Optimization of the dynamic transition in the continuous coloring problem

Abstract

Random constraint satisfaction problems (CSPs) can exhibit a phase where the number of constraints per variable α makes the system solvable in theory on the one hand, but also makes the search for a solution hard, meaning that common algorithms such as Monte Carlo (MC) method fail to find a solution. The onset of this hardness is deeply linked to the appearance of a dynamical phase transition where the phase space of the problem breaks into an exponential number of clusters. The exact position of this dynamical phase transition is not universal with respect to the details of the Hamiltonian one chooses to represent a given problem. In this paper, we develop some theoretical tools in order to find a systematic way to build a Hamiltonian that maximizes the dynamic α d threshold. To illustrate our techniques, we will concentrate on the problem of continuous coloring, where one tries to set an angle x i ∈ [0; 2π] on each node of a network in such a way that no adjacent nodes are closer than some threshold angle θ, that is cos(x i − x j )⩽ cos θ. This problem can be both seen as a continuous version of the discrete graph coloring problem or as a one-dimensional version of the Mari–Krzakala–Kurchan model. The relevance of this model stems from the fact that continuous CSPs on sparse random graphs remain largely unexplored in statistical physics. We show that for sufficiently small angle θ this model presents a random first order transition and compute the dynamical, condensation and Kesten–Stigum transitions; we also compare the analytical predictions with MC simulations for values of θ = 2π/q, q ∈ N . Choosing such values of q allows us to easily compare our results with the renowned problem of discrete coloring.

On the atypical solutions of the symmetric binary perceptron

“…[39,40]. In other problems, a prominent example being the binary perceptron (symmetric or not), it is known that certain efficient heuristics are able to find solutions for α small enough as a function of κ [6,7,16,18,19,23,24]. Statistical physics studies of the neighborhood of the solutions returned by efficient heuristics have put forward the intriguing observation that in the binary perceptron problem, a dense region of other solutions surrounds the ones which are returned [5,9,10].…”

Machine-learning-assisted Monte Carlo fails at sampling computationally hard problems

“…The coloring problem is specified by the number of sites N, the choice of the random graph G which is specified by its average connectivity c, the number of colors q, and temperature T. For the model to be a good representative of the RFOT class, we need q to be large enough, otherwise the transition is too close to a second order one, especially at finite N [54]. We thus chose q = 10, for which a finite RFOT dynamical transition T d (c) is present for all c > 39.02, and the corresponding phase diagram is reported in figure 4 (data were privately communicated by the authors of Cavaliere et al [53]).…”

“…It is also important to mention that the thermodynamics of the model can be solved, for T > T K (c) and for N → ∞, by a simple cavity computation [53][54][55]. In particular, the energy and entropy per spin are given by:…”

The closest vector problem and the zero-temperature p-spin landscape for lossy compression

“…the set of allowed configurations (defined as the solution set of the XORSAT instance) is rather well-connected, yet, and despite the simplicity of the optimization function (7), the optimization problem is in a glassy phase: the energy landscape presents many local minima that are separated by free-energy barriers. A similar situation has been recently encountered in other high-dimensional constrained optimization problems, see [22] for a study of the optimization of a quadratic function where the constraints are modeled by a perceptron constraint satisfaction problem.…”

“…Given that the search space is always the same (solutions to a XORSAT constraint satisfaction problem, CSP) the addition of the external field can be seen as a reweighting of the CSP solution space. It is well known that the reweighting of the solution space can induce ergodicity breaking phase transitions [5] and change the location of the phase transitions [6,7]. In the present model we are going to show how important the effects of the reweighting can be and how they can affect algorithms searching for optimal solutions, which are relevant is several common applications.…”