Most,however,are nonparametric approaches and so can be difficult to scale.

Moreover, the detailed balance condition is highly restrictive: energy based models corresponding to neural networks must have symmetric weights, unlike biological neural circuits.

We propose a novel method to directly learn a stochastic transition operator whose repeated application provides generated samples. Traditional undirected graphical models approach this problem indirectly by learning a Markov chain model whose stationary distribution obeys detailed balance with respect to a parameterized energy function. The energy function is then modified so the model and data distributions match, with no guarantee on the number of steps required for the Markov chain to converge. Moreover, the detailed balance condition is highly restrictive: energy based models corresponding to neural networks must have symmetric weights, unlike biological neural circuits. In contrast, we develop a method for directly learning arbitrarily parameterized transition operators capable of expressing nonequilibrium stationary distributions that violate detailed balance, thereby enabling us to learn more biologically plausible asymmetric neural networks and more general non-energy based dynamical systems. The proposed training objective, which we derive via principled variational methods, encourages the transition operator to "walk back" (prefer to revert its steps) in multi-step trajectories that start at datapoints, as quickly as possible back to the original data points. We present a series of experimental results illustrating the soundness of the proposed approach, Variational Walkback (VW), on the MNIST, CIFAR-10, SVHN and CelebA datasets, demonstrating superior samples compared to earlier attempts to learn a transition operator. We also show that although each rapid training trajectory is limited to a finite but variable number of steps, our transition operator continues to generate good samples well past the length of such trajectories, thereby demonstrating the match of its non-equilibrium stationary distribution to the data distribution.

Recent work has explored integrating autoregressive language models with energy-based models (EBMs) to enhance text generation capabilities. However, learning effective EBMs for text is challenged by the discrete nature of language. This work proposes an adversarial training strategy to address limitations in prior efforts. Specifically, an iterative adversarial attack algorithm is presented to generate negative samples for training the EBM by perturbing text from the autoregressive model. This aims to enable the EBM to suppress spurious modes outside the support of the data distribution. Experiments on an arithmetic sequence generation task demonstrate that the proposed adversarial training approach can substantially enhance the quality of generated sequences compared to prior methods. The results highlight the promise of adversarial techniques to improve discrete EBM training. Key contributions include: (1) an adversarial attack strategy tailored to text to generate negative samples, circumventing MCMC limitations; (2) an adversarial training algorithm for EBMs leveraging these attacks; (3) empirical validation of performance improvements on a sequence generation task.

The use of neural network parametrizations to represent the ground state in variational Monte Carlo (VMC) calculations has generated intense interest in recent years. However, as we demonstrate in the context of the periodic Heisenberg spin chain, this approach can produce unreliable wave function approximations. One of the most obvious signs of failure is the occurrence of random, persistent spikes in the energy estimate during training. These energy spikes are caused by regions of configuration space that are over-represented by the wave function density, which are called ``spurious modes'' in the machine learning literature. After exploring these spurious modes in detail, we demonstrate that a collective-variable-based penalization yields a substantially more robust training procedure, preventing the formation of spurious modes and improving the accuracy of energy estimates. Because the penalization scheme is cheap to implement and is not specific to the particular model studied here, it can be extended to other applications of VMC where a reasonable choice of collective variable is available.