# Unsupervised Layer-wise Score Aggregation for Textual OOD Detection

Maxime Darrin Guillaume Staerman Eduardo D. C. Gomes Jackie Chi Kit Cheung
Pablo Piantanida Pierre Colombo

###### Abstract

Out-of-distribution (OOD) detection is a rapidly growing field due to new
robustness and security requirements driven by an increased number of AI-based
systems. Existing OOD textual detectors often rely on anomaly scores (e.g.,
Mahalanobis distance) computed on the embedding output of the last layer of
the encoder. In this work, we observe that OOD detection performance varies
greatly depending on the task and layer output. More importantly, we show that
the usual choice (the last layer) is rarely the best one for OOD detection and
that far better results can be achieved provided that an oracle selects the
best layer. To leverage this observation, we propose a data-driven,
unsupervised method to combine layer-wise anomaly scores. In addition, we
extend classical textual OOD benchmarks by including classification tasks with
a greater number of classes (up to 77), which reflects more realistic
settings. On this augmented benchmark, we show that the proposed post-
aggregation methods achieve robust and consistent results comparable to using
the best layer according to an oracle while removing manual feature selection
altogether.

Machine Learning, ICML

##  1 Introduction

With the increasing deployment of ML tools and systems, the issue of their
safety and robustness is becoming more and more critical. Out-of-distribution
robustness and detection have emerged as an important research direction (Yang
et al., 2021; Liu et al., 2020; Winkens et al., 2020a; Hendrycks & Gimpel,
2016; Kirichenko et al., 2020; Liang et al., 2017; Vyas et al., 2018; Ren et
al., 2019; Serrà et al., 2019; Hendrycks et al., 2021; McAllister et al.,
2019). These OOD samples can cause the deployed AI system to fail as neural
models rely heavily on previously seen concepts or patterns (Jakubovitz et
al., 2019) and tend to struggle with anomalous samples (Berend et al., 2020;
Bulusu et al., 2020) or new concepts. These failures can affect user
confidence, or even rule out the adoption of AI in critical applications.

Distinguishing OOD samples (OUT) from in-distribution (IN) samples is a
challenge when working on complex data structures (e.g., text or image) due to
their high dimensionality. Although OOD detection has attracted much attention
in computer vision (Huang et al., ; Wang et al., ; Fang et al., 2022), few
studies focused on textual data. Furthermore, distortion and perturbation
methods for sensitivity analysis used in computer vision are not suitable due
to the discrete nature of text (Lee et al., 2022; Schwinn et al., 2021).

A fruitful line of research (Lee et al., 2018; Liang et al., 2018; Esmaeilpour
et al., 2022; Xu et al., 2020; Huang et al., 2020) focuses on filtering
methods to be added on top of pre-trained models without requiring retraining
the model. These methods are often easy to deploy in a real-world scenario and
thus lend themselves well to broad adoption. They include plug-in detectors
that rely on softmax-based- or hidden-layer-based- confidence scores (Lee et
al., 2018; Liang et al., 2018; Esmaeilpour et al., 2022; Xu et al., 2020;
Huang et al., 2020). Softmax-based detectors (Liu et al., 2020; Pearce et al.,
2021; Techapanurak et al., 2019) rely on the predicted probabilities to decide
whether a sample is OOD. In contrast, hidden-layer-based scores (e.g., cosine
similarity, data-depth (Colombo et al., 2022a), or Mahalanobis distance (Lee
et al., 2018)) rely on input embedding of the model encoder. In natural
language, these methods arbitrarily rely on either the embedding generated by
the last layer of encoder (Podolskiy et al., 2021) or on the logits (Wang et
al., 2022; Khalid et al., 2022) to compute anomaly scores. While Softmax-based
detectors can be applied in black-box scenarios, where one has only access to
the model’s output, they have a very narrow view of the model’s behavior. In
contrast, hidden-layer-based methods enable one to get deeper insights. They
tend to yield better performance at the cost of memory and compute overhead.

We argue that the choice of the penultimate layers (i.e., the last layer, or
the logits) ignores the multi-layer nature of the encoder and should be
questioned. We give evidence that these representations are (i) not always the
best choices (see Fig. 1) and (ii) that leveraging information from all layers
can be beneficial. We introduce a data-driven procedure to exploit the
information extracted from existing OOD scores across all the different layers
of the encoder.

Our contribution can be summarized as follows:

1.We introduce a new paradigm. One of the main weaknesses of the previous
methods is that they rely on a manual selection of the layer to be used, which
ignores the information in the other layers of the encoder. We propose an
automatic approach to aggregate information from all hidden layers without
human (supervised) intervention. Our method, does not require access to OOD
samples and harnesses information available in all layers of the model by
leveraging principled anomaly detection tools.

2.We conduct extensive experiments on our newly proposed benchmark: We
introduce MILTOOD-C A MultI Lingual Text OOD detection benchmark for
Classification tasks. MILTOOD-C alleviates two main limitations of previous
works: (i) contrary to previous work that relies on datasets involving a
limited number of classes (up to $5$), MILTOOD-C includes datasets with a
higher number of classes (up to $77$ classes); (ii) MILTOOD-C goes beyond the
English-centric setting and includes French, Spanish, and German datasets. Our
experiments involve four models and over $186$ pairs of IN and OUT datasets,
which show that our new aggregation procedures achieve high performance. At
the same time, previous methods tend to suffer a drop in performance in these
more realistic scenarios.

3\. Open-Science & Open-source code. We will publish our code and benchmark in
the _datasets_ library (Lhoest et al., 2021) to ensure reproducibility and
reduce computational costs.

##  2 OOD detection for text classification

###  2.1 Background and notations

We adopt a text classification setting and rely on the encoder section of a
model. Let $\Omega$ be a vocabulary and $\Omega^{*}$ its Kleene closure111The
Kleene closure corresponds to sequences of arbitrary size written with words
in $\Omega$. Formally:
$\Omega^{*}=\overset{\infty}{\underset{i=0}{\bigcup}}\Omega^{i}$.. We consider
$(X,Y)$ a random variable with values in $\mathcal{X}\times\mathcal{Y}$ such
that $\mathcal{X}\subseteq\Omega^{*}$ is the textual input space, and $P_{XY}$
is its joint probability distribution. The set
$\mathcal{Y}=\left\\{1,\ldots,C\right\\}$ represents the classes of a
classification task and
$\mathcal{P}(\mathcal{Y})=\left\\{\mathbf{p}\in[0,1]^{\left|\mathcal{Y}\right|}:\sum_{i=1}^{\left|\mathcal{Y}\right|}p_{i}=1\right\\}$
the probability simplex over the classes. It is assumed that we have access to
a training set $\mathcal{D}_{N}=\\{(\mathbf{x}_{i},y_{i})\\}_{i=1}^{N}$
composed of independent and identically distributed (i.i.d) realizations of
$P_{XY}$. The Out-Of-Distribution (OOD) detection problem consists of deciding
whether a new, previously unseen sample comes (or not) from the IN
distribution $P_{XY}$. The goal is to build a binary function
$g:\mathcal{X}\rightarrow\\{0,1\\}$ based on the thresholding of an anomaly
score $s:\mathcal{X}\rightarrow\mathbb{R}_{+}$ that separates IN samples from
OOD samples. Namely, for a threshold $\gamma\in\mathbb{R}_{+}$, we have:

$g(\mathbf{x},\gamma)~{}=~{}\left\\{\begin{array}[]{ll}1&\text{if
}s\left(\mathbf{x}\right)>\gamma,\\\ 0&\text{if
}s\left(\mathbf{x}\right)\leqslant\gamma.\end{array}\right.$

###  2.2 Building an OOD detector

We assume that we have given a classifier
$f_{\theta}:\mathcal{X}\rightarrow\mathcal{P}(\mathcal{Y}$):

$f_{\theta}=\operatorname{softmax}\circ~{}h\circ~{}f^{\theta}_{L}\circ
f^{\theta}_{L-1}\circ\dots\circ f^{\theta}_{1},$ (1)

with $L>1$ layers222For the sake of brevity, we omit the parameters $\theta$
in the following., where
$f_{\ell}:\mathbb{R}^{d_{\ell-1}}\rightarrow\mathbb{R}^{d_{\ell}}$ is the
$\ell$-th layer of the encoder with $d_{\ell}$ being the dimension of the
latent space after the $\ell$-th layer ($d_{0}=d$). It is worth noting that in
the case of transformers (Vaswani et al., 2017), all latent spaces have the
same dimension. Finally, $h$ represents the logit function of the classifier.

To compute the anomaly score $s$ from $f_{\theta}$, OOD approaches rely on the
hidden representations of the (multilayer) encoder. For
$\mathbf{x}\in\mathcal{X}$ an input sequence, we denote
$\mathbf{z}_{\ell}=(f_{\ell}\circ\dots\circ f_{1})(\mathbf{x})$ its latent
representation at layer $\ell$. The latent representation obtained after the
$\ell$-th layer of the training set is denoted as
$\mathcal{D}_{N}^{\ell}=\\{(\mathbf{z}_{\ell,i},y_{i})\\}_{i=1}^{N}$.
Furthermore, we denote by $\mathcal{D}_{N}^{\ell,y}$ the restriction of
$\mathcal{D}_{N}^{\ell}$ to the samples with label $y$, i.e.,
$\mathcal{D}_{N}^{\ell,y}=\\{(\mathbf{z}_{\ell,i},y_{i})\in\mathcal{D}^{\ell}_{N}:y_{i}=y\\}$
with $N_{y}=|\mathcal{D}_{N}^{\ell,y}|$ indicates the cardinal of this set.

Feature-based OOD detectors usually rely on three key elements:

  1. (i)

Selecting features: the layer $\ell$ whose representation is considered to be
the input of the anomaly score.

  2. (ii)

A notion of an anomaly (or novelty) score built on the mapping
$\mathcal{D}_{N}^{\ell}$ of the training set on the chosen feature space. We
can build such a score $s(\cdot,\mathcal{D}_{N}^{\ell})$ defined on
$\mathbb{R}^{d}\times\left(\mathbb{R}^{d}\right)^{N}$ for any notion of
abnormality.

  3. (iii)

Setting a threshold to build the final decision function.

######  Remark 1.

Choice of the threshold. To select $\gamma$, we follow previous work (Colombo
et al., 2022a) by selecting an amount of training samples (i.e., “outliers”)
the detector can wrongfully detect. A classical choice is to set this
proportion to $80\%$.

###  2.3 Popular Anomaly Scores

In what follows, we present three common anomaly scores for step (ii) of the
previously mentioned procedure.

Mahalanobis distance. Authors of Lee et al. (2018) (see also Podolskiy et al.,
2021) propose to compute the Mahalanobis distance on the abstract
representations of each layer and each class. Precisely, this distance is
given by:

$s_{M}(\mathbf{z}_{\ell},\mathcal{D}_{N}^{\ell,y})=\left(\mathbf{z}_{\ell}-\mu_{\ell,y}\right)^{\top}\Sigma_{\ell,y}^{-1}\left(\mathbf{z}_{\ell}-\mu_{\ell,y}\right)$

on each layer $\ell$ and each class $y$ where $\mu_{\ell,y}$ and
$\Sigma_{\ell,y}$ are the estimated class-conditional mean and covariance
matrix computed on $\mathcal{D}_{N}^{\ell,y}$, respectively. The final score
from Lee et al. (2018) is obtained by choosing the minimum of these scores
over the classes on the penultimate encoder layer.

Integrated Rank-Weighted depth. Colombo et al. (2022a) propose to leverage the
Integrated Rank-Weighted (IRW) depth (Ramsay et al., 2019; Staerman et al.,
2021a). Similar to the Mahalanobis distance, the IRW data depth measures the
centrality/distance of a point to a point cloud. For the $\ell$-th layer, a
Monte-Carlo approximation of the IRW depth can be defined as:

$\displaystyle
s_{\text{IRW}}(\mathbf{z}_{\ell},\mathcal{D}_{N}^{\ell,y})=\frac{1}{n_{\mathrm{proj}}}\sum_{k=1}^{n_{\mathrm{proj}}}\min$
$\displaystyle\Bigg{\\{}\frac{1}{n}\sum_{i=1}^{N_{y}}\mathbbm{1}\left\\{g_{k,i}(\mathbf{z}_{\ell})\leqslant
0\right\\},$
$\displaystyle\frac{1}{n}\sum_{i=1}^{N_{y}}\mathbbm{1}\left\\{g_{k,i}(\mathbf{z}_{\ell})>0\right\\}\Bigg{\\}},$

where $g_{k,i}(\mathbf{z}_{\ell})=\langle
u_{k},\mathbf{z}_{\ell,i}-\mathbf{z}_{\ell}\rangle$,
$u_{k}\in\mathbb{S}^{d-1},\mathbf{z}_{\ell,i}\in\mathcal{D}_{N}^{\ell,y}$
where $\mathbb{S}^{d-1}=\\{x\in\mathbb{R}^{d}\;:\;||x||=1\\}$ is the unit
hypersphere and $n_{\mathrm{proj}}$ is the number of directions sampled on the
sphere.

Cosine similarity. Zhou et al. (2021) propose to compute the maximum cosine
similarity between the embedded sample $\mathbf{z}_{\ell}$ and the training
set $\mathcal{D}_{N}^{\ell}$ at layer $\ell$:

$s_{C}(\mathbf{z}_{\ell},\mathcal{D}_{N}^{\ell})=-\underset{\mathbf{z}_{\ell,i}\in\mathcal{D}_{N}^{\ell}}{\max}\;\;~{}\frac{\langle\mathbf{z}_{\ell},\mathbf{z}_{\ell,i}\rangle}{\left|\left|\mathbf{z}_{\ell}\right|\right|\left|\left|\mathbf{z}_{\ell,i}\right|\right|},$

where $\langle\cdot,\cdot\rangle$ and $\left|\left|\cdot\right|\right|$ denote
the Euclidean inner product and norm, respectively. They also choose the
penultimate layer. It is worth noting they do not rely on a per-class
decision.

###  2.4 Limitations of Existing Methods

The choice of layer for step (i) in Sec. 2.2 is not usually a question. Most
work arbitrarily relies on the logits (Liang et al., 2018; Liu et al., 2020)
or the last layer of the encoder (Winkens et al., 2020b; Podolskiy et al.,
2021; Sun et al., 2022; Ren et al., 2019; Sastry & Oore, 2020; Gomes et al.,
2022; Yang et al., 2021; Hendrycks & Gimpel, 2016; Wang et al., 2022). We
argue that these choices are unjustified and that previous work gives up on
important information in other layers. Moreover, previous works have shown
that all the layers carry different information or type of abstraction and
thus are useful for different tasks (Ilin et al., 2017; Kozma et al., 2018;
Dara & Tumma, 2018).

