Scaling for Training Time and Post-hoc Out-of-distribution Detection Enhancement

While OOD is relevant for many domains, we follow previous works (Yang et al., 2022) and focus specifically on semantic shifts in image classification. During training, the classification model is trained with ID data that fall into a pre-defined set of $K$ semantic categories: $\forall({\bm{x}},y)\sim\mathcal{D}_{\text{ID}},y\in\mathcal{Y}_{\text{ID}}$. During inference, there are both ID and OOD samples; the latter are samples drawn from categories unobserved during training, i.e. $\forall({\bm{x}},y)\sim\mathcal{D}_{\text{OOD}},y\notin\mathcal{Y}_{\text{ID}}$.

Now consider a neural network consisting of two parts: a feature extractor $f(\cdot)$, and a linear classifier parameterized by weight matrix $\mathbf{W}\in\mathbb{R}^{K\times D}$ and a bias vector ${\bm{b}}\in\mathbb{R}^{D}$. The network logit can be mathematically represented as

where ${\bm{a}}\in\mathbb{R}^{D}$ is the $D$-dimensional feature vector in the penultimate layer of the network and ${\bm{z}}\in\mathbb{R}^{K}$ is the logit vector from which the class label can be estimated by $\hat{y}=\operatorname*{arg\,max}({\bm{z}})$. In line with other OOD literature (Sun et al., 2021), an individual dimension of feature ${\bm{a}}$, denoted with index $j$ as ${\bm{a}}_{j}$, is referred to as an “activation”.

For a given test sample ${\bm{x}}$, an OOD score can be calculated to indicate the confidence that ${\bm{x}}$ is in-distribution. By convention, scores above a threshold $\tau$ are ID, while those equal or below are considered OOD. A common setting is the energy-based OOD score $S_{\textit{EBO}}({\bm{x}})$ together with indicator function $G(\cdot)$ that applies the thresholding (Liu et al., 2020):

where $T$ is a temperature parameter, $k$ is the logit index for the $K$ classes.

A state-of-the-art method for OOD detection is ASH (Djurisic et al., 2023). ASH stands for activation shaping and is a simple post-hoc method that applies a rectified scaling to the feature vector ${\bm{a}}$. Activations in ${\bm{a}}$ up to the $p^{\text{th}}$ percentile across the $D$ dimensions are rectified (“pruned” in the original text); activations above the $p^{\text{th}}$ percentile are scaled. More formally, ASH introduces a shaping function $s_{f}$ that is applied to each activation ${\bm{a}}_{j}$ in a given sample. If we define $P_{p}({\bm{a}})$ as the $p^{\text{th}}$ percentile of the elements in ${\bm{a}}$, ASH produces the logit ${\bm{z}}_{\text{ASH}}$:

and $\circ$ denotes an element-wise matrix multiplication, and the scaling factor $r$ is defined as the ratio of the sum of all activations versus the sum of un-pruned activations in ${\bm{a}}$:

Since $Q_{p}\leq Q$, the factor $r\geq 1$; the higher the percentile $p$, i.e. the greater the extent of pruning, the smaller $Q_{p}$ is with respect to $Q$ and the larger the scaling factor $r$. To distinguish OOD data, ASH then passes the logit from Eq. 3 to score and indicator function as given in Eq. 2.

While ASH is highly effective, the original paper has no explanation of the working mechanism¹¹1In fact, the authors put forth a call for explanation in their Appendix L.. We analyze the rectification and scaling components of ASH below and reveal that scaling helps to separate ID versus OOD energy scores, while rectification has an adverse effect.

Assumptions: Our analysis is based on two assumptions. (1) The penultimate activations of ID and OOD samples follow two differing rectified Gaussian distributions parameterized by $(\mu^{\text{ID}},\sigma^{\text{ID}})$ and $(\mu^{\text{OOD}},\sigma^{\text{OOD}})$. The Gaussian assumption is commonly used in the literature (Sun et al., 2021)and we verify it in Tab. 1; the rectification follows naturally if a ReLU is applied as the final operation of the penultimate layer. (2) Normalized ID activations are higher than that of OOD activations; this assumption is supported by (Liu et al., 2020) , who suggested that well-trained networks have higher responses to samples resembling those seen in training. Fig. 2 and Fig. 3 visualize statistical corroboration of these assumptions.

