Variational Bayesian Inference with Stochastic Search

Mean-field variational inference is a method for approximate Bayesian posterior inference. It approximates a full posterior distribution with a factorized set of distributions by maximizing a lower bound on the marginal likelihood. This requires the ability to integrate a sum of terms in the log joint likelihood using this factorized distribution. Often not all integrals are in closed form, which is typically handled by using a lower bound. We present an alternative algorithm based on stochastic optimization that allows for direct optimization of the variational lower bound. This method uses control variates to reduce the variance of the stochastic search gradient, in which existing lower bounds can play an important role. We demonstrate the approach on two non-conjugate models: logistic regression and an approximation to the HDP.

💡 Research Summary

The paper addresses a fundamental limitation of mean‑field variational inference (MF‑VI), namely the inability to compute closed‑form expectations of certain terms in the log‑joint density when the model is non‑conjugate. Traditional MF‑VI circumvents this problem by introducing a tractable lower bound on the problematic terms, then maximising the resulting Evidence Lower Bound (ELBO). While this approach yields a closed‑form optimisation problem, the bound can be loose, leading to biased posterior approximations and slow convergence.

To overcome these drawbacks, the authors propose a stochastic‑search variational inference algorithm that directly optimises the true ELBO using Monte‑Carlo estimates of its gradient. The key challenge of stochastic gradient methods—high variance of the gradient estimator—is tackled by incorporating control variates. Specifically, any function whose expectation under the variational distribution can be evaluated analytically (or with low‑variance estimates) may serve as a control variate; the paper demonstrates that the conventional lower bound itself is an effective choice. The corrected gradient takes the form

 ĝ(θ) = ∇θ E_q