(a) Emotion

(b) 20-newsgroup

(c) IMDB

(d) Trec

Figure 1: OOD detection performance in terms of AUROC $\uparrow$ for each
features-based OOD score (Mahalanobis distance ($s_{M}$), Maximum cosine
similarity ($s_{C}$) and IRW ($s_{IRW}$)) computed at each layer of the
encoder for $4$ different OOD datasets (Emotion, 20-newsgroup, IMDB and Trec)
for a model fine-tuned on SST2. We observe that the performance of each metric
on each layer varies significantly with the OOD task.

To support our claim, we report in Fig. 1 the OOD performance of popular
detectors described in Sec. 2.3 applied at each layer of the encoder (Devlin
et al., 2018). We observe a high variability across different layers. The last
layer is rarely the best-performing layer, and there is room for improvement
if we could choose the best possible layer or gather useful information from
all of them. This observation is consistent with the literature, as neural
networks are known to extract different information and construct different
abstractions at each layer (Ilin et al., 2017; Kozma et al., 2018; Dara &
Tumma, 2018).

To our knowledge, only a few studies address this issue and attempt to
leverage information from all layers. Notably, recent work by Colombo et al.
(2022a) considers representations obtained by taking the average embedding
across the encoder layers. They then apply common OOD detection methods to
this new aggregated embedding.

We propose to compute common OOD scores on each layer of the encoder (and not
only on the logits or the representation generated by the last layer) and to
aggregate this score in an unsupervised fashion to select and combine the most
relevant following the task at hand.

##  3 Leveraging information from all layers

In this section, we describe our aggregation methods that use the information
available in the different layers of the encoder.

###  3.1 Problem Statement

For an input $\mathbf{x}\in\mathcal{X}$ and a training dataset
$\mathcal{D}_{N}$, we obtain their set of embedding representation sets:
$\\{\mathbf{z}_{\ell}\\}_{\ell=1}^{L}$ and
$\\{\mathcal{D}^{\ell}_{N}\\}_{\ell=1}^{L}$, respectively. Given an anomaly
score function
$s:\;\mathbb{R}^{d}\times\left(\mathbb{R}^{d}\right)^{N}\rightarrow\mathbb{R}$
(e.g., those described in Sec. 2.3), we define the OOD score set of an input
$\mathbf{x}$ as
$\mathcal{S}_{s}(\mathbf{x};\mathcal{D}_{N})=\\{\\{s(\mathbf{z}_{\ell};\mathcal{D}^{\ell,y}_{N})\\}_{\ell=1}^{L}\\}_{y=1}^{C}\in\mathbb{R}^{L\times
C}$. Similarly, it is possible to obtain a reference set of
$\mathcal{R}(\mathcal{D}_{N})=\\{\mathcal{S}_{s}(\mathbf{x};\mathcal{D}_{N}),\;\forall\;(\mathbf{x},y)\in\mathcal{D}_{N}\\}$
from the training data333When using the cosine similarity, which does not rely
on a per-class decision, $\mathcal{S}_{s}(\mathbf{x};\mathcal{D}_{N})$ is
reduced to $\\{s(\mathbf{z}_{\ell};\mathcal{D}^{\ell}_{N})\\}_{\ell=1}^{L}$..
In what follows, we aim to answer the following question.

Can we leverage all the information available in
$\mathcal{S}_{s}(\mathbf{x};\mathcal{D}_{N})$ and/or
$\mathcal{R}(\mathcal{D}_{N})$ to build an OOD detector?

###  3.2 Proposed Framework

Our framework aims at comparing the set of scores of a sample to the sets of
scores of a reference relying on principled anomaly detection algorithms.

The goal of this work is to propose a data-driven aggregation method of OOD
scores444We do not assume that we have access to OOD samples as they are often
not available., $\operatorname{Agg}$. $\operatorname{Agg}$ is defined as:

$\displaystyle\operatorname{Agg}:\quad\mathbb{R}^{L\times
C}\times\left(\mathbb{R}^{L\times C}\right)^{N}$
$\displaystyle\rightarrow\mathbb{R}$
$\displaystyle(\mathcal{S}_{s}(\mathbf{x};\mathcal{D}_{N}),\mathcal{R}(\mathcal{D}_{N}))$
$\displaystyle\rightarrow\operatorname{Agg}(\mathcal{S}_{s},\mathcal{R}),$

where $\mathbf{x}$ denotes the input sample.

Intuition. This framework allows us to consider the whole trace of a sample
through the model. This formulation has two main advantages: it avoids manual
layer selection and enables us to leverage information from all the encoder
layers.

We propose two families of approaches: (i) one solely relies on the score set
$\mathcal{S}_{s}(\mathbf{x};\mathcal{D}_{N})$ (corresponding to a no-reference
scenario and denoted as $\operatorname{Agg}_{\emptyset}$) and (ii) the second
one (named reference scenario) leverages the reference set
$\mathcal{R}(\mathcal{D}_{N})$.

######  Remark 2.

It is worth noting that our framework through
$\operatorname{Agg}_{{\emptyset}set}$ or $\operatorname{Agg}$ naturally
includes previous approaches (Lee et al., 2018; Zhou et al., 2021; Colombo et
al., 2022a). For example, the detector of Lee et al. (2018) can be obtained by
defining $\operatorname{Agg}_{{\emptyset}}$ as the minimum of the penultimate
line of the matrix $\mathcal{S}_{s}(\mathbf{x},\mathcal{D}_{N})$.

###  3.3 Detailed Aggregation Procedures

Figure 2: Schema of our aggregation procedure. (1) We extract the embeddings
at each layer of the encoder for every sample. (2) We compute the per-class
scores for a reference set and the new sample to be evaluated for each layer
embedding. (3) We aggregate the scores over every layer to get an aggregated
per-class score before taking the min score over the classes. (4) Finally, we
apply the threshold on this minimum.

Intuition. Our framework through $\operatorname{Agg}$ and
$\operatorname{Agg}_{\emptyset}$ requires two types of operations to extract a
single score from $\mathcal{S}_{s}(\mathbf{x},\mathcal{D}_{N})$ and
$\mathcal{R}_{s}(\mathcal{D}_{N})$: one aggregation operation over the layers
and one aggregation operation over the classes, where necessary.

Our framework in a nutshell. We assume we are given an anomaly score, $s$,
that we want to enhance by leveraging all the layers of the encoder. For a
given input $\mathbf{x}$, our framework follows $4$ steps (see Fig. 2 for a
depiction of the procedure):

  1. 1.

Compute the embeddings $\\{\mathbf{z}_{l}\\}_{l=1}^{L}$ for $\mathbf{x}$ and
every element of $\mathcal{D}_{N}$.

  2. 2.

Form $\mathcal{S}_{s}(\mathbf{x};\mathcal{D}_{N})$ and
$\mathcal{R}(\mathcal{D}_{N})$ using the score $s$.

  3. 3.

Perform $\operatorname{Agg}_{\emptyset}$ or $\operatorname{Agg}$:

    1. (a)

(_per layer_) Aggregate score information over the layers to obtain a vector
composed of $C$ scores.

    2. (b)

(_per class_) Take the minimum value of this vector.

  4. 4.

Apply a threshold $\gamma$ on that value.

Step (3.b). is inspired by the OOD literature (Lee et al., 2018; Colombo et
al., 2022a). It relies on the observation that if the input sample is IN-
distribution, it is expected to have at least one low score in the class
vector, whereas an OOD sample should only have high scores equivalent to a
high minimum score.

####  3.3.1 No-reference scenario ($\operatorname{Agg}_{\emptyset}$)

In the no-reference scenario, we have access to a limited amount of
information. We thus propose to rely on simple statistics to aggregate the OOD
scores available in $\mathcal{S}_{s}(\mathbf{x};\mathcal{D}_{N})$ to compute
step (3). of the proposed procedure. Precisely, we use the average, the
minimum (min), the median (med), and coordinate (see Remark 2) operators on
the column of the matrix $\mathcal{S}_{s}(\mathbf{x};\mathcal{D}_{N})$.

####  3.3.2 Data-driven scenario ($\operatorname{Agg}$)

In the data-driven scenario, $\operatorname{Agg}$ also has access to the set
of reference OOD scores (i.e., $\mathcal{R}_{s}(\mathcal{D}_{N})$) for the
given OOD score $s$. The goal, then, is to compare the score set
$\mathcal{S}_{s}(\mathbf{x};\mathcal{D}_{N})$ of the input with this reference
set $\mathcal{R}_{s}(\mathcal{D}_{N})$ to obtain a score vector of size $C$.
In the following, we propose an original solution for the layer operation.

For the _per layer_ operation we rely on an anomaly detection algorithm for
each class $\mathcal{A}_{y}$ defined as:

$\displaystyle\mathcal{A}_{y}:\mathbb{R}^{L}\times(\mathbb{R}^{L})^{N_{y}}$
$\displaystyle\rightarrow\mathbb{R}$
$\displaystyle\mathbf{s}_{y}\times\mathcal{R}_{y}$
$\displaystyle\mapsto\mathcal{A}_{y}(\mathbf{s}_{y},\mathcal{R}_{y}),$ (2)

where
$\mathbf{s}_{y}=\\{s(\mathbf{z}_{\ell};\mathcal{D}^{\ell,y}_{N})\\}_{\ell=1}^{L}$
and $\mathcal{R}_{y}=\mathcal{R}(\mathcal{D}_{N}^{y})$.

######  Remark 3.

$\mathcal{A}_{y}$ is trained on the reference set $\mathcal{R}_{y}$ for each
class and thus does not involve any OOD samples. The score returned for a
vector $\mathbf{s}_{y}$ is the prediction score associated with the trained
algorithm.

######  Remark 4.

We define a per-class decision for $\operatorname{Agg}$ since it has been
shown to be significantly more effective than global scores (Huang & Li,
2021). It is the approach chosen by most state-of-the-art-methods. We have
validated this approach by conducting extensive experiments. We refer the
reader to Ap. C for further discussion.

We propose four popular anomaly detection algorithms. First, we propose to
reuse common OOD scores ($s_{M}$, $s_{C}$, $s_{\text{IRW}}$) as aggregation
methods(See Ap. C for more details): they are now trained on the reference set
of sets of OOD scores $\mathcal{R}_{s}(\mathcal{D}_{N})$ and provide a notion
of anomaly for the trace of a sample through the model. In addition, we used
Isolation Forest (IF; Liu et al., 2008, see also Hariri et al., 2019; Staerman
et al., 2019 for its extensions), see also Amer et al., 2013) and the Local
Outlier Factor (LOF; Breunig et al., 2000). Below, we briefly recall the
general insights of each of these algorithms. It is important to emphasize
that our framework can accommodate any anomaly detection algorithms (further
details are given in Ap. C).

Local Outlier Factor. This method compares a sample’s density with its
neighbors’ density. Any sample with a lower density than its neighbors is
regarded as an outlier.

Isolation Forest. This popular algorithm is built on the idea that anomalous
instances should be easily distinguished from normal instances. It leads to a
score that measures the complexity of separating a sample from others based on
the number of necessary decision trees required to isolate a data point. It is
computationally efficient, benefits from stable hyper-parameters, and is well
suited to the unsupervised setting.

###  3.4 Comparison to Baseline Methods

Current State-of-the-art methods for OOD detection on textual data have been
recently provided in Colombo et al. (2022a) (PW). They aggregate the hidden
layers using Power means (Hardy et al., 1952; Rücklé et al., 2018) and then
apply an OOD score on this aggregated representation. They achieved previous
SOTA performance by coupling it with the IRW depth and proposed a comparison
with Mahalanobis and Cosine versions. We reproduce these results as it is a
natural baseline for aggregation algorithms.

Last Layer. Considering that the model’s last layer or logits should output
the most abstract representation of an input, it has been the primary focus of
attention for OOD detection. It is a natural choice for any architecture or
model and therefore removes the hurdle of selecting features for different
tasks and architectures. For this heuristic, we obtain OOD scores using the
Mahanalobis distance (as in Lee et al., 2018), the IRW score (as in Colombo et
al., 2022a), and the cosine similarity (as in Winkens et al., 2020b).

Additional methods. It is common on OOD detection methods to report the
Maximum Softmax Prediction (MSP; Hendrycks & Gimpel, 2016) as well as the
Energy Score (E; Liu et al., 2020).

##  4 MILTOOD-C: A more realistic benchmark for OOD detection

In this section, we highlight the limitations of existing benchmarks and
introduce our own: MILTOOD-C A MultI Lingual Text OOD for classification
tasks.

###  4.1 Limitation of Existing Benchmarks

Number of classes. Text classification benchmarks for OOD detection often
consist of sentiment analysis tasks involving a small number of classes (Fang
& Zhan, 2015; Kharde et al., 2016). Those tasks with a larger number of
classes have been mostly ignored in previous OOD detection benchmarks (Colombo
et al., 2022a; Li et al., 2021; Zhou et al., 2021). However, real-world
problems do involve vastly multi-class classification tasks (Casanueva et al.,
2020). Previous work in computer vision found that these problems require
newer and carefully tuned methods to enable OOD detection in this more
realistic setting (Deng et al., 2009; Le & Yang, 2015).

Monolingual datasets. Most methods have been tested on architectures tailored
for the English language (Colombo et al., 2022a; Li et al., 2021; Arora et
al., 2021). With inclusivity and diversity in mind (Ruder, 2022; van Esch et
al., 2022), it is necessary to assess the performance of old and new OOD
detection methods on a variety of languages (Srinivasan et al., 2021; de Vries
et al., 2020; Baheti et al., 2021; Zhang et al., 2022).

###  4.2 Benchmark

We now present MILTOOD-C, which addresses the aforementioned limitations. It
consists of more than $25$ datasets involving up to $77$ classes, and $4$
languages.

Dataset selection. We gathered a large and diverse benchmark in terms of shift
typology, tasks, and languages. It covers $25$ datasets in $4$ different
languages (i.e., English, German, Spanish, and French) and classifications
tasks involving $2$ to $77$ classes. Following standard protocol (Hendrycks et
al., 2020), we train a classifier for each in-distribution dataset (IN-DS)
while the OOD dataset (OUT-DS) is coming from a different dataset. We provide
a comprehensive list of the $180$ pairs we considered in Sec. A.1. It is an
order of magnitude larger than recent concurrent work from Colombo et al.
(2022a).

English benchmark. We relied on the benchmark proposed by Zhou et al. (2021);
Hendrycks et al. (2020). It features three types of IN-DS: sentiment analysis
(i.e., SST2 (Socher et al., 2013), IMDB (Maas et al., 2011)), topic
classification (i.e., 20Newsgroup (Joachims, 1996)) and question answering
(i.e., TREC-10 and TREC-50 (Li & Roth, 2002)). We also included the Massive
(FitzGerald et al., 2022) dataset and the Banking (Casanueva et al., 2020) for
a larger number of classes and NLI datasets (i.e., RTE (Burger & Ferro, 2005;
Hickl et al., 2006) and MNLI (Williams et al., 2018)) following Colombo et al.
(2022a). We form IN and OOD pairs between the aforementioned tasks.

