On the convergence of an improved discrete simulated annealing via landscape modification

In this paper, we propose new Metropolis-Hastings and simulated annealing algorithms on finite state space via modifying the energy landscape. The core idea of landscape modification relies on introducing a parameter $c$, in which the landscape is modified once the algorithm is above this threshold parameter. We illustrate the power and benefits of landscape modification by investigating its effect on the classical Curie-Weiss model with Glauber dynamics and external magnetic field in the subcritical regime. This leads to a landscape-modified mean-field equation, and with appropriate choice of $c$ the free energy landscape can be transformed from a double-well into a single-well, while the location of the global minimum is preserved on the modified landscape. Consequently, running algorithms on the modified landscape can improve the convergence to the ground-state in the Curie-Weiss model. In the setting of simulated annealing, we demonstrate that landscape modification can yield improved mean tunneling time between global minima, and give convergence guarantee using an improved logarithmic cooling schedule with reduced critical height. Finally, we discuss connections between landscape modification and other acceleration techniques such as Catoni's energy transformation algorithm, preconditioning, importance sampling and quantum annealing. We stress that the technique developed in this paper is applicable to any difference-based discrete optimization algorithm.