When Is It Safe to Borrow Across Treatments? Target-Aware Learning for Combinatorial Interventions

Combinatorial interventions create an exponential treatment space from a modest number of components. The resulting sparsity motivates cross-treatment borrowing, but smoothing between neighboring bundles can attenuate interactions of interest. We introduce Treatment-Space Graph Selection (TSGS), which separates an immutable graph of feasible interventions from a target-indexed graph used only for statistical borrowing. The outcome surface is decomposed into prespecified structured variation and its orthogonal complement; only the latter is graph-regularized, and the graph and penalty are selected by an independent augmented inverse-propensity-weighted score for the desired contrast family. Our theory gives (i) an exact finite-sample contrast-risk decomposition, (ii) a sharp characterization of uniform dominance over all zero-sum causal contrasts by Loewner ordering on the contrast subspace, (iii) a Boolean spectral theorem explaining why edge effects and interaction effects weight the same smoothing error differently, and (iv) a finite-candidate oracle inequality. In projected graph-sequence experiments across 36 independently simulated support–roughness regimes, TSGS beats no borrowing in every cell, whereas indiscriminate full-graph smoothing is significantly harmful in five rough regimes; edge and synergy targets select different rules in 65.8% of repetitions. In the randomized ACTG 175 trial, target-selected borrowing reduces held-out contrast score by 0.7–1.7% relative to no borrowing across five clinical targets. The central conclusion is that treatment geometry specifies feasible interventions, but borrowing rules should be selected for the causal contrasts of interest.