An accelerated variant of simulated annealing that converges under fast cooling
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Given a target function U to minimize on a finite state space X, a proposal chain with generator Q and a cooling schedule T(t) that depends on time t, in this paper we study two types of simulated annealing (SA) algorithms with generators M_1,t(Q,U,T(t)) and M_2,t(Q,U,T(t)) respectively. While M_1,t is the classical SA algorithm, we introduce a simple and greedy variant that we call M_2,t which provably converges faster. Under any T(t) that converges to 0 and mild conditions on Q, the Markov chain generated by M_2,t is weakly ergodic. When T(t) > c_M_2/(t+1) follows the logarithmic cooling schedule, our proposed algorithm is strongly ergodic both in total variation and in relative entropy, and converges to the set of global minima, where c_M_2 is a constant that we explicitly identify. If c_M_1 is the optimal hill-climbing constant that appears in logarithmic cooling of M_1,t, we show that c_M_1≥ c_M_2 and give simple conditions under which c_M_1 > c_M_2. Our proposed M_2,t thus converges under a faster logarithmic cooling in this regime. The other situation that we investigate corresponds to c_M_1 > c_M_2 = 0, where we give a class of fast and non-logarithmic cooling schedule that works for M_2,t (but not for M_1,t). To the best of our knowledge this is the first instance where strong ergodicity and convergence in relative entropy are proved for faster than logarithmic cooling. Finally, we give an algorithm to simulate M_2,t by uniformization of Markov chains.
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