Quantum Algorithms for the Pathwise Lasso

• URL: http://arxiv.org/abs/2312.14141v2
• Date: Mon, 17 Jun 2024 08:55:21 GMT
• Title: Quantum Algorithms for the Pathwise Lasso
• Authors: Joao F. Doriguello, Debbie Lim, Chi Seng Pun, Patrick Rebentrost, Tushar Vaidya,
• Abstract summary: We present a novel quantum high-dimensional linear regression algorithm based on the classical LARS (Least Angle Regression) pathwise algorithm. Our quantum algorithm provides the full regularisation path as the penalty term varies, but quadratically faster per iteration under specific conditions. We prove, via an approximate version of the KKT conditions and a duality gap, that the LARS algorithm is robust to errors.
• Score: 1.8058773918890538
• License: http://creativecommons.org/licenses/by/4.0/
• 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 quantum minimum-finding routine from D\"urr 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 routine from Chen and de Wolf (ICALP'23). As one of our main contributions, we construct a quantum unitary to approximately compute the joining times to be searched over by the approximate quantum minimum finding. Since the joining times are no longer exactly computed, it is no longer clear that the resulting approximate quantum algorithm obtains a good solution. As our second main contribution, we prove, via an approximate version of the KKT conditions and a duality gap, that the LARS algorithm (and thus 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 approximately computed. Moreover, 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 $d$. Finally, we propose a dequantised algorithm that also retains the polylogarithmic dependence on $n$, albeit with the linear scaling on $d$ from the standard LARS algorithm.

Related papers

• A quantum-classical hybrid algorithm with Ising model for the learning with errors problem [13.06030390635216]
  We propose a quantum-classical hybrid algorithm with Ising model (HAWI) to address the Learning-With-Errors (LWE) problem. We identify the low-energy levels of the Hamiltonian to extract the solution, making it suitable for implementation on current noisy intermediate-scale quantum (NISQ) devices. Our algorithm is iterations, and its time complexity depends on the specific quantum algorithm employed to find the Hamiltonian's low-energy levels.
  arXiv  Detail & Related papers  (2024-08-15T05:11:35Z)
• Sum-of-Squares inspired Quantum Metaheuristic for Polynomial Optimization with the Hadamard Test and Approximate Amplitude Constraints [76.53316706600717]
  Recently proposed quantum algorithm arXiv:2206.14999 is based on semidefinite programming (SDP) We generalize the SDP-inspired quantum algorithm to sum-of-squares. Our results show that our algorithm is suitable for large problems and approximate the best known classicals.
  arXiv  Detail & Related papers  (2024-08-14T19:04:13Z)
• Solving k-SAT problems with generalized quantum measurement [0.7373617024876725]
  We generalize the projection-based quantum measurement-driven $k$-SAT algorithm of Benjamin, Zhao, and Fitzsimons to arbitrary strength quantum measurements. We argue that the algorithm is most efficient with finite time and measurement resources in the continuum limit. For solvable $k$-SAT problems, the dynamics generated by the algorithm converge deterministically towards target dynamics in the long-time (Zeno) limit.
  arXiv  Detail & Related papers  (2024-06-19T15:05:18Z)
• Quantum algorithms for Hopcroft's problem [45.45456673484445]
  We study quantum algorithms for Hopcroft's problem which is a fundamental problem in computational geometry. The classical complexity of this problem is well-studied, with the best known algorithm running in $O(n4/3)$ time. Our results are two different quantum algorithms with time complexity $widetilde O(n5/6)$.
  arXiv  Detail & Related papers  (2024-05-02T10:29:06Z)
• Efficient quantum linear solver algorithm with detailed running costs [0.0]
  We introduce a quantum linear solver algorithm combining ideasdiabatic quantum computing with filtering techniques based on quantum signal processing. Our protocol reduces the cost of quantum linear solvers over state-of-the-art close to an order of magnitude for early implementations.
  arXiv  Detail & Related papers  (2023-05-19T00:07:32Z)
• Quantum Algorithms for Sampling Log-Concave Distributions and Estimating Normalizing Constants [8.453228628258778]
  We develop quantum algorithms for sampling logconcave distributions and for estimating their normalizing constants. We exploit quantum analogs of the Monte Carlo method and quantum walks. We also prove a $1/epsilon1-o(1)$ quantum lower bound for estimating normalizing constants.
  arXiv  Detail & Related papers  (2022-10-12T19:10:43Z)
• Quantum Goemans-Williamson Algorithm with the Hadamard Test and Approximate Amplitude Constraints [62.72309460291971]
  We introduce a variational quantum algorithm for Goemans-Williamson algorithm that uses only $n+1$ qubits. Efficient optimization is achieved by encoding the objective matrix as a properly parameterized unitary conditioned on an auxilary qubit. We demonstrate the effectiveness of our protocol by devising an efficient quantum implementation of the Goemans-Williamson algorithm for various NP-hard problems.
  arXiv  Detail & Related papers  (2022-06-30T03:15:23Z)
• Adaptive Algorithm for Quantum Amplitude Estimation [13.82667502131475]
  We propose an adaptive algorithm for interval estimation of amplitudes. The proposed algorithm uses a similar number of quantum queries to achieve the same level of precision. We rigorously prove that the number of oracle queries achieves $O(1/epsilon)$, i.e., a quadratic speedup over classical Monte Carlo sampling.
  arXiv  Detail & Related papers  (2022-06-16T21:11:15Z)
• Entanglement and coherence in Bernstein-Vazirani algorithm [58.720142291102135]
  Bernstein-Vazirani algorithm allows one to determine a bit string encoded into an oracle. We analyze in detail the quantum resources in the Bernstein-Vazirani algorithm. We show that in the absence of entanglement, the performance of the algorithm is directly related to the amount of quantum coherence in the initial state.
  arXiv  Detail & Related papers  (2022-05-26T20:32:36Z)
• Quantum algorithm for stochastic optimal stopping problems with applications in finance [60.54699116238087]
  The famous least squares Monte Carlo (LSM) algorithm combines linear least square regression with Monte Carlo simulation to approximately solve problems in optimal stopping theory. We propose a quantum LSM based on quantum access to a process, on quantum circuits for computing the optimal stopping times, and on quantum techniques for Monte Carlo.
  arXiv  Detail & Related papers  (2021-11-30T12:21:41Z)
• Quantum Gram-Schmidt Processes and Their Application to Efficient State Read-out for Quantum Algorithms [87.04438831673063]
  We present an efficient read-out protocol that yields the classical vector form of the generated state. Our protocol suits the case that the output state lies in the row space of the input matrix. One of our technical tools is an efficient quantum algorithm for performing the Gram-Schmidt orthonormal procedure.
  arXiv  Detail & Related papers  (2020-04-14T11:05:26Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.