Improving Classifier Confidence using Lossy Label-Invariant Transformations

Despite the success of machine learning predictions in various applications including image classifications (He et al.,, 2016; Zagoruyko and Komodakis,, 2016; Xie et al.,, 2017), speech recognition (Graves et al.,, 2013; Wang et al.,, 2019), games (Mnih et al.,, 2013; Silver et al.,, 2017), and medical research (Rajpurkar et al.,, 2018), estimating prediction confidence has a different story. As observed in Guo et al., (2017), many modern neural networks are miscalibrated, i.e., they are over-confident in their predictions. As machine learning expands to safety-critical applications such as self-driving cars, autonomous pilots, and autonomous medical systems, accurately estimating confidence becomes imperative for robust decision making.

Consequently, various approaches have been introduced to address the problem of estimating confidence. Bayesian techniques (Gal and Ghahramani,, 2016; Zhang et al.,, 2017; Khan et al.,, 2018; Chang et al.,, 2019) provide a means of computing the posterior distribution of models for estimating confidence, but suffer from computational limitations. Also proposed are techniques that change the original model estimates (Tran et al.,, 2019; Kumar et al.,, 2018; Pereyra et al.,, 2017; Seo et al.,, 2019), but these techniques have the disadvantage that they require re-training the model and do not guarantee the accuracy of the original model. Lastly, there have been many post-hoc approaches proposed that learn a model mapping uncalibrated confidence to calibrated confidence on a comparatively small validation set e.g., temperature scaling, vector scaling(Guo et al.,, 2017; Rahimi et al.,, 2020), using spline (Gupta et al.,, 2020), MS-ODIR and Dir-ODIR (Kull et al.,, 2019), mix-n-match (Zhang et al.,, 2020), and GPcalib (Wenger et al.,, 2020). While these techniques provide improved average confidence calibration, poorly calibrated examples remain.

To address this issue, techniques that utilize redundancy in the example space have been proposed (Bahat and Shakhnarovich,, 2020; Thulasidasan et al.,, 2019; Patel et al.,, 2019). The premise behind these techniques is that utilizing additional information on the current sample, its confidence calibration can be improved. Most of these techniques augment the training dataset with examples on the same manifold and re-train a model (Thulasidasan et al.,, 2019; Patel et al.,, 2019) on the augmented dataset. While other techniques, such as (Bahat and Shakhnarovich,, 2020), avoid retraining by assuming there exist well calibrated examples within the manifold and perform filtering of a sampling of the manifold confidences. While evaluating the confidence of an individual sample remains a challenge – since most benchmark datasets do not contain confidence labels (only class labels) – these techniques do generally show marked improvement in expected confidence calibration consistent with a reduced number of poorly calibrated samples. However, manifold-based techniques have severe shortcomings when applied to datasets with large input spaces. Augmenting the training dataset and retraining a model scales poorly as the input space (and manifold dimensionality) increases. Similarly, filtering sampled confidences assumes that the original classifier is calibrated for a majority of the manifold, which becomes less likely as manifold dimensionality increases. Consequently, manifold-based calibration of models for large input spaces remains a challenge.

In this work, we present Recursive Lossy Label-Invariant Calibration (or ReCal) as a scalable manifold-based post-hoc confidence calibration algorithm that maintains the accuracy of the original classifier and scales to large datasets (e.g. ImageNet). To overcome the scalability issues of other manifold-based techniques, we only consider label-invariant transformations that are expected to result in a decreased confidence due to discriminatory information loss – i.e., lossy label-invariant transformations. For example, consider zooming out an image of a dog with the scale factor of 0.5x as shown in Figure 1. After the transformation, the image still contains the dog, but the dog becomes smaller and harder to recognize. Therefore, we should be able to identify the dog but with less confidence. Considering this intuition in the context of estimating confidence, a (well-calibrated) classifier should return the same prediction with smaller confidence after applying a lossy label-invariant transformation. Likewise, if we group examples based on the prediction and confidence change after such transformations, we expect that the examples in the same group will have similar properties respect to the classifier and confidence estimation. In other words, examples in each group require a similar amount of calibration, which may be different than the calibration needed for examples in other groups. This intuition – lossy label-invariant grouping – forms the premise of ReCal, and is discussed in detail in Section 4.

Leveraging group-wise calibration, we propose ReCal as a scalable post-hoc calibration algorithm in Section 5. Specifically, the proposed algorithm recursively leverages lossy label-invariant transformations to re-group images and perform group-wise calibration. Different from other approaches that aim to retrain a model on the augmented training set  (Patel et al.,, 2019; Thulasidasan et al.,, 2019), our proposed algorithm does not change the predictions and thus retains the original prediction accuracy while adjusting the confidence of the predictions.

