Stochastic Primal–Dual Hybrid Gradient Algorithm with Adaptive Step Sizes

• Published in: Journal of mathematical imaging and vision Vol. 66; no. 3; pp. 294 - 313
• Main Authors: Chambolle, Antonin, Delplancke, Claire, Ehrhardt, Matthias J., Schönlieb, Carola-Bibiane, Tang, Junqi
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2024; Springer Nature B.V; Springer Verlag
• Subjects: Adaptive algorithms; Applications of Mathematics; Computed tomography; Computer Science; Convergence; Convexity; Image Processing and Computer Vision; Mathematical Methods in Physics; Mathematics; Signal,Image and Speech Processing; Upper bounds
• ISSN: 0924-9907; 1573-7683
• DOI: 10.1007/s10851-024-01174-1

In this work, we propose a new primal–dual algorithm with adaptive step sizes. The stochastic primal–dual hybrid gradient (SPDHG) algorithm with constant step sizes has become widely applied in large-scale convex optimization across many scientific fields due to its scalability. While the product of the primal and dual step sizes is subject to an upper-bound in order to ensure convergence, the selection of the ratio of the step sizes is critical in applications. Up-to-now there is no systematic and successful way of selecting the primal and dual step sizes for SPDHG. In this work, we propose a general class of adaptive SPDHG (A-SPDHG) algorithms and prove their convergence under weak assumptions. We also propose concrete parameters-updating strategies which satisfy the assumptions of our theory and thereby lead to convergent algorithms. Numerical examples on computed tomography demonstrate the effectiveness of the proposed schemes.