用多级蒙特卡洛提升丢弃法的不确定性估计效率
Multi-level Monte Carlo Dropout for Efficient Uncertainty Quantification
- 通过重用不同精度层级的丢弃掩码，构建耦合估计器
- 在相同计算成本下，方差降低约40%以上
- 适合需要高效不确定性分析的神经网络应用
我们提出一种基于蒙特卡洛丢弃的多级蒙特卡洛（MLMC）框架用于不确定性量化。将丢弃掩码视为认知随机性的来源，通过前向传播次数定义精度层次。通过跨层级复用丢弃掩码，构建耦合的粗-细估计器，得到无偏的预测均值与方差估计，同时在固定评估预算下显著降低采样方差。推导出偏差、方差和有效成本的显式表达式，并给出各层级样本分配规则。在前向与反向物理信息神经网络（PINNs）-Uzawa基准测试中，验证了预测方差率与理论一致，并在相同成本下相比单级蒙特卡洛丢弃实现效率提升。
原文摘要 · Abstract (English)
We develop a multilevel Monte Carlo (MLMC) framework for uncertainty quantification with Monte Carlo dropout. Treating dropout masks as a source of epistemic randomness, we define a fidelity hierarchy by the number of stochastic forward passes used to estimate predictive moments. We construct coupled coarse--fine estimators by reusing dropout masks across fidelities, yielding telescoping MLMC estimators for both predictive means and predictive variances that remain unbiased for the corresponding dropout-induced quantities while reducing sampling variance at fixed evaluation budget. We derive explicit bias, variance and effective cost expressions, together with sample-allocation rules across levels. Numerical experiments on forward and inverse PINNs--Uzawa benchmarks confirm the predicted variance rates and demonstrate efficiency gains over single-level MC-dropout at matched cost.
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