Refined α-Divergence Variational Inference via Rejection Sampling
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We present an approximate inference method, based on a synergistic combination of Rényi α-divergence variational inference (RDVI) and rejection sampling (RS). RDVI is based on minimization of Rényi α-divergence D_α(p||q) between the true distribution p(x) and a variational approximation q(x); RS draws samples from a distribution p(x) = p̃(x)/Z_p using a proposal q(x), s.t. Mq(x) ≥p̃(x), ∀ x. Our inference method is based on a crucial observation that D_∞(p||q) equals M(θ) where M(θ) is the optimal value of the RS constant for a given proposal q_θ(x). This enables us to develop a two-stage hybrid inference algorithm. Stage-1 performs RDVI to learn q_θ by minimizing an estimator of D_α(p||q), and uses the learned q_θ to find an (approximately) optimal M̃(θ). Stage-2 performs RS using the constant M̃(θ) to improve the approximate distribution q_θ and obtain a sample-based approximation. We prove that this two-stage method allows us to learn considerably more accurate approximations of the target distribution as compared to RDVI. We demonstrate our method's efficacy via several experiments on synthetic and real datasets. For reproducibility, we provide the code for our method in the supplementary material.
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