[Paper Review] Improved Quantum Boosting

Adam Izdebski, Ronald de Wolf|UvA-DARE (University of Amsterdam)|Sep 17, 2020

Machine Learning and Algorithms23 references4 citations

TL;DR

This paper presents QuantumSmoothBoost, a novel quantum boosting algorithm based on Servedio's SmoothBoost that achieves a significant speedup over classical and prior quantum boosting methods. By leveraging quantum approximate counting and efficient state preparation, it reduces the dependence on the weak learner's advantage $γ$ from $1/\gamma^{11}$ to $1/\gamma^5$, while maintaining $\tilde{O}(\sqrt{d})$ scaling with VC-dimension $d$, offering a polynomial quantum advantage in key parameters.

ABSTRACT

Boosting is a general method to convert a weak learner (which generates hypotheses that are just slightly better than random) into a strong learner (which generates hypotheses that are much better than random). Recently, Arunachalam and Maity gave the first quantum improvement for boosting, by combining Freund and Schapire's AdaBoost algorithm with a quantum algorithm for approximate counting. Their booster is faster than classical boosting as a function of the VC-dimension of the weak learner's hypothesis class, but worse as a function of the quality of the weak learner. In this paper we give a substantially faster and simpler quantum boosting algorithm, based on Servedio's SmoothBoost algorithm.

Motivation & Objective

• To develop a faster and simpler quantum boosting algorithm that improves upon the quantum boosting approach by Arunachalam and Maity (2020).
• To reduce the dependence on the weak learner's advantage $\gamma$, which previously limited quantum speedups in boosting.
• To achieve a polynomial quantum speedup in both $\gamma$ and VC-dimension $d$ while maintaining robustness to label noise.
• To explore the feasibility of quantum boosting in settings where the weak learner has high VC-dimension and moderate $\gamma$, where quantum speedups could be most impactful.

Proposed method

• The algorithm uses Servedio's SmoothBoost as a classical foundation, which is more robust to label noise and better suited for quantum speedup than AdaBoost.
• It applies quantum approximate counting (via amplitude estimation) to estimate the total misclassification weight in each iteration, replacing classical counting with quantum amplitude estimation.
• Quantum state preparation is used to efficiently prepare the distribution $D^t$ over examples, reducing the cost of distribution updates from $O(m)$ to $\tilde{O}(T/\sqrt{\varepsilon})$ per iteration.
• The algorithm maintains a weighted combination of weak hypotheses, with coefficients $\alpha_t$ derived from the estimated error, and uses quantum access to the weak learner's output to compute misclassification weights $M_i^t$.
• The number of iterations is kept at $T = \tilde{O}(1/\varepsilon\gamma^2)$, matching classical SmoothBoost, but each iteration is accelerated using quantum subroutines.
• The overall complexity is derived by combining quantum subroutines: $\tilde{O}(T(W/\sqrt{\varepsilon} + \sqrt{m}))$, leading to a final runtime of $\tilde{O}(W/\varepsilon^{2.5}\gamma^4 + \sqrt{d}/\varepsilon^{3.5}\gamma^5)$.

Experimental results

Research questions

• RQ1 Can a quantum boosting algorithm be designed that improves the dependence on the weak learner's advantage $\gamma$ compared to prior quantum boosting methods?
• RQ2 Is it possible to achieve a quantum speedup in boosting that scales favorably with the VC-dimension $d$ while maintaining robustness to label noise?
• RQ3 Can the number of iterations in quantum boosting be reduced by using a different quantum formulation of SmoothBoost or another boosting variant?
• RQ4 Does the quantum version of SmoothBoost retain the noise-robustness properties of classical SmoothBoost, particularly against malicious label noise?
• RQ5 Can quantum boosting be extended to agnostic learning or learning with structured noise models such as Massart or Tsybakov noise?

Key findings

• QuantumSmoothBoost achieves a runtime of $\tilde{O}(W/\varepsilon^{2.5}\gamma^4 + \sqrt{d}/\varepsilon^{3.5}\gamma^5)$, which is a polynomial improvement over the $\tilde{O}(W^{1.5}\sqrt{d}/\gamma^{11})$ complexity of the quantum AdaBoost variant by Arunachalam and Maity.
• The algorithm improves the dependence on $\gamma$ from $1/\gamma^{11}$ to $1/\gamma^5$, significantly enhancing performance for weak learners with small but non-negligible advantage.
• The dependence on VC-dimension $d$ scales as $\tilde{O}(\sqrt{d})$, matching the classical SmoothBoost and offering a quantum advantage in this parameter.
• For the case $\varepsilon = \delta = 1/3$, the complexity reduces to $\tilde{O}(W/\gamma^4 + \sqrt{d}/\gamma^5)$, which is substantially faster than the prior quantum boosting method.
• The method maintains the robustness of classical SmoothBoost to malicious label noise, as the distribution updates only weakly penalize mislabeled examples.
• The algorithm uses quantum amplitude estimation to estimate misclassification weights with multiplicative error 1.1, enabling efficient and accurate distribution updates in each iteration.