Rank-Conditioned Sample Reuse for the Plackett--Luce Best-of-$K$ Objective

Abstract

We study the coupled objective J_K^WOR = E_{S ~ PL-WOR_K}[max_{i in S} R_i]: the expected maximum reward of a size-K Plackett-Luce draw without replacement, the law of Gumbel-Top-K / Stochastic Beam Search decoding. This estimand differs from the conventional i.i.d. objective J_K^iid = E[max_{i<=K} R_i] targeted by existing sample-reuse Max@K estimators, and reusing their i.i.d. weights under the coupled sampler is provably biased (a closed-form three-item instance gives E[g_iid] = (4/5) grad J_K^WOR exactly; pass@K under the coupled sampler is the binary-reward special case). Generic joint-score REINFORCE is already unbiased for J_K^WOR; what it lacks is sample reuse. Our contribution is to instantiate standard rank-conditioned Horvitz-Thompson estimation for the J_K^WOR subset total: from one Gumbel-Top-n pool (n>K) and its observed priority threshold we build an estimator that reuses all C(n,K) embedded K-subsets, unbiased with an unbiased exact score-function surrogate gradient, plus a reward-sorted Max-specific dynamic program that collapses the C(n,K)-term subset sum (with K!-cost set probabilities) exactly to a one-dimensional integral. A fixed-Q quadrature evaluation costs O(n log n + nKQ) arithmetic and is numerically, not algebraically, exact; no epsilon-approximation rate is certified. Each nonzero degree-K Horvitz-Thompson term has finite second moment exactly when n >= 2K; under the same assumptions the full surrogate gradient has finite second moment whenever n >= 2K (sharpness there is open). At K=1 the construction recovers classical priority sampling. All quantities require only the values and differentiable computation graphs of the n+1 drawn items' probabilities, so finite structured sequence policies sampled by exact SBS are covered. A certified finite-Q quadrature bound and countably infinite support remain open. Validation code is included as ancillary files.