Landscape modification meets spin systems: from torpid to rapid mixing, tunneling and annealing in the low-temperature regime

Given a target Gibbs distribution $\pi^0_{\beta} \propto e^{-\beta \mathcal{H}}$ to sample from in the low-temperature regime on $\Sigma_N := \{-1,+1\}^N$, in this paper we propose and analyze Metropolis dynamics that instead target an alternative distribution $\pi^{f}_{\alpha,c,1/\beta} \propto e^{-\mathcal{H}^{f}_{\alpha,c,1/\beta}}$, where $\mathcal{H}^{f}_{\alpha,c,1/\beta}$ is a transformed Hamiltonian whose landscape is suitably modified and controlled by the parameters $f,\alpha,c$ and $\beta$ and shares the same set of stationary points as $\mathcal{H}$. With appropriate tuning of these parameters, the major advantage of the proposed Metropolis dynamics on the modified landscape is that it enjoys an $\mathcal{O}(1)$ critical height while its stationary distribution $\pi^{f}_{\alpha,c,1/\beta}$ maintains close proximity with the original target $\pi^0_{\beta}$ in the low-temperature. We prove rapid mixing and tunneling on the modified landscape with polynomial dependence on the system size $N$ and the inverse temperature $\beta$, while the original Metropolis dynamics mixes torpidly with exponential dependence on the critical height and $\beta$. In the setting of simulated annealing, we prove its long-time convergence under a power-law cooling schedule that is faster than the typical logarithmic cooling in the classical setup. We illustrate our results on a host of models including the Ising model on various deterministic and random graphs as well as Derrida's Random Energy Model. In these applications, the original dynamics mixes torpidly while the proposed dynamics on the modified landscape mixes rapidly with polynomial dependence on both $\beta$ and $N$ and find the approximate ground state provably in $\mathcal{O}(N^4)$ time. This paper highlights a novel use of the geometry and structure of the landscape to the design of accelerated samplers or optimizers.