A Multilevel Method for Self-Concordant Minimization

Nick Tsipinakis () and Panos Parpas ()
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Nick Tsipinakis: UniDistance Suisse
Panos Parpas: Imperial College London

Journal of Optimization Theory and Applications, 2024, vol. 203, issue 3, No 16, 2509-2559

Abstract: 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.

Keywords: Convex optimization; Randomized Newton method; Multilevel methods; Self-concordant functions; Machine learning; 49J53; 49K99; more (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10957-024-02509-z

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