Error-controlled non-additive interaction discovery in machine learning models

TL;DR

Diamond provides a rigorous, model-agnostic framework for discovering trustworthy non-additive feature interactions by integrating model-X knockoffs with a non-additivity distillation step to control the false discovery rate ($\text{FDR}$) across diverse ML models. The method yields a ranked set of interactions with an estimated $\text{FDR}$ and an interpretable distillation of true non-additive effects, enabling robust discovery even under model misspecification or varying knockoff schemes. Empirical results on simulated data and real biomedical datasets (diabetes progression, Drosophila enhancer activity, and mortality risk) demonstrate controlled error rates and practical utility for hypothesis generation, while highlighting limitations such as conservatism and challenges distinguishing direct versus transitive interactions. Diamond’s flexibility and rigorous error control make it applicable to high-stakes domains where explainability and statistical validity are essential, and it provides a foundation for future extensions to higher-order interactions and causal analysis.

Abstract

Machine learning (ML) models are powerful tools for detecting complex patterns within data, yet their "black box" nature limits their interpretability, hindering their use in critical domains like healthcare and finance. To address this challenge, interpretable ML methods have been developed to explain how features influence model predictions. However, these methods often focus on univariate feature importance, overlooking the complex interactions between features that ML models are capable of capturing. Recognizing this limitation, recent efforts have aimed to extend these methods to discover feature interactions, but existing approaches struggle with robustness and error control, especially under data perturbations. In this study, we introduce Diamond, a novel method for trustworthy feature interaction discovery. Diamond uniquely integrates the model-X knockoffs framework to control the false discovery rate (FDR), ensuring that the proportion of falsely discovered interactions remains low. A key innovation in Diamond is its non-additivity distillation procedure, which refines existing interaction importance measures to distill non-additive interaction effects, ensuring that FDR control is maintained. This approach addresses the limitations of off-the-shelf interaction measures, which, when used naively, can lead to inaccurate discoveries. Diamond's applicability spans a wide range of ML models, including deep neural networks, transformer models, tree-based models, and factorization-based models. Our empirical evaluations on both simulated and real datasets across various biomedical studies demonstrate Diamond's utility in enabling more reliable data-driven scientific discoveries. This method represents a significant step forward in the deployment of ML models for scientific innovation and hypothesis generation.

Figures (12)

• Figure 1: Overview of Diamond. (a) Diamond achieves false discovery rate (FDR) control by leveraging knockoffs -- dummy features that perfectly mimic the empirical dependence structure among the original features but are conditionally independent of the response given the original features. (b) Diamond trains generic ML models using both the original features and their knockoff counterparts as inputs. Diamond quantifies feature interactions from trained ML models and produces a ranked list of these interactions with estimated FDR, allowing users to confidently determine a cutoff threshold based on their desired confidence level. (c) Existing feature importance measures are unable to directly capture non-additive interactions. Geometrically, the marginal or interaction importance reported by current interpretation methods correspond to the projection of the total contribution to the prediction onto a one-dimensional feature axis or a two-dimensional feature-feature plane. Non-additive effects, however, represent the difference between the interaction effects and the marginal effects. These non-additive effects can manifest as either synergistic or repressive interactions. (d) Diamond distills non-additive effects from the interaction importance measures reported by existing methods, thereby maintaining FDR control at the target level. The distillation procedure is designed to remove both label-dependent marginal effects and label-independent feature biases from the reported feature interactions, leaving only the label-dependent non-additive interaction effects.
• Figure 2: Evaluating Diamond for FDR control on simulated datasets. (a) The evaluation is based on a test suite of 10 data-generating simulation functions proposed by tsang:detecting. (b) The reported interaction importance from existing methods in simulation function $F_1$ reveals a clear distribution disparity between original-only interactions and those involving knockoffs. The distilled non-additive interactions help mitigate distributional disparities. (c) Baseline methods fail to correctly control the FDR, thereby rendering the reported high power and AUROC invalid. (d) Diamond identifies important non-additive interactions with controlled FDR, compatible with various ML models. The non-additivity distillation procedure is critical; without it, the FDR cannot be controlled. Even in scenarios of model misspecification, such as using convolutional neural networks for tabular data, Diamond maintains FDR control, albeit with a loss of power.
• Figure 3: Evaluating Diamond for robustness on simulated datasets. (a) Diamond demonstrates robustness across knockoffs generated by various methods, including KnockoffsDiagnostics, KnockoffGAN, Deep knockoffs, and VAE knockoffs. (b) Diamond maintains FDR control when paired with invalid knockoffs, generated by independently permuting each feature across samples, albeit at the cost of reduced power. (c) Diamond demonstrates robustness across methodologically different interaction importance measures: Expected Hessian, Integrated Hessian, and model-specific measures.
• Figure 4: Evaluating Diamond on a real diabetes dataset. (a) Each feature contributes differently to predicting disease progression, as measured by the Expected Gradient scores in the MLP model. (b) Diamond is compared against three baseline methods. The blue stars indicate interactions supported by literature evidence, referenced by the accompanying PubMed identifiers. (c) Diamond is used with various ML models to identify important non-additive interactions. Each possible interaction is measured by the minimum FDR threshold cutoff at which it is selected, with the top interaction annotated. It is worth mentioning that two marginally important features do not necessarily result in important interactions, as anticipated in Diamond's design. (d) The top interaction, between body mass index and serum triglycerides level, is qualitatively evaluated from three aspects: the marginal importance measure, the interaction importance measure, and the contribution of the marginal importance measures to the interaction importance measure.
• Figure 5: Evaluating Diamond on a real Drosophila enhancer dataset. (a) Each feature contributes differently to predicting enhancer status, as measured by the Expected Gradient scores in the MLP model. (b) Diamond is compared against three baseline methods. The annotated transcription factors are labeled by their UniProt identifiers. The red stars indicate well-characterized physical interactions in early Drosophila embryos as ground truth. The blue stars indicate interactions supported by literature evidence, referenced by the accompanying PubMed identifiers. (c) Diamond is used with various ML models to identify important non-additive interactions. Each possible interaction is measured by the minimum FDR threshold cutoff at which it is selected, with the top five interaction annotated. The top interactions reported by the MLP model are qualitatively evaluated from three aspects: the contribution of feature values to the (d) marginal and (e) interaction importance measure, and (f) the contribution of the marginal importance measures to the interaction importance measure.

Theorems & Definitions (1)

• Definition 1: Model-X knockoff candes:panning