Self-supervised learning methods that mask parts of the input data and train models to predict the missing components have led to significant advances in machine learning. These approaches learn conditional distributions $p(x_T \mid x_S)$ simultaneously, where $x_S$ and $x_T$ are subsets of the observed variables. In this paper, we examine the core problem of when all these conditional distributions are consistent with some joint distribution, and whether common models used in practice can learn consistent conditionals. We explore this problem in two settings. First, for the complementary conditioning sets where $S \cup T$ is the complete set of variables, we introduce the concept of path consistency, a necessary condition for a consistent joint. Second, we consider the case where we have access to $p(x_T \mid x_S)$ for all subsets $S$ and $T$. In this case, we propose the concepts of autoregressive and swap consistency, which we show are necessary and sufficient conditions for a consistent joint. For both settings, we analyze when these consistency conditions hold and show that standard discriminative models \emph{may fail to satisfy them}. Finally, we corroborate via experiments that proposed consistency measures can be used as proxies for evaluating the consistency of conditionals $p(x_T \mid x_S)$, and common parameterizations may find it hard to learn true conditionals.

Despite the empirical success of foundation models, we do not have a systematic characterization of the representations that these models learn. In this paper, we establish the contexture theory. It shows that a large class of representation learning methods can be characterized as learning from the association between the input and a context variable. Specifically, we show that many popular methods aim to approximate the top-d singular functions of the expectation operator induced by the context, in which case we say that the representation learns the contexture. We demonstrate the generality of the contexture theory by proving that representation learning within various learning paradigms -- supervised, self-supervised, and manifold learning -- can all be studied from such a perspective. We prove that representations that learn the contexture are optimal on those tasks that are compatible with the context. One important implication of our theory is that once the model is large enough to approximate the top singular functions, scaling up the model size yields diminishing returns, so further improvement requires better contexts. To this end, we study how to evaluate a context without knowing the downstream tasks. We propose a metric and show by experiments that it correlates well with the actual performance of the encoder on many real datasets.

This paper addresses intervention-based causal representation learning (CRL) under a general nonparametric latent causal model and an unknown transformation that maps the latent variables to the observed variables. Linear and general transformations are investigated. The paper addresses both the identifiability and achievability aspects. Identifiability refers to determining algorithm-agnostic conditions that ensure the recovery of the true latent causal variables and the underlying latent causal graph. Achievability refers to the algorithmic aspects and addresses designing algorithms that achieve identifiability guarantees. By drawing novel connections between score functions (i.e., the gradients of the logarithm of density functions) and CRL, this paper designs a score-based class of algorithms that ensures both identifiability and achievability. First, the paper focuses on linear transformations and shows that one stochastic hard intervention per node suffices to guarantee identifiability. It also provides partial identifiability guarantees for soft interventions, including identifiability up to mixing with parents for general causal models and perfect recovery of the latent graph for sufficiently nonlinear causal models. Secondly, it focuses on general transformations and demonstrates that two stochastic hard interventions per node are sufficient for identifiability. This is achieved by defining a differentiable loss function whose global optima ensure identifiability for general CRL. Notably, one does not need to know which pair of interventional environments has the same node intervened. Finally, the theoretical results are empirically validated via experiments on structured synthetic data and image data.

This paper focuses on causal representation learning (CRL) under a general nonparametric latent causal model and a general transformation model that maps the latent data to the observational data. It establishes identifiability and achievability results using two hard uncoupled interventions per node in the latent causal graph. Notably, one does not know which pair of intervention environments have the same node intervened (hence, uncoupled). For identifiability, the paper establishes that perfect recovery of the latent causal model and variables is guaranteed under uncoupled interventions. For achievability, an algorithm is designed that uses observational and interventional data and recovers the latent causal model and variables with provable guarantees. This algorithm leverages score variations across different environments to estimate the inverse of the transformer and, subsequently, the latent variables. The analysis, additionally, recovers the identifiability result for two hard coupled interventions, that is when metadata about the pair of environments that have the same node intervened is known. This paper also shows that when observational data is available, additional faithfulness assumptions that are adopted by the existing literature are unnecessary.