Beyond English-centric tasks.555We did not work on language changes because
they were easily detected with all the methods considered. Instead, we focus
on intra-language drifts. For language-specific datasets, we added the same
tasks as for English when available and extended it with language-specific
datasets such as the PAWS-S datasets (Yang et al., 2019), film reviews in
French and Spanish (Blard, 2019). For French and German, we also added the
Swiss judgments datasets (Niklaus et al., 2022). Finally, we added different
tweet classification tasks for each language (English, German, Spanish and
French) (Zotova et al., 2020; Barbieri et al., 2022).

Model selection. To ensure that our results are consistent not only across
tasks and shifts, but also across model architectures, we train classifiers
based on $6$ different Transformer (Vaswani et al., 2017) decoders: BERT
(Devlin et al., 2018) (base, large and multilingual versions), DISTILBERT
(Sanh et al., 2019) and RoBERTa (Liu et al., 2019) (base and large versions)
fine-tuned on each task.

###  4.3 Assessing OOD detection performance

The OOD detection problem is a binary classification problem where the
positive class is OUT. To evaluate our detectors, we follow concurrent work
(Colombo et al., 2022a; Darrin et al., 2022) and evaluate our detector using
threshold-free metrics such as AUROC $\uparrow$, AUPR-IN/AUPR-OUT and
threshold based metrics such as FPR $\downarrow$ at $95\%$ and Err. For sake
of brevity, we report detailed definitions of the metrics in Sec. A.2.

##  5 Experimental Results

###  5.1 Quantifying Aggregation Gains

Overall results. Data driven aggregation methods (i.e., with reference)
consistently outperform any other baselines or tested methods by a significant
margin (see Tab. 1 and Tab. 2) on our extensive MILTOOD-C benchmark. According
to our experiments, the best combination of hidden feature-based OOD score and
aggregation function is to use the Maximum cosine similarity as the underlying
OOD score and to aggregate these scores using the IRW data depth
($s_{\text{IRW}}$). A first time to get the abnormality of the representations
of the input and a second time to assess the abnormality of the set of layer-
wise scores through the model. It reaches an average AUROC $\uparrow$ of
$0.99$ and a FPR $\downarrow$ of $0.02$. It is a gain of more than $5\%$
compared to the previous state-of-the-art methods.

Most versatile aggregation method. While the $s_{C}$ and $s_{\text{IRW}}$ used
as aggregation methods achieve excellent performance when paired with $s_{C}$
as the underlying OOD score, they fail to aggregate as well other underlying
scores. Whereas the isolation forest algorithm is a more versatile and
consistent data-driven aggregation method: it yields performance gain for
every underlying OOD score.

Performance of common baselines. We show that, on average, using the last
layer or the logits as features to perform OOD detection leads to poorer
results than almost every other method. It is interesting to point out that
this is not the case in computer vision (Yang et al., 2021). This finding
further motivates the development of OOD detection methods tailored for text.

| $\mathcal{A}$ |  AUROC $\uparrow$ |  FPR $\downarrow$ | Err
---|---|---|---|---
$\operatorname{Agg}$ | $s_{M}$ | 0.88 | ±0.16 | 0.32 | ±0.35 | 0.16 | ±0.19
$s_{C}$ | 0.91 | ±0.16 | 0.21 | ±0.35 | 0.13 | ±0.21
$s_{IRW}$ | 0.87 | ±0.18 | 0.30 | ±0.39 | 0.16 | ±0.21
IF | 0.92 | ±0.13 | 0.21 | ±0.31 | 0.10 | ±0.15
LOF | 0.87 | ±0.15 | 0.37 | ±0.36 | 0.19 | ±0.22
$\operatorname{Agg}_{\emptyset}$ | Mean | 0.84 | ±0.18 | 0.43 | ±0.40 | 0.24 | ±0.25
Median | 0.83 | ±0.17 | 0.46 | ±0.39 | 0.25 | ±0.25
PW | 0.82 | ±0.17 | 0.48 | ±0.39 | 0.23 | ±0.23
Bas. | $E$ | 0.83 | ±0.18 | 0.39 | ±0.30 | 0.19 | ±0.18
Last layer | 0.84 | ±0.17 | 0.42 | ±0.37 | 0.20 | ±0.22
Logits | 0.75 | ±0.16 | 0.60 | ±0.34 | 0.30 | ±0.25
MSP | 0.83 | ±0.17 | 0.39 | ±0.28 | 0.19 | ±0.18

Table 1: Overall average performance of each aggregation method for all
architectures, tasks, and OOD scores.

|  | $\mathcal{A}$ |  AUROC $\uparrow$ |  FPR $\downarrow$ | Err
---|---|---|---|---|---
$E$ | Bas. | $E$ | 0.83 | ±0.18 | 0.39 | ±0.31 | 0.20 | ±0.19
$s_{M}$ | $\operatorname{Agg}$ | $s_{M}$ | 0.90 | ±0.14 | 0.27 | ±0.33 | 0.16 | ±0.20
$s_{C}$ | 0.88 | ±0.17 | 0.32 | ±0.40 | 0.19 | ±0.24
$s_{IRW}$ | 0.81 | ±0.20 | 0.44 | ±0.42 | 0.23 | ±0.23
IF | 0.94 | ±0.10 | 0.19 | ±0.25 | 0.11 | ±0.13
LOF | 0.87 | ±0.15 | 0.39 | ±0.37 | 0.22 | ±0.25
$\operatorname{Agg}_{\emptyset}$ | Mean | 0.74 | ±0.18 | 0.59 | ±0.39 | 0.32 | ±0.27
Median | 0.75 | ±0.17 | 0.61 | ±0.35 | 0.33 | ±0.25
PW | 0.80 | ±0.17 | 0.61 | ±0.38 | 0.30 | ±0.25
Bas. | Last layer | 0.92 | ±0.11 | 0.25 | ±0.31 | 0.13 | ±0.17
Logits | 0.71 | ±0.14 | 0.65 | ±0.27 | 0.33 | ±0.23
$s_{C}$ | $\operatorname{Agg}$ | $s_{M}$ | 0.93 | ±0.11 | 0.20 | ±0.27 | 0.11 | ±0.15
$s_{C}$ | 0.98 | ±0.10 | 0.04 | ±0.19 | 0.03 | ±0.11
$s_{IRW}$ | 0.99 | ±0.07 | 0.02 | ±0.15 | 0.02 | ±0.08
IF | 0.94 | ±0.14 | 0.12 | ±0.29 | 0.04 | ±0.12
LOF | 0.93 | ±0.11 | 0.20 | ±0.26 | 0.11 | ±0.14
$\operatorname{Agg}_{\emptyset}$ | Mean | 0.93 | ±0.12 | 0.25 | ±0.33 | 0.16 | ±0.21
Median | 0.92 | ±0.12 | 0.27 | ±0.34 | 0.17 | ±0.23
PW | 0.93 | ±0.11 | 0.19 | ±0.27 | 0.11 | ±0.14
Bas. | Last layer | 0.92 | ±0.11 | 0.22 | ±0.26 | 0.11 | ±0.13
Logits | 0.81 | ±0.17 | 0.52 | ±0.42 | 0.28 | ±0.29
$s_{\text{IRW}}$ | $\operatorname{Agg}$ | $s_{M}$ | 0.81 | ±0.18 | 0.50 | ±0.38 | 0.21 | ±0.21
$s_{C}$ | 0.89 | ±0.17 | 0.28 | ±0.36 | 0.17 | ±0.21
$s_{IRW}$ | 0.82 | ±0.19 | 0.43 | ±0.39 | 0.24 | ±0.21
IF | 0.89 | ±0.15 | 0.34 | ±0.36 | 0.17 | ±0.19
LOF | 0.82 | ±0.15 | 0.54 | ±0.35 | 0.27 | ±0.24
$\operatorname{Agg}_{\emptyset}$ | Mean | 0.84 | ±0.18 | 0.47 | ±0.40 | 0.24 | ±0.24
Median | 0.82 | ±0.18 | 0.50 | ±0.39 | 0.26 | ±0.24
PW | 0.74 | ±0.17 | 0.64 | ±0.34 | 0.31 | ±0.24
Bas. | Last layer | 0.66 | ±0.14 | 0.79 | ±0.21 | 0.38 | ±0.23
Logits | 0.73 | ±0.16 | 0.64 | ±0.28 | 0.29 | ±0.20
MSP | Bas. | MSP | 0.83 | ±0.17 | 0.39 | ±0.28 | 0.20 | ±0.18

Table 2: Average performance of each considered metric over all the OOD pairs
and model architectures in terms of AUROC $\uparrow$, Err, and FPR
$\downarrow$ . For each common OOD score, we report the results obtained using
every aggregation method or choice of features to consider.

Impact of simple statistical aggregation. Interestingly, simple statistical
aggregations of the set of OOD scores, such as the average or the median,
achieve, in some cases, similar or even better results than baselines relying
on the score of a single layer.

Impact of data-driven aggregation. In almost all scenarios, aggregating the
score using a data-driven anomaly detection method leads to a significant gain
in performance compared to baseline methods. This supports our claim that
useful information is scattered across the layers currently ignored by most
methods. We show that this information can be retrieved and effectively
leveraged to improve OOD detection.

###  5.2 Post Aggregation Is More Stable Across Task, Language, Model
Architecture

Figure 3: Average performance of OOD detectors in terms of AUROC $\uparrow$
for tasks involving different numbers of classes.

Most OOD scores have been crafted and finetuned for specific settings. In the
case of NLP, they have usually been validated only on datasets involving a
small number of classes or on English tasks. In this section, we study the
stability and consistency of the performance of each score and aggregation
method in different settings.

Stability of performance across tasks. In Fig. 3, we plot the average AUROC
$\uparrow$ across our models and datasets per number of classes of the IN
dataset. It is, therefore, the number of classes output by the model. Our best
post-aggregation methods (i.e., Maximum cosine similarity and Integrated Rank-
Weigthed) produced more consistent results across all settings. It can
maintain excellent performance for all types of datasets, whereas the
performance of baselines and other aggregation methods tends to fluctuate from
one setting to another. More generally, we observe that data-driven
aggregation methods tend to perform consistently on all tasks, whereas
previous baselines’ performance tends to vary.

Features aggregation vs. OOD scores aggregation. Interestingly, we show that
while Power Means pre-aggregation of the features yields better results than
single-layer scores, they still follow the same trend, and the gain is
inconsistent.

Figure 4: Comparison of the best-performing aggregation methods and underlying
OOD scores in terms of AUROC $\uparrow$ and FPR $\downarrow$ trade-off. We can
see that most OOD detectors’ performance varies greatly from language to
language. The maximum cosine similarity scores aggregated with IRW (green
circle) can maintain its performance in all four studied languages and the
isolation forests.

Stability of results across languages. In Fig. 4, we show the relation between
AUROC $\uparrow$ and FPR $\downarrow$ for all our aggregation methods and
underlying metrics for each language we studied. In contrast, the performance
of most combinations varies with the language. We especially notice that
$s_{C}$ scores aggregated using either the Integrated rank-weighted or $s_{C}$
consistently achieve excellent performance across all languages.

###  5.3 Explainability and layer significance

Best Layer Selection (Oracle). In Fig. 1, we showed that high OOD detection
performance could be reached, provided that we know which is the best layer to
perform the OOD detection on. We compare our aggregation methods to an oracle
method that always uses the best layer. We show in Fig. 5 that our
aggregation’s methods outperform baselines and, in some cases, the performance
of the oracle. This means that our aggregation methods reach and even
outperform oracle performance.

Figure 5: Average performance difference in terms of AUROC between aggregation
methods and the oracle (best possible layer).

Retro-engineering and explainability. We propose an explainability analysis of
the learned aggregation algorithms to gain more insights into the layer
selection retained by our data-driven detectors. We report in Fig. 6 the SHAP
scores (Lundberg & Lee, 2017) of one of them to distinguish Go-Emotions
samples from RTE samples and 20 news-group samples. It outlines the different
importance of the layers for different tasks. Not surprisingly, we found that
different layers better separate different classes and tasks. We also confirm
that the last layer is not always the best suited for OOD data separation.

Figure 6: Importance (in terms of Shap score Lundberg & Lee, 2017) of each
layer in OOD detection for Go-Emotions as IN-DS with BERT for two different
OOD datasets. It represents the importance of each feature (Mahalanobis
distance computed at a given layer) for the isolation forest anomaly detection
algorithms. We can see that the significant layers are different from one OOD
dataset to another and that in both presented cases, the last layer ($12$) nor
the logits ($13$) are not the most useful ones.

##  6 Conclusion and future work

We proposed a new framework that allows aggregating OOD scores across all the
layers of the encoder of a text classifier instead of solely relying on scores
computed on the logits or output of the last layer to improve OOD detection.
We confirmed that all the layers of the encoder of a text classifier are not
equal when it comes to OOD detection and, more importantly, that the common
choices for OOD detection (logits and last layer) are, more often than not,
not the best choice. We validated our methods on an extended text OOD
classification benchmark MILTOOD-C that we introduced. We showed that our
aggregation methods are not only able to outperform previous baselines and
recent work, but they were also able to outperform an oracle that would be
able to choose the best layer to perform OOD detection for a given task. This
leads us to conclude that there is useful information for OOD detection
scattered across all the layers of the encoder of a text classifier and that,
when appropriately extracted, it can be leveraged to improve OOD detection
performance vastly.

