Quantum Regularized Least Squares

Shantanav Chakraborty1,2, Aditya Morolia1,3, and Anurudh Peduri4,1,2

1Center for Quantum Science and Technology, IIIT Hyderabad, Telangana 500032, India
2Center for Security, Theory and Algorithmic Research, IIIT Hyderabad, Telangana 500032, India
3Center for Computational Natural Sciences and Bioinformatics, IIIT Hyderabad, Telangana 500032, India
4Faculty of Computer Science, Ruhr University Bochum, 44801 Bochum, Germany

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Linear regression is a widely used technique to fit linear models and finds widespread applications across different areas such as machine learning and statistics. In most real-world scenarios, however, linear regression problems are often ill-posed or the underlying model suffers from $overfitting$, leading to erroneous or trivial solutions. This is often dealt with by adding extra constraints, known as regularization. In this paper, we use the frameworks of block-encoding and quantum singular value transformation (QSVT) to design the first quantum algorithms for quantum least squares with general $\ell_2$-regularization. These include regularized versions of quantum ordinary least squares, quantum weighted least squares, and quantum generalized least squares. Our quantum algorithms substantially improve upon prior results on $\textit{quantum ridge regression}$ (polynomial improvement in the condition number and an exponential improvement in accuracy), which is a particular case of our result.
To this end, we assume approximate block-encodings of the underlying matrices as input and use robust QSVT algorithms for various linear algebra operations. In particular, we develop a variable-time quantum algorithm for matrix inversion using QSVT, where we use quantum eigenvalue discrimination as a subroutine instead of gapped phase estimation. This ensures that substantially fewer ancilla qubits are required for this procedure than prior results. Owing to the generality of the block-encoding framework, our algorithms are applicable to a variety of input models and can also be seen as improved and generalized versions of prior results on standard (non-regularized) quantum least squares algorithms.

The problem of fitting a theoretical model to a large set of experimental data, known as linear regression, finds applications across disciplines, ranging from the natural sciences to machine learning and statistics. One popular method is the least squares regression technique, which constructs the best linear fit to the series of data points while minimizing the sum of squared errors. However, often, this approach runs into frequent problems leading to trivial or erroneous solutions. As a result, in most practical scenarios, the method of regularized least squares is employed, a generalization of the standard least squares approach and its variants. Indeed, regularized versions of ordinary, weighted, and generalized least squares have found wide applicability in the classical machine learning literature.

In this article, we develop the first quantum algorithms for ordinary, weighted, and generalized least squares, with general $\ell_2$-regularization. As with most problems in quantum machine learning, our quantum algorithms output a quantum state proportional to the corresponding classical solution of the problem. A particular case of our method is the problem of quantum ridge regression, for which our approach offers an exponential advantage in precision over prior quantum algorithms. Our results can also be seen as improved and generalized versions of prior results on quantum linear regression.

The main technical tools we make use of are robust versions of state-of-the-art techniques from quantum linear algebra, such as quantum singular value transformation and the framework of block-encoding. Additionally, we also develop quantum algorithmic primitives that are of independent interest. For instance, we develop a robust procedure for quantum singular value discrimination: this procedure determines whether the singular values of a matrix is above or below a certain threshold. We show that this subroutine drastically reduces the number of ancilla qubits required to perform variable-time amplitude amplification and, consequently, to solve quantum linear systems with optimal complexity. Our work serves as a concrete example of how techniques from quantum linear algebra can help solve useful problems in machine learning.

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Cited by

[1] Yanlin Chen and Ronald de Wolf, "Quantum Algorithms and Lower Bounds for Linear Regression with Norm Constraints", arXiv:2110.13086, (2021).

[2] Tong Ning, Youlong Yang, and Zhenye Du, "Quantum kernel logistic regression based Newton method", Physica A Statistical Mechanics and its Applications 611, 128454 (2023).

[3] Changpeng Shao, "Quantum speedup of leverage score sampling and its application", arXiv:2301.06107, (2023).

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