Julius von Kügelgen

Julius von Kügelgen, Michel Besserve, Liang Wendong et al. · eth-zurich

We study causal representation learning, the task of inferring latent causal variables and their causal relations from high-dimensional mixtures of the variables. Prior work relies on weak supervision, in the form of counterfactual pre- and post-intervention views or temporal structure; places restrictive assumptions, such as linearity, on the mixing function or latent causal model; or requires partial knowledge of the generative process, such as the causal graph or intervention targets. We instead consider the general setting in which both the causal model and the mixing function are nonparametric. The learning signal takes the form of multiple datasets, or environments, arising from unknown interventions in the underlying causal model. Our goal is to identify both the ground truth latents and their causal graph up to a set of ambiguities which we show to be irresolvable from interventional data. We study the fundamental setting of two causal variables and prove that the observational distribution and one perfect intervention per node suffice for identifiability, subject to a genericity condition. This condition rules out spurious solutions that involve fine-tuning of the intervened and observational distributions, mirroring similar conditions for nonlinear cause-effect inference. For an arbitrary number of variables, we show that at least one pair of distinct perfect interventional domains per node guarantees identifiability. Further, we demonstrate that the strengths of causal influences among the latent variables are preserved by all equivalent solutions, rendering the inferred representation appropriate for drawing causal conclusions from new data. Our study provides the first identifiability results for the general nonparametric setting with unknown interventions, and elucidates what is possible and impossible for causal representation learning without more direct supervision.

Cian Eastwood, Alexander Robey, Shashank Singh et al. · eth-zurich

Domain generalization (DG) seeks predictors which perform well on unseen test distributions by leveraging data drawn from multiple related training distributions or domains. To achieve this, DG is commonly formulated as an average- or worst-case problem over the set of possible domains. However, predictors that perform well on average lack robustness while predictors that perform well in the worst case tend to be overly-conservative. To address this, we propose a new probabilistic framework for DG where the goal is to learn predictors that perform well with high probability. Our key idea is that distribution shifts seen during training should inform us of probable shifts at test time, which we realize by explicitly relating training and test domains as draws from the same underlying meta-distribution. To achieve probable DG, we propose a new optimization problem called Quantile Risk Minimization (QRM). By minimizing the $α$-quantile of predictor's risk distribution over domains, QRM seeks predictors that perform well with probability $α$. To solve QRM in practice, we propose the Empirical QRM (EQRM) algorithm and provide: (i) a generalization bound for EQRM; and (ii) the conditions under which EQRM recovers the causal predictor as $α\to 1$. In our experiments, we introduce a more holistic quantile-focused evaluation protocol for DG and demonstrate that EQRM outperforms state-of-the-art baselines on datasets from WILDS and DomainBed.

Ronan Perry, Julius von Kügelgen, Bernhard Schölkopf · eth-zurich

Machine learning approaches commonly rely on the assumption of independent and identically distributed (i.i.d.) data. In reality, however, this assumption is almost always violated due to distribution shifts between environments. Although valuable learning signals can be provided by heterogeneous data from changing distributions, it is also known that learning under arbitrary (adversarial) changes is impossible. Causality provides a useful framework for modeling distribution shifts, since causal models encode both observational and interventional distributions. In this work, we explore the sparse mechanism shift hypothesis, which posits that distribution shifts occur due to a small number of changing causal conditionals. Motivated by this idea, we apply it to learning causal structure from heterogeneous environments, where i.i.d. data only allows for learning an equivalence class of graphs without restrictive assumptions. We propose the Mechanism Shift Score (MSS), a score-based approach amenable to various empirical estimators, which provably identifies the entire causal structure with high probability if the sparse mechanism shift hypothesis holds. Empirically, we verify behavior predicted by the theory and compare multiple estimators and score functions to identify the best approaches in practice. Compared to other methods, we show how MSS bridges a gap by both being nonparametric as well as explicitly leveraging sparse changes.

Bernhard Schölkopf, Julius von Kügelgen · eth-zurich

We describe basic ideas underlying research to build and understand artificially intelligent systems: from symbolic approaches via statistical learning to interventional models relying on concepts of causality. Some of the hard open problems of machine learning and AI are intrinsically related to causality, and progress may require advances in our understanding of how to model and infer causality from data.

Patrik Reizinger, Luigi Gresele, Jack Brady et al. · eth-zurich

Variational autoencoders (VAEs) are a popular framework for modeling complex data distributions; they can be efficiently trained via variational inference by maximizing the evidence lower bound (ELBO), at the expense of a gap to the exact (log-)marginal likelihood. While VAEs are commonly used for representation learning, it is unclear why ELBO maximization would yield useful representations, since unregularized maximum likelihood estimation cannot invert the data-generating process. Yet, VAEs often succeed at this task. We seek to elucidate this apparent paradox by studying nonlinear VAEs in the limit of near-deterministic decoders. We first prove that, in this regime, the optimal encoder approximately inverts the decoder -- a commonly used but unproven conjecture -- which we refer to as {\em self-consistency}. Leveraging self-consistency, we show that the ELBO converges to a regularized log-likelihood. This allows VAEs to perform what has recently been termed independent mechanism analysis (IMA): it adds an inductive bias towards decoders with column-orthogonal Jacobians, which helps recovering the true latent factors. The gap between ELBO and log-likelihood is therefore welcome, since it bears unanticipated benefits for nonlinear representation learning. In experiments on synthetic and image data, we show that VAEs uncover the true latent factors when the data generating process satisfies the IMA assumption.

