Boosted Density Estimation Remastered

There has recently been a steady increase in the number iterative approaches to density estimation. However, an accompanying burst of formal convergence guarantees has not followed; all results pay the price of heavy assumptions which are often unrealistic or hard to check. The Generative Adversarial Network (GAN) literature --- seemingly orthogonal to the aforementioned pursuit --- has had the side effect of a renewed interest in variational divergence minimisation (notably $f$-GAN). We show that by introducing a weak learning assumption (in the sense of the classical boosting framework) we are able to import some recent results from the GAN literature to develop an iterative boosted density estimation algorithm, including formal convergence results with rates, that does not suffer the shortcomings other approaches. We show that the density fit is an exponential family, and as part of our analysis obtain an improved variational characterisation of $f$-GAN.