Best Subset Selection: Statistical Computing Meets Quantum Computing

TL;DR

This paper introduces a quantum algorithm for best subset selection that significantly reduces computational complexity without needing oracle information, and demonstrates its effectiveness through theoretical analysis and empirical experiments.

Contribution

It proposes a novel non-oracular quantum adaptive search method for subset selection, applicable to various statistical learning problems, and enhances it with a hybrid strategy for improved success probability.

Findings

QAS reduces complexity from O(D) to O(√D)logD

QAS attains success probability q in O(logD) iterations

Hybrid strategy improves success probability through majority voting

Abstract

With the rapid development of quantum computers, quantum algorithms have been studied extensively. However, quantum algorithms tackling statistical problems are still lacking. In this paper, we propose a novel non-oracular quantum adaptive search (QAS) method for the best subset selection problems. QAS performs almost identically to the naive best subset selection method but reduces its computational complexity from $O(D)$ to $O(\sqrt{D}\log_2D)$, where $D=2^p$ is the total number of subsets over $p$ covariates. Unlike existing quantum search algorithms, QAS does not require the oracle information of the true solution state and hence is applicable to various statistical learning problems with random observations. Theoretically, we prove QAS attains any arbitrary success probability $q \in (0.5, 1)$ within $O(\log_2D)$ iterations. When the underlying regression model is linear, we propose a quantum linear prediction method that is faster than its classical counterpart. We further introduce a hybrid quantum-classical strategy to avoid the capacity bottleneck of existing quantum computing systems and boost the success probability of QAS by majority voting. The effectiveness of this strategy is justified by both theoretical analysis and extensive empirical experiments on quantum and classical computers.