MAD-Bayes: MAP-based Asymptotic Derivations from Bayes

TL;DR

This paper introduces a novel framework applying small-variance asymptotics directly to Bayesian nonparametric models, enabling scalable algorithms for complex grouping tasks beyond traditional clustering.

Contribution

It generalizes small-variance asymptotics to Bayesian nonparametric models, leading to new scalable algorithms for feature learning and grouping.

Findings

New objective function for feature learning beyond clustering

Algorithms are scalable and simple to implement

Empirical results show improved performance

Abstract

The classical mixture of Gaussians model is related to K-means via small-variance asymptotics: as the covariances of the Gaussians tend to zero, the negative log-likelihood of the mixture of Gaussians model approaches the K-means objective, and the EM algorithm approaches the K-means algorithm. Kulis & Jordan (2012) used this observation to obtain a novel K-means-like algorithm from a Gibbs sampler for the Dirichlet process (DP) mixture. We instead consider applying small-variance asymptotics directly to the posterior in Bayesian nonparametric models. This framework is independent of any specific Bayesian inference algorithm, and it has the major advantage that it generalizes immediately to a range of models beyond the DP mixture. To illustrate, we apply our framework to the feature learning setting, where the beta process and Indian buffet process provide an appropriate Bayesian nonparametric prior. We obtain a novel objective function that goes beyond clustering to learn (and penalize new) groupings for which we relax the mutual exclusivity and exhaustivity assumptions of clustering. We demonstrate several other algorithms, all of which are scalable and simple to implement. Empirical results demonstrate the benefits of the new framework.