Learning False Discovery Rate Control via Model-Based Neural Networks

Technical Summary: Learning False Discovery Rate Control via Model-Based Neural Networks

Problem Statement
In high-dimensional variable selection (where the number of variables $p$ exceeds the number of samples $n$), controlling the False Discovery Rate (FDR) while maximizing the True Positive Rate (TPR) is a critical challenge. Existing rigorous methods, such as the Terminating-Random Experiments (T-Rex) Selector, provide provable FDR guarantees but rely on analytical estimators for the False Discovery Proportion (FDP). These estimators are inherently conservative to ensure theoretical validity, often resulting in a significant gap between the realized FDP and the target FDR level. This over-cautiousness leads to a loss of statistical power, causing the method to miss true discoveries, particularly in low signal-to-noise ratio (SNR) regimes.

Methodology
The authors propose a learning-augmented enhancement to the T-Rex Selector framework. The core innovation is replacing the T-Rex's conservative analytical FDP estimator with a data-driven approximation learned by a model-based neural network.

1. Framework Integration: The T-Rex Selector operates by running $K$ parallel random experiments where dummy variables are appended to the predictor matrix. A forward selection algorithm (e.g., LARS) is terminated once a specific number of dummies enter the model. Variables are selected based on their relative frequency of selection ($\Phi$) across these experiments.
2. Neural Network Architecture: The proposed method utilizes a Multi-Layer Perceptron (MLP) with three ReLU-activated hidden layers (128, 64, and 32 neurons) and a final sigmoid output to constrain predictions to $[0, 1]$. The input consists of the flattened vector of T-Rex statistics, including the relative frequency matrix $\Phi$, and calibration parameters ($v, T, L$). To handle varying dimensions, $\Phi$ is zero-padded to a fixed maximum dimension.
3. Training Strategy: The network is trained exclusively on diverse synthetic datasets. This approach is chosen because real-world labeled data (with known ground-truth active sets) is rarely available for exploratory variable selection, and training on real data could introduce unfair advantages by compressing domain-specific information. The synthetic pipeline generates feature matrices from 14 different distributions (e.g., Gaussian, Student's t, Cauchy) with varying SNRs to ensure the model learns a generalizable calibration function rather than memorizing specific structures.
4. Loss Function: To maintain approximate control without formal guarantees, an asymmetric loss function is employed: $L(\hat{\xi}, \xi) = \max(0, \hat{\xi} - \xi)^2 + w \cdot \max(0, \xi - \hat{\xi})^2$ with $w > 1$. This penalizes underestimation of the FDP more heavily than overestimation, encouraging conservative behavior.
5. Calibration: During inference, the trained network replaces the analytical estimator in the "Calibrate & Fuse" stage. A grid search is performed to find optimal parameters $(v^*, T^*)$ that maximize the number of selected variables while satisfying the learned FDP constraint $\le \alpha$.

Key Contributions
The paper outlines three primary contributions:

1. Novel Framework: The integration of a deep neural network into the T-Rex Selector's calibration stage to provide a more accurate, data-driven FDP estimate.
2. Performance Validation: Extensive experiments demonstrating that the method, trained solely on synthetic data, significantly boosts statistical power (TPR) compared to the original T-Rex Selector while maintaining approximate FDR control.
3. Generalization: Evidence that the trained model generalizes effectively to realistic genomics data, specifically in a challenging synthetic Genome-Wide Association Study (GWAS) setting.

Results

• Synthetic Benchmarks: In experiments with $p=150$, $n=75$, and varying SNRs, the proposed method consistently achieved a higher TPR than the original T-Rex Selector across all SNR levels. This improvement was driven by the selection of less conservative calibration parameters (specifically lower voting thresholds), enabled by the tighter FDP approximation provided by the neural network.
• Genomics Application: On a simulated GWAS dataset (generated via HAPGEN2 to mimic human genomic correlation structures), the method increased the TPR from $14.9\% \pm 9.4\%$ (original T-Rex) to $17.3\% \pm 8.2\%$. The empirical FDR was $20.8\% \pm 27.3\%$, which remained close to the $20\%$ target, demonstrating effective approximate control despite the distributional shift from the training data.

Significance and Claims
The authors claim that their approach successfully addresses the trade-off between rigorous error control and statistical power. By trading the formal theoretical guarantee of the analytical estimator for a significant empirical increase in power, the method allows the procedure to operate much closer to the desired FDR boundary. The paper emphasizes that this data-driven calibration is a practical solution for applications where strict theoretical guarantees can be relaxed slightly in favor of discovering more true active variables. The exclusive reliance on diverse synthetic training data is highlighted as a key advantage, ensuring fairness, generalizability, and scalability without the need for labeled real-world datasets. The authors conclude that data-driven calibration serves as a powerful complement to traditional model-based methods in high-dimensional variable selection.