PAC-Causal Learnability: From Identification to Finite-Sample Guarantees

We provide explicit, finite-sample guarantees for learning causal representations from data with a sublinear number of environments. Causal representation learning seeks to provide a rigourous foundation for the general representation learning problem by bridging causal models with latent factor models in order to learn interpretable representations with causal semantics. Despite a blossoming theory of identifiability in causal representation learning, estimation and finite-sample bounds are less well understood. We show that causal representations can be learned with only a logarithmic number of unknown, multi-node interventions, and that the intervention targets need not be carefully designed in advance. Through a careful perturbation analysis, we provide a new analysis of this problem that guarantees consistent recovery of (a) the latent causal graph, (b) the mixing matrix and representations, and (c) unknown intervention targets.

Identification conditions describe the computability of a target query or parameter of interest as a function of the type and amount of information available. In causal identification, this information is often expressed in the form of a causal graph, and data are observed or collected for some subset of variables in the graph. Target queries may be for a single effect alone or for a class of effects in a given model. The derivation of an identification algorithm then defines mathematically the process by which the desired causal effect(s) can be uniquely determined, theoretically, in expectation. Identifiability in expectation, or 'theoretical identifiability,' generally assumes asymptotic properties, infinite data, or other mathematically idealized conditions. In this paper, we explore a fundamental distinction between this theoretical, idealized notion of identifiability and a proposed alternative that is computation-bound. The framework we propose - 'computational identifiability' - is to instead define a finite computational search procedure for an empirical estimator. If this process finds an estimator empirically, within a desired error tolerance, then identifiability is satisfied, conditional on the specified assumptions of the search (i.e., a prior distribution over the parameters) and conditional on the search procedure itself. Through several experiments, we demonstrate how this framework allows us to answer fine-grained, practical identification questions, such as identification with small finite samples, with ambiguous graphical criteria, with mixed observational-interventional data, and across counterfactual data and estimands. Code is available at https://github.com/lbynum/metadentify.

What determines the unavoidable sample cost of learning cyclic causal structure? For cyclic linear non-Gaussian models, we study exact condensation recovery from observational data: identifying the strongly connected component (SCC) partition and all edges between components. We establish the first information-theoretic lower bounds on sample complexity for this target. For $p$ variables, maximum SCC size $s_{\max}$, and maximum external-parent count $d_B$, any estimator requires order $s_{\max}\log(ep/s_{\max})+d_B\log(ep/d_B)$ samples in the worst case over a regular model class. These bounds distinguish the costs of SCC membership and external-parent selection. Under principal invertibility and without correlation faithfulness, we establish a population block-exogeneity principle that identifies unknown root SCCs through residual independence and inclusion minimality. A sparse-adjustment characterization shows that small adjustment sets suffice to identify SCCs and their direct external parents, without regressing on all previously recovered variables. These characterizations yield BlockExo, which attains a structurally matching sample bound without knowing $s_{\max}$ or $d_B$ under suitable conditions. Simulations support the structural dependence of our sample bound and demonstrate BlockExo's sample-efficient recovery in comparisons with other methods for cyclic causal discovery.