Quantum Algorithms for Projection-Free Sparse Convex Optimization

• URL: http://arxiv.org/abs/2507.08543v1
• Date: Fri, 11 Jul 2025 12:43:58 GMT
• Title: Quantum Algorithms for Projection-Free Sparse Convex Optimization
• Authors: Jianhao He, John C. S. Lui,
• Abstract summary: For the vector domain, we propose two quantum algorithms for sparse constraints that find a $varepsilon$-optimal solution with the query complexity of $O(sqrtd/varepsilon)$.<n>For the matrix domain, we propose two quantum algorithms for nuclear norm constraints that improve the time complexity to $tildeO(rd/varepsilon2)$ and $tildeO(sqrtrd/varepsilon3)$.
• Score: 32.34794896079469
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $d$.

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