| # 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. |
| |