Cian Eastwood, Shashank Singh, Andrei Liviu Nicolicioiu et al. · eth-zurich, mila

To avoid failures on out-of-distribution data, recent works have sought to extract features that have an invariant or stable relationship with the label across domains, discarding "spurious" or unstable features whose relationship with the label changes across domains. However, unstable features often carry complementary information that could boost performance if used correctly in the test domain. In this work, we show how this can be done without test-domain labels. In particular, we prove that pseudo-labels based on stable features provide sufficient guidance for doing so, provided that stable and unstable features are conditionally independent given the label. Based on this theoretical insight, we propose Stable Feature Boosting (SFB), an algorithm for: (i) learning a predictor that separates stable and conditionally-independent unstable features; and (ii) using the stable-feature predictions to adapt the unstable-feature predictions in the test domain. Theoretically, we prove that SFB can learn an asymptotically-optimal predictor without test-domain labels. Empirically, we demonstrate the effectiveness of SFB on real and synthetic data.

Cian Eastwood, Andrei Liviu Nicolicioiu, Julius von Kügelgen et al. · eth-zurich, mila

In representation learning, a common approach is to seek representations which disentangle the underlying factors of variation. Eastwood & Williams (2018) proposed three metrics for quantifying the quality of such disentangled representations: disentanglement (D), completeness (C) and informativeness (I). In this work, we first connect this DCI framework to two common notions of linear and nonlinear identifiability, thereby establishing a formal link between disentanglement and the closely-related field of independent component analysis. We then propose an extended DCI-ES framework with two new measures of representation quality - explicitness (E) and size (S) - and point out how D and C can be computed for black-box predictors. Our main idea is that the functional capacity required to use a representation is an important but thus-far neglected aspect of representation quality, which we quantify using explicitness or ease-of-use (E). We illustrate the relevance of our extensions on the MPI3D and Cars3D datasets.

Cian Eastwood, Julius von Kügelgen, Linus Ericsson et al. · eth-zurich

Self-supervised representation learning often uses data augmentations to induce some invariance to "style" attributes of the data. However, with downstream tasks generally unknown at training time, it is difficult to deduce a priori which attributes of the data are indeed "style" and can be safely discarded. To deal with this, current approaches try to retain some style information by tuning the degree of invariance to some particular task, such as ImageNet object classification. However, prior work has shown that such task-specific tuning can lead to significant performance degradation on other tasks that rely on the discarded style. To address this, we introduce a more principled approach that seeks to disentangle style features rather than discard them. The key idea is to add multiple style embedding spaces where: (i) each is invariant to all-but-one augmentation; and (ii) joint entropy is maximized. We formalize our structured data-augmentation procedure from a causal latent-variable-model perspective, and prove identifiability of both content and individual style variables. We empirically demonstrate the benefits of our approach on both synthetic and real-world data.

Klaus-Rudolf Kladny, Julius von Kügelgen, Bernhard Schölkopf et al. · eth-zurich

Counterfactuals answer questions of what would have been observed under altered circumstances and can therefore offer valuable insights. Whereas the classical interventional interpretation of counterfactuals has been studied extensively, backtracking constitutes a less studied alternative where all causal laws are kept intact. In the present work, we introduce a practical method called deep backtracking counterfactuals (DeepBC) for computing backtracking counterfactuals in structural causal models that consist of deep generative components. We propose two distinct versions of our method--one utilizing Langevin Monte Carlo sampling and the other employing constrained optimization--to generate counterfactuals for high-dimensional data. As a special case, our formulation reduces to methods in the field of counterfactual explanations. Compared to these, our approach represents a causally compliant, versatile and modular alternative. We demonstrate these properties experimentally on a modified version of MNIST and CelebA.

Ankur Garg, Michael Stettler, Aaron Schein et al.

Causal representation learning aims to infer the high-level latent causal concepts that give rise to observed low-level measurements. This is particularly relevant for heterogeneous data from different environments or domains since distribution shifts often arise through sparse, localized changes in some of the underlying causal mechanisms, while other parts of the generative process remain unchanged. Whereas identifiability of causal representations has been studied extensively, practical uncertainty-aware methods and real-world use cases remain less explored. In this work, we propose a Bayesian approach to learning causal representations from multi-environment data, focusing on the case of discrete causal concepts and unknown multi-node soft interventions. To this end, we translate causal assumptions and interpretability desiderata into suitable priors and parametric choices within a hierarchical model. We then devise an inference scheme based on sequential Monte Carlo sampling to approximate the resulting multimodal posterior. We showcase our approach through case studies on social survey data, where latent causal concepts correspond to cultural values or political opinions, measurements to survey responses, and environments to different countries or states. Our model infers meaningful high-level concepts and plausible causal relations among them, demonstrating its utility for learning causal representations of complex real-world data.

