Quantum Algorithms for Projection-Free Sparse Convex Optimization

TL;DR

This paper introduces quantum algorithms for sparse convex optimization that significantly reduce query complexity and runtime compared to classical methods, applicable to vector and matrix domains with sparse and nuclear norm constraints.

Contribution

The paper presents novel quantum algorithms for projection-free sparse convex optimization, achieving improved query and time complexities over classical algorithms in both vector and matrix settings.

Findings

Quantum algorithms reduce query complexity by factors of up to O(√d) and O(d).

Quantum methods improve runtime for matrix nuclear norm constraints by factors of at least O(√d).

Algorithms outperform classical methods in high-dimensional sparse convex optimization tasks.

Abstract

This paper considers the projection-free sparse convex optimization problem for the vector domain and the matrix domain, which covers a large number of important applications in machine learning and data science. For the vector domain $\mathcal{D} \subset \mathbb{R}^d$, we propose two quantum algorithms for sparse constraints that finds a $\varepsilon$-optimal solution with the query complexity of $O(\sqrt{d}/\varepsilon)$ and $O(1/\varepsilon)$ by using the function value oracle, reducing a factor of $O(\sqrt{d})$ and $O(d)$ over the best classical algorithm, respectively, where $d$ is the dimension. For the matrix domain $\mathcal{D} \subset \mathbb{R}^{d\times d}$, we propose two quantum algorithms for nuclear norm constraints that improve the time complexity to $\tilde{O}(rd/\varepsilon^2)$ and $\tilde{O}(\sqrt{r}d/\varepsilon^3)$ for computing the update step, reducing at least a factor of $O(\sqrt{d})$ over the best classical algorithm, where $r$ is the rank of the gradient matrix. Our algorithms show quantum advantages in projection-free sparse convex optimization problems as they outperform the optimal classical methods in dependence on the dimension $d$.