Sparse Neural SIV: Learning Nonlinear Causal Relationships with Synthetic Instruments

We consider learning nonlinear causal relationships between multiple treatments and an outcome under unobserved confounding. We leverage information in the joint treatment distribution to learn these relationships without external instruments, assuming that only a sparse, unknown subset of treatments has a causal effect on the outcome. We develop a latent-factor framework under which valid instrumental variables can be derived from the treatments themselves. These synthetic instruments (SIVs) enable identification of nonlinear causal relationships through functional sparsity and a plurality rule. For estimation, we propose Sparse Neural SIV, which incorporates treatment-level sparsity into neural instrumental variable regression to jointly select active treatments and estimate the causal relationship. We fit the network by matching outcomes to predictions averaged over treatments sampled conditional on the synthetic instruments. Under suitable regularity conditions, we establish an estimation-error bound for a global minimizer of the regularized fitting criterion and give conditions for recovering the active treatments. Simulations show lower estimation error than the competing methods in the interaction settings studied under Gaussian and mixture treatment distributions. A mouse-genomics application illustrates nonlinear fitted relationships between gene expression and body weight.