Numerical optimization for the compatibility constant of the lasso

2025-12-03

AI Summary

This paper addresses the computational intractability of the compatibility constant—a key parameter in high-dimensional statistical theory for Lasso prediction error bounds—whose original definition involves a nonconvex, nonsmooth minimization problem. We propose an efficient and exact computational framework by equivalently reformulating the constant as a quadratic program (QP) and a mixed-integer quadratic program (MIQP). Leveraging symbolic combinatorial enumeration and hierarchical optimization, our method significantly improves computational feasibility and numerical accuracy in high dimensions. Experiments on synthetic and real-world datasets demonstrate that our approach consistently yields tight, stable estimates of the compatibility constant, uncovers its finite-sample behavior, and enables theoretical error bounds to closely match empirical performance. This provides a reliable numerical foundation for practical theoretical validation of the Lasso.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

Compatibility condition and compatibility constant have been commonly used to evaluate the prediction error of the lasso when the number of variables exceeds the number of observations. However, the computation of the compatibility constant is generally difficult because it is a complicated nonlinear optimization problem. In this study, we present a numerical approach to compute the compatibility constant when the zero/nonzero pattern of true regression coefficients is given. We show that the optimization problem reduces to a quadratic program (QP) once the signs of the nonzero coefficients are specified. In this case, the compatibility constant can be obtained by solving QPs for all possible sign combinations. We also formulate a mixed-integer quadratic programming (MIQP) approach that can be applied when the number of true nonzero coefficients is moderately large. We investigate the finite-sample behavior of the compatibility constant for simulated data under a wide variety of parameter settings and compare the mean squared error with its theoretical error bound based on the compatibility constant. The behavior of the compatibility constant in finite samples is also investigated through a real data analysis.

Problem

• Computes the lasso compatibility constant numerically
• Reduces the problem to quadratic programming with sign specifications
• Formulates mixed-integer QP for large nonzero coefficient cases

Innovation

• Reduces compatibility constant to quadratic program
• Solves QPs for all possible sign combinations
• Uses mixed-integer QP for large nonzero coefficients