Scaling-up Split-Merge MCMC with Locality Sensitive Sampling (LSS)
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Split-Merge MCMC (Monte Carlo Markov Chain) is one of the essential and popular variants of MCMC for problems when an MCMC state consists of an unknown number of components or clusters. It is well known that state-of-the-art methods for split-merge MCMC do not scale well. Strategies for rapid mixing requires smart and informative proposals to reduce the rejection rate. However, all known smart proposals involve cost at least linear in the size of the data > O(N), to suggest informative transitions. Thus, the cost of each iteration is prohibitive for massive scale datasets. It is further known that uninformative but computationally efficient proposals, such as random split-merge, leads to extremely slow convergence. This tradeoff between mixing time and per update cost seems hard to get around. In this paper, we get around this tradeoff by utilizing simple similarity information, such as cosine similarity, between the entity vectors to design a proposal distribution. Such information is readily available in almost all applications. We show that the recent use of locality sensitive hashing for efficient adaptive sampling can be leveraged to obtain a computationally efficient pseudo-marginal MCMC. The new split-merge MCMC has constant time update, just like random split-merge, and at the same time the proposal is informative and needs significantly fewer iterations than random split-merge. Overall, we obtain a sweet tradeoff between convergence and per update cost. As a direct consequence, our proposal, named LSHSM, is around 10x faster than the state-of-the-art sampling methods on both synthetic datasets and two large real datasets KDDCUP and PubMed with several millions of entities and thousands of cluster centers.
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