Notes on optimal approximations for importance sampling

In this manuscript, we derive optimal conditions for building function approximations that minimize variance when used as importance sampling estimators for Monte Carlo integration problems. Particularly, we study the problem of finding the optimal projection $g$ of an integrand $f$ onto certain classes of piecewise constant functions, in order to minimize the variance of the unbiased importance sampling estimator $E_g[f/g]$, as well as the related problem of finding optimal mixture weights to approximate and importance sample a target mixture distribution $f = \sum_i \alpha_i f_i$ with components $f_i$ in a family $\mathcal{F}$, through a corresponding mixture of importance sampling densities $g_i$ that are only approximately proportional to $f_i$. We further show that in both cases the optimal projection is different from the commonly used $\ell_1$ projection, and provide an intuitive explanation for the difference.