| # Biased-representative finite-policy experiment |
|
|
| ## Scope and frozen instance |
|
|
| For each pair, draw `A_i ~ Bernoulli(a)` independently of a mismatch vector |
| `E`, and set `B_i = A_i xor E_i`. The teacher has iid mismatches with rate |
| `tau = 1/4`. The oracle mismatch law is the fixed two-component mixture |
|
|
| ``` |
| R_E = (11681887 / 16777216) Bern(1/4)^n |
| + ( 5095329 / 16777216) Bern(1/2)^n. |
| ``` |
|
|
| These rational parameters are frozen from the phase-2 selected |
| `n8_B1_10_single_d1_4` model. They are not retuned at a new dimension, bias, or |
| round cap. Required cases are `n=4`, `a in {1/2,2/5,1/4}`, `K in {2,3,4}`. |
| The extension is `n=6`, the same three biases, and `K in {3,4}`, and is run |
| only if the prespecified resource gate succeeds. The gate is independent of |
| the numerical scientific outcome. |
|
|
| At every round, the policy may select any subset of currently masked |
| coordinates, using all past information and private randomization. Selected |
| coordinates are sampled independently from the oracle's exact singleton |
| conditionals at the **pre-round context**. All coordinates must be completed |
| by round K; empty batches allow earlier completion. This is a fixed-head, |
| fixed-oracle experiment, not optimization over oracle parameters. |
|
|
| ## Nine-type state and exact singleton conditionals |
|
|
| A state is the histogram of these pair types, in order: |
|
|
| ``` |
| empty, A0, A1, B0, B1, 00, 01, 10, 11. |
| ``` |
|
|
| Unlike in the fair case, A and B are not interchangeable, B-only observations |
| carry information about the latent component, and bit flips are not a valid |
| symmetry. The fair three-count state is therefore not used. |
|
|
| For component mismatch rate p, put `q_p = a + (1-2a)p`. The likelihoods of |
| the nine types are |
|
|
| ``` |
| 1, 1-a, a, 1-q_p, q_p, (1-a)(1-p), (1-a)p, ap, a(1-p). |
| ``` |
|
|
| For a labeled partial context with histogram h, its component posterior is |
| proportional to its component weight times the product of these likelihoods |
| raised to the histogram counts. The multinomial coefficient cancels between |
| components if one conditions on the histogram instead of a labeled context. |
|
|
| Within component p, the probabilities of value 1 for the six possible head |
| types are |
|
|
| ``` |
| empty A: a empty B: q_p |
| B | A=0: p B | A=1: 1-p |
| A | B=0: ap/(1-q_p) A | B=1: a(1-p)/q_p. |
| ``` |
|
|
| The oracle heads average these six quantities against the **full current |
| component posterior**. In particular, the nonlinear reverse heads must not |
| be obtained by substituting the posterior mean p into the displayed |
| fractions. That substitution would double-condition incorrectly. |
|
|
| For example, a=2/5, component weights (7/10,3/10), rates (1/4,1/2), and a |
| context observing only B=1 give `R(A=1 | B=1)=18/31`. This is an independent |
| reverse-head regression fixture. |
|
|
| ## Actions and transitions |
|
|
| An aggregate action is the seven-tuple |
|
|
| ``` |
| (empty_A, empty_B, empty_both, fill_A0, fill_A1, fill_B0, fill_B1). |
| ``` |
|
|
| The first three numbers partition a subset of empty pairs; the last four |
| select a subset from each singleton type. At the final round the unique |
| legal action completes all masked coordinates. In other rounds every legal |
| tuple, including the empty action, is enumerated. |
|
|
| For an empty pair selected in both coordinates, the two sampled bits are |
| independent Bernoulli draws with the frozen empty-A and empty-B heads. Their |
| joint law is **not** the oracle's pair law. All other selected-coordinate |
| draws are also conditionally independent in the round. Exact multinomial |
| laws within each selected type, followed by histogram convolution, give the |
| rational transition kernel. |
|
|
| For n pairs there are `C(n+8,8)` states and `C(n+3,3)` terminal histograms. |
| Strict positivity makes every histogram reachable after one round. A full |
| nonfinal layer has `C(n+15,15)` state-action pairs. Thus for K>=2 the layered |
| occupation LP has |
|
|
| ``` |
| C(n+3,3) + (K-2) C(n+15,15) + C(n+8,8) |
| ``` |
|
|
| action variables, before the additional terminal absolute-value variables. |
| The required n=4 cases have 530, 4406, and 8282 action variables for K=2,3,4. |
| The n=6 K=3,4 extension has 57351 and 111615 action variables. The extension |
| gate explicitly constructs all n=6 exact kernels at a=2/5, checks the K=4 |
| edge count, and enforces 120000 action variables, 3000000 transition entries, |
| and 120 seconds of kernel-building time. Each LP has a 90-second solve cap. |
|
|
| ## Why the terminal objective is full TV |
|
|
| The four completed pair types (00,01,10,11), not merely mismatch count, are |
| the terminal observable. Both the teacher and the uniformly lifted policies |
| are invariant under permutations of pairs. A terminal four-type histogram |
| is exactly an orbit of that permutation group. Both distributions are |
| constant on each orbit, so summing the labeled-coordinate L1 distance over |
| an orbit gives the L1 distance of its histogram masses. Consequently, |
| four-type histogram TV equals the full 2n-coordinate TV for the constructed |
| policy. A mismatch-count-only TV would in general be insufficient. |
|
|
| Every aggregate action can be lifted by uniformly choosing disjoint pair |
| subsets within each current type. This gives a legal, permutation-equivariant |
