Recently, a series of papers proposed deep learning-based approaches to sample from target distributions using controlled diffusion processes, being trained only on the unnormalized target densities without access to samples. Building on previous work, we identify these approaches as special cases of a generalized Schr\"odinger bridge problem, seeking a stochastic evolution between a given prior distribution and the specified target. We further generalize this framework by introducing a variational formulation based on divergences between path space measures of time-reversed diffusion processes. This abstract perspective leads to practical losses that can be optimized by gradient-based algorithms and includes previous objectives as special cases. At the same time, it allows us to consider divergences other than the reverse Kullback-Leibler divergence that is known to suffer from mode collapse. In particular, we propose the so-called log-variance loss, which exhibits favorable numerical properties and leads to significantly improved performance across all considered approaches.

翻译：近年来，一系列研究提出了基于深度学习的采样方法，这些方法利用受控扩散过程从目标分布中采样，且仅需基于未归一化的目标密度函数进行训练，而无需访问实际样本。在先前研究的基础上，我们将这些方法识别为广义薛定谔桥问题的特例，其核心在于寻找给定先验分布与指定目标分布之间的随机演化过程。我们进一步推广该框架，提出了一种基于时间反转扩散过程路径空间测度间散度的变分表述。这种抽象视角导出了可通过梯度优化算法求解的实用损失函数，并将先前目标函数纳入为特例。同时，该框架使我们能够考虑除反向Kullback-Leibler散度之外的其他散度度量——已知后者易受模态坍缩问题影响。特别地，我们提出了所谓的对数方差损失函数，该函数展现出优越的数值特性，并在所有考察方法中实现了显著提升的性能表现。