In this paper, we investigate the impact of numerical instability on the reliability of sampling, density evaluation, and evidence lower bound (ELBO) estimation in variational flows. We first empirically demonstrate that common flows can exhibit a catastrophic accumulation of error: the numerical flow map deviates significantly from the exact map -- which affects sampling -- and the numerical inverse flow map does not accurately recover the initial input -- which affects density and ELBO computations. Surprisingly though, we find that results produced by flows are often accurate enough for applications despite the presence of serious numerical instability. In this work, we treat variational flows as dynamical systems, and leverage shadowing theory to elucidate this behavior via theoretical guarantees on the error of sampling, density evaluation, and ELBO estimation. Finally, we develop and empirically test a diagnostic procedure that can be used to validate results produced by numerically unstable flows in practice.

翻译：本文研究了数值不稳定性对变分流中采样、密度评估及证据下界（ELBO）估计可靠性的影响。我们首先通过实验证明，常见流模型可能表现出灾难性的误差累积：数值流映射显著偏离精确映射（影响采样），且数值逆流映射无法准确恢复初始输入（影响密度与ELBO计算）。然而令人惊讶的是，尽管存在严重的数值不稳定性，流模型产生的结果在应用中通常仍足够精确。本文将变分流视为动力系统，并利用影子理论通过采样、密度评估和ELBO估计的误差理论保证来阐明这种行为。最后，我们开发并实验验证了一种诊断程序，可用于在实践中验证数值不稳定流模型所产生的结果。