This paper explores the connections between tempering (for Sequential Monte Carlo; SMC) and entropic mirror descent to sample from a target probability distribution whose unnormalized density is known. We establish that tempering SMC corresponds to entropic mirror descent applied to the reverse Kullback-Leibler (KL) divergence and obtain convergence rates for the tempering iterates. Our result motivates the tempering iterates from an optimization point of view, showing that tempering can be seen as a descent scheme of the KL divergence with respect to the Fisher-Rao geometry, in contrast to Langevin dynamics that perform descent of the KL with respect to the Wasserstein-2 geometry. We exploit the connection between tempering and mirror descent iterates to justify common practices in SMC and derive adaptive tempering rules that improve over other alternative benchmarks in the literature.

翻译：本文探讨了从已知未归一化密度的目标概率分布中采样时，退火（用于序列蒙特卡洛方法；SMC）与熵镜像下降之间的关联。我们证明了退火SMC对应于应用于反向Kullback-Leibler（KL）散度的熵镜像下降，并获得了退火迭代的收敛速率。我们的结果从优化角度阐释了退火迭代，表明退火可被视为KL散度在Fisher-Rao几何下的下降方案，这与在Wasserstein-2几何下执行KL散度下降的Langevin动力学形成对比。我们利用退火与镜像下降迭代之间的关联，为SMC中的常见实践提供了理论依据，并推导出自适应退火规则，这些规则在文献中优于其他替代基准方法。