On the Necessity of Non-Additive Modeling for Interaction Prediction: Theory and Application to Drug Synergy

Many prediction tasks require combining separately observed inputs, such as drugmolecules and a cell line, imaging modalities, or sensor channels, whose jointeffect cannot be captured by summing independent contributions. Whether inter-action modeling is theoretically necessary in such settings, or merely empiricallyuseful, has remained an open question. We prove a structural necessity theoremfor interaction prediction: any model in the functionally additive class, regard-less of the expressiveness of its per-condition components, suffers an irreducibleprediction gap when the target contains genuine cross-condition interaction terms.We derive an explicit, strictly positive lower bound for this gap that depends onlyon the magnitude of the target’s interaction component, thereby characterizing alimitation of the function class rather than of any particular architecture. We in-stantiate this result in drug synergy prediction, where the standard null modelsof pharmacological interaction are additive by construction, making the neces-sity gap concrete and measurable. We introduce Iψ, a structured bilinear operatorthat computes pairwise drug interactions as a low-rank product of individual com-pound embeddings, instantiating the non-additive class the theorem calls for. Onthe Therapeutics Data Commons (TDC) DrugComb benchmark, Iψcloses thepredicted gap across five synergy scores under a common training and evaluationprotocol, with each model receiving its own hyperparameter sweep. Iψuses abouta quarter of the parameter budget of FiLM, adapted here for drug synergy, andstill exceeds it at its full published size on the combo split. On an independentbenchmark (OncopolyPharmacology) the ordering holds. An ablation confirmsthe gain comes from the bilinear term itself. Hard generalization splits confirmthe theorem’s boundary condition.