How to Train Your Energy-Based Models (2101.03288v2)

Abstract: Energy-Based Models (EBMs), also known as non-normalized probabilistic models, specify probability density or mass functions up to an unknown normalizing constant. Unlike most other probabilistic models, EBMs do not place a restriction on the tractability of the normalizing constant, thus are more flexible to parameterize and can model a more expressive family of probability distributions. However, the unknown normalizing constant of EBMs makes training particularly difficult. Our goal is to provide a friendly introduction to modern approaches for EBM training. We start by explaining maximum likelihood training with Markov chain Monte Carlo (MCMC), and proceed to elaborate on MCMC-free approaches, including Score Matching (SM) and Noise Constrastive Estimation (NCE). We highlight theoretical connections among these three approaches, and end with a brief survey on alternative training methods, which are still under active research. Our tutorial is targeted at an audience with basic understanding of generative models who want to apply EBMs or start a research project in this direction.

• The paper compares key training methodologies for EBMs, including MLE with MCMC, Score Matching, and Noise Contrastive Estimation.
• It highlights trade-offs such as MCMC’s computational demands and SM refinements like Denoising and Sliced Score Matching for handling high-dimensional data.
• The study explores future directions, including adversarial training and KL divergence minimization, to advance the performance and efficiency of EBMs.

Insights into Training Energy-Based Models: A Comprehensive Examination

Yang Song and Diederik P. Kingma present an authoritative survey on the training methodologies of Energy-Based Models (EBMs). This paper dissects several nuanced approaches, including Maximum Likelihood Estimation (MLE) with Markov Chain Monte Carlo (MCMC), Score Matching (SM), and Noise Contrastive Estimation (NCE), elucidating their theoretical connections and practical implementations. The document is a vital resource for researchers seeking to understand the current landscape and future directions of EBM training.

Energy-Based Models: A Primer

EBMs, distinguished by their flexible functional forms, are unnormalized probabilistic models defined by the energy function. The lack of a tractable normalizing constant presents significant training challenges, necessitating advanced methods. Models can be adapted to specialized architectures for various data types, enabling applications across image generation, natural language processing, and reinforcement learning. Despite their potential, the intrinsic intractable likelihood complicates sample synthesis and likelihood computation.

Maximum Likelihood Training with MCMC

The authors delve into Maximum Likelihood Estimation (MLE) facilitated by MCMC sampling. MLE is a standard statistical method that approximates models to data distribution by optimizing log-likelihood functions. The incorporation of MCMC into MLE allows approximation of the otherwise intractable gradients of the log-likelihood of EBMs. The paper reviews several MCMC optimization enhancements, such as Langevin and Hamiltonian Monte Carlo methods. However, MCMC can be computationally demanding; hence, variations like Contrastive Divergence serve to mitigate these demands albeit with potential bias risks.

Score Matching Approaches

Score Matching (SM) offers an attractive alternative without the need for MCMC, focusing on matching the derivative (score) of the model's log probability to the score of the data log probability. This technique resolves the integral dependency associated with the likelihood's intractable normalizing constant. However, base SM can become computationally expensive for high-dimensional data due to second-derivative calculations. Denoising Score Matching (DSM) and Sliced Score Matching (SSM) present refinements for computational feasibility and consistency, respectively. DSM inherently links to the deficiencies in capturing mode weights, whilst SSM alleviates the computational burden without this concession.

Noise Contrastive Estimation

NCE approaches training by contrasting the model distribution against a noise distribution with a known density. The optimization of a classifier that distinguishes data from noise intuitively aligns with distinguishing between true and model distributions. As the authors note, NCE also uniquely estimates normalizing constants, a facet unexplored by SM and MCMC. Selecting an effective noise distribution is pivotal, particularly for structured data, and continues to be a subject of innovative strategies.

Future Trajectories and Conclusion

The instructive discourse extends into lesser-explored methodologies, such as adversarial training analogs and minimization strategies of KL divergence differences. Such techniques address the demanding computational cost associated with traditional MCMC iterations and provide flexible divergences for refined EBM training. These methods underscore potential future advancements, bridging current theoretical insights with practical applications in EBMs.

This paper effectively synthesizes modern methods, critically evaluating them while highlighting shared theoretical frameworks. The nuanced exploration into auxiliary models and kernelized perspectives suggests pathways to harness nonparametric generative capacities efficiently. Researchers are left with a detailed understanding of strategic methodological advancements as well as open questions and potential research vectors in enhancing EBM training efficacy.