Closing the gap between myopic lower bounds and polynomial particle requirements

Ascertain and close the gap between (i) the lower bound that any myopic particle filtering algorithm requires at least \Omega(log H / log log H) particles to achieve non-trivial sampling accuracy under constant-factor inaccuracies in the process reward model and (ii) the current polynomial-in-H particle requirements of Sequential Monte Carlo and its variants; determine the optimal scaling in H, possibly by allowing lookahead beyond myopic methods.

Background

The authors prove a lower bound showing that any myopic particle filtering method needs at least \Omega(log H / log log H) particles when the process reward model is only approximately correct up to a constant factor. On the other hand, their algorithms (SMC and variants) require maintaining poly(H) particles under similar assumptions, implying superlinear work.

This leaves a gap between the provable lower bound and the best known upper bounds. Establishing tight bounds—either by improving algorithms (potentially using lookahead) or by strengthening lower bounds—remains unresolved.