Determine the sharp full-support hard-cap constant
Determine the exact leading constant governing the least hard cap for achieving fixed output-law KL tolerance under the full-support Beta(1, 1)–Bernoulli target, whose currently established asymptotic constant lies between one-half and one when measured in units of log₂ log₂ n.
References
Open are the constant of the full-support hard cap (between 2 and 1), a common length on weighted slices without a linear-size equal-weight group, growing weight ratios, and sharp rates for vanishing tolerances.
Theorem 19 concerns expected executed rounds, but its part (a), a tail bound, also bounds hard caps. A policy in Ar completes, and has τ+ ≤ τ ≤ r, almost surely, so Pr(τ+ > R0(n)) = 0when r ≤ R0(n). For every ε ≥ 0 and ϑ > 4, the admissible (ξ, ω) with η∞ > 0 of the proof of Theorem 15(b) give an < 1 and ηn > 0 for all large n, so by (a) every π ∈ Ar with KL(p∥Qπ) ≤ εhas r > R0(n); as R0(n)/ log2 log2 n → ln 2/ ln ϑ, letting ϑ ↓ 4 shows that the least hard cap of causal policies with F ≤ ε is at least ( 2 − o(1)) log2 log2 n, as for mixtures. For fixed ε > 0, one uniformly relabelled profile of length L, a finite predrawn mixture and so a policy in AL, gives at most log2 log2 n + Oε(1) (Theorem 7(b)); the sharp constant between these bounds is open.