Structure-exploiting interior-point solver for high-dimensional entropy-sparsified regression learning

Abstract

The solution of high-dimensional nonlinear regression problems through standard machine learning approaches often relies on first-order information, due to the numerical and memory challenges arising from the computation of the Hessian matrix and of the higher-order derivatives. While this scenario seems not favorable to second-order methods, here we show that an efficient and modular structure-exploiting interior-point solver can be successfully applied to the recently introduced class of entropy-based methods for regression learning. Specifically, by exploiting the favorable structure of the problem and of the Hessian matrix, we suggest a robust solution strategy based on explicit low-rank updates combined with an iterative Symmetric Quasi-Minimal Residual (SQMR) algorithm to solve the underlying system of linear equations. The results show that the proposed structure-exploiting solver – which relies on the hybrid parallelism and distributed-memory computing paradigm – allows a significant solution time speed-up with respect to a naive solution strategy. Furthermore, through an adequate use of the Message Passing Interface (MPI) and of Open Multi-Processing (OpenMP), the proposed solver enables the solution of large-scale problems on high-performance computing architectures consisting of thousands of compute nodes. The accompanying detailed convergence and performance analyses demonstrate both numerical robustness and high-performance capabilities for increasingly high-dimensional problems.