Bound relating within-set predictability to incremental-weight variance

Derive a bound relating the within-set predictability measure \(\rho_k\) of future reward to the incremental-weight variance of finite-particle LatticeSMC.

Background

LatticeSMC uses exact reward potentials and effective-sample-size-gated resampling on the chunk axis of a chunked sequence generator. The paper introduces ρk\rho_k as a measure of how strongly the current prefix determines the future reward within a particle set, arguing that this quantity predicts whether content-axis lookahead or twisting is likely to improve resampling.

The theoretical propositions establish the target distribution and telescoping identities, but they do not characterize finite-particle behavior. In particular, the paper leaves unresolved how the within-set predictability of future reward quantitatively controls the variance of incremental particle weights, which would provide a principled guarantee for deciding when lookahead is beneficial.

References

The horizon study is on dance, since the music generator's 40 s context does not carry the prompted tempo over 80 s (Appendix~\ref{app:horizon}); the dance generators are small models trained for this study; the human study covers two rewards; gradient guidance, which composes with every placement, is not compared; and the propositions fix the target, not finite-particle behaviour, so a bound relating \rho_k to the incremental-weight variance is open.

— LatticeSMC: Where to Spend Inference-Time Compute in Chunked Sequence Generators  (2610.02774 - Wang et al., 2 Oct 2026) in Section Discussion and Conclusion, subsection “Limitations”