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 $VC(C)$.

本文研究了一种利用量子技术改进 AdaBoost 算法，实现在 Boolean 概念类中比传统算法具有二次量子改进的时间复杂度，为弱学习算法提供了新思路。

量子提升

Quantum Boosting

We extend the theory of boosting for regression problems to the online
learning setting. Generalizing from the batch setting for boosting, the notion
of a weak learning algorithm is modeled as an online learning algorithm with
linear loss functions that competes with a base class of regression functions,
while a strong learning algorithm is an online learning algorithm with convex
loss functions that competes with a larger class of regression functions. Our
main result is an online gradient boosting algorithm which converts a weak
online learning algorithm into a strong one where the larger class of functions
is the linear span of the base class. We also give a simpler boosting algorithm
that converts a weak online learning algorithm into a strong one where the
larger class of functions is the convex hull of the base class, and prove its
optimality.

本文将扩展反推理论以解决在线学习中的回归问题，提出了两种弱学习算法模型，并提出一种在线梯度推进算法将弱学习算法转化为强学习算法，同时介绍了一种更简单的反推算法，并证明了其最优性。