arXiv 2110.130862021-10quant-phcs.CCcs.DScs.LG

范数约束线性回归的量子算法及其下界

Quantum Algorithms and Lower Bounds for Linear Regression with Norm Constraints

QuSoft（量子软件联盟）；CWI（荷兰国家数学与计算机科学研究所）；University of Amsterdam（阿姆斯特丹大学）

Yanlin Chen, Ronald de Wolf

备注：: v2: Main changes are the addition of a tight classical lower bound for Lasso, and small improvements in the existing text. 38 pages LaTeX

摘要：

Lasso和Ridge是机器学习和统计学中的重要最小化问题。它们是具有平方损失的线性回归的版本，其中系数的向量$θ\in\mathbb{R}^d$被约束在$\ell_1$ -范数（对于Lasso）或$\ell_2$ -范数（对于Ridge）中。我们研究了量子算法的复杂性，寻找这些最小化问题的$\varepsilon$ -极小。我们表明，对于拉索，我们可以得到一个二次量子加速$d$方面，通过加快弗兰克-沃尔夫算法的每次迭代的成本，而脊最好的量子算法是线性的$d$，是最好的经典算法。作为Lasso的量子下界的一个副产品，我们还证明了Lasso的第一个经典下界，它紧到多项式因子。

英文摘要：

Lasso and Ridge are important minimization problems in machine learning and statistics. They are versions of linear regression with squared loss where the vector $θ\in\mathbb{R}^d$ 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.

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