Identifying causal relationships from observational data is a fundamental yet challenging problem. This paper focuses on learning the causal outcome model, namely identifying the direct causes of an outcome and estimating their effects. We leverage data collected from multiple heterogeneous environments, which enable causal discovery through the invariance principle that the causal outcome model remains invariant across environments. Based on this principle, we propose Negative Weighted Distributionally Robust Optimization (NegDRO), a framework that minimizes the worst-case weighted combination of risks across environments and enforces invariance by allowing negative weights.

Under the additive interventions regime, we establish three main contributions. From a statistical perspective, we provide sufficient and nearly necessary identification conditions under which the NegDRO solution coincides with the true causal outcome model. From an optimization perspective, despite the inherent nonconvexity of the NegDRO objective, we show that it admits a benign optimization landscape in which all stationary points lie close to the causal outcome model. From a computational perspective, we develop a gradient-based algorithm that provably converges to the causal outcome model with non-asymptotic rates in both sample size and number of iterations.

In particular, NegDRO avoids exhaustive combinatorial searches over subsets of covariates used in existing approaches and scales efficiently to high-dimensional settings. To our knowledge, this is the first causal invariance learning method that solves a nonconvex optimization problem to global optimality.