Branch-and-Bound Algorithms for Computing the Best-Subset Regression Models

Abstract

An efficient branch-and-bound algorithm for computing the best-subset regression models is proposed. The algorithm avoids the computation of the whole regression tree that generates all possible subset models. It is formally shown that if the branch-and-bound test holds, then the current subtree together with its right-hand side subtrees are cut. This reduces significantly the computational burden of the proposed algorithm when compared to an existing leaps-and-bounds method which generates two trees. Specifically, the proposed algorithm, which is based on orthogonal transformations, outperforms by O(n³) the leaps-and-bounds strategy. The criteria used in identifying the best subsets are based on monotone functions of the residual sum of squares (RSS) such as R², adjusted R², mean square error of prediction, and C_p. Strategies and heuristics that improve the computational performance of the proposed algorithm are investigated. A computationally efficient heuristic version of the branch-and-bound strategy which decides to cut subtrees using a tolerance parameter is proposed. The heuristic algorithm derives models close to the best ones. However, it is shown analytically that the relative error of the RSS, and consequently the corresponding statistic, of the computed subsets is smaller than the value of the tolerance parameter which lies between zero and one. Computational results and experiments on random and real data are presented and analyzed.

Key Words:

• Least squares
• QR decomposition
• Subset regression