Extend deterministic-policy lower bounds to randomized policies

Determine whether the lower bounds proved for deterministic value-adaptive policies by Wen et al. (2026) extend to arbitrary randomized policies in the same parallel-sampling interface.

Background

The paper studies parallel samplers that reveal coordinates in batches using independent draws from exact conditional marginals frozen at the beginning of each round. It distinguishes deterministic adaptive policies from causal randomized policies, in which a seed may influence batch choices and may be consulted throughout the run.

The cited prior work establishes a doubly logarithmic-squared lower-bound order for deterministic adaptive policies on the balanced slice under a hard round cap and under an appropriate mean cost. The paper notes that whether this lower-bound argument applies to arbitrary randomized policies was left unresolved, while subsequent work shows that randomization can sometimes outperform deterministic adaptive policies.