Benedikt Seiter, Anya Fries, Julius von Kügelgen et al.

Principal component analysis (PCA) is one of the most widely used unsupervised dimension reduction techniques. We study PCA for data from multiple related domains. Since principal components generally differ across domains, one way to obtain a shared low-rank embedding is to perform PCA on the pooled data. However, this approach can focus on spurious directions that exhibit high variation in only a few domains. To find a robust embedding that still explains most variance in unseen but similar domains, we propose instead to focus on shared directions of variation. To this end, we introduce Anchor PCA which trades off overall explained variance with agreement between the shared and domain-specific low-rank embeddings. Anchor PCA amounts to PCA on a modified target matrix and thus can be solved efficiently. Moreover, we show that Anchor PCA recovers a maximal invariant subspace and admits a minimax reconstruction interpretation under bounded domain-specific covariance inflations. On simulated and real-world gas sensor data with temporal drift, we demonstrate, respectively, that Anchor PCA recovers the maximally invariant subspace and yields embeddings that explain more variance on unseen domains than the pooling baseline and a worst-case alternative. Taken together, these findings establish Anchor PCA as a promising approach to robust unsupervised dimension reduction from multi-domain data.

Christian Toth, Lars Lorch, Christian Knoll et al. · eth-zurich

Causal discovery and causal reasoning are classically treated as separate and consecutive tasks: one first infers the causal graph, and then uses it to estimate causal effects of interventions. However, such a two-stage approach is uneconomical, especially in terms of actively collected interventional data, since the causal query of interest may not require a fully-specified causal model. From a Bayesian perspective, it is also unnatural, since a causal query (e.g., the causal graph or some causal effect) can be viewed as a latent quantity subject to posterior inference -- other unobserved quantities that are not of direct interest (e.g., the full causal model) ought to be marginalized out in this process and contribute to our epistemic uncertainty. In this work, we propose Active Bayesian Causal Inference (ABCI), a fully-Bayesian active learning framework for integrated causal discovery and reasoning, which jointly infers a posterior over causal models and queries of interest. In our approach to ABCI, we focus on the class of causally-sufficient, nonlinear additive noise models, which we model using Gaussian processes. We sequentially design experiments that are maximally informative about our target causal query, collect the corresponding interventional data, and update our beliefs to choose the next experiment. Through simulations, we demonstrate that our approach is more data-efficient than several baselines that only focus on learning the full causal graph. This allows us to accurately learn downstream causal queries from fewer samples while providing well-calibrated uncertainty estimates for the quantities of interest.

Dingling Yao, Danru Xu, Sébastien Lachapelle et al. · eth-zurich

We present a unified framework for studying the identifiability of representations learned from simultaneously observed views, such as different data modalities. We allow a partially observed setting in which each view constitutes a nonlinear mixture of a subset of underlying latent variables, which can be causally related. We prove that the information shared across all subsets of any number of views can be learned up to a smooth bijection using contrastive learning and a single encoder per view. We also provide graphical criteria indicating which latent variables can be identified through a simple set of rules, which we refer to as identifiability algebra. Our general framework and theoretical results unify and extend several previous works on multi-view nonlinear ICA, disentanglement, and causal representation learning. We experimentally validate our claims on numerical, image, and multi-modal data sets. Further, we demonstrate that the performance of prior methods is recovered in different special cases of our setup. Overall, we find that access to multiple partial views enables us to identify a more fine-grained representation, under the generally milder assumption of partial observability.

Klaus-Rudolf Kladny, Julius von Kügelgen, Bernhard Schölkopf et al. · eth-zurich

We study causal effect estimation from a mixture of observational and interventional data in a confounded linear regression model with multivariate treatments. We show that the statistical efficiency in terms of expected squared error can be improved by combining estimators arising from both the observational and interventional setting. To this end, we derive methods based on matrix weighted linear estimators and prove that our methods are asymptotically unbiased in the infinite sample limit. This is an important improvement compared to the pooled estimator using the union of interventional and observational data, for which the bias only vanishes if the ratio of observational to interventional data tends to zero. Studies on synthetic data confirm our theoretical findings. In settings where confounding is substantial and the ratio of observational to interventional data is large, our estimators outperform a Stein-type estimator and various other baselines.

