  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 non-equilibrium 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" in multi-step trajectories that start at data-points, 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. Source Code: http://github.com/anirudh9119/walkback_nips17