This paper investigates the robustness of causal bandits (CBs) in the face of temporal model fluctuations. This setting deviates from the existing literature's widely-adopted assumption of constant causal models. The focus is on causal systems with linear structural equation models (SEMs). The SEMs and the time-varying pre- and post-interventional statistical models are all unknown and subject to variations over time. The goal is to design a sequence of interventions that incur the smallest cumulative regret compared to an oracle aware of the entire causal model and its fluctuations. A robust CB algorithm is proposed, and its cumulative regret is analyzed by establishing both upper and lower bounds on the regret. It is shown that in a graph with maximum in-degree$d$, length of the largest causal path$L$, and an aggregate model deviation$C$, the regret is upper bounded by$\tilde{\mathrm{O}}(d^{L-\frac{1}{2}}(\sqrt{T}+C))$and lower bounded by$\Omega(d^{\frac{L}{2}-2}\max\{\sqrt{T}\,\ d^{2}C\})$. The proposed algorithm achieves nearly optimal$\tilde{\mathcal{O}}(\sqrt{T})$regret when$C$is$o(\sqrt{T})$, maintaining sub-linear regret for a broad range of C.

Consider a data-generation process that transforms low-dimensional _latent_ causally-related variables to high-dimensional _observed_ variables. Causal representation learning (CRL) is the process of using the observed data to recover the latent causal variables and the causal structure among them. Despite the multitude of identifiability results under various interventional CRL settings, the existing guarantees apply exclusively to the _infinite-sample_ regime (i.e., infinite observed samples). This paper establishes the first sample-complexity analysis for the finite-sample regime, in which the interactions between the number of observed samples and probabilistic guarantees on recovering the latent variables and structure are established. This paper focuses on _general_ latent causal models, stochastic _soft_ interventions, and a linear transformation from the latent to the observation space. The identifiability results ensure graph recovery up to ancestors and latent variables recovery up to mixing with parent variables. Specifically, ${\cal O}((\log \frac{1}{\delta})^{4})$ samples suffice for latent graph recovery up to ancestors with probability $1 - \delta$, and ${\cal O}((\frac{1}{\epsilon}\log \frac{1}{\delta})^{4})$ samples suffice for latent causal variables recovery that is $\epsilon$ close to the identifiability class with probability $1 - \delta$.

Despite the multifaceted recent advances in interventional causal representation learning (CRL), they primarily focus on the stylized assumption of single-node interventions. This assumption is not valid in a wide range of applications, and generally, the subset of nodes intervened in an interventional environment is *fully unknown*. This paper focuses on interventional CRL under unknown multi-node (UMN) interventional environments and establishes the first identifiability results for *general* latent causal models (parametric or nonparametric) under stochastic interventions (soft or hard) and linear transformation from the latent to observed space. Specifically, it is established that given sufficiently diverse interventional environments, (i) identifiability *up to ancestors* is possible using only *soft* interventions, and (ii) *perfect* identifiability is possible using *hard* interventions. Remarkably, these guarantees match the best-known results for more restrictive single-node interventions. Furthermore, CRL algorithms are also provided that achieve the identifiability guarantees. A central step in designing these algorithms is establishing the relationships between UMN interventional CRL and score functions associated with the statistical models of different interventional environments. Establishing these relationships also serves as constructive proof of the identifiability guarantees.

Causal interactions among a group of variables are often modeled by a single causal graph. In some domains, however, these interactions are best described by multiple co-existing causal graphs, e.g., in dynamical systems or genomics. This paper addresses the hitherto unknown role of interventions in learning causal interactions among variables governed by a mixture of causal systems, each modeled by one directed acyclic graph (DAG). Causal discovery from mixtures is fundamentally more challenging than single-DAG causal discovery. Two major difficulties stem from (i) an inherent uncertainty about the skeletons of the component DAGs that constitute the mixture and (ii) possibly cyclic relationships across these component DAGs. This paper addresses these challenges and aims to identify edges that exist in at least one component DAG of the mixture, referred to as the *true* edges. First, it establishes matching necessary and sufficient conditions on the size of interventions required to identify the true edges. Next, guided by the necessity results, an adaptive algorithm is designed that learns all true edges using ${\cal O}(n^2)$ interventions, where $n$ is the number of nodes. Remarkably, the size of the interventions is optimal if the underlying mixture model does not contain cycles across its components. More generally, the gap between the intervention size used by the algorithm and the optimal size is quantified. It is shown to be bounded by the *cyclic complexity number* of the mixture model, defined as the size of the minimal intervention that can break the cycles in the mixture, which is upper bounded by the number of cycles among the ancestors of a node.

