In this work, we improve upon the guarantees for sparse random embeddings, as they were recently provided and analyzed by Freksen at al. (NIPS'18) and Jagadeesan (NIPS'19). Specifically, we show that (a) our bounds are explicit as opposed to the asymptotic guarantees provided previously, and (b) our bounds are guaranteed to be sharper by practically significant constants across a wide range of parameters, including the dimensionality, sparsity and dispersion of the data. Moreover, we empirically demonstrate that our bounds significantly outperform prior works on a wide range of real-world datasets, such as collections of images, text documents represented as bags-of-words, and text sequences vectorized by neural embeddings. Behind our numerical improvements are techniques of broader interest, which improve upon key steps of previous analyses in terms of (c) tighter estimates for certain types of quadratic chaos, (d) establishing extreme properties of sparse linear forms, and (e) improvements on bounds for the estimation of sums of independent random variables.

Bayesian optimization techniques have been successfully applied to robotics, planning, sensor placement, recommendation, advertising, intelligent user interfaces and automatic algorithm configuration. Despite these successes, the approach is restricted to problems of moderate dimension, and several workshops on Bayesian optimization have identified its scaling to high dimensions as one of the holy grails of the field. In this paper, we introduce a novel random embedding idea to attack this problem. The resulting Random EMbedding Bayesian Optimization (REMBO) algorithm is very simple and applies to domains with both categorical and continuous variables. The experiments demonstrate that REMBO can effectively solve high-dimensional problems, including automatic parameter configuration of a popular mixed integer linear programming solver.