Improved Langevin Monte Carlo for stochastic optimization via landscape modification
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Given a target function H to minimize or a target Gibbs distribution π_β^0 ∝ e^-β H to sample from in the low temperature, in this paper we propose and analyze Langevin Monte Carlo (LMC) algorithms that run on an alternative landscape as specified by H^f_β,c,1 and target a modified Gibbs distribution π^f_β,c,1∝ e^-β H^f_β,c,1, where the landscape of H^f_β,c,1 is a transformed version of that of H which depends on the parameters f,β and c. While the original Log-Sobolev constant affiliated with π^0_β exhibits exponential dependence on both β and the energy barrier M in the low temperature regime, with appropriate tuning of these parameters and subject to assumptions on H, we prove that the energy barrier of the transformed landscape is reduced which consequently leads to polynomial dependence on both β and M in the modified Log-Sobolev constant associated with π^f_β,c,1. This yield improved total variation mixing time bounds and improved convergence toward a global minimum of H. We stress that the technique developed in this paper is not only limited to LMC and is broadly applicable to other gradient-based optimization or sampling algorithms.
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