In this paper, we quantify the impact of using non-convergent Markov chains to train Energy-Based models (EBMs). In particular, we show analytically that EBMs trained with non-persistent short runs to estimate the gradient can perfectly reproduce a set of empirical statistics of the data, not at the level of the equilibrium measure, but through a precise dynamical process. Our results provide a first-principles explanation for the observations of recent works proposing the strategy of using short runs starting from random initial conditions as an efficient way to generate high-quality samples in EBMs, and lay the groundwork for using EBMs as diffusion models. After explaining this effect in generic EBMs, we analyze two solvable models in which the effect of the non-convergent sampling in the trained parameters can be described in detail. Finally, we test these predictions numerically on a ConvNet EBM and a Boltzmann machine.

翻译：本文量化了使用非收敛马尔可夫链训练能量基模型（EBM）时产生的影响。具体而言，我们通过分析表明，使用非持续性短程运行来估计梯度的EBM能够精确复现数据的一组经验统计量——但并非通过平衡测度层面，而是通过精确的动力学过程实现。这一结果为近期研究提出的策略（即从随机初始条件出发采用短程运行作为在EBM中高效生成高质量样本的方法）提供了第一性原理解释，并为将EBM用作扩散模型奠定了基础。在阐释该效应在通用EBM中的表现后，我们分析了两个可解模型，从而能够详细描述非收敛采样对训练参数的影响。最后，我们在卷积网络EBM和玻尔兹曼机上对这些预测进行了数值验证。