Quantum Algorithms for the Pathwise Lasso
Joao F. Doriguello, Debbie Lim, Chi Seng Pun, Patrick Rebentrost, Tushar Vaidya
TL;DR
The paper addresses the computational bottleneck of obtaining the full regularisation path for high-dimensional Lasso regression by introducing quantum algorithms based on the LARS (LARS) path algorithm. It presents a Simple quantum LARS that exactly computes the path with a quadratic speedup in the number of features, and an Approximate quantum LARS that also speeds up with a quadratic factor in the number of observations by approximating joining times, supported by an approximate KKT and duality-gap analysis to guarantee a close-to-optimal path. A dequantised classical variant is provided, along with strong Gaussian-design matrix results showing polylogarithmic dependence on $n$ in favorable regimes, and rigorous lower bounds for both classical and quantum Lasso queries. The work demonstrates a robust pathwise Lasso framework under approximation, with meaningful speedups in typical high-dimensional regimes and explicit conditions under which the quantum advantages manifest. These developments offer a pathway to faster sparse regression in settings where $d\gg n$, with implications for quantum-accelerated model selection and high-dimensional statistics.
Abstract
We present a novel quantum high-dimensional linear regression algorithm with an $\ell_1$-penalty based on the classical LARS (Least Angle Regression) pathwise algorithm. Similarly to available classical algorithms for Lasso, our quantum algorithm provides the full regularisation path as the penalty term varies, but quadratically faster per iteration under specific conditions. A quadratic speedup on the number of features $d$ is possible by using the simple quantum minimum-finding subroutine from Dürr and Hoyer (arXiv'96) in order to obtain the joining time at each iteration. We then improve upon this simple quantum algorithm and obtain a quadratic speedup both in the number of features $d$ and the number of observations $n$ by using the approximate quantum minimum-finding subroutine from Chen and de Wolf (ICALP'23). In order to do so, we approximately compute the joining times to be searched over by the approximate quantum minimum-finding subroutine. As another main contribution, we prove, via an approximate version of the KKT conditions and a duality gap, that the LARS algorithm (and therefore our quantum algorithm) is robust to errors. This means that it still outputs a path that minimises the Lasso cost function up to a small error if the joining times are only approximately computed. Furthermore, we show that, when the observations are sampled from a Gaussian distribution, our quantum algorithm's complexity only depends polylogarithmically on $n$, exponentially better than the classical LARS algorithm, while keeping the quadratic improvement on $d$. Moreover, we propose a dequantised version of our quantum algorithm that also retains the polylogarithmic dependence on $n$, albeit presenting the linear scaling on $d$ from the standard LARS algorithm. Finally, we prove query lower bounds for classical and quantum Lasso algorithms.