具有许多输入变量的计算机实验的保序替代模型
Isotonic surrogate modeling for computer experiments with many input variables
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中文总结 AI 辅助
针对高维计算机实验，提出变换加性保序模型（TAIM），利用数据估计连接变换和 spike-and-slab 先验的单调基模型，实现维度无关的后验收缩率，并通过线性复杂度吉布斯采样高效推断，有效处理保序替代建模。
中文摘要 AI 辅助
虚拟模拟器被广泛用于研究复杂的物理现象，从粒子碰撞到火箭推进。这类“计算机实验”可能非常耗时，而贝叶斯替代模型可用于高效仿真，并提供可靠的不确定性量化。为了在有限的样本量 $n$ 下训练准确的替代模型，近期工作探索了纳入单调性（或保序性）信息，这些信息通常可以从物理系统中引出。然而，在具有许多输入变量的实际应用中，现有的贝叶斯保序模型可能面临统计和计算上的局限性，导致其性能可能不如不纳入保序性的模型。我们提出了一种新的变换加性保序模型（TAIM），旨在驯服这种“维数灾难”。TAIM 利用灵活的变换加性保序建模框架，该框架借助数据估计的连接变换和具有 spike-and-slab 先验的单调基模型（作用于基权重）。在预测方面，当真实黑盒函数具有温和光滑性条件的变换加性保序形式时，TAIM 实现了（最高达对数因子）$O(n^{-1/3})$ 的后验收缩率。该速率对于涉及 $n$ 的项不依赖于输入维度 $d$，从而减轻了维度对后验预测的影响。在计算方面，TAIM 通过精心设计的吉布斯采样器实现高效的后验推断，每次采样迭代仅需 $d$ 的线性工作量。我们进一步提出了 TAIM 的扩展，可以建模偏离变换加性结构的情况。数值实验和两个应用展示了 TAIM 在具有许多输入变量的保序替代建模中的有效性。
英文摘要
Virtual simulators are widely used for studying complex physical phenomena, from particle collisions to rocket propulsion. Such "computer experiments" can be highly time-intensive, and a Bayesian surrogate model can be used for efficient emulation with reliable uncertainty quantification. To train accurate surrogates with a limited sample size $n$, recent work has explored the incorporation of monotonicity (or isotonicity) information, which can often be elicited from physical systems. In practical applications with many input variables, however, existing Bayesian isotonic models can face statistical and computational limitations, which may result in worse performance compared to models that do not incorporate isotonicity. We propose a new transformed additive isotonic model (TAIM), which aims to tame this "curse-of-dimensionality". TAIM makes use of a flexible transformed additive isotonic modeling framework, which leverages a data-estimated link transformation and a monotone basis model with spike-and-slab priors on basis weights. Prediction-wise, TAIM achieves (up to log factors) a posterior contraction rate of $O(n^{-1/3})$ when the true black-box function is in a transformed additive isotonic form with mild smoothness conditions. Such a rate does not depend on the input dimension $d$ for terms involving $n$, which softens the effect of dimensionality on posterior predictions. Computation-wise, TAIM allows for efficient posterior inference via a carefully designed Gibbs sampler, where each sampling iteration requires only linear work in $d$. We further present an extension of TAIM that can model potential deviations from transformed additivity. Numerical experiments and two applications show the effectiveness of TAIM for isotonic surrogate modeling with many input variables.
发表机构
- Department of Statistical Science, Duke University（杜克大学统计科学系）
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