We demonstrate the scalability and performance of the proposed algorithm by applying it to ImageNet, and also compare ReCal with other calibration algorithms on CIFAR10/100, ImageNet in Section 6. On multiple models e.g., LeNet5, DenseNet, ReseNet, ResNet SD, and Wide ResNet, on the datasets, we compare Expected Calibration Error (ECE) (Naeini et al.,, 2015), Brier score (Brier,, 1950) and time for learning a calibration map. On the large scale image dataset, ImageNet, ReCal can be applied to the dataset in terms of time, and it outperforms other calibration algorithms such as temperature scaling, vector scaling (Guo et al.,, 2017), MS-ODIR, Dir-ODIR(Kull et al.,, 2019) on DenseNet161 and ResNet152 models. Besides ImageNet, ReCal shows the best performance or the second-best performance for seven of ten models on CIFAR10/100.

The contributions of this paper are summarized as follows:

• •

  introducing lossy label-invariant grouping and empirically demonstrating that each group needs different calibration;

• •

  presenting ReCal, a scalable post-hoc calibration algorithm based on lossy label-invariant transformation, which can be applied to a large-scale datasets;

• •

  evaluating ReCal in comparison to other publicly released post-hoc calibration algorithms using multiple datasets and models.

The remainder of this paper is structured as follows. In the next section, we present the related work on confidence calibration. In Section 3, we present the problem statement considered herein. Section 4 describes lossy label-invariant grouping and its effectiveness. We then propose ReCal in Section 5, present the experimental results in Section 6, and conclusions in Section 7.

While a complete review of all confidence calibration techniques is beyond the scope of this work, in this section we selectively review those techniques most related to the proposed approach. In the following, we consider confidence calibration techniques leveraging Bayesian uncertainty estimation, calibration via re-training, post-hoc calibration maps, and manifold-based calibration.

Bayesian uncertainty estimation. One approach to confidence calibration is to provide uncertainty estimation with Bayesian framework (Gal and Ghahramani,, 2016; Zhang et al.,, 2017; Khan et al.,, 2018; Chang et al.,, 2019). While Baysian techniques can provide very accurate calibration, they suffer from computational limitations associated with estimating the posterior distribution used for uncertainty estimation.

Calibration via re-training Another type of approach targets training a well-calibrated classifier (Kumar et al.,, 2018; Lakshminarayanan et al.,, 2017; Pereyra et al.,, 2017; Seo et al.,, 2019; Tran et al.,, 2019; Müller et al.,, 2019). A potential pitfall of calibration via re-training is that the accuracy of the prediction can change. Moreover, many of these approaches require training sophisticated networks on large training datasets, which may consume significant time and computational resources.

Post-hoc calibration maps. Post-hoc methods address the calibration problem without requiring model retraining. These approaches employ binning methods such as Histogram Binning (Guo et al.,, 2017), Bayesian Binning into Quantiles (Naeini et al.,, 2015), Mutual Information Maximization-based Binning (Patel et al.,, 2020) or train a function mapping from original confidence to calibrated one on validation data which is smaller compared to the training data. For training a mapping function, several techniques have been proposed (Platt et al.,, 1999; Zadrozny and Elkan,, 2002; Guo et al.,, 2017; Rahimi et al.,, 2020; Gupta et al.,, 2020; Kull et al.,, 2019; Zhang et al.,, 2020; Wenger et al.,, 2020). Most notably, Guo et al., (2017), introduces temperature scaling which transforms original logits to calibrated logits with a single parameter. Besides temperature scaling, intra-order preserving function (Rahimi et al.,, 2020), Dirichlet calibration with ODIR regularization (Kull et al.,, 2019), splines (Gupta et al.,, 2020), latent Gaussian function (GPcalib) (Wenger et al.,, 2020), and ETS, IRM, IROvA-TS (Zhang et al.,, 2020) have been proposed. Depending on the mapping function, some of the approaches such as temperature scaling, intra-order preserving function, splines, ETS and IRM preserve the accuracy, while the others like matrix scaling, vector scaling, IROvA-TS, GPcalib and Dirichlet calibration do not preserve the original model accuracy.

