Randomization Beyond Deterministic Adaptivity in Parallel Sampling

2026-10-02

AI Summary

This study addresses the precision bottleneck of deterministic adaptive strategies under finite rounds in parallel sampling. By leveraging forward KL divergence analysis and a density covering lemma, it investigates coordinate-wise single-shot parallel samplers and proposes a randomized mixture of fixed strategies requiring only a few bits for schedule selection, alongside constructing a separation certificate at dimension 389. Theoretically, this work demonstrates that randomization mechanisms can surpass the error upper bounds of deterministic adaptive strategies, achieving exponentially small divergence with established asymptotic constants. These findings reveal the fundamental superiority of randomization over deterministic approaches when operating within known finite structures.

Abstract

We study parallel samplers that reveal each coordinate once and draw the coordinates of each batch independently from their exact conditional marginals. Under the same sampling interface and a hard three-round cap, randomization can improve accuracy beyond every deterministic policy that adapts its batches to previously observed values. For an explicit full-support binary family, the optimal deterministic adaptive forward Kullback-Leibler divergence grows linearly with the dimension, while a uniform mixture of four fixed schedules has exponentially small divergence. Only two independent fair bits are needed to select the schedule. We determine the asymptotic additive constant in the deterministic optimum and show that its optimal total variation tends to one quarter. An outward-rounded finite-sum certificate establishes divergence separation at dimension 389. The mechanism combines an irreversible first-batch dependence loss with posterior concentration in the randomized components. A density-coverage lemma describes how their mixture restores the target. A complementary noisy-matching family exhibits a growing round gap for nonadaptive schedules, although uniform random coordinate order still has linear mixture error. These results concern known finite structures and exact conditional marginals.

Problem

• parallel sampling
• deterministic adaptivity
• randomization
• Kullback-Leibler divergence
• conditional marginals

Innovation

• Parallel Sampling
• Randomized Scheduling
• Deterministic Adaptivity
• KL Divergence Separation
• Posterior Concentration