How Many Initial Points Does Bayesian Optimization Need?

2026-07-05

AI Summary

This study addresses the lack of theoretical guidance in selecting the initial number of points $n_0$ in Bayesian optimization, a choice often made heuristically and leading to inefficient resource use. Through systematic empirical investigation, the work reveals—for the first time—a U-shaped relationship between $n_0$ and total optimization cost, demonstrating that both insufficient and excessive initial points increase computational expense. Experiments employing maximum likelihood estimation, Bayesian MCMC, and exact Gaussian process hyperparameters across multiple acquisition functions—including Thompson sampling—show that Thompson sampling exhibits strong robustness to variations in $n_0$, yielding slightly higher but significantly more stable costs. The study further links boundary variance issues to early sampling behavior and offers practical recommendations: prioritize multi-step lookahead Bayesian optimization when feasible; otherwise, adopt Thompson sampling for its reliability under uncalibrated settings.

Abstract

Bayesian Optimization (BO) generally begins with an initialization phase: a batch of $n_0$ uninformed evaluations. The choice of $n_0$ remains largely heuristic, and we empirically observe that the total cost (random initial points plus BO iterations needed to find the global optimum) is U-shaped in $n_0$, i.e., a practitioner wastes resources by selecting either too low or too high a value of $n_0$. We find this tradeoff persists across MLE, Bayesian MCMC, and exact GP hyperparameters, as well as across acquisition functions. Toward the latter, Thompson Sampling appears an exception, with both total cost and simple regret essentially $n_0$-agnostic, though higher in our experiments. We attribute this U-shape to the known boundary issue of variance-driven BO: BO burns early budget on corners of the hypercube before turning inward. We demonstrate this effect using a 3D BO trajectory where the exact hyperparameters are known. We conclude with practical recommendations: use multi-step lookahead BO where possible; otherwise use Thompson Sampling when $n_0$ cannot be tuned, and a generously large $n_0$ when it can.

Problem

• Bayesian Optimization
• initial points
• total cost
• hyperparameter tuning
• Thompson Sampling

Innovation

• Bayesian Optimization
• initialization
• Thompson Sampling
• boundary effect
• acquisition function