Sparse Bayesian Learning via Stepwise Regression
This work addresses sparse probabilistic modeling for researchers in machine learning and statistics, offering incremental improvements in algorithm guarantees and performance.
The authors tackled the problem of sparse Bayesian learning by proposing a coordinate ascent algorithm called Relevance Matching Pursuit (RMP), which connects to Stepwise Regression as noise variance decreases, and derived new guarantees for Stepwise Regression algorithms, including improved bounds for Forward Regression and a polynomial-time computable bound for Backward Regression. They reported that RMP and its variant are efficient and maintain strong performance with correlated features in numerical experiments.
Sparse Bayesian Learning (SBL) is a powerful framework for attaining sparsity in probabilistic models. Herein, we propose a coordinate ascent algorithm for SBL termed Relevance Matching Pursuit (RMP) and show that, as its noise variance parameter goes to zero, RMP exhibits a surprising connection to Stepwise Regression. Further, we derive novel guarantees for Stepwise Regression algorithms, which also shed light on RMP. Our guarantees for Forward Regression improve on deterministic and probabilistic results for Orthogonal Matching Pursuit with noise. Our analysis of Backward Regression on determined systems culminates in a bound on the residual of the optimal solution to the subset selection problem that, if satisfied, guarantees the optimality of the result. To our knowledge, this bound is the first that can be computed in polynomial time and depends chiefly on the smallest singular value of the matrix. We report numerical experiments using a variety of feature selection algorithms. Notably, RMP and its limiting variant are both efficient and maintain strong performance with correlated features.