The full proof is given in Appendix A. Above, $\varphi$ and $\Phi$ denote the probability density function and cumulative distribution function of the standard normal distribution, respectively. The factor $C(p)$, plotted in Fig. 4a, relates the percentile of activations that distinguishes ID from OOD data.

Rectification (Pruning) The relative reduction of activations can be expressed as:

Note that a reduction in activations also leads to a reduction in the OOD energy. Since ${Q_{p}^{\textit{ID}}}/{Q^{\textit{ID}}}<{Q_{p}^{\textit{OOD}}}/{Q^{\textit{OOD}}}$, it directly implies that the decrease in ID samples will be greater than that in OOD samples, denoted as $D^{\textit{Pruning}}_{\textit{ID}}>D^{\textit{Pruning}}_{\textit{OOD}}$. From this result, we can show that the expected value of the relative decrease in energy scores with rectification will be greater for ID samples than OOD samples following the Remark 2 in Sun et al. (2021), which illustrates that the changes in logits is proportional to the changes in activations.

Our result above shows that rectification or pruning creates a greater overlap in energy scores between ID and OOD samples, making it more difficult to distinguish them. Empirically, this result is shown in Fig. 4b, where AUROC steadily decreases with stand-alone pruning as the percentile $p$ increase.

Scaling on the other hand behaves in a manner opposite to the derivation above and enlarges the separation between ID and OOD scores.

Given ${Q_{p}^{\textit{ID}}}/{Q^{\textit{ID}}}<{Q_{p}^{\textit{OOD}}}/{Q^{\textit{OOD}}}$ and $r=Q/Q_{q}$, we have $r^{\textit{ID}}>r^{\textit{OOD}}$, which motivates the separation on $r$ between ID and OOD, Fig. 4c depicts the histograms for these respective distributions, they are well separated and therefore scale activations of ID and OOD samples differently. The relative increase on activation can be expressed as:

where we can get $I^{\text{scaling}}_{\text{ID}}>I^{\text{scaling}}_{\text{OOD}}$. This increase is then transferred to logit spaces ${\bm{z}}$ and energy-based scores $S_{\textit{EBO(ID)}}$ and $S_{\textit{EBO(OOD)}}$, which increase the gap between ID and OOD samples.

Discussion on percentile $p$:  Note that $C(p)$ does not monotonically increasing with respect to $p$ (see Fig. 4a). When $p\approx 0.95$, there is an inflection point and $C(p)$ decreases. A similar inflection follows on the AUROC for scaling (see Fig. 4b), though it is not exactly aligned to $C(p)$. The difference is likely due to the approximations made to estimate $C(p)$. Also, as $p$ gets progressively larger, fewer activations ($D=2048$ total activations) are considered for estimating $r$, leading to unreliable logits for the energy score. Curiously, pruning also drops off, which we believe to come similarly from the extreme reduction in activations.

From our analyses and findings above, we propose a new post-hoc model enhancement criterion, which we call SCALE. As the name suggests, it shapes the activation with (only) a scaling:

Fig. 5a illustrates how SCALE works. SCALE applies the same scaling factor $r$ as ASH, based on percentile $p$. Instead of pruning, it retains and scales all the activations. Doing so has two benefits. First, it enhances the separation in energy scores between ID and OOD samples. Secondly, scaling all activations equally preserve the ordinality of the logits ${\bm{z}}^{\prime}$ compared to ${\bm{z}}$. As such, the $\arg\max$ is not affected and there is no trade-off for ID accuracy; this is not the case with rectification, be it pruning, like in ASH or clipping, or like ReAct (see Fig. 1). Results in Tab. 2 and 3 verify that SCALE outperform ASH-S on all datasets and model architectures.

