Solving FDR-Controlled Sparse Regression Problems with Five Million Variables on a Laptop

Abstract: Currently, there is an urgent demand for scalable multivariate and high-dimensional false discovery rate (FDR)-controlling variable selection methods to ensure the repro-ducibility of discoveries. However, among existing methods, only the recently proposed Terminating-Random Experiments (T-Rex) selector scales to problems with millions of variables, as encountered in, e.g., genomics research. The T-Rex selector is a new learning framework based on early terminated random experiments with computer-generated dummy variables. In this work, we propose the Big T-Rex, a new implementation of T-Rex that drastically reduces its Random Access Memory (RAM) consumption to enable solving FDR-controlled sparse regression problems with millions of variables on a laptop. We incorporate advanced memory-mapping techniques to work with matrices that reside on solid-state drive and two new dummy generation strategies based on permutations of a reference matrix. Our nu-merical experiments demonstrate a drastic reduction in memory demand and computation time. We showcase that the Big T-Rex can efficiently solve FDR-controlled Lasso-type problems with five million variables on a laptop in thirty minutes. Our work empowers researchers without access to high-performance clusters to make reproducible discoveries in large-scale high-dimensional data.

• The paper introduces Big T-Rex, a novel implementation of the T-Rex selector that achieves FDR-controlled sparse regression on datasets with up to five million variables using standard laptop hardware.
• It employs memory mapping and innovative dummy permutation strategies to reduce RAM usage by up to 88 times and cut computation time by a factor of up to 6.
• Results show robust FDR control and high true positive rates, making Big T-Rex a valuable tool for scalable high-dimensional variable selection in scientific research.

Solving FDR-Controlled Sparse Regression Problems with Five Million Variables on a Laptop

The paper under consideration presents a novel implementation of the Terminating-Random Experiments (T-Rex) selector, named "Big T-Rex," designed to drastically reduce memory consumption while accommodating false discovery rate (FDR)-controlled sparse regression problems involving millions of variables using standard laptop hardware. This development is timely and essential given the increasing demand for scalable high-dimensional variable selection methods to enhance the reproducibility of scientific discoveries.

Overview of the T-Rex Selector

The T-Rex selector is an FDR-controlling variable selection method that operates efficiently even for ultra-high-dimensional data, where the number of predictors ($p$) significantly exceeds the number of observations ($n$). This approach leverages early terminated random experiments with dummy variables, aiming to maximize the number of selected variables while controlling the FDR at a user-defined level, $\alpha$. The core of the T-Rex selector involves:

1. Dummy Variable Generation: Multiple dummy matrices are generated and appended to the original predictor matrix.
2. Forward Variable Selection: A forward selection process is applied to enlarged predictor matrices, terminating once a predefined number of dummy variables are included.
3. Calibrated Fusion: The relative occurrence of variables across experiments is computed, followed by estimating the false discovery proportion and adjusting the selected variables to meet the FDR threshold.
4. Output: The final active set of variables is determined based on the calibrated fusion results.

Innovations in Big T-Rex Implementation

The Big T-Rex implementation introduces significant innovations primarily aimed at reducing the random access memory (RAM) demands of the T-Rex selector. Key enhancements include:

1. Memory Mapping: By adopting memory mapping techniques, the Big T-Rex stores the enlarged predictor matrices and T-LARS selector parameters on solid-state drives (SSDs) instead of RAM. This approach allows for online processing, where data is accessed and processed on-demand, thus circumventing memory limitations.
2. Dummy Permutation Strategies: Two new strategies for dummy generation based on permutations of a reference dummy matrix are proposed, significantly reducing both RAM usage and SSD storage requirements. These strategies ensure that the dummies retain their necessary statistical properties while being more memory-efficient.

Numerical Results and Performance

The Big T-Rex was rigorously tested through several simulation experiments, comparing its performance against the original T-Rex implementation. Key findings include:

• Memory Demand: The Big T-Rex reduced RAM usage by up to 88 times compared to the original T-Rex implementation, enabling the handling of larger datasets within the constraints of typical laptop hardware.
• Computation Time: The Big T-Rex, particularly with the dummy permutation strategy $\mathrm{S_1}$, achieved a reduction in computation time by a factor of up to 6.
• FDR and TPR: The Big T-Rex maintained robust FDR control and high true positive rates (TPR) across varying signal-to-noise ratios (SNR), ensuring its reliability for high-dimensional data analysis.

Implications and Future Developments

The Big T-Rex represents a significant advancement in the field of high-dimensional variable selection, particularly for applications in genomics, proteomics, and other fields dealing with large-scale data. By enabling efficient FDR-control on standard laptop hardware, this implementation democratizes access to advanced analytical tools, empowering researchers without access to high-performance computing (HPC) resources.

Future developments could explore further optimization of memory-mapping techniques, extensions to other model types beyond linear regression, and parallel processing capabilities to enhance scalability and reduce computation times further. Additionally, the open-source release of the Big T-Rex will likely spur community-driven improvements and broader adoption in various scientific domains.

In conclusion, the Big T-Rex not only extends the computational feasibility of the original T-Rex selector but also sets a new benchmark for efficiently tackling large-scale high-dimensional variable selection problems using commonly available hardware, thereby fostering reproducible and robust scientific discoveries.