Speed-accuracy relations for the diffusion models: Wisdom from nonequilibrium thermodynamics and optimal transport

We discuss a connection between a generative model, called the diffusion model, and nonequilibrium thermodynamics for the Fokker-Planck equation, called stochastic thermodynamics. Based on the techniques of stochastic thermodynamics, we derive the speed-accuracy relations for the diffusion models, which are inequalities that relate the accuracy of data generation to the entropy production rate, which can be interpreted as the speed of the diffusion dynamics in the absence of the non-conservative force. From a stochastic thermodynamic perspective, our results provide a quantitative insight into how best to generate data in diffusion models. The optimal learning protocol is introduced by the geodesic of space of the 2-Wasserstein distance in optimal transport theory. We numerically illustrate the validity of the speed-accuracy relations for the diffusion models with different noise schedules and the different data. We numerically discuss our results for the optimal and suboptimal learning protocols. We also show the inaccurate data generation due to the non-conservative force, and the applicability of our results to data generation from the real-world image datasets.