In practice, the semantic shift of ID versus OOD data may be ambiguous. For example, iNaturalist dataset features different species of plants; similar objects may be found in ImageNet. Our hypothesis is that, during training, we can emphasize the impact of samples possessing the most distinctive in-distribution characteristics, denoted as ”ID-ness”. Quantifying the ID-ness of specific samples is a challenging task, so we rely on a well-trained network to assist us in this endeavor. In particular, for a well-trained network, we can reacquire the activations of all training samples. We proceed on the assumption that the normalized ID activations are greater than those of out-of-distribution (OOD) activations. To measure the degree of ID-ness within the training data, we compute their scale factor, represented as ${Q}/{Q_{p}}$. Armed with this measurement of ID-ness, we can then undertake the process of re-optimizing the network using the high ID-ness data. Our approach draws inspiration from the concept of intermediate tensor compression found in memory-efficient training methods (Chen et al., 2023a), where modifications are exclusively applied to the backward pass, leaving the forward pass unchanged.

Fig. 5b illustrates our training time enhancement methods for OOD detection. We finetune a well-trained network, by introducing a modification to the gradient of the weights of the fully connected layer. The modified gradient is defined as follows:

where $i$ denotes sample index in the batch, $\nabla$ denotes the gradient regarding to the cross entropy loss, $t$ denotes the training step $t$, and $\eta$ represents the learning rate.

Modifying activations exclusively in the backward pass offers several advantages. Firstly, it leaves the forward pass unaffected, resulting in only a minimal loss in ID accuracy. Secondly, the model architecture remains exactly the same during inference, making this training strategy compatible with any OOD post-processing techniques. Since the saved activations in the backward pass are also referred to as intermediate tensors, we term this method as Intermediate tensor SHaping (ISH).

To verify SCALE as a post-hoc OOD method, we conduct experiments using CIFAR10, CIFAR100 (Krizhevsky, 2009), and ImageNet-1k (Deng et al., 2009) as in-distribution (ID) data sources.

CIFAR. We used SVHN (Netzer et al., 2011), LSUN-Crop (Yu et al., 2015), LSUN-Resize (Yu et al., 2015), iSUN (Xu et al., 2015), Places365 (Zhou et al., 2018), and Textures (Cimpoi et al., 2014) as OOD datasets, For consistency with previous work, we use the same model architecture and pretrained weights, namely, DenseNet-101 (Huang et al., 2017), in accordance with the other post-hoc approaches DICE, ReAct, and ASH. Table 3 compares the FPR@95 and AUROC averaged across all six datasets; detailed results are provided in Appendix B.

ImageNet. In our ImageNet experiments, we follow the OpenOOD v1.5 (Zhang et al., 2023) benchmark, which separates OOD datasets as near-OOD and far-OOD groups. We employed SSB-hard (Vaze et al., 2022) and NINCO (Bitterwolf et al., 2023) as near-OOD datasets and iNaturalist (Horn et al., 2018), Textures (Cimpoi et al., 2014), and OpenImage-O (Wang et al., 2022) as far-OOD datasets. Our reported metrics are the average FPR@95 and AUROC values across these categories; detailed results are given in Appendix B. The OpenOOD benchmark includes improved hyperparameter selection with a dedicated OOD validation set to prevent overfitting to the testing set. Additionally, we provide results following the same dataset and test/validation split settings as ASH and ReAct in the appendix. We adopted the ResNet50 (He et al., 2016) model architecture and obtained the pretrained network from the torchvision library.

Metrics. We evaluate with two measures. The first is FPR@95, which measures the false positive rate at a fixed true positive rate of 95%; lower scores are better). The second is AUROC (Area under the ROC curve). It represents the probability that a positive in-distribution (ID) sample will have a higher detection score than a negative out-of-distribution (OOD) sample; higher scores indicate superior discrimination.

Comparison of ODD score methods and post-hoc model enhancement methods (separated with a solid line) on the ImageNet and CIFAR are illustrated in the Table 2 and 3. Notably, SCALE attains the highest OOD detection scores.

