Extend lower bounds beyond myopic methods to query-limited algorithms

Prove a lower bound analogous to Theorem pf-lb-main—showing particle complexity barriers under constant-factor inaccuracies in the process reward model—that applies to all algorithms restricted to o(H^2) queries to the base transition kernel P and the process reward model \hat V, not only to myopic particle filtering algorithms.

Background

The formal lower bound proved in the paper applies to myopic particle filtering methods. The definition of "myopic" allows algorithms to be very powerful in terms of querying the reward model, so strengthening the lower bound to a broader class of algorithms would be informative.

The authors specifically ask for an analogous result that applies to any algorithm with subquadratic (o(H2)) query complexity to the transition kernel and process reward model, which would be formally incomparable but complementary to their myopic bound.

References

That said, it is an interesting open question to prove an analogous result to \cref{thm:pf-lb-main} that applies to all algorithms that make $o(H2)$ queries to $$ and $$. Such a result would be formally incomparable to \cref{thm:pf-lb-main}.

— Reject, Resample, Repeat: Understanding Parallel Reasoning in Language Model Inference  (2603.07887 - Golowich et al., 9 Mar 2026) in Appendix, Section "Myopic Particle Filtering Algorithms", Remark [Computational efficiency]