  We introduce a novel Entropy-driven Monte Carlo (EdMC) strategy to
efficiently sample solutions of random Constraint Satisfaction Problems (CSPs).
First, we extend a recent result that, using a large-deviation analysis, shows
that the geometry of the space of solutions of the Binary Perceptron Learning
Problem (a prototypical CSP), contains regions of very high-density of
solutions. Despite being sub-dominant, these regions can be found by optimizing
a local entropy measure. Building on these results, we construct a fast solver
that relies exclusively on a local entropy estimate, and can be applied to
general CSPs. We describe its performance not only for the Perceptron Learning
Problem but also for the random $K$-Satisfiabilty Problem (another prototypical
CSP with a radically different structure), and show numerically that a simple
zero-temperature Metropolis search in the smooth local entropy landscape can
reach sub-dominant clusters of optimal solutions in a small number of steps,
while standard Simulated Annealing either requires extremely long cooling
procedures or just fails. We also discuss how the EdMC can heuristically be
made even more efficient for the cases we studied.
