We propose and analyze random subspace variants of the second-order Adaptive Regularization using Cubics (ARC) algorithm. These methods iteratively restrict the search space to some random subspace of the parameters, constructing and minimizing a local model only within this subspace. Thus, our variants only require access to (small-dimensional) projections of first- and second-order problem derivatives and calculate a reduced step inexpensively. Under suitable assumptions, the ensuing methods maintain the optimal first-order, and second-order, global rates of convergence of (full-dimensional) cubic regularization, while showing improved scalability both theoretically and numerically, particularly when applied to low-rank functions. When applied to the latter, our adaptive variant naturally adapts the subspace size to the true rank of the function, without knowing it a priori.
@article{arxiv.2501.09734,
title = {Random Subspace Cubic-Regularization Methods, with Applications to Low-Rank Functions},
author = {Coralia Cartis and Zhen Shao and Edward Tansley},
journal= {arXiv preprint arXiv:2501.09734},
year = {2025}
}