Julius von Kügelgen, Abdirisak Mohamed, Sander Beckers · eth-zurich

Counterfactual reasoning -- envisioning hypothetical scenarios, or possible worlds, where some circumstances are different from what (f)actually occurred (counter-to-fact) -- is ubiquitous in human cognition. Conventionally, counterfactually-altered circumstances have been treated as "small miracles" that locally violate the laws of nature while sharing the same initial conditions. In Pearl's structural causal model (SCM) framework this is made mathematically rigorous via interventions that modify the causal laws while the values of exogenous variables are shared. In recent years, however, this purely interventionist account of counterfactuals has increasingly come under scrutiny from both philosophers and psychologists. Instead, they suggest a backtracking account of counterfactuals, according to which the causal laws remain unchanged in the counterfactual world; differences to the factual world are instead "backtracked" to altered initial conditions (exogenous variables). In the present work, we explore and formalise this alternative mode of counterfactual reasoning within the SCM framework. Despite ample evidence that humans backtrack, the present work constitutes, to the best of our knowledge, the first general account and algorithmisation of backtracking counterfactuals. We discuss our backtracking semantics in the context of related literature and draw connections to recent developments in explainable artificial intelligence (XAI).

Zhijing Jin, Julius von Kügelgen, Jingwei Ni et al.

The principle of independent causal mechanisms (ICM) states that generative processes of real world data consist of independent modules which do not influence or inform each other. While this idea has led to fruitful developments in the field of causal inference, it is not widely-known in the NLP community. In this work, we argue that the causal direction of the data collection process bears nontrivial implications that can explain a number of published NLP findings, such as differences in semi-supervised learning (SSL) and domain adaptation (DA) performance across different settings. We categorize common NLP tasks according to their causal direction and empirically assay the validity of the ICM principle for text data using minimum description length. We conduct an extensive meta-analysis of over 100 published SSL and 30 DA studies, and find that the results are consistent with our expectations based on causal insights. This work presents the first attempt to analyze the ICM principle in NLP, and provides constructive suggestions for future modeling choices. Code available at https://github.com/zhijing-jin/icm4nlp

Danru Xu, Dingling Yao, Sébastien Lachapelle et al. · eth-zurich

Causal representation learning aims at identifying high-level causal variables from perceptual data. Most methods assume that all latent causal variables are captured in the high-dimensional observations. We instead consider a partially observed setting, in which each measurement only provides information about a subset of the underlying causal state. Prior work has studied this setting with multiple domains or views, each depending on a fixed subset of latents. Here, we focus on learning from unpaired observations from a dataset with an instance-dependent partial observability pattern. Our main contribution is to establish two identifiability results for this setting: one for linear mixing functions without parametric assumptions on the underlying causal model, and one for piecewise linear mixing functions with Gaussian latent causal variables. Based on these insights, we propose two methods for estimating the underlying causal variables by enforcing sparsity in the inferred representation. Experiments on different simulated datasets and established benchmarks highlight the effectiveness of our approach in recovering the ground-truth latents.

Jack Brady, Julius von Kügelgen, Sébastien Lachapelle et al. · eth-zurich

Learning disentangled representations of concepts and re-composing them in unseen ways is crucial for generalizing to out-of-domain situations. However, the underlying properties of concepts that enable such disentanglement and compositional generalization remain poorly understood. In this work, we propose the principle of interaction asymmetry which states: "Parts of the same concept have more complex interactions than parts of different concepts". We formalize this via block diagonality conditions on the $(n+1)$th order derivatives of the generator mapping concepts to observed data, where different orders of "complexity" correspond to different $n$. Using this formalism, we prove that interaction asymmetry enables both disentanglement and compositional generalization. Our results unify recent theoretical results for learning concepts of objects, which we show are recovered as special cases with $n\!=\!0$ or $1$. We provide results for up to $n\!=\!2$, thus extending these prior works to more flexible generator functions, and conjecture that the same proof strategies generalize to larger $n$. Practically, our theory suggests that, to disentangle concepts, an autoencoder should penalize its latent capacity and the interactions between concepts during decoding. We propose an implementation of these criteria using a flexible Transformer-based VAE, with a novel regularizer on the attention weights of the decoder. On synthetic image datasets consisting of objects, we provide evidence that this model can achieve comparable object disentanglement to existing models that use more explicit object-centric priors.

Manuel Iglesias-Alonso, Felix Schur, Julius von Kügelgen et al.

We consider the problem of estimating a causal effect in a multi-domain setting. The causal effect of interest is confounded by an unobserved confounder and can change between the different domains. We assume that we have access to a proxy of the hidden confounder and that all variables are discrete or categorical. We propose methodology to estimate the causal effect in the target domain, where we assume to observe only the proxy variable. Under these conditions, we prove identifiability (even when treatment and response variables are continuous). We introduce two estimation techniques, prove consistency, and derive confidence intervals. The theoretical results are supported by simulation studies and a real-world example studying the causal effect of website rankings on consumer choices.

Shubhangi Ghosh, Luigi Gresele, Julius von Kügelgen et al. · eth-zurich

Independent Mechanism Analysis (IMA) seeks to address non-identifiability in nonlinear Independent Component Analysis (ICA) by assuming that the Jacobian of the mixing function has orthogonal columns. As typical in ICA, previous work focused on the case with an equal number of latent components and observed mixtures. Here, we extend IMA to settings with a larger number of mixtures that reside on a manifold embedded in a higher-dimensional than the latent space -- in line with the manifold hypothesis in representation learning. For this setting, we show that IMA still circumvents several non-identifiability issues, suggesting that it can also be a beneficial principle for higher-dimensional observations when the manifold hypothesis holds. Further, we prove that the IMA principle is approximately satisfied with high probability (increasing with the number of observed mixtures) when the directions along which the latent components influence the observations are chosen independently at random. This provides a new and rigorous statistical interpretation of IMA.