| policy with exactly the aggregate transition law. Conversely, after any |
| full labeled history, an arbitrary labeled action has an aggregate tuple |
| and the displayed aggregate kernel. This fact justifies the Bellman lower |
| bound for all history-dependent policies, without assuming their final |
| output is symmetric. Equivalently, pair-permutation symmetrization cannot |
| increase TV to the symmetric teacher, and occupation-measure Markovization |
| then yields the finite aggregate LP optimum. |
|
|
| ## Rational certificates |
|
|
| The LP minimizes half the terminal L1 distance using action occupation |
| measures. Mixtures are optimized before TV is evaluated; this is not an |
| average of deterministic-policy TVs. The numerical solver proposes a |
| bounded terminal test and a feasible policy candidate. Its floating-point |
| optimality report is not taken as a rigorous certificate. |
|
|
| For the lower certificate, clip the LP-derived test into [0,1], interpret |
| its printed decimals exactly, and round down to the dyadic grid 2^-40. |
| Starting from this test, perform backward Bellman minimization over every |
| legal aggregate action with the exact rational transition kernel, rounding |
| each expectation down to the same grid. The result is a lower enclosure |
| of the minimum expected test value over all policies. Subtract the exact |
| teacher mean of the dyadic test and truncate below at zero. Since TV |
| dominates `E_Q f - E_P f` for every f in [0,1], the result L is an all-policy |
| lower bound. Bellman rounding contributes at most K * 2^-40. |
|
|
| For the upper certificate, interpret nonnegative recovered action weights |
| as exact decimal rationals and normalize each state. Missing states use a |
| legal empty action, or the forced completion action in the last round. |
| Propagate state mass on a 2^-80 grid, rounding each edge contribution down. |
| This produces a componentwise lower terminal mass vector q_lo. Its missing |
| mass `e=1-sum(q_lo)` bounds the total omitted probability. Therefore |
|
|
| ``` |
| U = min(1, (sum_h |P(h)-q_lo(h)| + e)/2) |
| ``` |
|
|
| is an upper bound on the full TV of this explicitly legal lifted policy. |
| Every case reports exact rational L and U as authoritative fields, with |
| floating-point values for display. The interval `L <= inf_policy TV <= U` |
| is not a proof-assistant formalization. |
|
|
| The exact kernel is shared by LP assembly and rational certification; LP |
| assembly converts it to floating point. Independence comes from the |
| separate literal-coordinate kernel checks and literal Bellman/LP comparison, |
| not from claiming two independently written reduced kernels. |
|
|
| ## KL, model TV, and head errors |
|
|
| The teacher and oracle use the same A law. Under the invertible map |
| `(A,E) -> (A,B)`, their joint likelihood ratio depends only on mismatch |
| count. Thus their joint KL and TV are exactly the mismatch-count KL and TV, |
| independent of a at fixed n. The code additionally verifies model TV against |
| the full four-type histogram distribution. Natural logarithms are enclosed |
| using rational range reduction, a finite atanh series, and a rigorous tail |
| bound. The exact KL upper bound is checked against 1/10. |
|
|
| The n=8 calibration is not transferred as an equality to n=4 or n=6. Actual |
| KL is reported at each dimension; the oracle is not adjusted to use the |
| remaining budget. Similarly, bias changes can change the uniform head |
| error. Every valid partial histogram and every available head are enumerated |
| exactly. The maximum error and a witnessing context are reported, including |
| whether it exceeds 1/4. No point is excluded on that basis, and this is not |
| described as a common-head-budget comparison. In particular, a component |
| shift by 1/4 need not bound reverse-head error: a=2/5, tau=1/4, p=1/2 gives |
| an `A|B=1` head difference of 4/15. |
|
|
| Positive `L-TV(P,R)` would certify finite-depth cost beyond the serial |
| oracle-law baseline. Nonpositive values do not refute the theory. Positive |
| `L_K-U_(K+1)` certifies a finite-dimensional depth separation. Neither metric |
| is assumed positive when choosing the grid or deciding the resource gate. |
|
|
| ## Validation and command line |
|
|
| `--self-test` uses only the standard library and exhaustively compares n=2 |
| literal heads and all selected-coordinate subset kernels at all three |
| biases. It also checks exact-teacher two-round controls and the reverse-head |
| counterexample. The default scientific run additionally compares all n=2, |
| K=2,3, three-bias histogram LPs with independently assembled full-coordinate |
| LPs. Every a=1/2 scientific case is compared to the frozen phase-2 fair |
| three-count LP implementation in `vendor/policy_lp.py`. |
|
|
| ``` |
| python biased_policy.py --self-test |
| python biased_policy.py --config phase3_config.json --profile-only |
| python biased_policy.py --config phase3_config.json --gate-profile --output gate.json |
| python biased_policy.py --config phase3_config.json --lp-self-test --output lp_checks.json |
| python biased_policy.py --config phase3_config.json --output biased_results.json |
| ``` |
|
|
| The main result is checkpointed after every case. Failed cases are retained |
| with their errors; the resource-gated extension is explicitly marked run or |
| skipped. No failure or negative scientific outcome is silently removed. |
|
|