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[sections]

###### Contents

  1. 1 Introduction
  2. 2 OOD detection for text classification
    1. 2.1 Background and notations
    2. 2.2 Building an OOD detector
    3. 2.3 Popular Anomaly Scores
    4. 2.4 Limitations of Existing Methods
  3. 3 Leveraging information from all layers
    1. 3.1 Problem Statement
    2. 3.2 Proposed Framework
    3. 3.3 Detailed Aggregation Procedures
      1. 3.3.1 No-reference scenario ($\operatorname{Agg}_{\emptyset}$)
      2. 3.3.2 Data-driven scenario ($\operatorname{Agg}$)
    4. 3.4 Comparison to Baseline Methods
  4. 4 MILTOOD-C: A more realistic benchmark for OOD detection
    1. 4.1 Limitation of Existing Benchmarks
    2. 4.2 Benchmark
    3. 4.3 Assessing OOD detection performance
  5. 5 Experimental Results
    1. 5.1 Quantifying Aggregation Gains
    2. 5.2 Post Aggregation Is More Stable Across Task, Language, Model Architecture
    3. 5.3 Explainability and layer significance
  6. 6 Conclusion and future work
  7. A Benchmark details
    1. A.1 OOD pairs
    2. A.2 OOD detection performance metrics
  8. B Layer importance in OOD detection
  9. C Additional aggregation algorithms
    1. C.1 Details of anomaly detection algorithms
    2. C.2 Per-class scoring vs. Global aggregation
    3. C.3 Without reference statistical baselines
  10. D Computational cost
  11. E Explainability and variability
  12. F Experimental results
    1. F.1 Performance per tasks
      1. F.1.1 Language specific results
    2. F.2 Global averaged results per language
  13. G Futures Applications

[sections]l1

##  Appendix A Benchmark details

###  A.1 OOD pairs

|  | OUT-DS
---|---|---
Language | IN-DS |
English | 20ng | go-emotions,sst2,imdb,trec,mnli,snli,rte,b77,massive,trec-fine,emotion
b77 | go-emotions,sst2,imdb,20ng,trec,mnli,snli,rte,massive,trec-fine,emotion
emotion | go-emotions,sst2,imdb,20ng,trec,mnli,snli,rte,b77,massive,trec-fine
go-emotions | sst2,imdb,20ng,trec,mnli,snli,rte,b77,massive,trec-fine,emotion
imdb | go-emotions,sst2,20ng,trec,mnli,snli,rte,b77,massive,trec-fine,emotion
massive | go-emotions,sst2,imdb,20ng,trec,mnli,snli,rte,b77,trec-fine,emotion
rte | go-emotions,sst2,imdb,20ng,trec,mnli,snli,b77,massive,trec-fine,emotion
sst2 | go-emotions,imdb,20ng,trec,mnli,snli,rte,b77,massive,trec-fine,emotion
trec | go-emotions,sst2,imdb,20ng,mnli,snli,rte,b77,massive,trec-fine,emotion
trec-fine | go-emotions,sst2,imdb,20ng,trec,mnli,snli,rte,b77,massive,emotion
French | fr-allocine | fr-cls,fr-xnli,fr-pawsx,fr-xstance,fr-swiss-judgement,fr-tweet-sentiment
fr-cls | fr-xnli,fr-pawsx,fr-allocine,fr-xstance,fr-swiss-judgement,fr-tweet-sentiment
fr-pawsx | fr-cls,fr-xnli,fr-allocine,fr-xstance,fr-swiss-judgement,fr-tweet-sentiment
fr-swiss-judgement | fr-cls,fr-xnli,fr-pawsx,fr-allocine,fr-xstance,fr-tweet-sentiment
fr-tweet-sentiment | fr-cls,fr-xnli,fr-pawsx,fr-allocine,fr-xstance,fr-swiss-judgement
fr-xnli | fr-cls,fr-pawsx,fr-allocine,fr-xstance,fr-swiss-judgement,fr-tweet-sentiment
fr-xstance | fr-cls,fr-xnli,fr-pawsx,fr-allocine,fr-swiss-judgement,fr-tweet-sentiment
German | de-pawsx | de-xstance,de-swiss-judgement,de-tweet-sentiment
de-swiss-judgement | de-xstance,de-tweet-sentiment,de-pawsx
de-tweet-sentiment | de-xstance,de-swiss-judgement,de-pawsx
de-xstance | de-swiss-judgement,de-tweet-sentiment,de-pawsx
Spanish | es-cine | es-tweet-sentiment,es-pawsx,es-tweet-inde
es-pawsx | es-tweet-sentiment,es-cine,es-tweet-inde
es-tweet-inde | es-tweet-sentiment,es-pawsx,es-cine
es-tweet-sentiment | es-pawsx,es-cine,es-tweet-inde

Table 3: List of OOD pairs considered in our benchmark for each language.

| Dataset | Number of classes
---|---|---
Language |  |
English | go-emotions | 28
sst2 | 2
imdb | 2
20ng | 20
b77 | 77
massive | 60
trec-fine | 50
emotion | 6
trec | 6
rte | 2
mnli | 3
snli | 3
French | cls | 2
xnli | 3
pawsx | 2
allocine | 2
xstance | 2
swiss-judgement | 2
tweet-sentiment | 3
German | xstance | 2
swiss-judgement | 2
tweet-sentiment | 3
pawsx | 2
Spanish | tweet-sentiment | 3
pawsx | 2
cine | 5
tweet-inde | 3

Table 4: Details of the datasets composing MILTOOD-C.

###  A.2 OOD detection performance metrics

For evaluation we follow previous work in anomaly detection (Picot et al.,
2023b, a; Guerreiro et al., 2023; Gomes et al., ) and use AUROC $\uparrow$,
FPR $\downarrow$ , AUPR-IN/AUPR-OUT and Err. We do not aggregate the scores
using mean aggregation accross metrics (Colombo, 2021a; Colombo et al.,
2022b).

Area Under the Receiver Operating Characteristic curve (AUROC $\uparrow$;
Bradley, 1997). The Receiver Operating Characteristic curve is curve obtained
by plotting the True positive rate against the False positive rate. The area
under this curve is the probability that an in-distribution example
$\mathbf{X}_{in}$ has a anomaly score higher than an OOD sample
$\mathbf{x}_{out}$: AUROC
$\uparrow$$=\Pr(s(\mathbf{x}_{in})>s(\mathbf{x}_{out}))$. It is given by
$\gamma\mapsto(\Pr\big{(}s(\mathbf{x})>\gamma\,|\,Z=0\big{)},\Pr\big{(}s(\mathbf{x})>\gamma\,|\,Z=1\big{)})$.

False Positive Rate at $95\%$ True Positive Rate (FPR $\downarrow$ ). We
accept to allow only a given false positive rate $r$ corresponding to a
defined level of safety and we want to know what share of positive samples we
actually catch under this constraint. It leads to select a threshold
$\gamma_{r}$ such that the corresponding TPR equals $r$. At this threshold,
one then computes: $\Pr(s(\mathbf{x})>\gamma_{r}\,|\,Z=0)$ with $\gamma_{r}$
s.t. $\textrm{TPR}(\gamma_{r})=r$. $r$ is chosen depending on the difficulty
of the task at hand and the required level of safety.

Area Under the Precision-Recall curve (AUPR-IN/AUPR-OUT; Davis & Goadrich,
2006). The Precision-Recall curve plots the recall (true detection rate)
against the precision (actual proportion of OOD amongst the predicted OOD).
The area under this curve captures the trade-off between precision and recall
made by the model. A high value represents a high precision and a high recall
i.e. the detector captures most of the positive samples while having few false
positives.

Detection error (Err). It is simply the probability of miss-classification for
the best threshold.

##  Appendix B Layer importance in OOD detection

Our extensive experiments consistently show that there almost always exists a
layer that is excellent at separating OOD data.

|  |  AUROC $\uparrow$
---|---|---
Ours | Aggregation |  |
$\operatorname{Agg}$ | $s_{M}$ | 0.88 | ±0.16
$s_{C}$ | 0.91 | ±0.16
$s_{IRW}$ | 0.85 | ±0.19
IF | 0.92 | ±0.13
LOF | 0.87 | ±0.15
$\operatorname{Agg}_{\emptyset}$ | Mean | 0.84 | ±0.18
PW | 0.82 | ±0.17
Oracle | Oracle | 0.94 | ±0.12

Table 5: Average performance of aggregation scores along with the performance
of the oracle that is able to select the best layer to perform OOD detection.

In Tab. 6 we present the AUROC $\uparrow$ performance of all aggregation with
all metrics in comparison with the oracle’s performance. We see that there is
much room for improvement and that we are able to extract these improvements
and even go further.

|  |  |  AUROC $\uparrow$
---|---|---|---
Metric | Ours | Aggregation |  |
$s_{M}$ | $\operatorname{Agg}$ | $s_{IRW}$ | 0.82 | ±0.20
LOF | 0.87 | ±0.15
$s_{C}$ | 0.88 | ±0.17
$s_{M}$ | 0.91 | ±0.14
IF | 0.94 | ±0.10
$\operatorname{Agg}_{\emptyset}$ | Mean | 0.74 | ±0.18
PW | 0.80 | ±0.17
Oracle | Oracle | 0.95 | ±0.09
$s_{C}$ | $\operatorname{Agg}$ | $s_{M}$ | 0.93 | ±0.11
LOF | 0.93 | ±0.11
IF | 0.94 | ±0.14
$s_{C}$ | 0.98 | ±0.09
$s_{IRW}$ | 0.99 | ±0.07
$\operatorname{Agg}_{\emptyset}$ | PW | 0.93 | ±0.11
Mean | 0.93 | ±0.11
Oracle | Oracle | 0.97 | ±0.07
$s_{IRW}$ | $\operatorname{Agg}$ | $s_{M}$ | 0.81 | ±0.18
$s_{IRW}$ | 0.81 | ±0.19
LOF | 0.82 | ±0.15
$s_{C}$ | 0.88 | ±0.18
IF | 0.89 | ±0.15
$\operatorname{Agg}_{\emptyset}$ | PW | 0.74 | ±0.17
Mean | 0.84 | ±0.18
Oracle | Oracle | 0.90 | ±0.16

Table 6: Performance per metrics and aggregation with the oracle aggregation
for comparison

##  Appendix C Additional aggregation algorithms

We proposed in our work a framework that accommodates a wide range of
aggregation algorithms. We focused on unsupervised anomaly detection
algorithms and common statistics. However, many others options are available
in different flavors. For instance, we focused on per-class anomaly detection
but we can redo all our work for all classes at once or implement common OOD
scores as aggregation mechanisms. In this section, we propose additional
aggregation algorithms that are worth exploring. We provide their
formalization and results when available and propose direction for future
works.

###  C.1 Details of anomaly detection algorithms

Local Outlier Factor It measures the density of objects around a sample to
decide whether it is an inlier or an outlier. It relies on the $k$-distance of
a sample, e.g, its distance to its $k$th closest neighbor and considers
$N_{k}(\mathbf{x})$ the set of the $k$ nearest neighbors.

For stability reasons, the usual LOF method uses the reachability
distance666which is not a proper mathematical distance since it is not
symmetric which is defined as
$\operatorname{Rd}(\mathbf{x},\mathbf{x}^{\prime})=\max(k-\operatorname{distance}(\mathbf{x}^{\prime}),d(\mathbf{x},\mathbf{x}^{\prime}))$.
Intuitively, it is the distance between $\mathbf{x}$ and $\mathbf{x}^{\prime}$
if they are far enough from each other otherwise they are in the same $k$
nearest neighbors set and in that case the diameter of the set is used as
minimal distance. From these definitions, we can define the local density of a
sample $\mathbf{x}$,
$\operatorname{dens}(\mathbf{x})=\left(\frac{\sum_{\mathbf{x}^{\prime}\in
N_{k}(\mathbf{x})}\operatorname{Rd}_{k}(\mathbf{x},\mathbf{x}^{\prime})}{\left|N_{k}(\mathbf{x})\right|}\right)$.
The LOF score compare the density of a sample to the densities of its
neighbor: $\texttt{LOF}(\mathbf{x})=\frac{\sum_{\mathbf{x}^{\prime}\in
N_{k}(\mathbf{x})}\operatorname{dens}_{k}(\mathbf{x}^{\prime})}{\left|N_{k}(\mathbf{x})\right|\operatorname{dens}_{k}(\mathbf{x})}$.
If $\texttt{LOF}(\mathbf{x})=1$ the sample has the same density as his
neighbor, if it’s lower than $1$ it has a higher density than its neighbor and
thus is likely to be an inlier whereas if the score is higher than $1$ it has
a smaller density than its neighbor and should be an outlier.

Isolation Forest This popular algorithm is built on the idea that abnormal
instances should be easier to distinguish from normal instances. It leads to a
score that measures the complexity of separating samples based on the number
of necessary decision trees required to isolate a data point. In addition to
its computational ease, it benefits from stable hyper-parameters and is well
suited to the unsupervised setting. Formally speaking we consider a recursive
partitioning problem which is formalized as an isolation tree structure. The
number of trees required to isolate a point is the length of a path through
the tree from the root to a leaf.

###  C.2 Per-class scoring vs. Global aggregation

We reproduced the common strategy of relying on a per-class OOD score and then
using the minimum score over the classes as the OOD score. This strategy
relies on the intuition that an IN sample, belonging to a given class will at
least have a small anomaly score regarding this class whereas an OOD sample
would have only high scores.

However, using our aggregation tools we can imagine relying on per-class OOD
scores for the underlying scores but including them in the aggregation
mechanism. For comprehensiveness’ sake, we report here the results under this
setting but we found that in most cases per-class scores remain the better
solution.

Following our notations and framework, it means formally that the aggregation
algorithms are now not indexed by the classes but take as input a vector
containing all the scores per layer and per class:

$\displaystyle\mathcal{A}:$
$\displaystyle(\mathcal{R}^{L}\times\mathcal{R}^{C})\times(\mathcal{R}^{L}\times\mathcal{R}^{C})^{N}\rightarrow\mathbb{R}$
$\displaystyle\mathbf{s}\times\mathcal{R}\mapsto\mathcal{A}(\mathbf{s},\mathcal{R}),$
(3)

|  |  AUROC $\uparrow$ |  FPR $\downarrow$ | Err |
---|---|---|---|---|---
Ours | Aggregation |  |  |  |  |  |  |
$\operatorname{Agg}$ | $s_{C}*$ | 0.55 | ±0.09 | 0.92 | ±0.14 | 0.48 | ±0.27 |
$s_{M}*$ | 0.81 | ±0.18 | 0.45 | ±0.40 | 0.22 | ±0.23 |
$s_{IRW}*$ | 0.79 | ±0.19 | 0.55 | ±0.39 | 0.29 | ±0.26 |
IF* | 0.83 | ±0.16 | 0.36 | ±0.32 | 0.22 | ±0.22 |
LoF* | 0.50 | ±0.00 | 1.00 | ±0.02 | 0.45 | ±0.30 |

Table 7: Overall average performance of each aggregation method overall
architectures, tasks and underlying OOD scores when aggregating all layers and
classes at once.

|  |  |  AUROC $\uparrow$ |  FPR $\downarrow$ | Err
---|---|---|---|---|---
Metric | Ours | Aggregation |  |  |  |  |  |
$s_{M}$ | $\operatorname{Agg}$ | $s_{C}*$ | 0.57 | ±0.12 | 0.89 | ±0.19 | 0.48 | ±0.27
$s_{M}*$ | 0.75 | ±0.16 | 0.67 | ±0.35 | 0.33 | ±0.25
$s_{IRW}*$ | 0.67 | ±0.14 | 0.77 | ±0.27 | 0.38 | ±0.23
IF* | 0.75 | ±0.15 | 0.55 | ±0.29 | 0.34 | ±0.21
LoF* | 0.50 | ±0.01 | 0.99 | ±0.03 | 0.42 | ±0.28
$s_{C}$ | $\operatorname{Agg}$ | $s_{C}*$ | 0.51 | ±0.01 | 0.97 | ±0.02 | 0.50 | ±0.27
$s_{M}*$ | 0.75 | ±0.16 | 0.53 | ±0.34 | 0.28 | ±0.21
$s_{IRW}*$ | 0.82 | ±0.19 | 0.47 | ±0.38 | 0.24 | ±0.24
IF* | 0.84 | ±0.15 | 0.34 | ±0.32 | 0.23 | ±0.24
LoF* | 0.50 | ±0.00 | 1.00 | ±0.00 | 0.44 | ±0.29
$s_{\text{IRW}}$ | $\operatorname{Agg}$ | $s_{C}*$ | 0.58 | ±0.10 | 0.91 | ±0.14 | 0.45 | ±0.26
$s_{M}*$ | 0.93 | ±0.15 | 0.16 | ±0.31 | 0.07 | ±0.11
$s_{IRW}*$ | 0.86 | ±0.18 | 0.43 | ±0.40 | 0.26 | ±0.29
IF* | 0.91 | ±0.13 | 0.19 | ±0.25 | 0.11 | ±0.15
LoF* | 0.50 | ±0.00 | 1.00 | ±0.00 | 0.50 | ±0.30

Table 8: Average performance of the aggregation methods (i.e., for
$\operatorname{Agg}$) performed on the whole score matrix at once (per-class
and per-layer scores concatenated into a single vector) instead of aggregation
each trajectory separately.