Alice Bizeul, Thomas Sutter, Alain Ryser et al. · eth-zurich

Predicting masked from visible parts of an image is a powerful self-supervised approach for visual representation learning. However, the common practice of masking random patches of pixels exhibits certain failure modes, which can prevent learning meaningful high-level features, as required for downstream tasks. We propose an alternative masking strategy that operates on a suitable transformation of the data rather than on the raw pixels. Specifically, we perform principal component analysis and then randomly mask a subset of components, which accounts for a fixed ratio of the data variance. The learning task then amounts to reconstructing the masked components from the visible ones. Compared to local patches of pixels, the principal components of images carry more global information. We thus posit that predicting masked from visible components involves more high-level features, allowing our masking strategy to extract more useful representations. This is corroborated by our empirical findings which demonstrate improved image classification performance for component over pixel masking. Our method thus constitutes a simple and robust data-driven alternative to traditional masked image modeling approaches.

Julius von Kügelgen, Jakob Ketterer, Xinwei Shen et al.

We consider the problem of modelling the effects of unseen perturbations such as gene knockdowns or drug combinations on low-level measurements such as RNA sequencing data. Specifically, given data collected under some perturbations, we aim to predict the distribution of measurements for new perturbations. To address this challenging extrapolation task, we posit that perturbations act additively in a suitable, unknown embedding space. More precisely, we formulate the generative process underlying the observed data as a latent variable model, in which perturbations amount to mean shifts in latent space and can be combined additively. Unlike previous work, we prove that, given sufficiently diverse training perturbations, the representation and perturbation effects are identifiable up to affine transformation, and use this to characterize the class of unseen perturbations for which we obtain extrapolation guarantees. To estimate the model from data, we propose a new method, the perturbation distribution autoencoder (PDAE), which is trained by maximising the distributional similarity between true and predicted perturbation distributions. The trained model can then be used to predict previously unseen perturbation distributions. Empirical evidence suggests that PDAE compares favourably to existing methods and baselines at predicting the effects of unseen perturbations.

Julius von Kügelgen

Causal models provide rich descriptions of complex systems as sets of mechanisms by which each variable is influenced by its direct causes. They support reasoning about manipulating parts of the system and thus hold promise for addressing some of the open challenges of artificial intelligence (AI), such as planning, transferring knowledge in changing environments, or robustness to distribution shifts. However, a key obstacle to more widespread use of causal models in AI is the requirement that the relevant variables be specified a priori, which is typically not the case for the high-dimensional, unstructured data processed by modern AI systems. At the same time, machine learning (ML) has proven quite successful at automatically extracting useful and compact representations of such complex data. Causal representation learning (CRL) aims to combine the core strengths of ML and causality by learning representations in the form of latent variables endowed with causal model semantics. In this thesis, we study and present new results for different CRL settings. A central theme is the question of identifiability: Given infinite data, when are representations satisfying the same learning objective guaranteed to be equivalent? This is an important prerequisite for CRL, as it formally characterises if and when a learning task is, at least in principle, feasible. Since learning causal models, even without a representation learning component, is notoriously difficult, we require additional assumptions on the model class or rich data beyond the classical i.i.d. setting. By partially characterising identifiability for different settings, this thesis investigates what is possible for CRL without direct supervision, and thus contributes to its theoretical foundations. Ideally, the developed insights can help inform data collection practices or inspire the design of new practical estimation methods.

Liang Wendong, Armin Kekić, Julius von Kügelgen et al.

Independent Component Analysis (ICA) aims to recover independent latent variables from observed mixtures thereof. Causal Representation Learning (CRL) aims instead to infer causally related (thus often statistically dependent) latent variables, together with the unknown graph encoding their causal relationships. We introduce an intermediate problem termed Causal Component Analysis (CauCA). CauCA can be viewed as a generalization of ICA, modelling the causal dependence among the latent components, and as a special case of CRL. In contrast to CRL, it presupposes knowledge of the causal graph, focusing solely on learning the unmixing function and the causal mechanisms. Any impossibility results regarding the recovery of the ground truth in CauCA also apply for CRL, while possibility results may serve as a stepping stone for extensions to CRL. We characterize CauCA identifiability from multiple datasets generated through different types of interventions on the latent causal variables. As a corollary, this interventional perspective also leads to new identifiability results for nonlinear ICA -- a special case of CauCA with an empty graph -- requiring strictly fewer datasets than previous results. We introduce a likelihood-based approach using normalizing flows to estimate both the unmixing function and the causal mechanisms, and demonstrate its effectiveness through extensive synthetic experiments in the CauCA and ICA setting.

Jack Brady, Roland S. Zimmermann, Yash Sharma et al.

