Out of Distribution Reasoning by Weakly-Supervised Disentangled Logic Variational Autoencoder

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Out of Distribution Reasoning by

Weakly-Supervised Disentangled Logic Variational

Autoencoder

Zahra Rahiminasab

School of Computer Science

and Engineering

Nanyang Technological University

Singapore, Singapore

rahi0004@e.ntu.edu.sg

Michael Yuhas

Energy Research Institute

Nanyang Technological University

Singapore, Singapore

michaelj004@ntu.edu.sg

Arvind Easwaran

School of Computer Science

and Engineering

Nanyang Technological University

Singapore, Singapore

arvinde@ntu.edu.sg

Abstract—Out-of-distribution (OOD) detection, i.e., ﬁnding test

samples derived from a different distribution than the training

set, as well as reasoning about such samples (OOD reasoning),

are necessary to ensure the safety of results generated by

machine learning models. Recently there have been promising

results for OOD detection in the latent space of variational

autoencoders (VAEs). However, without disentanglement, VAEs

cannot perform OOD reasoning. Disentanglement ensures a one-

to-many mapping between generative factors of OOD (e.g., rain

in image data) and the latent variables to which they are encoded.

Although previous literature has focused on weakly-supervised

disentanglement on simple datasets with known and independent

generative factors. In practice, achieving full disentanglement

through weak supervision is impossible for complex datasets,

such as Carla, with unknown and abstract generative factors. As

a result, we propose an OOD reasoning framework that learns

a partially disentangled VAE to reason about complex datasets.

Our framework consists of three steps: partitioning data based

on observed generative factors, training a VAE as a logic tensor

network that satisﬁes disentanglement rules, and run-time OOD

reasoning. We evaluate our approach on the Carla dataset and

compare the results against three state-of-the-art methods. We

found that our framework outperformed these methods in terms

of disentanglement and end-to-end OOD reasoning.

Index Terms—Out-of-distribution reasoning, Weakly-

supervised disentanglement, Variational autoencoder, Logic

tensor network

I. INTRODUCTION

Since machine learning models are frequently used in

safety-critical applications such as autonomous driving, it

is important to identify whether the results generated by

machine learning models are safe. It has been shown that the

distribution of training and test samples can be different [1]

and as a result, it is important to identify test samples derived

from a different distribution than the training distribution, i.e.,

out of distribution (OOD) samples. In addition, identifying the

reason behind OOD behavior (OOD reasoning) can help to

This work is supported by MoE, Singapore, Tier-2 grant number MOE2019-

T2-2-040.

provide a safe-fail mechanism to prevent or alleviate damage

in safety-critical applications.

Consider a machine learning (ML) model that controls an

autonomous vehicle (AV). This model receives image data

and outputs steering and throttle set points. If the model’s

training set was gathered in urban environments with no

precipitation, both rural roads and rainy weather would be

OOD. However, the risk mitigation actions for each generative

factor (background, weather) should also be different, and

detecting that a sample is OOD is insufﬁcient to ensure safety.

For example, samples coming from the rural road should return

control to a human driver if the hazards of rural operations

have not been sufﬁciently analyzed. Conversely, rainy samples

should trigger the switch to a more conservative ML model

to compensate for reduced traction if this poses less risk than

returning control to an unaware human driver.

For identifying OOD samples, different models such as a

variational autoencoder (VAE) [2] can be used. A VAE learns

a data distribution by encoding data in a lower-dimensional

representation (latent space) and regenerating original sam-

ples from encoded representations. In general, there are two

approaches for identifying OOD samples with VAEs. In the

ﬁrst approach, the sample likelihood is calculated in the

output space of a VAE, and samples with lower likelihood are

identiﬁed as OOD samples. However, in [3], it is shown that

OOD samples can get a higher likelihood than in-distribution

samples; therefore, OOD detection in output space is unreli-

able. One solution for this problem is using the latent space of

a VAE rather than output space and comparing distributions

of a test sample and training samples in latent space [4, 5, 6].

OOD reasoning focuses on ﬁnding the source of OOD

behavior by analyzing one-to-many maps between generative

factors of data and corresponding latent dimensions that en-

code them. Generative factors are speciﬁc data characteristics

essential for data reconstruction, such as the rain intensity in

an image. Disentanglement of latent space is the process of

establishing such one-to-many maps between given generative

arXiv:2210.09959v1 [cs.LG] 18 Oct 2022

factors and their corresponding latent dimensions. However,

without inductive bias in the data or model, learning disen-

tangled latent space for a VAE is theoretically impossible [7].

Therefore, we should train the VAE with a degree of super-

vision on data or apply inductive bias to the model to learn

disentangled latent space.

In complex datasets such as Carla dataset [8], generative

factors are deﬁned at a more abstract level. In addition, not

all generative factors are known, and the domain of observed

generative factors can be continuous. Therefore, providing

labels based on generative factors for each image can be

expensive. As a result, using match pairing [9] for a complex

dataset is more practical. However, structuring the latent space

of VAE without labels and just based on partitions for more

than one generative factor can be challenging. As shown

in [10], the disentanglement performance can decrease when

changes in other factors affect the learned distribution for a

given factor.

