Piecewise Bounds for Logistic-Gaussian Integrals
By Benjamin Marlin, University of British Columbia
Abstract:
We propose the application of a class of piecewise linear
and quadratic bounds for parameter estimation in models involving
intractable logistic-Gaussian integrals. We build on past work for
optimal piece-wise linear bounds to the log-sum-exp function by
introducing an optimization framework for minimax fitting of piecewise
quadratic bounds. We show that piecewise quadratic bounds can reduce
the maximum error in the bound by more than a factor of ten relative
to a piecewise linear bound with the same number of pieces. Finally,
we show that the uniform minimax error guarantees that our bounds
possess lead to much more accurate estimation of covariance parameters
in binary latent Gaussian graphical models where existing variational
quadratic bounds fail. This is joint work with Emtiyaz Khan and Kevin
Murphy.