Learning in Markov Random Fields with Contrastive Free Energies

Learning Markov random field (MRF) models is notoriously hard due to the presence of a global normalization factor. In this paper we present a new framework for learning MRF models based on the contrastive free energy (CF) objective function. In this scheme the parameters are updated in an attempt to match the average statistics of the data distribution and a distribution which is (partially or approximately) "relaxed" to the equilibrium distribution. We show that maximum likelihood, mean field, contrastive divergence and pseudo-likelihood objectives can be understood in this paradigm. Moreover, we propose and study a new learning algorithm: the "kstep Kikuchi/Bethe approximation". This algorithm is then tested on a conditional random field model with "skip-chain" edges to model long range interactions in text data. It is demonstrated that with no loss in accuracy, the training time is brought down on average from 19 hours (BP based learning) to 83 minutes, an order of magnitude improvement

A simple analytical description of the non-stationary dynamics in Ising spin systems

The analytical description of the dynamics in models with discrete variables (e.g. Isingspins) is a notoriously difficult problem, that can be tackled only undersome approximation.Recently a novel variational approach to solve the stationary dynamical regime has beenintroduced by Pelizzola [Eur. Phys. J. B, 86 (2013) 120], where simpleclosed equations arederived under mean-field approximations based on the cluster variational method. Here wepropose to use the same approximation based on the cluster variational method also for thenon-stationary regime, which has not been considered up to now within this framework. Wecheck the validity of this approximation in describing the non-stationary dynamical regime ofseveral Ising models defined on Erdos-R ́enyi random graphs: westudy ferromagnetic modelswith symmetric and partially asymmetric couplings, models with randomfields and also spinglass models. A comparison with the actual Glauber dynamics, solvednumerically, showsthat one of the two studied approximations (the so-called ‘diamond’approximation) providesvery accurate results in all the systems studied. Only for the spin glass models we find somesmall discrepancies in the very low temperature phase, probably due to the existence of alarge number of metastable states. Given the simplicity of the equations to be solved, webelieve the diamond approximation should be considered as the ‘minimalstandard’ in thedescription of the non-stationary regime of Ising-like models: any new method pretending toprovide a better approximate description to the dynamics of Ising-like models should performat least as good as the diamond approximation

Replica Cluster Variational Method: the Replica Symmetric solution for the 2D random bond Ising model

We present and solve the Replica Symmetric equations in the context of the Replica Cluster Variational Method for the 2D random bond Ising model (including the 2D Edwards-Anderson spin glass model). First we solve a linearized version of these equations to obtain the phase diagrams of the model on the square and triangular lattices. In both cases the spin-glass transition temperatures and the tricritical point estimations improve largely over the Bethe predictions. Moreover, we show that this phase diagram is consistent with the behavior of inference algorithms on single instances of the problem. Finally, we present a method to consistently find approximate solutions to the equations in the glassy phase. The method is applied to the triangular lattice down to T=0, also in the presence of an external field.Comment: 22 pages, 11 figure

Message passing and Monte Carlo algorithms: connecting fixed points with metastable states

Mean field-like approximations (including naive mean field, Bethe and Kikuchi and more general Cluster Variational Methods) are known to stabilize ordered phases at temperatures higher than the thermodynamical transition. For example, in the Edwards-Anderson model in 2-dimensions these approximations predict a spin glass transition at finite $T$. Here we show that the spin glass solutions of the Cluster Variational Method (CVM) at plaquette level do describe well actual metastable states of the system. Moreover, we prove that these states can be used to predict non trivial statistical quantities, like the distribution of the overlap between two replicas. Our results support the idea that message passing algorithms can be helpful to accelerate Monte Carlo simulations in finite dimensional systems.Comment: 6 pages, 6 figure