In Bayesian Optimization (BO), additive assumptions can mitigate the twin difficulties of modeling and searching a complex function in high dimension. However, common acquisition functions, like the Additive Lower Confidence Bound, ignore pairwise covariances between dimensions, which we'll call \textit{bilateral uncertainty} (BU), imposing a second layer of approximations. While theoretical results indicate that asymptotically not much is lost in doing so, little is known about the practical effects of this assumption in small budgets. In this article, we show that by exploiting conditional independence, Thompson Sampling respecting BU can be efficiently conducted. We use this fact to execute an empirical investigation into the loss incurred by ignoring BU, finding that the additive approximation to Thompson Sampling does indeed have, on balance, worse performance than the exact method, but that this difference is of little practical significance. This buttresses the theoretical understanding and suggests that the BU-ignoring approximation is sufficient for BO in practice, even in the non-asymptotic regime.

翻译：在贝叶斯优化（BO）中，加性假设可以缓解高维复杂函数建模与搜索的双重困难。然而，常见的采集函数（如加性下置信界）忽略了维度间的两两协方差——我们称之为**双边不确定性**（BU），这引入了第二层近似。虽然理论结果表明，在渐近意义上这样做损失不大，但对于有限预算下的实际影响却知之甚少。本文证明，通过利用条件独立性，可以高效地执行考虑BU的汤普森采样。基于这一事实，我们对忽略BU所造成的损失进行了实证研究，发现加性近似的汤普森采样总体上确实比精确方法性能更差，但这一差异的实际意义很小。这支撑了理论理解，并表明即使在非渐近情况下，忽略BU的近似对于实际贝叶斯优化应用已然足够。