Bayesian brain theory suggests that the brain employs generative models to understand the external world.

This paper studies the fundamental learning problem of the energy-based model (EBM).

As the world's largest companies pour hundreds of billions of dollars into large language models, San Francisco-based Logical Intelligence is trying something different in pursuit of AI that can mimic the human brain. If you ask Yann LeCun, Silicon Valley has a groupthink problem. Since leaving Meta in November, the researcher and AI luminary has taken aim at the orthodox view that large language models (LLMs) will get us to artificial general intelligence (AGI), the threshold where computers match or exceed human smarts. Everyone, he declared in a recent interview, has been "LLM-pilled." On January 21, San Francisco-based startup Logical Intelligence appointed LeCun to its board .

However, training EBMs on data in discrete or mixed state spaces poses significant challenges due to the lack of robust and fast sampling methods. In this work, we propose to train discrete EBMs with Energy Discrepancy, a loss function which only requires the evaluation of the energy function at data points and their perturbed counterparts, thus eliminating the need for Markov chain Monte Carlo. We introduce perturbations of the data distribution by simulating a diffusion process on the discrete state space endowed with a graph structure. This allows us to inform the choice of perturbation from the structure of the modelled discrete variable, while the continuous time parameter enables fine-grained control of the perturbation. Empirically, we demonstrate the efficacy of the proposed approaches in a wide range of applications, including the estimation of discrete densities with non-binary vocabulary and binary image modelling. We also introduce the first application of EBMs to tabular data sets with applications in synthetic data generation and calibrated classification.

This paper studies the fundamental learning problem of the energy-based model (EBM). Learning the EBM can be achieved using the maximum likelihood estimation (MLE), which typically involves the Markov Chain Monte Carlo (MCMC) sampling, such as the Langevin dynamics. However, the noise-initialized Langevin dynamics can be challenging in practice and hard to mix. This motivates the exploration of joint training with the generator model where the generator model serves as a complementary model to bypass MCMC sampling. However, such a method can be less accurate than the MCMC and result in biased EBM learning.

In domains with interdependent data, such as graphs, quantifying the epistemic uncertainty of a Graph Neural Network (GNN) is challenging as uncertainty can arise at different structural scales. Existing techniques neglect this issue or only distinguish between structure-aware and structure-agnostic uncertainty without combining them into a single measure. We propose GEBM, an energy-based model (EBM) that provides high-quality uncertainty estimates by aggregating energy at different structural levels that naturally arise from graph diffusion. In contrast to logit-based EBMs, we provably induce an integrable density in the data space by regularizing the energy function. We introduce an evidential interpretation of our EBM that significantly improves the predictive robustness of the GNN. Our framework is a simple and effective post hoc method applicable to any pre-trained GNN that is sensitive to various distribution shifts. It consistently achieves the best separation of in-distribution and out-of-distribution data on 6 out of 7 anomaly types while having the best average rank over shifts on datasets.

However, due to the intractable partition function, they are typically trained via contrastive divergence for maximum likelihood estimation. In this paper, we propose pseudo-spherical contrastive divergence (PS-CD) to generalize maximum likelihood learning of EBMs. PS-CD is derived from the maximization of a family of strictly proper homogeneous scoring rules, which avoids the computation of the intractable partition function and provides a generalized family of learning objectives that include contrastive divergence as a special case. Moreover, PS-CD allows us to flexibly choose various learning objectives to train EBMs without additional computational cost or variational minimax optimization. Theoretical analysis on the proposed method and extensive experiments on both synthetic data and commonly used image datasets demonstrate the effectiveness and modeling flexibility of PS-CD, as well as its robustness to data contamination, thus showing its superiority over maximum likelihood and $f$-EBMs.

We present a maximum entropy inverse reinforcement learning (IRL) approach for improving the sample quality of diffusion generative models, especially when the number of generation time steps is small. Similar to how IRL trains a policy based on the reward function learned from expert demonstrations, we train (or fine-tune) a diffusion model using the log probability density estimated from training data. Since we employ an energy-based model (EBM) to represent the log density, our approach boils down to the joint training of a diffusion model and an EBM. Our IRL formulation, named Diffusion by Maximum Entropy IRL (DxMI), is a minimax problem that reaches equilibrium when both models converge to the data distribution. The entropy maximization plays a key role in DxMI, facilitating the exploration of the diffusion model and ensuring the convergence of the EBM. We also propose Diffusion by Dynamic Programming (DxDP), a novel reinforcement learning algorithm for diffusion models, as a subroutine in DxMI. DxDP makes the diffusion model update in DxMI efficient by transforming the original problem into an optimal control formulation where value functions replace back-propagation in time. Our empirical studies show that diffusion models fine-tuned using DxMI can generate high-quality samples in as few as 4 and 10 steps. Additionally, DxMI enables the training of an EBM without MCMC, stabilizing EBM training dynamics and enhancing anomaly detection performance.

This work proposes a method for using any generator network as the foundation of an Energy-Based Model (EBM). Our formulation posits that observed images are the sum of unobserved latent variables passed through the generator network and a residual random variable that spans the gap between the generator output and the image manifold. One can then define an EBM that includes the generator as part of its forward pass, which we call the Hat EBM. The model can be trained without inferring the latent variables of the observed data or calculating the generator Jacobian determinant. This enables explicit probabilistic modeling of the output distribution of any type of generator network. Experiments show strong performance of the proposed method on (1) unconditional ImageNet synthesis at 128$\times$128 resolution, (2) refining the output of existing generators, and (3) learning EBMs that incorporate non-probabilistic generators.

Autoregressive models (ARMs) currently constitute the dominant paradigm for large language models (LLMs). Energy-based models (EBMs) represent another class of models, which have historically been less prevalent in LLM development, yet naturally characterize the optimal policy in post-training alignment. In this paper, we provide a unified view of these two model classes. Taking the chain rule of probability as a starting point, we establish an explicit bijection between ARMs and EBMs in function space, which we show to correspond to a special case of the soft Bellman equation in maximum entropy reinforcement learning. Building upon this bijection, we derive the equivalence between supervised learning of ARMs and EBMs. Furthermore, we analyze the distillation of EBMs into ARMs by providing theoretical error bounds. Our results provide insights into the ability of ARMs to plan ahead, despite being based on the next-token prediction paradigm.