Year: 2019
Type: preprint
Abstract: Gaussian processes (GP) are a well studied Bayesian approach for the optimization of black-box functions. Despite their effectiveness in simple problems, GP-based algorithms hardly scale to high-dimensional functions, as their per-iteration time and space cost is at least quadratic in the number of dimensions $d$ and iterations $t$. Given a set of $A$ alternatives to choose from, the overall runtime $O(t^3A)$ is prohibitive. In this paper we introduce BKB (budgeted kernelized bandit), a new approximate GP algorithm for optimization under bandit feedback that achieves near-optimal regret (and hence near-optimal convergence rate) with near-constant per-iteration complexity and remarkably no assumption on the input space or covariance of the GP. We combine a kernelized linear bandit algorithm (GP-UCB) with randomized matrix sketching based on leverage score sampling, and we prove that randomly sampling inducing points based on their posterior variance gives an accurate low-rank approximation of the GP, preserving variance estimates and confidence intervals. As a consequence, BKB does not suffer from variance starvation, an important problem faced by many previous sparse GP approximations. Moreover, we show that our procedure selects at most $\tilde{O}(d_{eff})$ points, where $d_{eff}$ is the effective dimension of the explored space, which is typically much smaller than both $d$ and $t$. This greatly reduces the dimensionality of the problem, thus leading to a $O(TAd_{eff}^2)$ runtime and $O(A d_{eff})$ space complexity.
Source: arXiv (Cornell University)
Authors: Calandriello, Daniele, Carratino, Luigi, Lazaric, Alessandro, Valko, Michal, Rosasco, Lorenzo
Language: English
Field: Decision Sciences
Domain: Social Sciences
Open Access status: green
Funders: National Science Foundation, Agence Nationale de la Recherche, Ministère de l'Education Nationale, de l'Enseignement Superieur et de la Recherche, Fondation Mathématique Jacques Hadamard, Companhia Brasileira de Metalurgia e Mineração, Horizon 2020 Framework Programme, Air Force Office of Scientific Research, Air Force Research Laboratory, CHIST-ERA
OpenAlex ID: W2920859117