Variational Russian Roulette

Statistical machine learning is all about gambling.

How to estimate an infinite sum \(S = \sum_{k=1}^\infty T_k\) that is convergent?

Spin a roulette via \(K \sim P(\tau) = (1 - \rho_{\tau+1})\prod_{s=1}^\tau \rho_s\), where \(P(\tau)\) represents an arbitrary discrete distribution.

Then your unbiased estimate of the sum is \(\hat{S}_K = \sum_{k=1}^K \frac{T_k}{p_k}\), where \(p_k = \prod_{j=1}^k \rho_j\).

The trick is used in my method roulette-based amortized variational expectation (RAVE) to perform amortized inference in deep models with an Indian buffet process prior.

Check out our paper on RAVE if you are interested in how to spin roulettes in Indian buffets.

PS: Gambling is risky and, therefore, is not recommended.