ON DISENTANGLED REPRESENTATIONS LEARNED FROM CORRELATED DATA Anonymous authors Paper under double-blind review

Abstract

Despite impressive progress in the last decade, it still remains an open challenge to build models that generalize well across multiple tasks and datasets. One path to achieve this is to learn meaningful and compact representations, in which different semantic aspects of data are structurally disentangled. The focus of disentanglement approaches has been on separating independent factors of variation despite the fact that real-world observations are often not structured into meaningful independent causal variables. In this work, we bridge the gap to real-world scenarios by analyzing the behavior of most prominent methods and disentanglement scores on correlated data in a large scale empirical study (including 4260 models). We show that systematically induced correlations in the dataset are being learned and reflected in the latent representations, while widely used disentanglement scores fall short of capturing these latent correlations. Finally, we demonstrate how to disentangle these latent correlations using weak supervision, even if we constrain this supervision to be causally plausible. Our results thus support the argument to learn independent mechanisms rather than independent factors of variations.

1. INTRODUCTION

Figure 1 : While in principle we consider the presence of the objects (coffee cup, table, chair) to be independent mechanisms, they tend to appear together in observed data. Due to the induced structure, disentangled representations promise generalization to unseen scenarios (Higgins et al., 2017b) , increased interpretability (Adel et al., 2018; Higgins et al., 2018) and faster learning on downstream tasks (van Steenkiste et al., 2019; Locatello et al., 2019a) . While the advantages of disentangled representations have been well established, they generally assume the existence of natural factors that vary independently within the given dataset, which is rarely the case in real-world settings. As an example, consider a scene with a table and some chairs (see Fig. 1 ). The higher-level factors of this representation are in fact correlated and what we actually want to infer are independent (causal) mechanisms (Peters et al., 2017; Parascandolo et al., 2018; Suter et al., 2019; Goyal et al., 2019) . A complex generative model can be thought of as the composition of independent mechanisms or "causal" modules, which generate highdimensional observations (such as images or videos). In the causality community, this is often considered a prerequisite to achieve representations which are robust to interventions upon variables determined by such models (Peters et al., 2017) . One particular instantiation of this idea in the machine learning community is the notion of disentangled representations (Bengio et al., 2013) . The goal of disentanglement learning is to find a representation of the data which captures all the ground-truth factors of variation (FoV) independently. Despite the recent growth of the field, the performance of state-of-the-art disentanglement learners remains unknown for more realistic settings where FoV are correlated during training. Given the potential societal impact in the medical domain (Chartsias et al., 2018) or fair decision making (Locatello et al., 2019a; Madras et al., 2018; Creager et al., 2019) , the evaluation of the usefulness of disentangled representations trained on correlated data is of high importance. To go beyond the highly idealized settings considered thus far, we conducted a large scale empirical study to systematically assess the effect of induced correlations between pairs of factors of variation in training data on the learned representations. To provide a qualitative and quantitative evaluation, we investigate multiple datasets with access to ground-truth labels. Moreover, we study the generalization abilities of the representations learned on correlated data as well as their performance in particular for the downstream task of fair decision making. Contributions. Our main contributions can be summarized as follows: • We present the first large-scale empirical study (4260 models)foot_0 that examines how modern disentanglement learners perform when ground truth factors of the observational data are correlated. • We find that factorization-based inductive biases are insufficient to learn disentangled representations from observational data. Existing methods fail to disentangle correlated factors of variation, resulting in correlated latent space dimensions. Moreover, standard disentanglement metrics do not reveal these persisting correlations. • We investigate the usefulness of semi-supervised and weakly-supervised approaches to resolve latent entanglement. For the latter setting, we focus on multiple observational and interventional distributions.

2. BACKGROUND AND RELATED WORK

Disentanglement. Current state-of-the-art disentanglement approaches use the framework of variational auto-encoders (VAEs) (Kingma & Welling, 2014; Rezende et al., 2014) . The (high-dimensional) observations x are modelled as being generated from some latent features z with chosen prior p(z) according to the probabilistic model p θ (x|z)p(z). The generative model p θ (x|z) as well as the proxy posterior q φ (z|x) can be parameterized by neural networks, which are optimized by maximizing the variational lower bound (ELBO) of log p(x 1 , . . . , x N ). L V AE = N i=1 E q φ (z|x (i) ) [log p θ (x (i) |z)] -D KL (q φ (z|x (i) ) p(z)) The above objective does not enforce any structure on the latent space, except for similarity (in KL-divergence) to the prior p(z) (typically chosen as an isotropic Gaussian). However, the structure and semantic meaning of latent representations can be relevant to study generation properties. Consequently, various proposals for structure-imposing regularization and commonly used evaluation metrics measuring different notions of disentanglement of the learned representations have been made (Higgins et al., 2017a; Kim & Mnih, 2018; Burgess et al., 2018; Kumar et al., 2018; Chen et al., 2018; Eastwood & Williams, 2018; Mathieu et al., 2018) . Recently, it has been shown that unsupervised disentangling by optimising marginal likelihood in a generative model is impossible without further inductive bias (Locatello et al., 2019b) . To address this theoretical limitation, methods have been proposed that do not require explicitly labelled data but only some weak labeling information (Locatello et al., 2020; Shu et al., 2019) . Ideas related to disentangling the factors of variations date back to the non-linear ICA literature (Bach & Jordan, 2002; Comon, 1994; Jutten & Karhunen, 2003; Hyvärinen & Pajunen, 1999; Hyvarinen et al., 2019; Hyvarinen & Morioka, 2016; Gresele et al., 2019) . Recent work combines non-linear ICA with disentanglement (Khemakhem et al., 2020; Sorrenson et al., 2020; Klindt et al., 2020) Correlations. A set of random variables X i=1,...,n is not independent, if and only if their joint distribution does not factorize P (X 1 , X 2 , . . . , X n ) = n i=1 P (X i ). (2) In this case, we speak of dependence between the random variables, also commonly referred to as correlation.foot_1 Correlation between two variables can either stem from a direct causal relationship (one



Each model has been trained for 300,000 iterations on Tesla V100 GPUs. Reproducing these experiments requires approximately 0.79 GPU years. We use the term correlation here in a broad sense of any statistical association, not just linear dependencies.