Learning structured representations of the visual world in terms of objects promises to significantly improve the generalization abilities of current machine learning models. While recent efforts to this end have shown promising empirical progress, a theoretical account of when unsupervised object-centric representation learning is possible is still lacking. Consequently, understanding the reasons for the success of existing object-centric methods as well as designing new theoretically grounded methods remains challenging. In the present work, we analyze when object-centric representations can provably be learned without supervision. To this end, we first introduce two assumptions on the generative process for scenes comprised of several objects, which we call compositionality and irreducibility. Under this generative process, we prove that the ground-truth object representations can be identified by an invertible and compositional inference model, even in the presence of dependencies between objects. We empirically validate our results through experiments on synthetic data. Finally, we provide evidence that our theory holds predictive power for existing object-centric models by showing a close correspondence between models' compositionality and invertibility and their empirical identifiability.

Armin Kekić, Jonas Dehning, Luigi Gresele et al.

Early on during a pandemic, vaccine availability is limited, requiring prioritisation of different population groups. Evaluating vaccine allocation is therefore a crucial element of pandemics response. In the present work, we develop a model to retrospectively evaluate age-dependent counterfactual vaccine allocation strategies against the COVID-19 pandemic. To estimate the effect of allocation on the expected severe-case incidence, we employ a simulation-assisted causal modelling approach which combines a compartmental infection-dynamics simulation, a coarse-grained, data-driven causal model and literature estimates for immunity waning. We compare Israel's implemented vaccine allocation strategy in 2021 to counterfactual strategies such as no prioritisation, prioritisation of younger age groups or a strict risk-ranked approach; we find that Israel's implemented strategy was indeed highly effective. We also study the marginal impact of increasing vaccine uptake for a given age group and find that increasing vaccinations in the elderly is most effective at preventing severe cases, whereas additional vaccinations for middle-aged groups reduce infections most effectively. Due to its modular structure, our model can easily be adapted to study future pandemics. We demonstrate this flexibility by investigating vaccine allocation strategies for a pandemic with characteristics of the Spanish Flu. Our approach thus helps evaluate vaccination strategies under the complex interplay of core epidemic factors, including age-dependent risk profiles, immunity waning, vaccine availability and spreading rates.

Shubhangi Ghosh, Luigi Gresele, Julius von Kügelgen et al.

Model identifiability is a desirable property in the context of unsupervised representation learning. In absence thereof, different models may be observationally indistinguishable while yielding representations that are nontrivially related to one another, thus making the recovery of a ground truth generative model fundamentally impossible, as often shown through suitably constructed counterexamples. In this note, we discuss one such construction, illustrating a potential failure case of an identifiability result presented in "Desiderata for Representation Learning: A Causal Perspective" by Wang & Jordan (2021). The construction is based on the theory of nonlinear independent component analysis. We comment on implications of this and other counterexamples for identifiable representation learning.

Luigi Gresele, Julius von Kügelgen, Jonas M. Kübler et al.

We introduce an approach to counterfactual inference based on merging information from multiple datasets. We consider a causal reformulation of the statistical marginal problem: given a collection of marginal structural causal models (SCMs) over distinct but overlapping sets of variables, determine the set of joint SCMs that are counterfactually consistent with the marginal ones. We formalise this approach for categorical SCMs using the response function formulation and show that it reduces the space of allowed marginal and joint SCMs. Our work thus highlights a new mode of falsifiability through additional variables, in contrast to the statistical one via additional data.

Matthias Tangemann, Steffen Schneider, Julius von Kügelgen et al.

Learning generative object models from unlabelled videos is a long standing problem and required for causal scene modeling. We decompose this problem into three easier subtasks, and provide candidate solutions for each of them. Inspired by the Common Fate Principle of Gestalt Psychology, we first extract (noisy) masks of moving objects via unsupervised motion segmentation. Second, generative models are trained on the masks of the background and the moving objects, respectively. Third, background and foreground models are combined in a conditional "dead leaves" scene model to sample novel scene configurations where occlusions and depth layering arise naturally. To evaluate the individual stages, we introduce the Fishbowl dataset positioned between complex real-world scenes and common object-centric benchmarks of simplistic objects. We show that our approach allows learning generative models that generalize beyond the occlusions present in the input videos, and represent scenes in a modular fashion that allows sampling plausible scenes outside the training distribution by permitting, for instance, object numbers or densities not observed in the training set.

Osama Makansi, Julius von Kügelgen, Francesco Locatello et al.

Predicting the future trajectory of a moving agent can be easy when the past trajectory continues smoothly but is challenging when complex interactions with other agents are involved. Recent deep learning approaches for trajectory prediction show promising performance and partially attribute this to successful reasoning about agent-agent interactions. However, it remains unclear which features such black-box models actually learn to use for making predictions. This paper proposes a procedure that quantifies the contributions of different cues to model performance based on a variant of Shapley values. Applying this procedure to state-of-the-art trajectory prediction methods on standard benchmark datasets shows that they are, in fact, unable to reason about interactions. Instead, the past trajectory of the target is the only feature used for predicting its future. For a task with richer social interaction patterns, on the other hand, the tested models do pick up such interactions to a certain extent, as quantified by our feature attribution method. We discuss the limits of the proposed method and its links to causality

Lukas Schott, Julius von Kügelgen, Frederik Träuble et al.

