Gaussian-Bernoulli RBMs Without Tears (2210.10318v1)

Abstract: We revisit the challenging problem of training Gaussian-Bernoulli restricted Boltzmann machines (GRBMs), introducing two innovations. We propose a novel Gibbs-Langevin sampling algorithm that outperforms existing methods like Gibbs sampling. We propose a modified contrastive divergence (CD) algorithm so that one can generate images with GRBMs starting from noise. This enables direct comparison of GRBMs with deep generative models, improving evaluation protocols in the RBM literature. Moreover, we show that modified CD and gradient clipping are enough to robustly train GRBMs with large learning rates, thus removing the necessity of various tricks in the literature. Experiments on Gaussian Mixtures, MNIST, FashionMNIST, and CelebA show GRBMs can generate good samples, despite their single-hidden-layer architecture. Our code is released at: \url{https://github.com/lrjconan/GRBM}.

• The paper introduces a novel Gibbs-Langevin sampling algorithm that merges Gibbs sampling with Langevin dynamics to improve GRBM training.
• The modified contrastive divergence method enables GRBMs to start from Gaussian noise, enhancing generative quality and aligning with deep models.
• Gradient clipping paired with large learning rates stabilizes training, achieving competitive FID scores on datasets like MNIST and CelebA.

Gaussian-Bernoulli RBMs Without Tears

The paper investigates the longstanding challenges associated with training Gaussian-Bernoulli Restricted Boltzmann Machines (GRBMs) and proposes methodological advancements to enhance their training efficiency and generative performance. This work is situated within the field of energy-based generative models, specifically focusing on Restricted Boltzmann Machines (RBMs) extended for continuous data. The authors introduce two pivotal contributions: a novel Gibbs-Langevin sampling algorithm and a modified contrastive divergence (CD) approach that colligates GRBMs more naturally with contemporary deep generative models.

Contributions and Methodologies

1. Gibbs-Langevin Sampling Algorithm: The paper introduces a hybrid sampling algorithm which integrates the strengths of Gibbs sampling with Langevin dynamics, tailored for GRBMs. The Gibbs-Langevin sampling is innovative in that it leverages the computational tractability of Langevin methods while preserving effective sampling within the discrete-continuous domain of GRBMs.
2. Modified Contrastive Divergence (CD) Algorithm: A modified form of CD is proposed, allowing GRBMs to generate samples starting from Gaussian noise rather than from observed data. This modification aligns GRBM training with techniques used in other deep generative models and ensures better generative analysis and comparison.
3. Gradient Clipping and Large Learning Rates: By incorporating gradient clipping, the authors argue that their methods stabilize training even with large learning rates—traditionally a challenge in GRBM training due to unstable gradients stemming from the intractable log-partition function.

Numerical Experiments and Results

The empirical results are robust, demonstrating the feasibility of training GRBMs effectively using the proposed methods. The experiments involve datasets such as Gaussian Mixtures, MNIST, FashionMNIST, and CelebA, showcasing the GRBMs' capacity to generate high-quality samples. Specifically, using Gibbs-Langevin sampling, GRBMs achieve competitive Frechet Inception Distance (FID) scores comparable to more complex deep generative models, despite the simplicity of their single-hidden-layer architecture.

Practical and Theoretical Implications

The implications of this work extend both practically and theoretically. Practically, GRBMs can serve as a bridge to convert real-valued data into a form amenable to stochastic binary processing, expanding the utility of GRBMs in neural architectures and hybrid generative models. Theoretically, the research provides insights into sampling methodologies for energy-based models, potentially impacting the broader domain of probabilistic inference and Monte Carlo methods in machine learning.

Speculation on Future Developments

Given the advancements presented, future developments could include extending the GRBMs to convolutional architectures, thus leveraging spatial hierarchies in data for potentially improved generative performance. Additionally, exploring Gaussian deep belief networks (GDBNs) could provide an avenue for deeper, more expressive models that still benefit from the proposed sampling and gradient methods.

In conclusion, this work marks a significant step forward in the practical training and application of GRBMs by resolving key challenges and aligning them more closely with prevailing deep learning paradigms. The introduction of Gibbs-Langevin sampling and modifications to CD suggest a promising direction for future research in energy-based generative modeling.