Randomized Primal-Dual Methods with Adaptive Step Sizes
Erfan Yazdandoost Hamedani, Afrooz Jalilzadeh, Necdet S. Aybat
Abstract
In this paper we propose a class of randomized primal-dual methods incorporating line search to contend with large-scale saddle point (SP) problems defined by a convex-concave function $\mathcal L(\mathbf{x},y) = \sum_{i=1}^M f_i(x_i)+\Phi(\mathbf{x},y)-h(y)$. We analyze the convergence rate of the proposed method under mere convexity and strong convexity assumptions of $\mathcal L$ in $\mathbf{x}$-variable. In particular, assuming $\nabla_y\Phi(\cdot,\cdot)$ is Lipschitz and $\nabla_{\mathbf{x}}\Phi(\cdot,y)$ is coordinate-wise Lipschitz for any fixed $y$, the ergodic sequence generated by the algorithm achieves the $\mathcal O(M/k)$ convergence rate in the expected primal-dual gap. Furthermore, assuming that $\mathcal L(\cdot,y)$ is strongly convex for any $y$, and that $\Phi(\mathbf{x},\cdot)$ is affine for any $\mathbf{x}$, the scheme enjoys a faster rate of $\mathcal O(M/k^2)$ in terms of primal solution suboptimality. We implemented the proposed algorithmic framework to solve kernel matrix learning problem, and tested it against other state-of-the-art first-order methods.
BibTeX
@InProceedings{pmlr-v206-yazdandoost-hamedani23a,
title = {Randomized Primal-Dual Methods with Adaptive Step Sizes},
author = {Yazdandoost Hamedani, Erfan and Jalilzadeh, Afrooz and Aybat, Necdet S.},
booktitle = {Proceedings of The 26th International Conference on Artificial Intelligence and Statistics},
pages = {11185--11212},
year = {2023},
editor = {Ruiz, Francisco and Dy, Jennifer and van de Meent, Jan-Willem},
volume = {206},
series = {Proceedings of Machine Learning Research},
month = {25--27 Apr},
publisher = {PMLR},
pdf = {https://proceedings.mlr.press/v206/yazdandoost-hamedani23a/yazdandoost-hamedani23a.pdf},
url = {https://proceedings.mlr.press/v206/yazdandoost-hamedani23a.html},
abstract = {In this paper we propose a class of randomized primal-dual methods incorporating line search to contend with large-scale saddle point (SP) problems defined by a convex-concave function $\mathcal L(\mathbf{x},y) = \sum_{i=1}^M f_i(x_i)+\Phi(\mathbf{x},y)-h(y)$. We analyze the convergence rate of the proposed method under mere convexity and strong convexity assumptions of $\mathcal L$ in $\mathbf{x}$-variable. In particular, assuming $\nabla_y\Phi(\cdot,\cdot)$ is Lipschitz and $\nabla_{\mathbf{x}}\Phi(\cdot,y)$ is coordinate-wise Lipschitz for any fixed $y$, the ergodic sequence generated by the algorithm achieves the $\mathcal O(M/k)$ convergence rate in the expected primal-dual gap. Furthermore, assuming that $\mathcal L(\cdot,y)$ is strongly convex for any $y$, and that $\Phi(\mathbf{x},\cdot)$ is affine for any $\mathbf{x}$, the scheme enjoys a faster rate of $\mathcal O(M/k^2)$ in terms of primal solution suboptimality. We implemented the proposed algorithmic framework to solve kernel matrix learning problem, and tested it against other state-of-the-art first-order methods.}
}