Causal representation learning (CRL) is the process of disentangling the *latent* low-dimensional causally-related generating factors underlying high-dimensional observable data. Extensive recent studies have characterized CRL identifiability and *perfect* recovery of the latent variables and their attendant causal graph. This paper introduces the notion of *reward-oriented* CRL, the purpose of which is to move away from perfectly learning the latent representation and instead learning it to the extent needed for optimizing a desired downstream task (reward). In reward-oriented CRL, perfectly learning the latent representation can be excessive; instead, it must be learned at the *coarsest* level sufficient for optimizing the desired task. Reward-oriented CRL is formalized as the optimization of a desired function of the observable data over the space of all possible interventions and focuses on linear causal and transformation models. To sequentially identify the optimal subset of interventions, an adaptive exploration algorithm is designed that learns the latent causal graph and the variables needed to identify the best intervention. It is shown that for an $n$-dimensional latent space and a $d$-dimensional observation space, over a horizon $T$ the algorithm's regret scales as $\tilde O(d^{\frac{1}{3}}n^{\frac{1}{3}}u^{\frac{2}{3}}T^{\frac{2}{3}} + u\sqrt{T})$, where $u$ measures total uncertainty in the graph estimates. Furthermore, an almost-matching lower bound is shown to scale as $\Omega(d^{\frac{1}{3}}n^{\frac{1}{3}}p^{\frac{2}{3}}T^{\frac{2}{3}} + p\sqrt{T})$, in which $u$ is replaced by $p$ that counts the number of causal paths in the graph.

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.

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.