Quantum Algorithms and Lower Bounds for Linear Regression with Norm Constraints

TL;DR

This paper investigates the quantum computational complexity of solving Lasso and Ridge linear regression problems with norm constraints, demonstrating quantum speedups for Lasso and establishing classical lower bounds.

Contribution

It introduces quantum algorithms with quadratic speedup for Lasso and provides tight classical lower bounds, advancing understanding of quantum advantages in constrained linear regression.

Findings

Quantum algorithms achieve quadratic speedup for Lasso.

Classical lower bounds for Lasso are established, tight up to polylog factors.

Quantum algorithms for Ridge match classical complexity, showing no speedup.

Abstract

Lasso and Ridge are important minimization problems in machine learning and statistics. They are versions of linear regression with squared loss where the vector $\theta\in\mathbb{R}^d$ of coefficients is constrained in either $\ell_1$-norm (for Lasso) or in $\ell_2$-norm (for Ridge). We study the complexity of quantum algorithms for finding $\varepsilon$-minimizers for these minimization problems. We show that for Lasso we can get a quadratic quantum speedup in terms of $d$ by speeding up the cost-per-iteration of the Frank-Wolfe algorithm, while for Ridge the best quantum algorithms are linear in $d$, as are the best classical algorithms. As a byproduct of our quantum lower bound for Lasso, we also prove the first classical lower bound for Lasso that is tight up to polylog-factors.