Year: 2024
Type: article
Abstract: The analysis of second-order optimization methods based either on sub-sampling, randomization or sketching has two serious shortcomings compared to the conventional Newton method. The first shortcoming is that the analysis of the iterates has only been shown to be scale-invariant only under specific assumptions on the problem structure. The second shortfall is that the fast convergence rates of second-order methods have only been established by making assumptions regarding the input data. In this paper, we propose a randomized Newton method for self-concordant functions to address both shortfalls. We propose a Self-concordant Iterative-minimization-Galerkin-based Multilevel Algorithm (SIGMA) and establish its super-linear convergence rate using the theory of self-concordant functions. Our analysis is based on the connections between multigrid optimization methods, and the role of coarse-grained or reduced-order models in the computation of search directions. We take advantage of the insights from the analysis to significantly improve the performance of second-order methods in machine learning applications. We report encouraging initial experiments that suggest SIGMA outperforms other state-of-the-art sub-sampled/sketched Newton methods for both medium and large-scale problems.
Authors: Nick Tsipinakis, Panos Parpas
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
OpenAlex ID: W3176634865