On the Computational Complexity of Geometric Langevin Monte Carlo
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Manifold Markov chain Monte Carlo algorithms have been introduced to sample more effectively from challenging target densities exhibiting multiple modes or strong correlations. Such algorithms exploit the local geometry of the parameter space, thus enabling chains to achieve a faster convergence rate when measured in number of steps. However, often acquiring local geometric information increases computational complexity per step to the extent that sampling from high-dimensional targets becomes inefficient in terms of total computational time. This paper analyzes the computational complexity of manifold Langevin Monte Carlo and proposes a manifold adaptive Monte Carlo sampler aimed at balancing the benefits of exploiting local geometry with computational requirements to achieve a high effective sample size for a given computational cost. The suggested strategy randomly switches between a local geometric and an adaptive proposal kernel via a schedule to regulate the frequency of manifold-based updates. An exponentially decaying schedule is put forward that enables more frequent updates of geometric information in early transient phases of the chain, while saving computational time in late stationary phases. The average complexity can be manually set depending on the need for geometric exploitation posed by the underlying model.
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