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arXiv:2312.14141 · Quantum

Quantum Algorithms for the Pathwise Lasso

João Fernando Doriguello, Debbie Lim, Chi Seng Pun, Patrick Rebentrost, Tushar S. Vaidya

Published Mar 25, 2025• 5 citations
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Abstract

We present a novel quantum high-dimensional linear regression algorithm with an ℓ1 -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.

DOI
10.22331/q-2025-03-25-1674

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We present a novel quantum high-dimensional linear regression algorithm with an ℓ1 -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.
Lasso and Ridge are important minimization problems in machine learning and statistics. They are versions of linear regression with squared loss where the vector θ ∈ R^d of coefficients is constrained in either ℓ1-norm (for Lasso) or in ℓ2-norm (for Ridge). We study the complexity of quantum algorithms for finding ε-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.
Lasso and Ridge are important minimization problems in machine learning and statistics. They are versions of linear regression with squared loss where the vector θ∈Rdθ\in\mathbb{R}^dθ∈Rd of coefficients is constrained in either ℓ1\ell_1ℓ1​-norm (for Lasso) or in ℓ2\ell_2ℓ2​-norm (for Ridge). We study the complexity of quantum algorithms for finding ε\varepsilonε-minimizers for these minimization problems. We show that for Lasso we can get a quadratic quantum speedup in terms of ddd by speeding up the cost-per-iteration of the Frank-Wolfe algorithm, while for Ridge the best quantum algorithms are linear in ddd, 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.
Linear regression is one of the most fundamental linear algebra problems. Given a dense matrix A∈Rn×dA \in \mathbb{R}^{n \times d}A∈Rn×d and a vector bbb, the goal is to find x′x'x′ such that $ \| Ax' - b \|_2^2 \leq (1+ε) \min_{x} \| A x - b \|_2^2 $. The best classical algorithm takes O(nd)+poly(d/ε)O(nd) + \mathrm{poly}(d/ε)O(nd)+poly(d/ε) time [Clarkson and Woodruff STOC 2013, Nelson and Nguyen FOCS 2013]. On the other hand, quantum linear regression algorithms can achieve exponential quantum speedups, as shown in [Wang Phys. Rev. A 96, 012335, Kerenidis and Prakash ITCS 2017, Chakraborty, Gily{é}n and Jeffery ICALP 2019]. However, the running times of these algorithms depend on some quantum linear algebra-related parameters, such as κ(A)κ(A)κ(A), the condition number of AAA. In this work, we develop a quantum algorithm that runs in O~(ε−1nd1.5)+poly(d/ε)\widetilde{O}(ε^{-1}\sqrt{n}d^{1.5}) + \mathrm{poly}(d/ε)O(ε−1n​d1.5)+poly(d/ε) time. It provides a quadratic quantum speedup in nnn over the classical lower bound without any dependence on data-dependent parameters. In addition, we also show our result can be generalized to multiple regression and ridge linear regression.
Regression is a cornerstone of statistics and machine learning, with applications spanning science, engineering, and economics. While quantum algorithms for regression have attracted considerable attention, most existing work has focused on linear regression, leaving many more complex yet practically important variants unexplored. In this work, we present a unified quantum framework for accelerating a broad class of regression tasks -- including linear and multiple regression, Lasso, Ridge, Huber, ℓp\ell_pℓp​-, and δpδ_pδp​-type regressions -- achieving up to a quadratic improvement in the number of samples mmm over the best classical algorithms. This speedup is achieved by extending the recent classical breakthrough of Jambulapati et al. (STOC'24) using several quantum techniques, including quantum leverage score approximation (Apers &Gribling, 2024) and the preparation of many copies of a quantum state (Hamoudi, 2022). For problems of dimension nnn, sparsity r<nr < nr<n, and error parameter εεε, our algorithm solves the problem in O~(rmn/ε+poly(n,1/ε))\widetilde{O}(r\sqrt{mn}/ε+ \mathrm{poly}(n,1/ε))O(rmn​/ε+poly(n,1/ε)) quantum time, demonstrating both the applicability and the efficiency of quantum computing in accelerating regression tasks.
This dissertation presents a study of quantum algorithms for problems that can be posed as matrix function tasks. In Chapter 1 we demonstrate a simple unifying framework for implementing of smooth functions of matrices on a quantum computer. This framework captures a variety of problems that can be solved by evaluating properties of some function of a matrix, and we identify speedups over classical algorithms for some problem classes. The analysis combines ideas from the classical theory of function approximation with the quantum algorithmic primitive of implementing linear combinations of unitary operators. In Chapter 2 we continue this study by looking at the role of sparsity of input matrices in constructing efficient quantum algorithms. We show that classically pre-processing an input matrix by spectral sparsification can be profitable for quantum Hamiltonian simulation algorithms, without compromising the simulation error or complexity. Such preprocessing incurs a one time cost linear in the size of the matrix, but can be exploited to exponentially speed up subsequent subroutines such as inversion. In Chapter 3, we give an application of this theory of matrix functions to the problem of estimating the Renyi entropy of an unknown quantum state. We combine matrix function techniques with mixed state quantum computation in the one-clean qubit model, and are able to bound of the expected runtime of our algorithm in terms of the unknown target quantity. In addition to the theme of analysing the complexity of our algorithms, we also identify instances that are of practical relevance, leading us to some problems of machine learning. In Chapter 4 we investigate kernel based learning methods using random features. We work with the QRAM input model suitable for big data, and show how matrix functions and the quantum Fourier transform can be used to devise a quantum algorithm for sampling random features that are optimised for given input data and choice of kernel. We obtain a potential exponential speedup over the best known classical algorithm even without explicit assumptions of sparsity or low rank. Finally in Chapter 5 we consider the technique of beamsearch decoding used in natural language processing. We work in the query model, and show how quantum search with advice can be used to construct a quantum search decoder that can find the optimal parse (which may for instance be a best translation, or text-to-speech transcript) at least quadratically faster than the best known classical algorithms, and obtain super-quadratic speedups in the expected runtime.
Quantum computing has the potential to speed up machine learning methods. One major direction is using quantum linear algebra to solve linear system problems or optimization problems in the machine learning area. Quantum approaches in the literature for different types of least squares problems demonstrate speedups w.r.t. dimension but have disadvantages w.r.t. precision and condition number. In this paper, we discuss how an iterative refinement scheme can deliver an accurate solution without the excessive cost of Quantum Linear System Algorithms (QLSAs). In addition, we propose an adaptive regularization approach that can mitigate the effect of condition number on solution time. In the second part of this paper, we investigate how state-of-the-art Quantum Interior Point Methods (QIPMs) can solve more sophisticated regression problems such as Lasso regression and support vector machine problems, which can be reformulated as quadratic optimization problems.