Although theoretically achieving full disentanglement with

weak supervision is possible, in practice, based on the size of

latent space, incomplete knowledge about generative factors,

level of abstraction in deﬁning generative factors, etc., total

disentanglement may not be achieved. For example, if a ﬁxed

number of latent dimensions are selected for a rain generative

factor, selected dimensions may not represent all the informa-

tion regarding that factor. However, these dimensions majorly

encode the rain information. Therefore, a mechanism is needed

to learn partial disentanglement for complex datasets.

Logic tensor networks (LTN) [11] distill knowledge in the

network weights based on a set of rules during training. For

this purpose, the loss function is deﬁned as a set of rules. For

LTN, training is the process of optimizing network parameters

to minimize the satisfaction of the loss rule. Since LTN uses

ﬁrst-order fuzzy logic semantics (real-logic), the rules can be

satisﬁed partially. Therefore, LTNs’ characteristics make them

suitable for deﬁning partial disentanglement.

Currently, OOD detection and reasoning approaches [4]

try to achieve partial disentanglement through model-based

inductive bias. However, they cannot guarantee the mapping

between generative factors and speciﬁc latent dimensions.

Our contribution: To solve the aforementioned issues, we

propose an OOD reasoning framework that consists of three

phases: data partitioning, training OOD reasoners, and run-

time OOD reasoning. Data partitions are formed based on ob-

served values for generative factors, and OOD reasoners (latent

dimensions of a weakly-disentangled VAE) are designed with

match pairing supervision and LTN. Using the LTN version

of a VAE allows us to deﬁne disentanglement formally based

on given data partition samples. Inspired by [9] we deﬁne the

adaptation and isolation rules for achieving disentanglement.

The adaptation rule ensures that the change in generative

factor values is reﬂected in the distribution learned for the

corresponding latent dimensions. The isolation rule guarantees

that the change in a given factor is only reﬂected in its corre-

sponding latent dimensions. Since LTN uses ﬁrst-order fuzzy

logic semantics (real-logic), adaptation and isolation rules can

be partially satisﬁed. As a result, the VAE can achieve a

proper level of disentanglement even when latent space size

is small, and some generative factors are not observed during

training. Finally, we use the corresponding dimension for a

given factor to identify OOD samples based on a given factor.

We show the effect of deﬁned constraints on disentanglement

by visualization, and also mutual information [12]. We also

show that our approach achieves an AUROC of 0.98 and 0.89

on the Carla dataset for rain and city reasoners, respectively.

II. BACKGROUND

Our framework is built based on three concepts. In this

section, we introduce these concepts.

Variational autoencoder: A VAE is a machine learning

model formed by two attached neural networks: the encoder

and the decoder. Given an input x, encoder qφ(z|x)with

parameters φmaps the input to latent representation z. The

decoder pθ(x|z)with parameters θregenerates data from z

representation. Equation 1 shows the ELBO loss of a VAE.

loss =Eqφ(z|x)[log pθ(x|z)] −KL(qφ(z|x)||p(z)) (1)

The ﬁrst and second terms are reconstruction loss and

regularization losses, respectively. The reconstruction loss

ensures that the distribution learned for data reﬂects the main

factors required for data reconstruction. The regularization loss

evaluates the similarity between the learned distribution and

the prior distribution by using KL-divergence between learned

and prior (usually standard Gaussian distribution N(0,1))

distributions [13].

Logic tensor network: A logic tensor network (LTN)

uses logical constraints to distill knowledge about data and

a model in model weights during training. The knowledge

is formally described by ﬁrst-order fuzzy logic named real

logic. Real logic uses a set of functions, predicates, variables,

and constants to form logical terms. These elements, alongside

operators such as negation, conjunction, etc., form the syntax

of real logic. The fuzzy semantic is deﬁned for real logic

so that the rules can be satisﬁed partially. The operators

are semantically deﬁned based on product real logic. Table

I, summarizes the operator deﬁnition. In this table a,b,

and a1, ..., anare predicates with values that fall in [0,1].

Learning for a logic tensor network is the process of ﬁnding

a set of parameters that maximize the satisfaction of rules

or minimizing the satisfaction of a loss rule deﬁned by real

logic [11].

Weakly supervised disentanglement: Disentanglement is

deﬁned by two concepts: consistency and restrictiveness.

Given a generative factor sencoded in dimensions i∈Iof

latent space, consistency means changes in the distributions

of dimensions outside the speciﬁed set (I) do not affect given

factor s. Restrictiveness means other factors s0∈S\ {s}are

immune to changes in the distributions of speciﬁed dimensions

(i∈I) that encode generative factor s[9]. We can use different

levels of weak supervision to attain disentanglement, such as

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OutofDistributionReasoningbyWeakly-SupervisedDisentangledLogicVariationalAutoencoderZahraRahiminasabSchoolofComputerScienceandEngineeringNanyangTechnologicalUniversitySingapore,Singaporerahi0004@e.ntu.edu.sgMichaelYuhasEnergyResearchInstituteNanyangTechnologicalUniversitySingapore,Singaporemichaelj0...

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Out of Distribution Reasoning by Weakly-Supervised Disentangled Logic Variational Autoencoder

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