Overall aggregation vs per-class aggregation. Consistently with previous work,
our aggregation methods do not perform as well when used to produce directly a
single overall score instead of being used class-wise and then taking the
minimum score over the classes. In Tab. 8 we report the OOD detection
performance of this setting.

###  C.3 Without reference statistical baselines

The simplest way to aggregate OOD scores is to consider statistical
aggregation over the layers and the classes. We showed that even basic
aggregations such as taking the median score enable significant gains with
respect to the last layer baselines.

##  Appendix D Computational cost

Time complexity. While the addition of an aggregation method induces obvious
additional computational costs they are actually quite limited (they do not
slow down the process significantly. They require only to compute the usual
OOD scores on each layer (which does not change the asymptotic complexity) and
then to perform the inference of common anomaly detection algorithms. For
example, Isolation forests are known to have a linear complexity in the number
of samples and to be able to perform well and fast with numerous and very high
dimensional data.

Memory footprint. Perhaps most of the overhead is a memory overhead: for
underlying OOD scores relying on a reference set we have to store one trained
score for each layer. In the case of the Mahalanobis distance, it means
storing $L$ covariance matrices instead of one in addition of the trained
aggregation algorithm.

##  Appendix E Explainability and variability

Isolation forests are constructed by choosing at random separating plans and
thus each run might give different importance to features. We benchmarked the
methods over $10$ seeds to alleviate variability and validate our results. It
showed that while some features could be permuted the overall trend were
consistent: features that are not relevant for a run do not significantly gain
in importance.

(a) RTE as Out-DS

(b) 20ng as Out-DS

Figure 7: Importance (in terms of shap score (Lundberg & Lee, 2017)) of each
layer in OOD detection for Go-Emotions as IN-DS with a bert architecture for
two different OOD datasets. The first row represents the samples from a
dataset containing IN and OUT samples. Each line corresponds to a feature and
shows how a given feature’s value influences the model to rule the sample as
an anomaly (negative) or as a normal (positive) sample. The features are
ranked by importance in the decision. The second row represents the importance
of each feature (OOD score computed at a given layer) for the isolation forest
anomaly detection algorithms. We can see that the significant layers are
different from one OOD dataset to another and that in both presented cases,
the last layer (Layer 12) is not the most useful one.

##  Appendix F Experimental results

###  F.1 Performance per tasks

####  F.1.1 Language specific results

In Tab. 9, Tab. 10, Tab. 12, Tab. 11 we present the average performance of
each aggregation methods on each language.

|  |  |  AUROC $\uparrow$ |  FPR $\downarrow$ | Err | AUPR-IN | AUPR-OUT
---|---|---|---|---|---|---|---
Metric | Ours | Agg. |  |  |  |  |  |  |  |  |  |
$E$ | Bas. | $E$ | 0.86 | ±0.16 | 0.35 | ±0.29 | 0.17 | ±0.17 | 0.82 | ±0.25 | 0.81 | ±0.22
$s_{M}$ | Bas. | Last layer | 0.92 | ±0.11 | 0.24 | ±0.30 | 0.12 | ±0.17 | 0.92 | ±0.16 | 0.86 | ±0.20
Logits | 0.72 | ±0.14 | 0.61 | ±0.26 | 0.30 | ±0.22 | 0.66 | ±0.29 | 0.67 | ±0.27
$\operatorname{Agg}_{\emptyset}$ | PW | 0.82 | ±0.17 | 0.57 | ±0.40 | 0.26 | ±0.24 | 0.84 | ±0.22 | 0.67 | ±0.30
Mean | 0.77 | ±0.18 | 0.54 | ±0.40 | 0.30 | ±0.27 | 0.68 | ±0.30 | 0.76 | ±0.28
Median | 0.76 | ±0.17 | 0.60 | ±0.37 | 0.31 | ±0.25 | 0.70 | ±0.29 | 0.70 | ±0.29
Max | 0.75 | ±0.19 | 0.55 | ±0.40 | 0.31 | ±0.27 | 0.67 | ±0.30 | 0.75 | ±0.29
Min | 0.50 | ±0.00 | 1.00 | ±0.00 | 0.41 | ±0.29 | 0.80 | ±0.15 | 0.70 | ±0.15
$\operatorname{Agg}$ | IF | 0.95 | ±0.09 | 0.16 | ±0.23 | 0.09 | ±0.12 | 0.93 | ±0.15 | 0.91 | ±0.16
$s_{M}$ | 0.93 | ±0.11 | 0.20 | ±0.29 | 0.12 | ±0.17 | 0.91 | ±0.19 | 0.90 | ±0.17
$s_{C}$ | 0.91 | ±0.16 | 0.24 | ±0.36 | 0.15 | ±0.23 | 0.89 | ±0.20 | 0.93 | ±0.15
LOF | 0.90 | ±0.14 | 0.32 | ±0.36 | 0.17 | ±0.23 | 0.88 | ±0.20 | 0.85 | ±0.21
$s_{M}*$ | 0.78 | ±0.15 | 0.63 | ±0.37 | 0.30 | ±0.25 | 0.75 | ±0.27 | 0.68 | ±0.31
IF* | 0.78 | ±0.14 | 0.49 | ±0.27 | 0.31 | ±0.20 | 0.55 | ±0.28 | 0.84 | ±0.22
$s_{C}*$ | 0.56 | ±0.12 | 0.89 | ±0.19 | 0.49 | ±0.28 | 0.51 | ±0.31 | 0.59 | ±0.29
LoF* | 0.50 | ±0.01 | 0.99 | ±0.03 | 0.43 | ±0.30 | 0.71 | ±0.23 | 0.69 | ±0.19
$s_{C}$ | Bas. | Last layer | 0.92 | ±0.11 | 0.22 | ±0.26 | 0.11 | ±0.13 | 0.90 | ±0.18 | 0.88 | ±0.18
Logits | 0.85 | ±0.16 | 0.44 | ±0.41 | 0.24 | ±0.28 | 0.83 | ±0.24 | 0.81 | ±0.23
$\operatorname{Agg}_{\emptyset}$ | Mean | 0.94 | ±0.11 | 0.22 | ±0.31 | 0.15 | ±0.20 | 0.91 | ±0.17 | 0.93 | ±0.16
PW | 0.94 | ±0.11 | 0.17 | ±0.26 | 0.09 | ±0.13 | 0.92 | ±0.16 | 0.91 | ±0.17
Min | 0.93 | ±0.11 | 0.24 | ±0.31 | 0.15 | ±0.18 | 0.90 | ±0.17 | 0.91 | ±0.18
Median | 0.93 | ±0.12 | 0.26 | ±0.34 | 0.17 | ±0.23 | 0.88 | ±0.19 | 0.92 | ±0.16
Max | 0.57 | ±0.15 | 0.90 | ±0.29 | 0.35 | ±0.29 | 0.82 | ±0.18 | 0.71 | ±0.18
$\operatorname{Agg}$ | $s_{C}$ | 0.98 | ±0.10 | 0.04 | ±0.20 | 0.03 | ±0.11 | 0.98 | ±0.12 | 0.98 | ±0.12
IF | 0.94 | ±0.14 | 0.11 | ±0.29 | 0.03 | ±0.11 | 0.95 | ±0.16 | 0.85 | ±0.23
LOF | 0.94 | ±0.10 | 0.17 | ±0.25 | 0.09 | ±0.12 | 0.92 | ±0.16 | 0.91 | ±0.17
$s_{M}$ | 0.94 | ±0.10 | 0.18 | ±0.25 | 0.10 | ±0.13 | 0.92 | ±0.16 | 0.90 | ±0.17
IF* | 0.88 | ±0.12 | 0.25 | ±0.26 | 0.18 | ±0.21 | 0.67 | ±0.27 | 0.93 | ±0.11
$s_{M}*$ | 0.78 | ±0.16 | 0.47 | ±0.33 | 0.25 | ±0.20 | 0.64 | ±0.28 | 0.78 | ±0.27
$s_{C}*$ | 0.51 | ±0.01 | 0.97 | ±0.02 | 0.48 | ±0.28 | 0.53 | ±0.30 | 0.49 | ±0.30
LoF* | 0.50 | ±0.00 | 1.00 | ±0.00 | 0.40 | ±0.29 | 0.80 | ±0.15 | 0.70 | ±0.15
$s_{\text{IRW}}$ | Bas. | Logits | 0.73 | ±0.17 | 0.65 | ±0.30 | 0.28 | ±0.21 | 0.75 | ±0.25 | 0.68 | ±0.26
Last layer | 0.66 | ±0.13 | 0.79 | ±0.21 | 0.38 | ±0.24 | 0.65 | ±0.29 | 0.64 | ±0.26
$\operatorname{Agg}_{\emptyset}$ | Max | 0.84 | ±0.18 | 0.45 | ±0.40 | 0.22 | ±0.22 | 0.85 | ±0.19 | 0.83 | ±0.22
Mean | 0.84 | ±0.18 | 0.45 | ±0.41 | 0.22 | ±0.23 | 0.85 | ±0.20 | 0.83 | ±0.21
Median | 0.82 | ±0.18 | 0.49 | ±0.39 | 0.25 | ±0.24 | 0.82 | ±0.22 | 0.82 | ±0.21
PW | 0.75 | ±0.17 | 0.61 | ±0.35 | 0.29 | ±0.23 | 0.75 | ±0.26 | 0.73 | ±0.25
Min | 0.50 | ±0.00 | 1.00 | ±0.00 | 0.41 | ±0.29 | 0.80 | ±0.15 | 0.70 | ±0.15
$\operatorname{Agg}$ | $s_{M}*$ | 0.99 | ±0.02 | 0.03 | ±0.05 | 0.03 | ±0.03 | 0.96 | ±0.09 | 0.99 | ±0.04
IF* | 0.96 | ±0.04 | 0.10 | ±0.10 | 0.06 | ±0.05 | 0.89 | ±0.16 | 0.96 | ±0.07
IF | 0.89 | ±0.14 | 0.32 | ±0.36 | 0.15 | ±0.18 | 0.89 | ±0.18 | 0.80 | ±0.25
$s_{C}$ | 0.88 | ±0.18 | 0.29 | ±0.38 | 0.18 | ±0.22 | 0.88 | ±0.19 | 0.93 | ±0.11
LOF | 0.84 | ±0.15 | 0.51 | ±0.35 | 0.24 | ±0.23 | 0.84 | ±0.23 | 0.71 | ±0.27
$s_{M}$ | 0.81 | ±0.18 | 0.48 | ±0.39 | 0.19 | ±0.19 | 0.84 | ±0.21 | 0.75 | ±0.24
$s_{C}*$ | 0.60 | ±0.10 | 0.90 | ±0.15 | 0.43 | ±0.26 | 0.60 | ±0.29 | 0.53 | ±0.31
LoF* | 0.50 | ±0.00 | 1.00 | ±0.00 | 0.51 | ±0.31 | 0.75 | ±0.16 | 0.62 | ±0.28
MSP | Bas. | MSP | 0.86 | ±0.15 | 0.35 | ±0.26 | 0.17 | ±0.16 | 0.81 | ±0.24 | 0.81 | ±0.20

Table 9: Average performance on the english benchamrk.

