arXiv - CS - Computational Complexity Pub Date : 2020-02-12 , DOI: arxiv-2002.05056
Srinivasan Arunachalam; Reevu Maity

Suppose we have a weak learning algorithm $\mathcal{A}$ for a Boolean-valued problem: $\mathcal{A}$ produces hypotheses whose bias $\gamma$ is small, only slightly better than random guessing (this could, for instance, be due to implementing $\mathcal{A}$ on a noisy device), can we boost the performance of $\mathcal{A}$ so that $\mathcal{A}$'s output is correct on $2/3$ of the inputs? Boosting is a technique that converts a weak and inaccurate machine learning algorithm into a strong accurate learning algorithm. The AdaBoost algorithm by Freund and Schapire (for which they were awarded the G\"odel prize in 2003) is one of the widely used boosting algorithms, with many applications in theory and practice. Suppose we have a $\gamma$-weak learner for a Boolean concept class $C$ that takes time $R(C)$, then the time complexity of AdaBoost scales as $VC(C)\cdot poly(R(C), 1/\gamma)$, where $VC(C)$ is the $VC$-dimension of $C$. In this paper, we show how quantum techniques can improve the time complexity of classical AdaBoost. To this end, suppose we have a $\gamma$-weak quantum learner for a Boolean concept class $C$ that takes time $Q(C)$, we introduce a quantum boosting algorithm whose complexity scales as $\sqrt{VC(C)}\cdot poly(Q(C),1/\gamma);$ thereby achieving a quadratic quantum improvement over classical AdaBoost in terms of $VC(C)$.

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