Stochastic Gradient Riemannian Langevin Dynamics on the Probability Simplex
In this paper we investigate the use of Langevin Monte Carlo methods on the probability simplex and propose a new method, Stochastic gradient Riemannian Langevin dynamics, which is simple to implement and can be applied to large scale data. We apply this method to latent Dirichlet allocation in an online minibatch setting, and demonstrate that it achieves substantial performance improvements over the state of the art online variational Bayesian methods.
This paper combines the recently proposed stochastic 
      gradient Langevin dynamics with Riemannian Langevin dynamics. The new 
      sampling algorithm is able to use a locally adaptive preconditioning 
      matrix for a faster convergence rate. It is applied to draw samples in the 
      posterior distribution of the LDA model under the online learning setting, 
      and achieves a better predictive performance than online variational 
      Bayesian methods. The combination of SGLD with RLD provides a 
      promising direction to improve the application of MCMC methods on large 
      scale data set. It is usually difficult to choose a suitable and 
      easy-to-compute Riemannian metric. But the Dirichlet model is one of the 
      exceptions where the expected Fisher information can be computed exactly 
      and efficiently. How likely is it that SGRLD would be applied to a more 
      general area or probably with some extra approximation? In order 
      to converge to the exact posterior distribution, SGLD has to anneal the 
      step size towards 0, which will slow down the mixing of the Markov chain 
      even with the help of RLD. Is it possible to run SGRLD with a fixed step 
      size for a faster mixing at the price of introducing bias in the 
      stationary distribution? Also, is there any strategy/heuristic to choose 
      appropriate step size parameters instead of doing a grid search? 
      Among the four parameterization, RLD + expanded-mean is shown to 
      obtain the best performance. However, with the mirroring trick, the 
      log-likelihood of the model is not differentiable at 0. Would that result 
      in extra error of Langevin dynamics? In Table 1, what does “n.” 
      mean? Also, it seems there are some errors in G(\theta). E.g. in 
      reduced-mean, there is a factor of n, but in expanded-mean, G(\theta) does 
      not depend on n. So is reduced-nature vs. expanded-natural. Besides, 
      \theta in reduced-nature should be replaced with exp(\theta). The 
      text and markers in all the figures are too small to read. It would be 
      better to put the description of the parameter values in the caption. 
      Other typos: Line 42, are based a batch → missing “on” 
      Line 79, a SGLD → an SGLD Figure 1(b), Phi → \pi Line 399, The 
      algorithms’ performances … was assessed Line 402, in vocabulary size → 
      missing “a”