An important component for generalization in machine learning is to uncover underlying latent factors of variation as well as the mechanism through which each factor acts in the world. In this paper, we test whether 17 unsupervised, weakly supervised, and fully supervised representation learning approaches correctly infer the generative factors of variation in simple datasets (dSprites, Shapes3D, MPI3D) from controlled environments, and on our contributed CelebGlow dataset. In contrast to prior robustness work that introduces novel factors of variation during test time, such as blur or other (un)structured noise, we here recompose, interpolate, or extrapolate only existing factors of variation from the training data set (e.g., small and medium-sized objects during training and large objects during testing). Models that learn the correct mechanism should be able to generalize to this benchmark. In total, we train and test 2000+ models and observe that all of them struggle to learn the underlying mechanism regardless of supervision signal and architectural bias. Moreover, the generalization capabilities of all tested models drop significantly as we move from artificial datasets towards more realistic real-world datasets. Despite their inability to identify the correct mechanism, the models are quite modular as their ability to infer other in-distribution factors remains fairly stable, providing only a single factor is out-of-distribution. These results point to an important yet understudied problem of learning mechanistic models of observations that can facilitate generalization.

Frederik Träuble, Julius von Kügelgen, Matthäus Kleindessner et al.

When machine learning systems meet real world applications, accuracy is only one of several requirements. In this paper, we assay a complementary perspective originating from the increasing availability of pre-trained and regularly improving state-of-the-art models. While new improved models develop at a fast pace, downstream tasks vary more slowly or stay constant. Assume that we have a large unlabelled data set for which we want to maintain accurate predictions. Whenever a new and presumably better ML models becomes available, we encounter two problems: (i) given a limited budget, which data points should be re-evaluated using the new model?; and (ii) if the new predictions differ from the current ones, should we update? Problem (i) is about compute cost, which matters for very large data sets and models. Problem (ii) is about maintaining consistency of the predictions, which can be highly relevant for downstream applications; our demand is to avoid negative flips, i.e., changing correct to incorrect predictions. In this paper, we formalize the Prediction Update Problem and present an efficient probabilistic approach as answer to the above questions. In extensive experiments on standard classification benchmark data sets, we show that our method outperforms alternative strategies along key metrics for backward-compatible prediction updates.

Julius von Kügelgen, Nikita Agarwal, Jakob Zeitler et al.

Algorithmic recourse aims to provide actionable recommendations to individuals to obtain a more favourable outcome from an automated decision-making system. As it involves reasoning about interventions performed in the physical world, recourse is fundamentally a causal problem. Existing methods compute the effect of recourse actions using a causal model learnt from data under the assumption of no hidden confounding and modelling assumptions such as additive noise. Building on the seminal work of Balke and Pearl (1994), we propose an alternative approach for discrete random variables which relaxes these assumptions and allows for unobserved confounding and arbitrary structural equations. The proposed approach only requires specification of the causal graph and confounding structure and bounds the expected counterfactual effect of recourse actions. If the lower bound is above a certain threshold, i.e., on the other side of the decision boundary, recourse is guaranteed in expectation.

Luigi Gresele, Julius von Kügelgen, Vincent Stimper et al.

Independent component analysis provides a principled framework for unsupervised representation learning, with solid theory on the identifiability of the latent code that generated the data, given only observations of mixtures thereof. Unfortunately, when the mixing is nonlinear, the model is provably nonidentifiable, since statistical independence alone does not sufficiently constrain the problem. Identifiability can be recovered in settings where additional, typically observed variables are included in the generative process. We investigate an alternative path and consider instead including assumptions reflecting the principle of independent causal mechanisms exploited in the field of causality. Specifically, our approach is motivated by thinking of each source as independently influencing the mixing process. This gives rise to a framework which we term independent mechanism analysis. We provide theoretical and empirical evidence that our approach circumvents a number of nonidentifiability issues arising in nonlinear blind source separation.

Julius von Kügelgen, Yash Sharma, Luigi Gresele et al.

Self-supervised representation learning has shown remarkable success in a number of domains. A common practice is to perform data augmentation via hand-crafted transformations intended to leave the semantics of the data invariant. We seek to understand the empirical success of this approach from a theoretical perspective. We formulate the augmentation process as a latent variable model by postulating a partition of the latent representation into a content component, which is assumed invariant to augmentation, and a style component, which is allowed to change. Unlike prior work on disentanglement and independent component analysis, we allow for both nontrivial statistical and causal dependencies in the latent space. We study the identifiability of the latent representation based on pairs of views of the observations and prove sufficient conditions that allow us to identify the invariant content partition up to an invertible mapping in both generative and discriminative settings. We find numerical simulations with dependent latent variables are consistent with our theory. Lastly, we introduce Causal3DIdent, a dataset of high-dimensional, visually complex images with rich causal dependencies, which we use to study the effect of data augmentations performed in practice.