|  |  |  AUROC $\uparrow$ |  FPR $\downarrow$ | Err | AUPR-IN | AUPR-OUT
---|---|---|---|---|---|---|---
Metric | Ours | Agg. |  |  |  |  |  |  |  |  |  |
$E$ | Bas. | $E$ | 0.69 | ±0.16 | 0.62 | ±0.25 | 0.31 | ±0.19 | 0.62 | ±0.26 | 0.69 | ±0.27
$s_{M}$ | Bas. | Logits | 0.61 | ±0.08 | 0.91 | ±0.07 | 0.52 | ±0.24 | 0.55 | ±0.26 | 0.61 | ±0.28
$\operatorname{Agg}_{\emptyset}$ | Median | 0.72 | ±0.15 | 0.69 | ±0.25 | 0.41 | ±0.21 | 0.61 | ±0.27 | 0.73 | ±0.26
PW | 0.69 | ±0.12 | 0.83 | ±0.18 | 0.43 | ±0.24 | 0.70 | ±0.24 | 0.61 | ±0.29
Mean | 0.62 | ±0.10 | 0.80 | ±0.16 | 0.43 | ±0.23 | 0.55 | ±0.28 | 0.63 | ±0.26
Max | 0.60 | ±0.08 | 0.84 | ±0.15 | 0.45 | ±0.25 | 0.56 | ±0.29 | 0.60 | ±0.26
Min | 0.50 | ±0.00 | 1.00 | ±0.00 | 0.50 | ±0.28 | 0.75 | ±0.14 | 0.75 | ±0.14
$\operatorname{Agg}$ | IF | 0.88 | ±0.14 | 0.35 | ±0.31 | 0.18 | ±0.19 | 0.85 | ±0.19 | 0.84 | ±0.24
$s_{M}$ | 0.78 | ±0.16 | 0.59 | ±0.34 | 0.31 | ±0.25 | 0.75 | ±0.24 | 0.73 | ±0.29
$s_{C}$ | 0.74 | ±0.19 | 0.64 | ±0.41 | 0.33 | ±0.20 | 0.79 | ±0.15 | 0.79 | ±0.15
LOF | 0.73 | ±0.14 | 0.74 | ±0.23 | 0.39 | ±0.24 | 0.67 | ±0.29 | 0.66 | ±0.27
$s_{C}*$ | 0.59 | ±0.12 | 0.89 | ±0.15 | 0.45 | ±0.27 | 0.59 | ±0.31 | 0.59 | ±0.24
$s_{M}*$ | 0.57 | ±0.05 | 0.89 | ±0.06 | 0.37 | ±0.22 | 0.66 | ±0.25 | 0.44 | ±0.25
IF* | 0.56 | ±0.05 | 0.89 | ±0.05 | 0.47 | ±0.23 | 0.52 | ±0.28 | 0.56 | ±0.28
LoF* | 0.50 | ±0.01 | 1.00 | ±0.01 | 0.41 | ±0.27 | 0.70 | ±0.20 | 0.61 | ±0.21
$s_{C}$ | Bas. | Logits | 0.64 | ±0.12 | 0.96 | ±0.10 | 0.50 | ±0.26 | 0.62 | ±0.26 | 0.61 | ±0.27
$\operatorname{Agg}_{\emptyset}$ | Median | 0.92 | ±0.13 | 0.26 | ±0.32 | 0.17 | ±0.23 | 0.90 | ±0.21 | 0.91 | ±0.16
PW | 0.89 | ±0.14 | 0.29 | ±0.30 | 0.17 | ±0.20 | 0.85 | ±0.20 | 0.88 | ±0.20
Mean | 0.89 | ±0.15 | 0.36 | ±0.35 | 0.21 | ±0.23 | 0.89 | ±0.20 | 0.85 | ±0.21
Min | 0.84 | ±0.15 | 0.56 | ±0.33 | 0.29 | ±0.24 | 0.85 | ±0.19 | 0.76 | ±0.24
Max | 0.61 | ±0.16 | 0.92 | ±0.23 | 0.48 | ±0.29 | 0.70 | ±0.26 | 0.74 | ±0.19
$\operatorname{Agg}$ | $s_{C}$ | 1.00 | ±0.00 | 0.00 | ±0.00 | 0.02 | ±0.01 | 1.00 | ±0.00 | 1.00 | ±0.00
IF | 0.92 | ±0.13 | 0.13 | ±0.22 | 0.07 | ±0.14 | 0.86 | ±0.25 | 0.85 | ±0.19
LOF | 0.87 | ±0.14 | 0.33 | ±0.30 | 0.19 | ±0.20 | 0.82 | ±0.21 | 0.87 | ±0.21
$s_{M}$ | 0.87 | ±0.15 | 0.33 | ±0.31 | 0.20 | ±0.22 | 0.81 | ±0.23 | 0.87 | ±0.20
IF* | 0.61 | ±0.06 | 0.81 | ±0.11 | 0.47 | ±0.21 | 0.49 | ±0.27 | 0.65 | ±0.25
$s_{M}*$ | 0.59 | ±0.08 | 0.86 | ±0.12 | 0.45 | ±0.23 | 0.54 | ±0.28 | 0.57 | ±0.29
$s_{C}*$ | 0.51 | ±0.01 | 0.99 | ±0.00 | 0.59 | ±0.26 | 0.42 | ±0.27 | 0.60 | ±0.26
LoF* | 0.50 | ±0.00 | 1.00 | ±0.00 | 0.50 | ±0.28 | 0.75 | ±0.14 | 0.75 | ±0.14
$s_{\text{IRW}}$ | Bas. | Logits | 0.74 | ±0.10 | 0.59 | ±0.21 | 0.31 | ±0.19 | 0.64 | ±0.25 | 0.74 | ±0.21
$\operatorname{Agg}_{\emptyset}$ | Median | 0.83 | ±0.14 | 0.59 | ±0.35 | 0.34 | ±0.26 | 0.80 | ±0.24 | 0.77 | ±0.24
Mean | 0.82 | ±0.15 | 0.60 | ±0.35 | 0.37 | ±0.28 | 0.76 | ±0.27 | 0.79 | ±0.22
Max | 0.80 | ±0.15 | 0.63 | ±0.30 | 0.38 | ±0.25 | 0.73 | ±0.27 | 0.77 | ±0.22
PW | 0.66 | ±0.09 | 0.82 | ±0.18 | 0.42 | ±0.24 | 0.64 | ±0.28 | 0.60 | ±0.24
Min | 0.50 | ±0.00 | 1.00 | ±0.00 | 0.50 | ±0.28 | 0.75 | ±0.14 | 0.75 | ±0.14
$\operatorname{Agg}$ | $s_{C}$ | 0.90 | ±0.13 | 0.27 | ±0.31 | 0.16 | ±0.15 | 0.89 | ±0.15 | 0.91 | ±0.12
IF | 0.85 | ±0.15 | 0.44 | ±0.36 | 0.22 | ±0.21 | 0.83 | ±0.24 | 0.79 | ±0.27
$s_{M}$ | 0.77 | ±0.17 | 0.60 | ±0.34 | 0.28 | ±0.23 | 0.78 | ±0.25 | 0.69 | ±0.30
LOF | 0.73 | ±0.14 | 0.76 | ±0.27 | 0.40 | ±0.25 | 0.70 | ±0.26 | 0.66 | ±0.29
IF* | 0.65 | ±0.08 | 0.69 | ±0.16 | 0.38 | ±0.20 | 0.54 | ±0.28 | 0.69 | ±0.22
$s_{M}*$ | 0.60 | ±0.07 | 0.85 | ±0.12 | 0.28 | ±0.15 | 0.74 | ±0.19 | 0.40 | ±0.21
$s_{C}*$ | 0.52 | ±0.01 | 0.96 | ±0.01 | 0.55 | ±0.26 | 0.46 | ±0.28 | 0.57 | ±0.28
LoF* | 0.50 | ±0.00 | 1.00 | ±0.00 | 0.53 | ±0.28 | 0.73 | ±0.14 | 0.64 | ±0.26
MSP | Bas. | MSP | 0.69 | ±0.16 | 0.62 | ±0.25 | 0.31 | ±0.19 | 0.63 | ±0.25 | 0.69 | ±0.27

Table 10: Average performance on the french benchamrk.

|  |  |  AUROC $\uparrow$ |  FPR $\downarrow$ | Err | AUPR-IN | AUPR-OUT
---|---|---|---|---|---|---|---
Metric | Ours | Agg. |  |  |  |  |  |  |  |  |  |
$E$ | Bas. | $E$ | 0.69 | ±0.15 | 0.67 | ±0.29 | 0.35 | ±0.18 | 0.67 | ±0.16 | 0.70 | ±0.17
$s_{M}$ | Bas. | Logits | 0.62 | ±0.08 | 0.85 | ±0.07 | 0.48 | ±0.12 | 0.55 | ±0.16 | 0.65 | ±0.12
$\operatorname{Agg}_{\emptyset}$ | Median | 0.71 | ±0.15 | 0.78 | ±0.22 | 0.43 | ±0.17 | 0.69 | ±0.20 | 0.69 | ±0.20
PW | 0.70 | ±0.13 | 0.83 | ±0.15 | 0.46 | ±0.16 | 0.69 | ±0.19 | 0.69 | ±0.13
Mean | 0.58 | ±0.07 | 0.90 | ±0.07 | 0.47 | ±0.11 | 0.56 | ±0.16 | 0.57 | ±0.15
Max | 0.57 | ±0.06 | 0.91 | ±0.07 | 0.48 | ±0.12 | 0.55 | ±0.17 | 0.56 | ±0.14
Min | 0.50 | ±0.00 | 1.00 | ±0.00 | 0.50 | ±0.14 | 0.75 | ±0.07 | 0.75 | ±0.07
$\operatorname{Agg}$ | IF | 0.85 | ±0.13 | 0.45 | ±0.27 | 0.24 | ±0.13 | 0.83 | ±0.15 | 0.83 | ±0.18
LOF | 0.74 | ±0.07 | 0.80 | ±0.11 | 0.42 | ±0.11 | 0.72 | ±0.13 | 0.70 | ±0.13
$s_{M}$ | 0.65 | ±0.09 | 0.83 | ±0.10 | 0.45 | ±0.14 | 0.62 | ±0.18 | 0.65 | ±0.09
$s_{C}$ | 0.61 | ±0.10 | 0.91 | ±0.12 | 0.46 | ±0.05 | 0.64 | ±0.10 | 0.65 | ±0.10
IF* | 0.56 | ±0.04 | 0.89 | ±0.05 | 0.42 | ±0.11 | 0.60 | ±0.13 | 0.51 | ±0.11
$s_{C}*$ | 0.56 | ±0.08 | 0.94 | ±0.09 | 0.45 | ±0.12 | 0.63 | ±0.14 | 0.54 | ±0.17
$s_{M}*$ | 0.53 | ±0.02 | 0.93 | ±0.02 | 0.49 | ±0.13 | 0.52 | ±0.14 | 0.53 | ±0.14
LoF* | 0.50 | ±0.00 | 1.00 | ±0.00 | 0.47 | ±0.14 | 0.68 | ±0.17 | 0.65 | ±0.14
$s_{C}$ | Bas. | Logits | 0.66 | ±0.10 | 0.87 | ±0.09 | 0.46 | ±0.12 | 0.64 | ±0.13 | 0.64 | ±0.17
$\operatorname{Agg}_{\emptyset}$ | PW | 0.85 | ±0.13 | 0.43 | ±0.32 | 0.23 | ±0.17 | 0.84 | ±0.16 | 0.84 | ±0.17
Median | 0.85 | ±0.13 | 0.48 | ±0.37 | 0.28 | ±0.21 | 0.84 | ±0.16 | 0.84 | ±0.16
Mean | 0.85 | ±0.14 | 0.49 | ±0.36 | 0.28 | ±0.21 | 0.84 | ±0.16 | 0.83 | ±0.18
Min | 0.82 | ±0.14 | 0.58 | ±0.30 | 0.33 | ±0.18 | 0.82 | ±0.16 | 0.80 | ±0.18
Max | 0.52 | ±0.04 | 1.00 | ±0.00 | 0.49 | ±0.14 | 0.68 | ±0.17 | 0.74 | ±0.07
$\operatorname{Agg}$ | $s_{C}$ | 1.00 | ±0.00 | 0.00 | ±0.00 | 0.01 | ±0.01 | 1.00 | ±0.00 | 1.00 | ±0.00
IF | 0.96 | ±0.11 | 0.16 | ±0.38 | 0.09 | ±0.21 | 0.95 | ±0.18 | 0.92 | ±0.16
$s_{M}$ | 0.84 | ±0.14 | 0.45 | ±0.32 | 0.24 | ±0.18 | 0.82 | ±0.18 | 0.83 | ±0.16
LOF | 0.83 | ±0.14 | 0.45 | ±0.31 | 0.24 | ±0.17 | 0.81 | ±0.18 | 0.83 | ±0.16
IF* | 0.58 | ±0.07 | 0.84 | ±0.10 | 0.44 | ±0.14 | 0.55 | ±0.15 | 0.59 | ±0.15
$s_{M}*$ | 0.55 | ±0.02 | 0.91 | ±0.03 | 0.48 | ±0.12 | 0.52 | ±0.14 | 0.56 | ±0.14
$s_{C}*$ | 0.51 | ±0.01 | 0.99 | ±0.00 | 0.47 | ±0.13 | 0.54 | ±0.15 | 0.47 | ±0.14
LoF* | 0.50 | ±0.00 | 1.00 | ±0.00 | 0.50 | ±0.14 | 0.75 | ±0.07 | 0.75 | ±0.07
$s_{\text{IRW}}$ | Bas. | Logits | 0.69 | ±0.08 | 0.63 | ±0.17 | 0.35 | ±0.13 | 0.59 | ±0.15 | 0.75 | ±0.14
$\operatorname{Agg}_{\emptyset}$ | Mean | 0.79 | ±0.16 | 0.67 | ±0.32 | 0.37 | ±0.21 | 0.80 | ±0.18 | 0.75 | ±0.17
Median | 0.78 | ±0.17 | 0.64 | ±0.41 | 0.37 | ±0.26 | 0.77 | ±0.22 | 0.77 | ±0.16
Max | 0.76 | ±0.15 | 0.74 | ±0.25 | 0.41 | ±0.18 | 0.77 | ±0.18 | 0.74 | ±0.16
PW | 0.68 | ±0.13 | 0.87 | ±0.11 | 0.45 | ±0.14 | 0.68 | ±0.21 | 0.63 | ±0.13
Min | 0.50 | ±0.00 | 1.00 | ±0.00 | 0.50 | ±0.14 | 0.75 | ±0.07 | 0.75 | ±0.07
$\operatorname{Agg}$ | $s_{C}$ | 0.83 | ±0.14 | 0.50 | ±0.33 | 0.28 | ±0.17 | 0.80 | ±0.15 | 0.83 | ±0.14
IF | 0.80 | ±0.15 | 0.53 | ±0.31 | 0.29 | ±0.17 | 0.77 | ±0.16 | 0.79 | ±0.19
$s_{M}$ | 0.69 | ±0.16 | 0.73 | ±0.28 | 0.35 | ±0.12 | 0.69 | ±0.11 | 0.65 | ±0.24
IF* | 0.63 | ±0.08 | 0.74 | ±0.15 | 0.41 | ±0.13 | 0.55 | ±0.15 | 0.69 | ±0.14
LOF | 0.61 | ±0.09 | 0.91 | ±0.04 | 0.47 | ±0.12 | 0.63 | ±0.12 | 0.57 | ±0.15
$s_{M}*$ | 0.57 | ±0.05 | 0.89 | ±0.08 | 0.37 | ±0.07 | 0.66 | ±0.09 | 0.46 | ±0.10
$s_{C}*$ | 0.51 | ±0.01 | 0.96 | ±0.02 | 0.53 | ±0.13 | 0.48 | ±0.13 | 0.55 | ±0.14
LoF* | 0.50 | ±0.00 | 0.99 | ±0.00 | 0.46 | ±0.13 | 0.77 | ±0.07 | 0.62 | ±0.18
MSP | Bas. | MSP | 0.68 | ±0.15 | 0.69 | ±0.28 | 0.36 | ±0.17 | 0.65 | ±0.16 | 0.69 | ±0.17

Table 11: Average performance on the spanish benchamrk.