Manifold-based calibration. Several manifold-based confidence calibration have been proposed (Bahat and Shakhnarovich,, 2020; Thulasidasan et al.,, 2019; Patel et al.,, 2019; Lee et al.,, 2017; Verma et al.,, 2019). Bahat and Shakhnarovich, (2020) augments test data using transformations to calibrate confidence, while Thulasidasan et al., (2019) and Patel et al., (2019) augment data by interpolating existing data and using an auto-encoder based model, respectively. Other techniques augment the training data with samples from the manifold and retrain the model (Lee et al.,, 2017; Verma et al.,, 2019). Manifold-based algorithms can improve worst-case calibration errors as shown by their ability to address over-confident prediction on out-of-distribution samples. However, they generally suffer from scalability issues as discussed in Section 1.

In this paper, we aim to develop a post-hoc calibration algorithm which addresses the worst-case confidence error that does not change accuracy on a multi-class classification task. Consider a multi-class classification task on data, $D=\{(x_{n},y_{n})\}^{N}\sim\mathcal{X}\times\mathcal{Y}$, where $\mathcal{X}$ is an input space and $\mathcal{Y}$ is a label set, $\{1,2,\dots,K\}$. Let $f:\mathcal{X}\to\mathbb{R}^{K}$ denote a multi-class classifier. A neural network classifier typically has a softmax output layer as a final layer, which returns a vector $z$ for the given input $x$. Here, $z=f(x)=\{z_{1},z_{2},\dots,z_{k}\}$. Each $z_{i}$ is the estimated probability of a label $i$, and the classifier chooses the label whose probability is the maximum. Consequently, the prediction $\hat{y}=\text{argmax}_{i}\{z\}$ has confidence $p=\text{max}_{i}\{z\}$.

A classifier $f$ is calibrated if confidence is equal to accuracy given the confidence. More formally,

where, $z=f(x),k=\text{argmax}_{i}\{z\}$ for all $(x,y)\in D$ and for all $p\in[0,1]$. Here, the difference between the both sides is defined as calibration error. This error is estimated by Expected Calibration Error (ECE) (Naeini et al.,, 2015) which is the weighted average of the differences over bins. ECE is computed by first splitting the confidence range with equal size bins, and calculating the difference between average confidence and average accuracy of each bins, and finally, computing an average of those differences weighted by the number of samples in each bins. More formally,

where, $N$ is the number of bins, $B_{1},\dots,B_{N}$ are bins which equally divides the interval [0, 1], and $accuracy(B_{i})$ is the average accuracy of examples in bin $B_{i}$, and $confidence(B_{i})$ is the average confidence of examples in bin $B_{i}$.

Ideally, we would like to minimize the worst-case confidence error, however, this is impossible to quantify with the current datasets that lack ground truth calibration values for the labels. As a surrogate, and consistent with other works in the literature, we rather aim to minimize ECE. Therefore, we would like to learn a calibration map which transforms the original confidence (or logits) to calibrated one which minimize ECE without affecting original accuracy.

It is reasonable to assume that different examples may need different level of calibration, i.e., some examples require more calibration than others. Consequently, we would like to group inputs based on some measure of calibration needed so that we can apply different level of calibration to each group. In other words, we would like to apply more calibration when predictions are very mis-calibrated, and calibrate less when predictions near calibration.

We utilize a subset of transformations that do not change the label called label-invariant transformations. Specifically, we choose label-invariant transformations which induce a loss in discriminatory information – i.e., lossy label-invariant transformations. As an example, consider an image classification task. The zoom-out transformation and brightness transformation are the examples of lossy label-invariant transformations. These two transformation do not change label, but reduce discriminatory information by making an image smaller or darker. Therefore, after such transformations, a (well-calibrated) classifier should not change its prediction but should become less confident on its prediction.

Our approach, as described in Algorithm 1, begins by applying a lossy label-invariant transformation to the inputs, and group based on the observed prediction and confidence changes after the transformation. There can be two possible outcomes to each observation, i.e., prediction change vs not change, and confidence increase vs. not increase, and in total there can be four possible combinations as shown in Table 1. We perform lossy label-invariant grouping by comparing the prediction and confidence of the transformed input with the original input, and group based on the comparison result. More formally, group number $k$ for an input is

where, $\hat{y},\hat{p}$ are the prediction and confidence for the original input, $\hat{y}_{t},\hat{p}_{t}$ are the prediction and confidence for the transformed input, and $\mathbbm{1}_{(\cdot)}$ is 1 if $(\cdot)$ is true, and 0, otherwise.

To demonstrate the effectiveness of our Lossy Label-Invariant Grouping Algorithm we consider an image classification task using ResNet152 model on ImageNet. We choose two image transformations, zoom-out and brightness. These two transformations have one parameter which determines how much transformation will be applied. A zoom-out transformation with smaller parameter value will return the smaller image and a brightness transformation with smaller parameter value will yield the darker image.

