Kernel Exponential Family Estimation via Doubly Dual Embedding

We investigate penalized maximum log-likelihood estimation for exponential family distributions whose natural parameter resides in a reproducing kernel Hilbert space. Key to our approach is a novel technique, doubly dual embedding, that avoids computation of the partition function. This technique also allows the development of a flexible sampling strategy that amortizes the cost of Monte-Carlo sampling in the inference stage. The resulting estimator can be easily generalized to kernel conditional exponential families. We furthermore establish a connection between infinite-dimensional exponential family estimation and MMD-GANs, revealing a new perspective for understanding GANs. Compared to current score matching based estimators, the proposed method improves both memory and time efficiency while enjoying stronger statistical properties, such as fully capturing smoothness in its statistical convergence rate while the score matching estimator appears to saturate. Finally, we show that the proposed estimator can empirically outperform state-of-the-art methods in both kernel exponential family estimation and its conditional extension.