Thermodynamic Bayesian Inference (2410.01793v1)

Abstract: A fully Bayesian treatment of complicated predictive models (such as deep neural networks) would enable rigorous uncertainty quantification and the automation of higher-level tasks including model selection. However, the intractability of sampling Bayesian posteriors over many parameters inhibits the use of Bayesian methods where they are most needed. Thermodynamic computing has emerged as a paradigm for accelerating operations used in machine learning, such as matrix inversion, and is based on the mapping of Langevin equations to the dynamics of noisy physical systems. Hence, it is natural to consider the implementation of Langevin sampling algorithms on thermodynamic devices. In this work we propose electronic analog devices that sample from Bayesian posteriors by realizing Langevin dynamics physically. Circuit designs are given for sampling the posterior of a Gaussian-Gaussian model and for Bayesian logistic regression, and are validated by simulations. It is shown, under reasonable assumptions, that the Bayesian posteriors for these models can be sampled in time scaling with $\ln(d)$, where $d$ is dimension. For the Gaussian-Gaussian model, the energy cost is shown to scale with $ d \ln(d)$. These results highlight the potential for fast, energy-efficient Bayesian inference using thermodynamic computing.

• The paper proposes thermodynamic computing to sample Bayesian posteriors using Langevin dynamics in electronic analog devices.
• It details circuit designs for Gaussian-Gaussian and logistic regression models, achieving efficient sampling with reduced computational time.
• Simulations show that sample time scales logarithmically with feature space dimensions, highlighting potential improvements in energy efficiency and speed.

Essay: Thermodynamic Bayesian Inference

The paper "Thermodynamic Bayesian Inference" presents an innovative approach to Bayesian inference using thermodynamic computing, combining concepts from statistical physics and probabilistic machine learning. The authors propose the use of electronic analog devices that leverage Langevin dynamics to sample from Bayesian posteriors. This method promises a significant improvement in computational efficiency for complex predictive models such as deep neural networks by implementing these dynamics physically.

Bayesian Inference and Computational Challenges

Bayesian statistics offer a robust framework for uncertainty quantification, which is crucial for model selection and predictions in machine learning. Despite its advantages, the computational demands of sampling from Bayesian posteriors are often prohibitive, particularly for models with many parameters. Traditional solutions, including the Laplace approximation and variational inference, fail to capture the full complexity of Bayesian posteriors, especially for models like Bayesian neural networks.

Thermodynamics and Langevin Dynamics

Thermodynamic computing is proposed as a solution to these challenges. By mapping Bayesian sampling tasks onto physical systems using Langevin dynamics, the approach allows for efficient energy and time utilization. The Langevin sampling algorithm is well-suited for such tasks because it uses the dynamics of noisy physical systems to mimic the statistical process necessary for Bayesian inference.

Circuit Design and Implementation

The paper introduces circuit designs for two specific models: the Gaussian-Gaussian model and Bayesian logistic regression. For the Gaussian-Gaussian model, the circuit facilitates the posterior computation by implementing mathematically tractable multivariate distributions, thereby overcoming computational bottlenecks typical in digital processing. The logistics regression circuit design, on the other hand, introduces the first thermodynamic approach to non-Gaussian sampling, opening new possibilities for complex and traditionally difficult computing tasks.

Simulation and Results

The proposed circuits were validated using simulations, demonstrating their ability to sample Bayesian posteriors efficiently. The authors show that the time required to obtain samples scales logarithmically with the dimension of the feature space, which is an improvement over previous thermodynamic algorithms for linear algebra primitives. Energy-wise, the circuit operations were shown to be potentially more resource-efficient compared to classical computational methods.

Implications and Future Directions

This work illustrates the potential of thermodynamic computing to transform Bayesian inference, making previously computationally intensive tasks feasible. The compatibility of these designs with CMOS technology suggests that scalable, energy-efficient Bayesian inference could be integrated into current silicon-based computing systems. Future work could explore the adaptation of this methodology to a wider range of machine learning models and refine the energy analysis to further optimize the designs.

Moreover, the framework established here paves the way for further exploration into the theoretical limits imposed by thermodynamics on computational efficiency. Extending these findings to encompass a broader class of Bayesian models and establishing more detailed complexity bounds in terms of energy could significantly advance the field of probabilistic machine learning.

In summary, "Thermodynamic Bayesian Inference" presents a compelling synthesis of Bayesian statistics and thermodynamic principles, offering practical solutions to longstanding computational challenges and establishing a foundation for future breakthroughs in AI and machine learning domains.