Closing the order-T gap for normalising-constant complexity in waste-free SMC

Determine whether the order-T gap between the lower bound Omega(T^2/(gamma epsilon^2)) implied by the central limit theorem for the waste-free Sequential Monte Carlo estimator of log(\widehat{Z}_T/Z_T) and the current upper bound O(T^3/(gamma epsilon^2) log(T/eta)) for the product-of-medians estimator of the normalising constant can be closed, either by sharpening the upper bound to O(T^2/(gamma epsilon^2) polylog) or by proving a matching lower bound.

Background

In the analysis of normalising constant estimation with waste-free SMC under spectral gap assumptions, a lower bound on the required chain length P follows from a central limit theorem for log(\widehat{Z}_T/Z_T), yielding P = Omega(sigma_infty^2/epsilon^2) with sigma_infty^2 typically Theta(sum_t 1/gamma_t), which leads to an overall complexity of order Omega(T^2/(gamma epsilon^2)) when M is fixed.

A boosted product-of-medians estimator achieves an upper bound complexity O(T^3/(gamma epsilon^2) log(T/eta)). This results in an order-T gap between lower and upper bounds that is not currently explained by the analysis.