Variational Representations of Annealing Paths: Bregman Information under Monotonic Embedding
This work provides theoretical insights for researchers in computational statistics and machine learning, but it is incremental as it builds on prior variational representations without introducing new methods or applications.
The paper tackles the analysis of annealing paths in Markov Chain Monte Carlo methods by extending results on Bregman divergences to quasi-arithmetic means under monotonic embeddings, showing how these means minimize expected divergences and associate common divergence functionals with intermediate densities.
Markov Chain Monte Carlo methods for sampling from complex distributions and estimating normalization constants often simulate samples from a sequence of intermediate distributions along an annealing path, which bridges between a tractable initial distribution and a target density of interest. Prior works have constructed annealing paths using quasi-arithmetic means, and interpreted the resulting intermediate densities as minimizing an expected divergence to the endpoints. To analyze these variational representations of annealing paths, we extend known results showing that the arithmetic mean over arguments minimizes the expected Bregman divergence to a single representative point. In particular, we obtain an analogous result for quasi-arithmetic means, when the inputs to the Bregman divergence are transformed under a monotonic embedding function. Our analysis highlights the interplay between quasi-arithmetic means, parametric families, and divergence functionals using the rho-tau representational Bregman divergence framework, and associates common divergence functionals with intermediate densities along an annealing path.