|  |  |  AUROC $\uparrow$ |  FPR $\downarrow$ | Err | AUPR-IN | AUPR-OUT
---|---|---|---|---|---|---|---
Metric | Ours | Agg. |  |  |  |  |  |  |  |  |  |
$E$ | Bas. | $E$ | 0.55 | ±0.29 | 0.67 | ±0.28 | 0.36 | ±0.27 | 0.54 | ±0.34 | 0.65 | ±0.33
$s_{M}$ | Bas. | Logits | 0.66 | ±0.11 | 0.82 | ±0.20 | 0.54 | ±0.31 | 0.51 | ±0.35 | 0.71 | ±0.29
$\operatorname{Agg}_{\emptyset}$ | Median | 0.78 | ±0.15 | 0.58 | ±0.32 | 0.38 | ±0.27 | 0.64 | ±0.27 | 0.82 | ±0.27
PW | 0.76 | ±0.13 | 0.72 | ±0.16 | 0.43 | ±0.26 | 0.72 | ±0.24 | 0.73 | ±0.28
Max | 0.65 | ±0.10 | 0.82 | ±0.17 | 0.24 | ±0.18 | 0.83 | ±0.16 | 0.39 | ±0.26
Mean | 0.63 | ±0.10 | 0.83 | ±0.14 | 0.38 | ±0.27 | 0.66 | ±0.31 | 0.53 | ±0.30
Min | 0.50 | ±0.00 | 1.00 | ±0.00 | 0.50 | ±0.34 | 0.75 | ±0.17 | 0.75 | ±0.17
$\operatorname{Agg}$ | IF | 0.90 | ±0.15 | 0.24 | ±0.33 | 0.07 | ±0.06 | 0.96 | ±0.04 | 0.81 | ±0.32
$s_{M}$ | 0.74 | ±0.15 | 0.62 | ±0.30 | 0.35 | ±0.28 | 0.71 | ±0.29 | 0.73 | ±0.30
LOF | 0.74 | ±0.11 | 0.70 | ±0.27 | 0.41 | ±0.28 | 0.63 | ±0.34 | 0.69 | ±0.28
$s_{C}$ | 0.70 | ±0.15 | 0.75 | ±0.32 | 0.39 | ±0.15 | 0.73 | ±0.13 | 0.73 | ±0.15
$s_{C}*$ | 0.63 | ±0.11 | 0.96 | ±0.09 | 0.25 | ±0.21 | 0.84 | ±0.15 | 0.40 | ±0.19
IF* | 0.57 | ±0.04 | 0.87 | ±0.05 | 0.53 | ±0.28 | 0.44 | ±0.33 | 0.64 | ±0.30
$s_{M}*$ | 0.57 | ±0.06 | 0.90 | ±0.06 | 0.26 | ±0.18 | 0.78 | ±0.20 | 0.30 | ±0.22
LoF* | 0.50 | ±0.00 | 1.00 | ±0.01 | 0.31 | ±0.27 | 0.76 | ±0.20 | 0.50 | ±0.26
$s_{C}$ | Bas. | Logits | 0.61 | ±0.08 | 0.99 | ±0.02 | 0.58 | ±0.32 | 0.51 | ±0.31 | 0.65 | ±0.31
$\operatorname{Agg}_{\emptyset}$ | Median | 0.93 | ±0.12 | 0.23 | ±0.35 | 0.22 | ±0.32 | 0.89 | ±0.22 | 0.97 | ±0.02
Mean | 0.93 | ±0.11 | 0.28 | ±0.40 | 0.26 | ±0.35 | 0.90 | ±0.18 | 0.97 | ±0.03
PW | 0.91 | ±0.14 | 0.18 | ±0.24 | 0.06 | ±0.04 | 0.96 | ±0.03 | 0.86 | ±0.23
Min | 0.90 | ±0.12 | 0.39 | ±0.37 | 0.32 | ±0.34 | 0.86 | ±0.21 | 0.93 | ±0.06
Max | 0.56 | ±0.07 | 0.98 | ±0.04 | 0.38 | ±0.31 | 0.69 | ±0.31 | 0.60 | ±0.25
$\operatorname{Agg}$ | $s_{C}$ | 1.00 | ±0.00 | 0.00 | ±0.00 | 0.02 | ±0.01 | 1.00 | ±0.00 | 1.00 | ±0.00
IF | 0.93 | ±0.13 | 0.11 | ±0.18 | 0.02 | ±0.03 | 0.95 | ±0.07 | 0.84 | ±0.16
$s_{M}$ | 0.92 | ±0.12 | 0.17 | ±0.20 | 0.06 | ±0.04 | 0.96 | ±0.03 | 0.88 | ±0.19
LOF | 0.91 | ±0.12 | 0.19 | ±0.19 | 0.06 | ±0.04 | 0.96 | ±0.03 | 0.87 | ±0.19
IF* | 0.61 | ±0.06 | 0.78 | ±0.10 | 0.51 | ±0.23 | 0.40 | ±0.31 | 0.73 | ±0.27
$s_{M}*$ | 0.56 | ±0.03 | 0.90 | ±0.03 | 0.43 | ±0.29 | 0.58 | ±0.34 | 0.49 | ±0.33
$s_{C}*$ | 0.51 | ±0.00 | 0.99 | ±0.00 | 0.75 | ±0.21 | 0.24 | ±0.21 | 0.76 | ±0.20
LoF* | 0.50 | ±0.00 | 1.00 | ±0.00 | 0.50 | ±0.34 | 0.75 | ±0.17 | 0.75 | ±0.17
$s_{\text{IRW}}$ | Bas. | Logits | 0.69 | ±0.09 | 0.75 | ±0.21 | 0.48 | ±0.29 | 0.54 | ±0.30 | 0.75 | ±0.25
$\operatorname{Agg}_{\emptyset}$ | Mean | 0.92 | ±0.09 | 0.35 | ±0.39 | 0.24 | ±0.32 | 0.89 | ±0.19 | 0.87 | ±0.15
Median | 0.91 | ±0.13 | 0.40 | ±0.43 | 0.24 | ±0.31 | 0.88 | ±0.23 | 0.83 | ±0.21
Max | 0.88 | ±0.13 | 0.40 | ±0.33 | 0.28 | ±0.31 | 0.84 | ±0.24 | 0.84 | ±0.19
PW | 0.79 | ±0.14 | 0.86 | ±0.25 | 0.47 | ±0.35 | 0.77 | ±0.25 | 0.65 | ±0.26
Min | 0.50 | ±0.00 | 1.00 | ±0.00 | 0.50 | ±0.34 | 0.75 | ±0.17 | 0.75 | ±0.17
$\operatorname{Agg}$ | $s_{C}$ | 0.96 | ±0.07 | 0.13 | ±0.19 | 0.09 | ±0.09 | 0.94 | ±0.10 | 0.97 | ±0.06
IF | 0.90 | ±0.18 | 0.23 | ±0.34 | 0.16 | ±0.24 | 0.85 | ±0.26 | 0.91 | ±0.19
$s_{M}$ | 0.87 | ±0.17 | 0.34 | ±0.40 | 0.25 | ±0.31 | 0.77 | ±0.30 | 0.90 | ±0.20
LOF | 0.84 | ±0.17 | 0.47 | ±0.40 | 0.33 | ±0.33 | 0.75 | ±0.29 | 0.86 | ±0.23
IF* | 0.67 | ±0.09 | 0.66 | ±0.19 | 0.41 | ±0.21 | 0.44 | ±0.31 | 0.75 | ±0.30
$s_{M}*$ | 0.65 | ±0.09 | 0.80 | ±0.14 | 0.21 | ±0.12 | 0.83 | ±0.15 | 0.36 | ±0.20
$s_{C}*$ | 0.51 | ±0.01 | 0.96 | ±0.01 | 0.65 | ±0.28 | 0.34 | ±0.29 | 0.68 | ±0.28
LoF* | 0.50 | ±0.00 | 1.00 | ±0.00 | 0.59 | ±0.33 | 0.70 | ±0.16 | 0.72 | ±0.26
MSP | Bas. | MSP | 0.56 | ±0.28 | 0.67 | ±0.27 | 0.36 | ±0.27 | 0.55 | ±0.34 | 0.65 | ±0.33

Table 12: Average performance on the german benchamrk.

In addition to the global performance, we provide the average OOD detection
performance per IN-DS in Tab. 13 Tab. 15 Tab. 19 and Tab. 17 and per OUT-DS in
Tab. 14 Tab. 16 Tab. 20 and Tab. 18.

|  | in | 20ng | emotion | go-emotions | imdb | massive | rte | sst2 | trec | trec-fine
---|---|---|---|---|---|---|---|---|---|---|---
Metric | Ours | Agg. |  |  |  |  |  |  |  |  |
$s_{C}$ | $\operatorname{Agg}$ | IF | 0.95 | 0.95 | 0.95 | 0.91 | 0.95 | 0.82 | 0.91 | 0.95 | 0.95
LOF | 0.99 | 0.99 | 0.81 | 0.97 | 0.92 | 0.81 | 0.94 | 0.93 | 0.90
$\operatorname{Agg}_{\emptyset}$ | Mean | 0.99 | 0.99 | 0.82 | 0.92 | 0.92 | 0.84 | 0.96 | 0.93 | 0.90
Median | 0.89 | 0.98 | 0.81 | 0.96 | 0.92 | 0.91 | 0.90 | 0.91 | 0.88
PW | 0.98 | 0.99 | 0.79 | 0.96 | 0.92 | 0.84 | 0.96 | 0.93 | 0.90
Bas. | Last layer | 0.96 | 0.96 | 0.83 | 0.95 | 0.89 | 0.83 | 0.87 | 0.93 | 0.88
Logits | 0.97 | 0.96 | 0.69 | 0.78 | 0.88 | 0.57 | 0.67 | 0.91 | 0.86
$s_{\text{IRW}}$ | $\operatorname{Agg}$ | IF | 0.99 | 0.95 | 0.81 | 0.98 | 0.85 | 0.91 | 0.77 | 0.90 | 0.88
LOF | 0.92 | 0.90 | 0.66 | 0.78 | 0.75 | 0.84 | 0.78 | 0.89 | 0.85
$\operatorname{Agg}_{\emptyset}$ | Mean | 0.98 | 0.91 | 0.84 | 0.99 | 0.80 | 0.95 | 0.82 | 0.90 | 0.50
Median | 0.95 | 0.89 | 0.82 | 0.99 | 0.78 | 0.91 | 0.85 | 0.86 | 0.50
PW | 0.87 | 0.84 | 0.73 | 0.91 | 0.67 | 0.79 | 0.77 | 0.72 | 0.50
Bas. | Last layer | 0.85 | 0.79 | 0.72 | 0.87 | 0.71 | 0.80 | 0.81 | 0.77 | 0.50
Logits | 0.88 | 0.90 | 0.61 | 0.88 | 0.62 | 0.57 | 0.84 | 0.60 | 0.50
$s_{M}$ | $\operatorname{Agg}$ | IF | 1.00 | 0.98 | 0.80 | 1.00 | 0.92 | 0.91 | 0.94 | 0.90 | 0.89
LOF | 0.94 | 0.92 | 0.73 | 0.85 | 0.87 | 0.85 | 0.84 | 0.86 | 0.92
$\operatorname{Agg}_{\emptyset}$ | Mean | 0.91 | 0.83 | 0.65 | 0.72 | 0.72 | 0.68 | 0.72 | 0.74 | 0.84
Median | 0.79 | 0.73 | 0.66 | 0.71 | 0.71 | 0.76 | 0.68 | 0.72 | 0.86
PW | 0.85 | 0.85 | 0.65 | 0.87 | 0.77 | 0.76 | 0.72 | 0.83 | 0.90
Bas. | Last layer | 0.93 | 0.85 | 0.64 | 0.77 | 0.79 | 0.75 | 0.72 | 0.83 | 0.88
Logits | 0.85 | 0.71 | 0.65 | 0.73 | 0.62 | 0.61 | 0.66 | 0.65 | 0.76

Table 13: Performance (AUROC $\uparrow$) of different aggregation per english
IN-DS .

|  |  |  AUROC $\uparrow$
---|---|---|---
|  | out | 20ng | amazon-reviews-multi | b77 | emotion | go-emotions | imdb | massive | rte | snli | sst2 | trec | trec-fine
Metric | Ours | Agg. |  |  |  |  |  |  |  |  |  |  |  |
$s_{C}$ | $\operatorname{Agg}$ | IF | 0.54 | 0.41 | 0.52 | 0.53 | 0.53 | 0.54 | 0.53 | 0.53 | 0.54 | 0.54 | 0.52 | 0.54
LOF | 0.52 | 0.52 | 0.50 | 0.51 | 0.51 | 0.52 | 0.51 | 0.50 | 0.53 | 0.51 | 0.52 | 0.52
$\operatorname{Agg}_{\emptyset}$ | Mean | 0.52 | 0.53 | 0.51 | 0.51 | 0.52 | 0.53 | 0.51 | 0.50 | 0.53 | 0.51 | 0.52 | 0.52
Median | 0.51 | 0.53 | 0.50 | 0.51 | 0.52 | 0.52 | 0.53 | 0.51 | 0.52 | 0.52 | 0.52 | 0.52
PW | 0.52 | 0.53 | 0.51 | 0.51 | 0.52 | 0.52 | 0.52 | 0.51 | 0.53 | 0.52 | 0.52 | 0.52
Bas. | Last layer | 0.53 | 0.52 | 0.49 | 0.51 | 0.51 | 0.53 | 0.52 | 0.50 | 0.52 | 0.51 | 0.51 | 0.52
Logits | 0.50 | 0.51 | 0.48 | 0.50 | 0.51 | 0.51 | 0.51 | 0.50 | 0.52 | 0.51 | 0.49 | 0.50
$s_{\text{IRW}}$ | $\operatorname{Agg}$ | IF | 0.51 | 0.50 | 0.49 | 0.52 | 0.49 | 0.51 | 0.51 | 0.51 | 0.52 | 0.49 | 0.50 | 0.51
LOF | 0.53 | 0.49 | 0.47 | 0.48 | 0.45 | 0.52 | 0.50 | 0.50 | 0.48 | 0.46 | 0.48 | 0.49
$\operatorname{Agg}_{\emptyset}$ | Mean | 0.54 | 0.52 | 0.49 | 0.50 | 0.49 | 0.54 | 0.50 | 0.53 | 0.51 | 0.50 | 0.49 | 0.50
Median | 0.53 | 0.51 | 0.48 | 0.49 | 0.48 | 0.52 | 0.50 | 0.52 | 0.50 | 0.49 | 0.50 | 0.50
PW | 0.48 | 0.44 | 0.45 | 0.46 | 0.43 | 0.47 | 0.46 | 0.47 | 0.46 | 0.46 | 0.47 | 0.48
Bas. | Last layer | 0.49 | 0.47 | 0.44 | 0.45 | 0.44 | 0.48 | 0.46 | 0.47 | 0.47 | 0.45 | 0.46 | 0.45
Logits | 0.44 | 0.47 | 0.45 | 0.42 | 0.44 | 0.44 | 0.47 | 0.45 | 0.45 | 0.44 | 0.46 | 0.47
$s_{M}$ | $\operatorname{Agg}$ | IF | 0.51 | 0.50 | 0.51 | 0.51 | 0.50 | 0.52 | 0.51 | 0.50 | 0.53 | 0.51 | 0.53 | 0.52
LOF | 0.52 | 0.51 | 0.50 | 0.51 | 0.50 | 0.52 | 0.51 | 0.53 | 0.52 | 0.50 | 0.51 | 0.51
$\operatorname{Agg}_{\emptyset}$ | Mean | 0.52 | 0.45 | 0.46 | 0.47 | 0.41 | 0.52 | 0.45 | 0.47 | 0.50 | 0.42 | 0.46 | 0.42
Median | 0.52 | 0.47 | 0.42 | 0.45 | 0.41 | 0.51 | 0.47 | 0.44 | 0.49 | 0.42 | 0.47 | 0.43
PW | 0.51 | 0.49 | 0.47 | 0.48 | 0.47 | 0.51 | 0.50 | 0.50 | 0.52 | 0.46 | 0.49 | 0.47
Bas. | Last layer | 0.52 | 0.50 | 0.48 | 0.49 | 0.48 | 0.52 | 0.49 | 0.51 | 0.51 | 0.48 | 0.48 | 0.47
Logits | 0.44 | 0.44 | 0.43 | 0.44 | 0.43 | 0.43 | 0.45 | 0.42 | 0.43 | 0.43 | 0.43 | 0.44

Table 14: Performance (AUROC $\uparrow$) of different aggregation methods per
english OUT-DS .