This paper studies the problem of designing an optimal sequence of interventions in a causal graphical model to minimize cumulative regret with respect to the best intervention in hindsight. This is, naturally, posed as a causal bandit problem. The focus is on causal bandits for linear structural equation models (SEMs) and soft interventions. It is assumed that the graph's structure is known and has $N$ nodes. Two linear mechanisms, one soft intervention and one observational, are assumed for each node, giving rise to $2^N$ possible interventions. The majority of the existing causal bandit algorithms assume that at least the interventional distributions of the reward node's parents are fully specified. However, there are $2^N$ such distributions (one corresponding to each intervention), acquiring which becomes prohibitive even in moderate-sized graphs. This paper dispenses with the assumption of knowing these distributions or their marginals. Two algorithms are proposed for the frequentist (UCB-based) and Bayesian (Thompson sampling-based) settings. The key idea of these algorithms is to avoid directly estimating the $2^N$ reward distributions and instead estimate the parameters that fully specify the SEMs (linear in $N$) and use them to compute the rewards. In both algorithms, under boundedness assumptions on noise and the parameter space, the cumulative regrets scale as $\tilde{\cal O} (d^{L+\frac{1}{2}} \sqrt{NT})$, where $d$ is the graph's maximum degree, and $L$ is the length of its longest causal path. Additionally, a minimax lower of $\Omega(d^{\frac{L}{2}-2}\sqrt{T})$ is presented, which suggests that the achievable and lower bounds conform in their scaling behavior with respect to the horizon $T$ and graph parameters $d$ and $L$.

This paper considers the problem of estimating unknown intervention targets in causal directed acyclic graphs from observational and interventional data in the presence of latent variables. The focus is on linear structural equation models with soft interventions. The existing approaches to this problem involve performing extensive conditional independence tests, and they estimate the unknown intervention targets alongside learning the structure of the causal model in its entirety. This joint learning approach results in algorithms that are not scalable as graph sizes grow. This paper proposes an approach that does not necessitate learning the entire causal model and focuses on learning only the intervention targets. The key idea of this approach is leveraging the property that interventions impose sparse changes in the precision matrix of a linear model. The proposed framework consists of a sequence of precision difference estimation steps. Furthermore, the necessary knowledge to refine an observational Markov equivalence class (MEC) to an interventional MEC is inferred. Simulation results are provided to illustrate the scalability of the proposed algorithm and compare it with those of the existing approaches.

Probabilistic graphical models (PGMs) are effective for capturing the statistical dependencies in stochastic databases. In many domains (e.g., working with multimodal data), one faces multiple information layers that can be modeled by structurally similar PGMs. While learning the structures of PGMs in isolation is well-investigated, the algorithmic design and performance limits of learning from multiple coupled PGMs are investigated far less. This paper considers learning the structural similarities shared by a pair of Ising PGMs. The objective is learning the shared structure with no regard for the structures exclusive to either of the graphs, and significantly different from the existing approaches that focus on entire structure of the graphs. We propose an algorithm for the shared structure learning objective, evaluate its performance empirically, and compare with existing approaches on structure learning of single graphs.

This paper considers the problem of estimating the unknown intervention targets in a causal directed acyclic graph from observational and interventional data. The focus is on soft interventions in linear structural equation models (SEMs). Current approaches to causal structure learning either work with known intervention targets or use hypothesis testing to discover the unknown intervention targets even for linear SEMs. This severely limits their scalability and sample complexity. This paper proposes a scalable and efficient algorithm that consistently identifies all intervention targets. The pivotal idea is to estimate the intervention sites from the difference between the precision matrices associated with the observational and interventional datasets. It involves repeatedly estimating such sites in different subsets of variables. The proposed algorithm can be used to also update a given observational Markov equivalence class into the interventional Markov equivalence class. Consistency, Markov equivalency, and sample complexity are established analytically. Finally, simulation results on both real and synthetic data demonstrate the gains of the proposed approach for scalable causal structure recovery. Implementation of the algorithm and the code to reproduce the simulation results are available at \url{https://github.com/bvarici/intervention-estimation}.