A note on the complexity of random subspace model-based methods for derivative-free optimization

Abstract

We demonstrate that, with a suitable rescaling, using Johnson-Lindenstrauss transforms (JLTs) in the random subspace model-based derivative-free optimization (DFO) algorithm from [Cartis & Roberts, Math. Prog. 199 (2023)] achieves an improved first-order worst-case evaluation complexity bound compared to 'unscaled' JLTs. This improved bound does not require any modification to the original algorithm or complexity analysis, and matches the best-known bound in terms of dimension dependence for model-based DFO (e.g. achieved by [Scheinberg & Chaudhry, ICM 2026] using Haar matrices in a similar framework). A similarly refined analysis of the analogous second-order subspace method from [Cartis & Roberts, Opt. Meth. & Softw. (2026)] shows that using suitably scaled JLTs to generate the subspaces gives a better complexity bound, which improves over full-space methods only if the problem has low effective rank.