperature scalings. While our method is also based on transformations, the choice of random transformations allows us to obtain a theoretical guarantee for perfect calibration.

Calibration with theoretical guarantee. Kumar et al. [23] propose a scaling-binning calibrator which combines Platt Scaling and histogram binning, and provide a bound on the calibration error. Park et al. [24] propose a calibrated prediction method which provides per-prediction confidence bound using Clopper-Pearson interval based on histogram binning. With this method, examples in the same bin are assigned the same confidence, whereas Hoki assigns a different confidence to each example, while also aiming to achieve the sufficient condition for perfect calibration within the bin.

Let $\mathcal{X}$ be a feature space, $\mathcal{Y}=\{1,\dots,C\}$ be a set of labels, and $\mathcal{D}$ be a distribution over $\mathcal{X}\times\mathcal{Y}$. We are given a classifier $f:\mathcal{X}\to\mathcal{Y}$ and a corresponding calibrator $g:\mathcal{X}\to[0,1]$ such that, for a given example $x$, $(f(x),g(x))=(\hat{y},\hat{p})$ is the label prediction $\hat{y}$ with a corresponding confidence $\hat{p}$. In what follows, we say that the sets $\mathcal{P}_{1},\dots,\mathcal{P}_{J}$ form a bin partition of the confidence space, $[0,1]$, if $\cup_{i=1}^{J}\mathcal{P}_{i}=[0,1]$ and $\forall i\neq j,\;\mathcal{P}_{i}\cap\mathcal{P}_{j}=\emptyset$. Furthermore, given a dataset $Z=\{(x_{1},y_{1}),\dots,(x_{N},y_{N})\}$, we say $g$ induces an index partition $\mathcal{I}_{1},\dots,\mathcal{I}_{J}$ of $\{1,\dots,N\}$ such that $g(x_{n})\in\mathcal{P}_{j}\iff n\in\mathcal{I}_{j},\forall(x_{n},y_{n})\in Z$.

Before formally stating the problem considered in this work, we define calibration error (CE) and expected calibration error (ECE) [20].

Intuitively, the CE of a classifier-calibrator pair in a given partition is the expected difference between the classifier’s accuracy and the calibrator’s confidence. To get the CE over the entire space, we sum up all the individual partition CEs, weighted by the probability mass of each partition (i.e., the probability of an example falling in that partition).

Thus, the ECE is the sampled version of the CE. Note that Definition 2 is equivalent to the standard ECE definition, as used in prior work [9]. We are now ready to state the problem addressed in this work, namely find a calibrator $\hat{g}$ that minimizes the ECE over a validation set.

This section provides the intuition and theory of using transformations for the purpose of calibration. We begin by providing high level intuition, followed by a sufficient condition for perfect calibration in expectation, which leads to a natural implementation as well.

As discussed in Section 4, the choice of transformations greatly affects the benefit of the result presented in Theorem 1. In particular, if a certain transformation results in a label switch for all examples, then it does not provide any useful confidence information. Thus, the most beneficial transformations are those that separate different examples into different partitions, as measured by the proportion of label switches caused by those transformations.

Based on the theory described in Section 4, we propose Hoki, an iterative algorithm for confidence calibration. Hoki operates differently during design time and runtime. During design time, Hoki samples random transformations and learns the $\hat{\alpha}_{j}$ and $\hat{\beta}_{j}$ parameters for each bin. These parameters are then used at runtime, on test data, to estimate the confidence for new examples.

We compare Hoki with state-of-the-art calibration algorithms using several standard datasets and models. For each model and dataset, we compute the ECE for the uncalibrated model and the calibrated confidence by the algorithms. The experimental setup, baseline algorithms and the evaluation metrics are explained in the following subsections.

This work proposed a confidence calibration algorithm based on the intuition that we can partition examples based on the neural network’s sensitivity to transformations. Based on this intuition, we provided a sufficient condition for perfect calibration in terms of ECE. We performed an extensive experimental comparison and demonstrated that Hoki outperforms state-of-the-art approaches in multiple datasets and models, and the benefits are especially pronounced on the challenging ImageNet. For future work, we plan to explore the benefits of combining different transformations, particularly a mix of input and logit transformations. If those transformations are chosen carefully in order to identify input sensitivity, we expect that more accurate calibration is possible.