Sparse ridge regression problems play a significant role across various domains.

Sparse ridge regression problems play a significant role across various domains. To solve sparse ridge regression, Liu et al. (2023) recently propose an advanced algorithm, Scalable Optimal $K$-Sparse Ridge Regression (OKRidge), which is both faster and more accurate than existing approaches. However, the absence of theoretical analysis on the error of OKRidge impedes its large-scale applications. In this paper, we reframe the estimation error of OKRidge as a Primary Optimization ($\textbf{PO}$) problem and employ the Convex Gaussian min-max theorem (CGMT) to simplify the $\textbf{PO}$ problem into an Auxiliary Optimization ($\textbf{AO}$) problem. Subsequently, we provide a theoretical error analysis for OKRidge based on the $\textbf{AO}$ problem. This error analysis improves the theoretical reliability of OKRidge. We also conduct experiments to verify our theorems and the results are in excellent agreement with our theoretical findings.

We consider an important problem in scientific discovery, namely identifying sparse governing equations for nonlinear dynamical systems. This involves solving sparse ridge regression problems to provable optimality in order to determine which terms drive the underlying dynamics. We propose a fast algorithm, OKRidge, for sparse ridge regression, using a novel lower bound calculation involving, first, a saddle point formulation, and from there, either solving (i) a linear system or (ii) using an ADMM-based approach, where the proximal operators can be efficiently evaluated by solving another linear system and an isotonic regression problem. We also propose a method to warm-start our solver, which leverages a beam search. Experimentally, our methods attain provable optimality with run times that are orders of magnitude faster than those of the existing MIP formulations solved by the commercial solver Gurobi.

Sparse ridge regression problems play a significant role across various domains.

We consider an important problem in scientific discovery, namely identifying sparse governing equations for nonlinear dynamical systems. This involves solving sparse ridge regression problems to provable optimality in order to determine which terms drive the underlying dynamics. We propose a fast algorithm, OKRidge, for sparse ridge regression, using a novel lower bound calculation involving, first, a saddle point formulation, and from there, either solving (i) a linear system or (ii) using an ADMM-based approach, where the proximal operators can be efficiently evaluated by solving another linear system and an isotonic regression problem. We also propose a method to warm-start our solver, which leverages a beam search. Experimentally, our methods attain provable optimality with run times that are orders of magnitude faster than those of the existing MIP formulations solved by the commercial solver Gurobi.