OOD Detection Accuracy. Compared to the current state-of-the-art ASH-S, SCALE demonstrates significant improvements on ImageNet – 1.73 on Near-OOD and 0.26 on far-OOD when considering AUROC. For FPR@95, it outperforms ASH-S by 2.27 and 0.33. On CIFAR10 and CIFAR100, SCALE has even greater improvements of 2.48 and 2.41 for FPR@95, as well as 0.66 and 0.72 for AUROC, respectively.

ID Accuracy. One of SCALE’s key advantages is it only applies linear transformations on features, so ID accuracy is guaranteed to stay the same. This differentiates it from other post-hoc enhancement methods that rectify or prune activations, thereby modifying inference and invariably compromises the ID accuracy. SCALE’s performance surpasses ASH-S by a substantial margin of 0.67 on the ID dataset, ImageNet-1k. This capability is pivotal for establishing a unified pipeline that excels for ID and OOD.

Comparison with TempScale. Temperature scaling (TempScale) is widely used for confidence calibration (Guo et al., 2017). SCALE and TempScale both leverage scaling for OOD detection, but with two distinctions. Firstly, TempScale directly scales logits for calibration, whereas SCALE applies scaling at the penultimate layer. Secondly, TempScale employs a uniform scaling factor for all samples, whereas SCALE applies a sample-specific scaling factor based on the sample’s activation statistics. The sample-specific scaling is a crucial differentiator that enables the discrimination between ID and OOD samples. Notably, our SCALE model significantly outperforms TempScale in both Near-OOD and Far-OOD scenarios.

SCALE with different percentiles $p$. Table 2 uses $p=0.85$ for SCALE and ASH-S, which is verified on the validation set. As detailed in Section 3.2, in order to ensure the validity of scaling, it is essential for the percentile value $p$ to fall within a specific range where the parameter $C(p)$ exhibits a sufficiently high value to meet the required condition. Our experimental observations align with this theoretical premise. Specifically, we have empirically observed that, up to the 85% percentile threshold, the AUROC values for both Near-OOD and Far-OOD scenarios consistently show an upward trend. However, a noticeable decline becomes apparent beyond this percentile threshold. This empirical finding corroborates our theoretical insight, indicating that the parameter $C(p)$ experiences a reduction in magnitude as $p$ approaches the 90%.

We used the same dataset splits as the post-hoc experiments in Sec. 4.1. For training, we fine-tuned the torchvision pretrained model with ISH for 10 epochs with a cosine annealing learning rate schedule initiated at 0.003 and a minimum of 0. We additionally observed that using a smaller weight decay value (5e-6) enhances OOD detection performance. The results are presented in Table 5. We compare ISH with other training time model enhancement methods.

Comparison with OOD traning methods.

The work LogitNorm(Wei et al., 2022) focuses on diagnosing the gradual narrowing of the gap between the logit magnitudes of ID and OOD distributions during later stages of training. Their proposed approach involves normalizing logits, and the scaling factor is applied within the logits space during the backward pass.

The key distinction between their LogitNorm method and our ISH approach lies in the purpose of scaling. LogitNorm scales logits primarily for confidence calibration, aiming to align the model’s confidence with the reliability of its predictions. In contrast, ISH scales activations to prioritize weighted optimization, emphasizing the impact of high ID-ness data on the fine-tuning process.

Comparisons with data augmentation-based methods.  Zhang et al. (2023) indicates that data augmentation methods, while not originally designed for OOD detection improvement, can simultaneously enhance both ID and OOD accuracy.

In comparison to AugMix and RegMixup, our ISH approach, while slightly reducing ID accuracy, delivers superior OOD performance with significantly fewer computational resources. When compared to AugMix, ISH achieves substantial improvements, enhancing AUROC by 0.46 and 0.8 for Near-OOD and Far-OOD, respectively, with just 0.1x the extended training epochs. Notably, ISH sets the highest AUROC records, reaching 84.01% on Near-OOD scores and 96.79% on Far-OOD scores among all methods on OpenOODv1.5 benchmark.