Quantum Algorithms for Structured Prediction

TL;DR

This paper presents two quantum algorithms that significantly speed up structured prediction tasks by reducing the complexity related to the label space size, with practical implications demonstrated through image tagging simulations.

Contribution

The paper introduces novel quantum algorithms for structured prediction that achieve quadratic speedup in label space size and incorporate robust stochastic gradient methods with quantum subroutines.

Findings

Quantum algorithms reduce runtime complexity for structured prediction.

The proposed methods maintain convergence despite gradient approximation errors.

Numerical simulations show improved performance in image tagging tasks.

Abstract

We introduce two quantum algorithms for solving structured prediction problems. We first show that a stochastic gradient descent that uses the quantum minimum finding algorithm and takes its probabilistic failure into account solves the structured prediction problem with a runtime that scales with the square root of the size of the label space, and in $\widetilde O\left(1/\epsilon\right)$ with respect to the precision, $\epsilon$, of the solution. Motivated by robust inference techniques in machine learning, we then introduce another quantum algorithm that solves a smooth approximation of the structured prediction problem with a similar quantum speedup in the size of the label space and a similar scaling in the precision parameter. In doing so, we analyze a variant of stochastic gradient descent for convex optimization in the presence of an additive error in the calculation of the gradients, and show that its convergence rate does not deteriorate if the additive errors are of the order $O(\sqrt\epsilon)$. This algorithm uses quantum Gibbs sampling at temperature $\Omega (\epsilon)$ as a subroutine. Based on these theoretical observations, we propose a method for using quantum Gibbs samplers to combine feedforward neural networks with probabilistic graphical models for quantum machine learning. Our numerical results using Monte Carlo simulations on an image tagging task demonstrate the benefit of the approach.