Fixed-schedule benchmark: exact full-output TV
Scope and frozen inputs
The benchmark reuses all nine phase-2 rational oracles and their certified all-policy intervals. No mixture parameter or KL budget is recalibrated. For each oracle, total rounds are K = 3 and K = 4. These are finite, oracle-specific comparisons; optimizing a batch schedule against each known oracle is not a claim about choosing a schedule uniformly before the oracle is known.
The teacher has independent fair representatives A_i and mismatches E_i = A_i xor B_i that are iid Bernoulli(tau). The oracle has the same independent fair representatives and a positive exchangeable mixture of iid mismatch laws.
Deterministic fixed-label schedules
Round 1 samples every A_i. The remaining r = K - 1 rounds sample disjoint, preassigned labeled groups of B coordinates, with ordered group sizes (b_1,...,b_r), summing to n. Empty groups are legal and allow schedules with fewer effective rounds. Heads within a round are frozen before that round.
If c pairs and m mismatches have already been completed, the oracle's mismatch prediction is exactly
p(c,m) = sum_l w_l p_l^(m+1) (1-p_l)^(c-m)
/ sum_l w_l p_l^m (1-p_l)^(c-m).
Thus a new group of size b has Binomial(b,p(c,m)) mismatches. Let Z_j denote the mismatch count in group j. The full joint probability of the vector Z is the product of these sequential binomial probabilities, with the previous cumulative counts supplied to each p(c,m). The teacher's corresponding vector law is the product of Binomial(b_j,tau) laws.
Both teacher and generated law are uniform over representative vectors, and, conditional on Z, uniform over mismatch configurations inside each preassigned group. Consequently the full 2n-coordinate output TV equals TV of the joint vector (Z_1,...,Z_r). It generally does not equal TV of the sum of those counts. The implementation sums positive differences of the vector probabilities in exact Fraction arithmetic; this equals one-half the L1 difference since both laws are normalized.
Random pair-label permutation is a different policy
For a separate, explicitly labeled comparison, draw a uniform random permutation of pair labels privately before running the same batch sizes. Averaging the output law over this randomness makes the mismatch law exchangeable. Its total mismatch-count distribution is the same as for the fixed-label schedule, but conditional on the total it is uniform over all mismatch strings. Hence its full-output TV equals total-count TV. This TV is no greater than the fixed-label vector TV by data processing / convexity. The public output contains only generated coordinates, not the private permutation. If the permutation were included in the observed output, this improvement would not follow.
The best fixed-label schedule and best random-permuted schedule are optimized and reported separately. They need not have the same minimizing batch sizes. The latter optimization chooses one fixed batch-size vector and then uniformly permutes pair labels; it does not allow an ex-ante random mixture of multiple batch-size vectors. Its K=4 gap from the general policy class therefore does not establish a separation from all non-adaptive strategies.
Compared schedules and optimization
- Geometric schedule from the manuscript: cumulative sizes c_0=0, c_j=floor(n^(j/r)) for 1 <= j < r, c_r=n. Integer powers and integer binary search implement the floor exactly, avoiding floating-root rounding errors.
- Equal schedule: c_j=floor(j*n/r).
- Oracle-specific best fixed-label schedule: enumerate every ordered weak composition of n into r groups; minimize the exact vector TV.
- Oracle-specific best random-permuted schedule: use the same enumeration, minimizing the exact total-count TV of the marginalized random-permuted law.
All exact ties are recorded; a displayed representative is the lexicographically first batch vector. The full enumeration is retained, not merely the winner.
Verification
For each computed composition, the exact generated distribution normalizes; total-count TV is no larger than joint-vector TV; and both policies' exact TVs are no smaller than the supplied phase-2 certified all-policy lower bound. Input oracle parameters must match the certificate exactly, and its KL and head-error constraints must already be certified.
The self-test separately constructs full labeled P and R tables with 2^(2n) entries for n=2,3. It computes literal conditional heads by summing R over compatible complete strings, multiplies frozen heads for each literal batch, and enumerates every pair-label permutation for the randomized policy. These full-output TVs must equal the count-vector and marginalized count calculations, respectively. This check does not reuse the posterior or binomial transition formula. Additional controls check zero TV for R=P, one-nonempty-batch count sufficiency, and exact integer geometric roots.
All reported distance values include their exact rational representation. Floating values are only presentation conveniences. These are independently cross-checked computational calculations, not proof-assistant formalization.