A new study was conducted by Kotaro Ikeda, a fourth-year undergraduate student in the Department of Mathematical Engineering and Information Physics in the Faculty of Engineering; Tomoya Uda, a fourth-year undergraduate student in the Department of Earth and Planetary Physics in the Faculty of Science; and Daisuke Okanohara, the CTO of Preferred Networks, Inc.; and Sosuke Ito, an associate professor in the Graduate School of Science, revealed that the accuracy of data generated by diffusion models (*1) can be evaluated through an analogy with thermodynamic trade-off relations (*2) in nonequilibrium thermodynamics (*3).
This study derived thermodynamic inequalities linking estimation error to the amount of thermodynamic dissipation (*4) for diffusion models. Selecting the noise schedule (*5)—that is, the time dependence of parameters in diffusion dynamics—is important for improving the quality of data generation. Empirically, a noise schedule based on optimal transport theory (*6) minimizes estimation error. However, this empirical fact lacked a theoretical basis. This work provides a theoretical foundation for selecting noise schedules from the perspective of nonequilibrium thermodynamics.
To derive inequalities for the relationship between thermodynamic dissipation and data generation accuracy, the researchers discuss optimal diffusion dynamics for achieving maximum accuracy. In nonequilibrium thermodynamics, optimal transport dynamics are known to minimize dissipation. Thus, the researchers confirmed that optimal transport dynamics are optimal for data generation based on the newly derived inequalities. This result offers a new theoretical approach to explaining the rationale behind empirical methods in generative modeling from a nonequilibrium thermodynamic perspective. The study suggests that this perspective could improve and propose methods for generative models, including diffusion models.
This research was conducted in part as a seminar offered by the Universal Biology Institute (UBI) to undergraduate students. The findings were published in Physical Review X.
Caption: Analyzing diffusion models using nonequilibrium thermodynamics and optimal transport theory.
Footnote:
(*1) Diffusion models: Diffusion models are machine learning models that learn diffusion dynamics in order to generate new data points by reconstructing the time-reversal process from the learned dynamics. Recently, the use of diffusion models in generative AI for images and videos has attracted attention due to the high quality of the generated data. This method was originally inspired by nonequilibrium thermodynamics, in which thermodynamic dissipation is discussed in terms of the statistical difference between time-forward and time-reversed dynamics.
(*2) Thermodynamic trade-off relations: This generic term refers to inequalities connecting an amount of thermodynamic dissipation to performance-related values such as speed, fluctuations, and the accuracy of information processing. Speed limits, for example, are a type of thermodynamic trade-off relation connecting thermodynamic dissipation in finite-time dynamics to the speed of changes in the system via optimal transport theory.
(*3) Nonequilibrium thermodynamics: It is an area of statistical mechanics that discusses thermodynamic dissipation and state changes in irreversible systems. Recently, relationships with various indices in information theory have been discovered in nonequilibrium thermodynamics, particularly in stochastic thermodynamics, which is a branch of nonequilibrium thermodynamics that deals with diffusion dynamics.
(*4) Amount of thermodynamic dissipation: In nonequilibrium thermodynamics, entropy production rates are a well-known measure of irreversibility. Entropy production rates are always nonnegative; this property is known as the second law of thermodynamics. The inequality derived in this paper uses a value obtained by multiplying the entropy production rate by the temperature and integrating it over time.
(*5) Noise schedule: The forward process of diffusion models is determined by the intensity of the noise over time. The selection of this noise intensity is referred to as a "noise schedule." It is well known that the results of generation depend on noise schedules, and the literature contains various discussions about selecting noise schedules.
(*6) Optimal transport theory: Optimal transport theory, a branch of mathematics, emerged from the problem of minimizing transport costs. It has proposed distance metrics for probability distributions, such as the Wasserstein distance, as well as optimal protocols for transporting distributions. In machine learning, metrics from optimal transport theory are used to evaluate generated data quality. Additionally, optimal transport dynamics for minimizing transport costs have been introduced as noise schedules in diffusion models. Recently, the close relationship between nonequilibrium thermodynamics and optimal transport theory has been recognized. Researchers have found that minimum dissipation is achieved through optimal transport dynamics.
Papers
Journal: Physical Review X
Title: Speed-accuracy relations for diffusion models: Wisdom from nonequilibrium thermodynamics and optimal transport
Authors: Kotaro Ikeda, Tomoya Uda, Daisuke Okanohara, and Sosuke Ito*
DOI: 10.1103/x5vj-8jq9
Revealing the Effects of Symmetry on Thermodynamic Trade-off Relations: Toward elucidating the design principles of thermodynamic devices that achieve high-speed operation and low energetic costs
