Soft Labeling Affects Out-of-Distribution Detection of Deep Neural Networks

Outlier exposure (Hendrycks et al., 2019) finetunes a model with some OOD samples to predict uniform distribution for the OOD training samples.

where $p_{i}$ is a prediction of a ID sample, $p_{o}$ is a prediction of a OOD sample, $q$ is an one-hot represented ground truth, $\lambda$ is a hyper-parameter, $\mathcal{H}$ is cross-entropy, and $\mathcal{U}(K)$ is uniform distribution over all $K$ classes.

Despite a significant improvement of OOD detection, training additional OOD samples has two drawbacks. First, original test accuracy is often degraded after outlier exposure as a trade-off between OOD detection and the original task. Second, we cannot consider all possible OOD samples in training, because there are infinitely many OOD samples.

In this study, soft label prevents the degradation of classification accuracy and often improves the test accuracy (Table 1). In addition, we show that a soft label can make DNNs robust to OOD without any OOD training sample and model modification (Figure 2).

Given an one-hot represented ground truth $q$ of a training sample $x$, soft labeling is defined as

where $\alpha$ is a hyper-parameter for soft labeling, $q^{\prime}\in[0,1]^{K}$ is a soft target that satisfies $\mathrm{argmax}{(\tilde{q})}=\mathrm{argmax}{(q)}$ and $\sum_{i=1}^{K}{\tilde{q}_{i}}=1$, and $K$ is the number of classes. Then, the training loss with soft labeling is

Note that a soft labeling is a regularization of the predictions including incorrect classes.

The training objective of both label smoothing (Szegedy et al., 2016) and knowledge distillation (Hinton et al., 2015) are represented by Eq (3) (Yuan et al., 2019). Label smoothing (Szegedy et al., 2016) is a soft labeling that regularizes DNNs to predict an uniform distribution $\mathcal{U}(K)$ over all $K$ classes:

In knowledge distillation, a soft target of a student model $\tilde{q}$ consists of a prediction of its teacher model:

where $p_{t}$ is the prediction of the teacher model. Knowledge distillation is a kind of output regularization of student models by the teacher’s predictions (Yuan et al., 2019) for model compression (Hinton et al., 2015) or generalization (Xie et al., 2019).

In this paper, we train WRN-40-2 (Zagoruyko & Komodakis, 2016) with the SVHN, CIFAR-10, and CIFAR-100 datasets (ID). We follow the experimental setting in the official code of outlier exposure¹¹1https://github.com/hendrycks/outlier-exposure except that we use 150 epochs for training. In addition, we follow the hyper-parameter settings of knowledge distillation in (Müller et al., 2019). For evaluation of OOD detection, we use the MNIST, Fashion-MNIST, SVHN (or CIFAR-10), LSUN, and TinyImageNet datasets for OOD samples, and AUROC for the evaluation measure.

Figure 1 shows the effects of label smoothing with different $\alpha$ on test accuracy, expected calibration error (ECE) (Guo et al., 2017), and detection of the OOD datasets. As shown in (Lukasik et al., 2020), ECE starts to increase when the label smoothing $\alpha$ is larger than the optimal values (red dot lines). Although the test accuracy on CIFAR-10 with $\alpha=0.001,0.1$ deteriorates, the test accuracy is always improved when ECE is minimized (Müller et al., 2019).

Even though label smoothing improves test accuracy and ECE (Müller et al., 2019), label smoothing makes DNNs vulnerable to out-of-distribution and disable to distinguish ID and OOD datasets. Label smoothing always deteriorates OOD detection regardless of the magnitude of $\alpha$, and larger $\alpha$ results in more degradation of OOD detection. In particular, WRN models, trained with SVHN and CIFAR-10, show significant AUROC drops, when the ECE starts to increase (after the red dot line).

We can infer the reason why label smoothing hurts OOD detection of DNNs from two perspectives. First, combining Eq (1) and (4), we can interpret the output regularization of label smoothing as an outlier exposure of ID samples. Then, label smoothing can deteriorate OOD detection of DNNs, making DNNs disable to discriminate OOD samples from ID samples. When the magnitude of $\alpha$ increases, the effect of output regularization in Eq (4) increases and deteriorates the OOD detection performance as outlier exposure of the ID datasets.

Meanwhile, knowledge distillation is the other view to interpret the negative effect of label smoothing on OOD detection. Note that label smoothing is a knowledge distillation with a teacher model, which perfectly learns the ID samples as OOD and predicts uniform distribution for all ID samples. Thus, we assume that Soft labels of incorrect classes, generated by a teacher model, determines OOD detection performance of its student model, and empirically verify the assumption in section 3.2.

In this section, we show that OOD detection performance is determined by the soft labels. Specifically, soft labels that are generated by a teacher model determine the performance of its student model. Figure 2 shows OOD detection performance of teacher models and their student models in various settings. For a student model, we use the same architecture (WRN-40-2) with its teacher, because our concern is to analyze the effects of soft labeling, not a model compression.

Figure 2 (top) shows OOD detection AUROCs of the WRN-40-2 models (SVHN, CIFAR-10, and CIFAR-100), and their student models. The teacher and its student model have similar AUROCs regardless of test datasets (OOD).

In Figure 2 (bottom), we finetune the teacher models with various OOD samples (MNIST, TinyImageNet, and MNIST+TinyImageNet) to improve OOD detection by outlier exposure. The OOD detection of the teacher models is improved in different OOD datasets, according to the exposed OOD samples. We find that OOD detection performance of student models is always consistent with their teacher models (OE), regardless of the choice of training OOD samples for the teacher.

Especially, when we use MNIST+TinyImageNet for outlier exposure of the teacher model, both the teacher and its student almost perfectly detect the test OOD samples. Exposing various OOD samples in training time is an unrealistic setting, because there are infinitely many cases of OOD. However, the experimental results is worth noting, because the student model is trained only using ID samples with soft labels, and any OOD sample is not directly used to train the student model. Although how to generate soft labels without a perfectly OOD-robust teacher remains in an open question, the result show the existence of soft labeling for OOD-robust DNNs to various OOD datasets without OOD training.

OOD detection performance is also distilled into a student model that has a different architecture from its teacher model. In Table 2, we use DenseNet (Huang et al., 2017) with 40 hidden layers and 12 growth rates as the student model of WRN-40-2. Note that the number of trainable parameters of DenseNet (1.1 M) is twice less than WRN-40-2 (2.2 M). Even though the size and model architecture of the student are different from those of its teacher, OOD detection AUROCs of the teacher and student are consistent. The results imply that the effect of soft labeling on OOD detection is model-agnostic. Then, if we find a soft labeling method for OOD-robust DNNs, the soft labeling can be generally used for various DNN architectures.

Orthogonal to the OOD detection, one disadvantage of post-training with OOD samples is a degradation of original classification accuracy (Hendrycks et al., 2019; Hein et al., 2019). However, we find that both test accuracy and ECE of the student models (+OD) are similar to or better than the original model before outlier exposure (baseline) in (Table 1). The improvement of test accuracy results from soft labeling, because soft labels can help model prevent an overfitting problem regardless of the type of soft labeling (Yuan et al., 2019).