We randomly select parameter values between 0.1 and 0.9, we observe label prediction change and confidence change for validation images, and group images into four different groups as shown in Table 1. For each parameter, we compute and rank the ECE values among four groups, and draw the distribution of the ranks for each transformation as shown in Figure 2 . For the reference, ECE values and number of images of each group for the two transformations are displayed in the supplementary material.

As shown in Figure 2 (Left), with zoom-out transformation, for about 80% of parameters, Group 4 which represents that prediction does not change and confidence does not increase, has the best ECE, i.e., rank one. On the other hand, Group 1 which corresponds to the case that prediction changes and confidence increases shows the worst ECE, i.e., rank four. The similar pattern appears on the brightness transformation as shown in Figure 2 (Right). With brightness transformation, Group 4 has the best ECE for about 80% of parameters, and Group 1 has the worst ECE for about 50% of parameters. Group 2 and Group 3 show a little different pattern between two transformations. For zoom-out transformation, Group 2 has better rank than Group 3 with about 80% of each group is either rank 2 or 3. On the other hand, for brightness transformation, Group 2 has worse rank than Group 3. It is hard to decide which one requires more calibration, but in general, these two groups should be calibrated differently.

These results empirically demonstrate that lossy label-invariant grouping partitions the inputs into groups that require different amounts of calibration. Group 4 inputs which match our intuition tend to have the best ECE, i.e., requires the least calibration, while Group 1 inputs which opposite to our intuition show the worst ECE, i.e., requires the most calibration. Furthermore, input grouping differs depending on the transformation, as shown by the input distribution over groups for different transformations in the supplementary Tables. Consequently, in the following section, we design an algorithm that utilizes different lossy label-invariant transformations at each iteration to perturb the groupings and perform recursive calibration.

As illustrated in Figure 3 and Algorithm 2, ReCal consists of 3 steps: initialization, iterative group-wise calibration, final calibration. In the following we describe each of these steps in detail. We conclude this section by presenting a runtime implementation of ReCal and a discussion of its limitations.

We apply ReCal to multiple models on three datasets to compare its calibration performance and scalability. In detail, we train or obtain models for each dataset, and calibrate confidence using ReCal and other baselines. We then compare the calibration performance using two metrics and the time for learning a calibration map to evaluate the scalability. The details of datasets, model, baselines, metrics and results are described in the following subsections.

In this paper, we propose an accuracy preserving post-hoc calibration method based on a label-invariant image transformation. ReCal exploits the properties of label-invariant transformations to group inputs, and applies different temperature scaling to each group. Because ReCal is based on temperature scaling, it preserves the original classifier accuracy. In addition, it has more expressiveness compared to original temperature scaling because it uses multiple temperature scaling coefficients. Experiments on CIFAR10/100 and ImageNet datasets show that ReCal can be applied to the large-scale ImageNet, and outperforms other methods on those dataset including ImageNet.

For the future work, incorporating multiple types of transformation type may improve the calibration performance. In this paper, we use one type of transformation at a time, but, different transformation utilizes different information in example space, and combination of multiple transformation type may results in improvements. Additionally, ReCal can be extended to other types of dataset as long as appropriate transformation exists. For example, we can apply ReCal to time-series data classification. We can consider a transformation of eliminating some data at random time point. This transformation is a label-invariant transformation which decrease confidence, and ReCal can be applied to calibrate confidence.

This work was supported by the Air Force Research Laboratory and the Defense Advanced Research Projects Agency under Contract No. FA8750-18-C-0090, and by the Army Research Office under Grant Number W911NF-20-1-0080. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the Air Force Research Laboratory (AFRL), the Army Research Office (ARO), the Defense Advanced Research Projects Agency (DARPA), or the Department of Defense, or the United States Government.

Improving Classifier Confidence using Lossy Label-Invariant Transformations:
Supplementary Materials

Brier score (Brier,, 1950) is defined as Equation 5.

where, $N$ is the dataset size, $K$ is the number of classes, $p_{ik}$ is the confidence for the label k of the $i^{th}$ data, and $y_{ik}$ is 1 if the true label for $i^{th}$ data is k otherwise 0. Here, we normalized Brier score by dividing it by the number of classes as used in Kull et al., (2019). The formal definition of the normalized Brier score is shown in Equation 6.

In this section, we display more results for Section 4 and Section 6. For Section 4, we present the detail ECE and number of images of each group for each transformation type and parameter, and for Section 6, we show the test error rate and learning time of a calibration map.