|  | in | fr-allocine | fr-cls | fr-pawsx | fr-swiss-judgement | fr-tweet-sentiment | fr-xnli | fr-xstance
---|---|---|---|---|---|---|---|---|---
Metric | Ours | Agg. |  |  |  |  |  |  |
$s_{C}$ | $\operatorname{Agg}$ | IF | 0.87 | 1.00 | 0.84 | 0.95 | 1.00 | 1.00 | 0.81
LOF | 0.91 | 1.00 | 0.95 | 0.87 | 0.66 | 0.82 | 0.89
$\operatorname{Agg}_{\emptyset}$ | Mean | 0.92 | 1.00 | 0.98 | 0.90 | 0.67 | 0.82 | 0.93
Median | 0.89 | 1.00 | 0.99 | 0.98 | 0.73 | 0.92 | 0.95
PW | 0.91 | 1.00 | 0.98 | 0.88 | 0.69 | 0.85 | 0.93
Bas. | Last layer | 0.89 | 1.00 | 0.91 | 0.86 | 0.63 | 0.75 | 0.76
Logits | 0.82 | 0.67 | 0.64 | 0.56 | 0.57 | 0.58 | 0.62
$s_{\text{IRW}}$ | $\operatorname{Agg}$ | IF | 0.73 | 0.74 | 0.95 | 0.99 | 0.86 | 0.89 | 0.80
LOF | 0.63 | 0.77 | 0.68 | 0.85 | 0.72 | 0.77 | 0.67
$\operatorname{Agg}_{\emptyset}$ | Mean | 0.71 | 0.76 | 0.88 | 0.93 | 0.79 | 0.83 | 0.83
Median | 0.71 | 0.75 | 0.96 | 0.86 | 0.84 | 0.83 | 0.84
PW | 0.67 | 0.59 | 0.65 | 0.74 | 0.66 | 0.71 | 0.60
Bas. | Last layer | 0.81 | 0.68 | 0.75 | 0.58 | 0.74 | 0.74 | 0.63
Logits | 0.85 | 0.73 | 0.78 | 0.66 | 0.74 | 0.76 | 0.66
$s_{M}$ | $\operatorname{Agg}$ | IF | 0.89 | 0.94 | 0.95 | 0.93 | 0.67 | 0.86 | 0.92
LOF | 0.70 | 0.87 | 0.84 | 0.60 | 0.78 | 0.56 | 0.73
$\operatorname{Agg}_{\emptyset}$ | Mean | 0.63 | 0.58 | 0.57 | 0.65 | 0.71 | 0.62 | 0.54
Median | 0.75 | 0.82 | 0.65 | 0.63 | 0.71 | 0.67 | 0.80
PW | 0.69 | 0.69 | 0.66 | 0.62 | 0.76 | 0.68 | 0.69
Bas. | Last layer | 0.85 | 0.90 | 0.90 | 0.82 | 0.63 | 0.72 | 0.82
Logits | 0.64 | 0.59 | 0.59 | 0.58 | 0.64 | 0.62 | 0.62

Table 15: Performance (AUROC $\uparrow$) of different aggregation per french
IN-DS .

|  |  |  AUROC $\uparrow$
---|---|---|---
|  | out | fr-allocine | fr-cls | fr-pawsx | fr-swiss-judgement | fr-tweet-sentiment | fr-xnli | fr-xstance
Metric | Ours | Agg. |  |  |  |  |  |  |
$s_{C}$ | $\operatorname{Agg}$ | IF | 0.51 | 0.54 | 0.50 | 0.51 | 0.54 | 0.53 | 0.50
LOF | 0.50 | 0.46 | 0.51 | 0.51 | 0.50 | 0.49 | 0.49
$\operatorname{Agg}_{\emptyset}$ | Mean | 0.52 | 0.48 | 0.52 | 0.51 | 0.50 | 0.51 | 0.51
Median | 0.52 | 0.51 | 0.50 | 0.50 | 0.50 | 0.52 | 0.52
PW | 0.51 | 0.48 | 0.50 | 0.51 | 0.50 | 0.50 | 0.50
Bas. | Last layer | 0.48 | 0.43 | 0.50 | 0.50 | 0.51 | 0.48 | 0.49
Logits | 0.28 | 0.29 | 0.43 | 0.41 | 0.40 | 0.37 | 0.37
$s_{\text{IRW}}$ | $\operatorname{Agg}$ | IF | 0.51 | 0.50 | 0.48 | 0.52 | 0.50 | 0.45 | 0.47
LOF | 0.38 | 0.38 | 0.44 | 0.50 | 0.46 | 0.40 | 0.37
$\operatorname{Agg}_{\emptyset}$ | Mean | 0.46 | 0.46 | 0.48 | 0.51 | 0.50 | 0.42 | 0.41
Median | 0.47 | 0.47 | 0.46 | 0.51 | 0.50 | 0.43 | 0.43
PW | 0.35 | 0.35 | 0.41 | 0.37 | 0.41 | 0.37 | 0.37
Bas. | Last layer | 0.40 | 0.40 | 0.42 | 0.41 | 0.43 | 0.40 | 0.39
Logits | 0.39 | 0.39 | 0.45 | 0.43 | 0.45 | 0.42 | 0.40
$s_{M}$ | $\operatorname{Agg}$ | IF | 0.48 | 0.48 | 0.51 | 0.51 | 0.50 | 0.50 | 0.49
LOF | 0.41 | 0.40 | 0.44 | 0.50 | 0.46 | 0.38 | 0.40
$\operatorname{Agg}_{\emptyset}$ | Mean | 0.35 | 0.36 | 0.37 | 0.42 | 0.34 | 0.31 | 0.34
Median | 0.38 | 0.39 | 0.46 | 0.48 | 0.51 | 0.33 | 0.37
PW | 0.35 | 0.36 | 0.45 | 0.46 | 0.44 | 0.34 | 0.35
Bas. | Last layer | 0.43 | 0.42 | 0.48 | 0.50 | 0.48 | 0.46 | 0.47
Logits | 0.34 | 0.34 | 0.37 | 0.38 | 0.33 | 0.29 | 0.33

Table 16: Performance (AUROC $\uparrow$) of different aggregation methods per
french OUT-DS .

|  | in | es-cine | es-pawsx | es-tweet-inde | es-tweet-sentiment
---|---|---|---|---|---|---
Metric | Ours | Agg. |  |  |  |
$s_{C}$ | $\operatorname{Agg}$ | IF | 1.00 | 0.87 | 0.98 | 1.00
LOF | 0.77 | 0.97 | 0.84 | 0.74
$\operatorname{Agg}-{\emptyset}$ | Mean | 0.79 | 0.99 | 0.85 | 0.74
Median | 0.86 | 0.99 | 0.82 | 0.72
PW | 0.84 | 0.99 | 0.85 | 0.73
Bas. | Last layer | 0.63 | 0.97 | 0.75 | 0.69
Logits | 0.60 | 0.74 | 0.70 | 0.58
$s_{\text{IRW}}$ | $\operatorname{Agg}$ | IF | 0.84 | 0.98 | 0.77 | 0.62
LOF | 0.60 | 0.67 | 0.61 | 0.58
$\operatorname{Agg}_{\emptyset}$ | Mean | 0.89 | 0.96 | 0.67 | 0.64
Median | 0.86 | 0.99 | 0.66 | 0.62
PW | 0.72 | 0.83 | 0.57 | 0.59
Bas. | Last layer | 0.66 | 0.69 | 0.61 | 0.59
Logits | 0.72 | 0.73 | 0.74 | 0.58
$s_{M}$ | $\operatorname{Agg}$ | IF | 0.92 | 0.96 | 0.83 | 0.69
LOF | 0.77 | 0.80 | 0.69 | 0.72
$\operatorname{Agg}_{\emptyset}$ | Mean | 0.53 | 0.57 | 0.55 | 0.66
Median | 0.90 | 0.63 | 0.73 | 0.56
PW | 0.67 | 0.71 | 0.74 | 0.70
Bas. | Last layer | 0.69 | 0.85 | 0.83 | 0.70
Logits | 0.58 | 0.67 | 0.55 | 0.66

Table 17: Performance (AUROC $\uparrow$) of different aggregation per spanish
IN-DS .

|  |  |  AUROC $\uparrow$
---|---|---|---
|  | out | es-cine | es-pawsx | es-tweet-inde | es-tweet-sentiment
Metric | Ours | Agg. |  |  |  |
$s_{C}$ | $\operatorname{Agg}$ | IF | 0.50 | 0.50 | 0.50 | 0.53
LOF | 0.50 | 0.49 | 0.48 | 0.48
$\operatorname{Agg}-{\emptyset}$ | Mean | 0.51 | 0.49 | 0.49 | 0.49
Median | 0.50 | 0.49 | 0.49 | 0.49
PW | 0.51 | 0.49 | 0.49 | 0.49
Bas. | Last layer | 0.49 | 0.43 | 0.48 | 0.47
Logits | 0.38 | 0.38 | 0.38 | 0.39
$s_{\text{IRW}}$ | $\operatorname{Agg}$ | IF | 0.47 | 0.43 | 0.49 | 0.49
LOF | 0.26 | 0.35 | 0.37 | 0.37
$\operatorname{Agg}_{\emptyset}$ | Mean | 0.45 | 0.43 | 0.51 | 0.51
Median | 0.48 | 0.41 | 0.51 | 0.50
PW | 0.41 | 0.31 | 0.43 | 0.46
Bas. | Last layer | 0.32 | 0.32 | 0.34 | 0.37
Logits | 0.37 | 0.42 | 0.37 | 0.39
$s_{M}$ | $\operatorname{Agg}$ | IF | 0.47 | 0.50 | 0.49 | 0.50
LOF | 0.36 | 0.41 | 0.41 | 0.42
$\operatorname{Agg}_{\emptyset}$ | Mean | 0.35 | 0.31 | 0.33 | 0.32
Median | 0.36 | 0.48 | 0.45 | 0.44
PW | 0.38 | 0.44 | 0.38 | 0.38
Bas. | Last layer | 0.45 | 0.47 | 0.43 | 0.44
Logits | 0.40 | 0.35 | 0.30 | 0.32

Table 18: Performance (AUROC $\uparrow$) of different aggregation methods per
spanish OUT-DS .

|  | in | de-pawsx | de-swiss-judgement | de-tweet-sentiment | de-xstance
---|---|---|---|---|---|---
Metric | Ours | Agg. |  |  |  |
$s_{C}$ | $\operatorname{Agg}$ | IF | 0.88 | 0.99 | 0.99 | 0.85
LOF | 0.94 | 0.99 | 0.74 | 0.99
$\operatorname{Agg}-{\emptyset}$ | Mean | 0.97 | 1.00 | 0.76 | 0.99
Median | 0.99 | 0.99 | 0.76 | 0.99
PW | 0.96 | 0.99 | 0.70 | 0.99
Bas. | Last layer | 0.93 | 0.99 | 0.61 | 0.97
Logits | 0.67 | 0.53 | 0.55 | 0.67
$s_{\text{IRW}}$ | $\operatorname{Agg}$ | IF | 0.95 | 1.00 | 0.68 | 0.97
LOF | 0.84 | 0.98 | 0.68 | 0.84
$\operatorname{Agg}_{\emptyset}$ | Mean | 0.93 | 0.99 | 0.85 | 0.93
Median | 0.97 | 0.99 | 0.77 | 0.90
PW | 0.75 | 0.94 | 0.63 | 0.83
Bas. | Last layer | 0.61 | 0.91 | 0.80 | 0.58
Logits | 0.71 | 0.71 | 0.66 | 0.68
$s_{M}$ | $\operatorname{Agg}$ | IF | 0.95 | 1.00 | 0.66 | 0.99
LOF | 0.85 | 0.67 | 0.74 | 0.69
$\operatorname{Agg}_{\emptyset}$ | Mean | 0.63 | 0.74 | 0.60 | 0.55
Median | 0.73 | 0.79 | 0.66 | 0.95
PW | 0.70 | 0.87 | 0.68 | 0.79
Bas. | Last layer | 0.74 | 0.99 | 0.61 | 0.93
Logits | 0.68 | 0.73 | 0.61 | 0.61

Table 19: Performance (AUROC $\uparrow$) of different aggregation per german
IN-DS .

|  |  |  AUROC $\uparrow$
---|---|---|---
|  | out | de-pawsx | de-swiss-judgement | de-tweet-sentiment | de-xstance
Metric | Ours | Agg. |  |  |  |
$s_{C}$ | $\operatorname{Agg}$ | IF | 0.54 | 0.51 | 0.50 | 0.50
LOF | 0.52 | 0.52 | 0.50 | 0.49
$\operatorname{Agg}-{\emptyset}$ | Mean | 0.51 | 0.53 | 0.50 | 0.51
Median | 0.51 | 0.53 | 0.50 | 0.53
PW | 0.52 | 0.54 | 0.50 | 0.51
Bas. | Last layer | 0.53 | 0.51 | 0.50 | 0.52
Logits | 0.33 | 0.33 | 0.37 | 0.31
$s_{\text{IRW}}$ | $\operatorname{Agg}$ | IF | 0.53 | 0.50 | 0.50 | 0.52
LOF | 0.49 | 0.49 | 0.50 | 0.47
$\operatorname{Agg}_{\emptyset}$ | Mean | 0.50 | 0.50 | 0.50 | 0.49
Median | 0.51 | 0.51 | 0.50 | 0.52
PW | 0.48 | 0.39 | 0.47 | 0.47
Bas. | Last layer | 0.45 | 0.41 | 0.42 | 0.50
Logits | 0.35 | 0.40 | 0.39 | 0.42
$s_{M}$ | $\operatorname{Agg}$ | IF | 0.52 | 0.54 | 0.50 | 0.52
LOF | 0.36 | 0.49 | 0.44 | 0.35
$\operatorname{Agg}_{\emptyset}$ | Mean | 0.34 | 0.38 | 0.40 | 0.34
Median | 0.46 | 0.48 | 0.49 | 0.35
PW | 0.44 | 0.37 | 0.45 | 0.46
Bas. | Last layer | 0.52 | 0.46 | 0.50 | 0.48
Logits | 0.34 | 0.36 | 0.37 | 0.46

Table 20: Performance (AUROC $\uparrow$) of different aggregation methods per
german OUT-DS .

###  F.2 Global averaged results per language

##  Appendix G Futures Applications

For future work, we plan to study OOD in sequence labelling tasks (Witon* et
al., 2018; Colombo* et al., 2020; Chapuis* et al., 2020a, b; Colombo et al.,
2021a), sequence generation (Colombo* et al., 2019; Jalalzai* et al., 2020;
Modi et al., 2020; Colombo et al., 2021e) and fair classification (Colombo et
al., 2021d, b; Pichler et al., 2022) and multimodal scenario (Garcia* et al.,
2019; Dinkar* et al., 2020) as well as automatic evaluation (Colombo et al.,
2021c; Colombo, 2021a, b; Staerman et al., 2021b).