Julius von Kügelgen, Amir-Hossein Karimi, Umang Bhatt et al.

Algorithmic fairness is typically studied from the perspective of predictions. Instead, here we investigate fairness from the perspective of recourse actions suggested to individuals to remedy an unfavourable classification. We propose two new fairness criteria at the group and individual level, which -- unlike prior work on equalising the average group-wise distance from the decision boundary -- explicitly account for causal relationships between features, thereby capturing downstream effects of recourse actions performed in the physical world. We explore how our criteria relate to others, such as counterfactual fairness, and show that fairness of recourse is complementary to fairness of prediction. We study theoretically and empirically how to enforce fair causal recourse by altering the classifier and perform a case study on the Adult dataset. Finally, we discuss whether fairness violations in the data generating process revealed by our criteria may be better addressed by societal interventions as opposed to constraints on the classifier.

Amir-Hossein Karimi, Julius von Kügelgen, Bernhard Schölkopf et al.

Recent work has discussed the limitations of counterfactual explanations to recommend actions for algorithmic recourse, and argued for the need of taking causal relationships between features into consideration. Unfortunately, in practice, the true underlying structural causal model is generally unknown. In this work, we first show that it is impossible to guarantee recourse without access to the true structural equations. To address this limitation, we propose two probabilistic approaches to select optimal actions that achieve recourse with high probability given limited causal knowledge (e.g., only the causal graph). The first captures uncertainty over structural equations under additive Gaussian noise, and uses Bayesian model averaging to estimate the counterfactual distribution. The second removes any assumptions on the structural equations by instead computing the average effect of recourse actions on individuals similar to the person who seeks recourse, leading to a novel subpopulation-based interventional notion of recourse. We then derive a gradient-based procedure for selecting optimal recourse actions, and empirically show that the proposed approaches lead to more reliable recommendations under imperfect causal knowledge than non-probabilistic baselines.

Julius von Kügelgen, Ivan Ustyuzhaninov, Peter Gehler et al.

Learning how to model complex scenes in a modular way with recombinable components is a pre-requisite for higher-order reasoning and acting in the physical world. However, current generative models lack the ability to capture the inherently compositional and layered nature of visual scenes. While recent work has made progress towards unsupervised learning of object-based scene representations, most models still maintain a global representation space (i.e., objects are not explicitly separated), and cannot generate scenes with novel object arrangement and depth ordering. Here, we present an alternative approach which uses an inductive bias encouraging modularity by training an ensemble of generative models (experts). During training, experts compete for explaining parts of a scene, and thus specialise on different object classes, with objects being identified as parts that re-occur across multiple scenes. Our model allows for controllable sampling of individual objects and recombination of experts in physically plausible ways. In contrast to other methods, depth layering and occlusion are handled correctly, moving this approach closer to a causal generative scene model. Experiments on simple toy data qualitatively demonstrate the conceptual advantages of the proposed approach.

Julius von Kügelgen, Paul K Rubenstein, Bernhard Schölkopf et al.

We study the problem of causal discovery through targeted interventions. Starting from few observational measurements, we follow a Bayesian active learning approach to perform those experiments which, in expectation with respect to the current model, are maximally informative about the underlying causal structure. Unlike previous work, we consider the setting of continuous random variables with non-linear functional relationships, modelled with Gaussian process priors. To address the arising problem of choosing from an uncountable set of possible interventions, we propose to use Bayesian optimisation to efficiently maximise a Monte Carlo estimate of the expected information gain.

Julius von Kügelgen, Alexander Mey, Marco Loog et al.

While the success of semi-supervised learning (SSL) is still not fully understood, Schölkopf et al. (2012) have established a link to the principle of independent causal mechanisms. They conclude that SSL should be impossible when predicting a target variable from its causes, but possible when predicting it from its effects. Since both these cases are somewhat restrictive, we extend their work by considering classification using cause and effect features at the same time, such as predicting disease from both risk factors and symptoms. While standard SSL exploits information contained in the marginal distribution of all inputs (to improve the estimate of the conditional distribution of the target given inputs), we argue that in our more general setting we should use information in the conditional distribution of effect features given causal features. We explore how this insight generalises the previous understanding, and how it relates to and can be exploited algorithmically for SSL.

Julius von Kügelgen, Alexander Mey, Marco Loog

Current methods for covariate-shift adaptation use unlabelled data to compute importance weights or domain-invariant features, while the final model is trained on labelled data only. Here, we consider a particular case of covariate shift which allows us also to learn from unlabelled data, that is, combining adaptation with semi-supervised learning. Using ideas from causality, we argue that this requires learning with both causes, $X_C$, and effects, $X_E$, of a target variable, $Y$, and show how this setting leads to what we call a semi-generative model, $P(Y,X_E|X_C,θ)$. Our approach is robust to domain shifts in the distribution of causal features and leverages unlabelled data by learning a direct map from causes to effects. Experiments on synthetic data demonstrate significant improvements in classification over purely-supervised and importance-weighting baselines.