server/docs/policy_lp_reduction.md

E3: fixed-oracle optimal randomized execution policies

This experiment computes a floating-point optimum for each fixed oracle. It does not optimize the oracle over a KL ball, numerically establish a minimax exponent, or certify exact optimality with rational or interval arithmetic. The oracle is held fixed when the number of permitted rounds changes.

Model and admissible interface

There are n pairs. Under the target, A_i are independent fair bits, E_i ~ Bernoulli(tau) independently, and B_i = A_i xor E_i. The oracle has the same independent fair representative bits, while the mismatch vector is a positive mixture of iid Bernoulli laws. More generally the reduction below applies to a positive finite-exchangeable mismatch law.

A policy chooses the entire next subset of hidden coordinates before any new outcome is drawn. Chosen coordinates are sampled independently from the oracle's conditional heads at the pre-round context. Values are irrevocable. The policy may depend on its full observed history, use private randomness, complete open pairs, expose either orientation of fresh pairs, or co-sample both endpoints. Empty rounds are allowed; a completed output is absorbing. A policy must finish by round K. This is not an adaptive-within-batch, rejection, remasking, or head-correction interface.

Controlled aggregation

The aggregate state is (c,m,o): the number of completed pairs, their mismatch count, and the number of pairs with exactly one revealed coordinate. There are f = n-c-o fresh pairs. An action (u,v,b) completes u open pairs, co-samples v fresh pairs, and reveals one endpoint in b other fresh pairs. The constraints are 0 <= u <= o, v >= 0, b >= 0, and v+b <= f.

The posterior mismatch probability in an open pair is

p(c,m) = sum_j omega_j theta_j^(m+1) (1-theta_j)^(c-m)
         / sum_j omega_j theta_j^m (1-theta_j)^(c-m).

Single revealed endpoints are fair and convey no mismatch information. A fresh co-sampled pair has mismatch probability 1/2, since both heads are fair and sampled independently. Conditional on the chosen action, the frozen-head rule therefore gives independent X ~ Binomial(u,p(c,m)) and Y ~ Binomial(v,1/2), and

(c',m',o') = (c+u+v, m+X+Y, o-u+b).

This formula holds conditional on any literal history and action inducing the aggregate state and action; it is not merely a target-distribution identity. For the last round, completion requires (u,v,b) = (o,f,0).

Why this preserves the optimum, including private randomization

1. For any admissible full-history policy, condition its current aggregate action on its current aggregate state and round. The resulting randomized Markov policy preserves the distributions of all aggregate states by induction, because the transition formula above is controlled-lumpable. In particular it preserves the terminal mismatch-count distribution. Data processing says count TV is at most the original full-output TV.
2. Conversely, lift a count-state policy by selecting uniformly among the literal subsets with the requested pair counts, and choosing the endpoint of each singleton birth by an independent fair coin. The target and oracle are invariant under pair permutations, simultaneous flips of both endpoints of any pair, and swaps of a pair's endpoints. The lifted dynamics are equivariant under this group, starting from the empty context. Consequently its full-output distribution is invariant. The full-output group orbits are characterized exactly by the total mismatch count. Both target and lifted output are uniform within each orbit, so their full TV equals the TV of their mismatch-count distributions.
3. These two inequalities show equality between the optimum over aggregate randomized Markov policies and the optimum over all allowed full-history, privately randomized literal policies, for this fixed fair oracle.

The result is about marginal output laws. Averaging the TV errors of deterministic policies would generally give a different and incorrect optimization problem.

Occupation LP and independent checks

There is a nonnegative variable for each reachable (round,state,action). State-flow equalities impose the initial empty context and the transition law. The terminal probabilities q_m are linear functions of final-round occupancies. Auxiliary nonnegative d_m obey d_m >= q_m-P_m and d_m >= P_m-q_m, where P_m = Binomial(n,tau)(m). Minimize sum_m d_m/2.

The implementation records solver status, primal flow residual, nonnegativity, terminal mass, L1-constraint residual, objective recalculated from terminal probabilities, dual slack violation, and primal-dual objective gap. The dual objective uses equality and inequality right-hand sides and the solver's dual marginals; all variable lower bounds are zero and all upper bounds infinite. These are floating-point diagnostics, not an exact lower-bound certificate. The native HiGHS option threads=1 is passed through SciPy (some SciPy versions emit a warning that this native option is forwarded rather than recognized by the wrapper). BLAS/OpenMP environment thread limits should also be set by the runner.

Positive-mass state occupancies are normalized into an explicit randomized policy. Its output is recomputed by a separate forward routine that uses direct mixture powers and explicit binomial-product sums, rather than reusing the LP's cached log-posterior/convolution transition arrays. Unvisited states use the first lexicographic legal action. The recovered output must match the LP output within a stated numerical tolerance.

For n <= 2 the independent literal-coordinate LP enumerates all complete bit vectors, forms the entire oracle law, obtains each coordinate head by conditioning that full law on a partial bit assignment, and enumerates all hidden-coordinate subsets. It never uses (c,m,o) to construct states, heads, or transitions. The terminal objective is full-bit-vector TV. Its optimum is checked against the aggregate LP. It shares only the generic occupation-LP assembly and solver.

A separate regression uses n=1, K=2, target mismatch probability 1/4, and oracle mismatch probability 1/8. Serial completion and co-sampling have errors 1/8 and 1/4 separately. Their mixture with probabilities 2/3 and 1/3 has exact target marginal output and zero TV. Both LP formulations must recover zero error; aggregate action occupancies must imply those mixture weights.

Frozen experiment grid and baselines

The default grid is n in {2,4,6,8} and K in {2,3,4}, tau=1/4, w=1/8, Delta=1/4, lambda=0.2. Two fixed oracles are used:

• Single alternative: weights (1-w,w) at (tau,tau+Delta).
• Fair two-level ladder: weights (1-w,w/2,w/2) at (tau,tau+Delta,tau+Delta*(w/2)*lambda).

The two-level ladder is not rebuilt from K. Actual joint KL(P||R) and joint TV are reported for every n; neither is inferred from the contamination weight. Matching w and Delta across the two models does not match their KL or initial drift, and the output includes the initial drift explicitly.

Baselines expose one representative endpoint of every pair in round one, then complete uniformly selected open pairs in subsequent rounds. An all-A first round has the same full-output law here: the representative bits are independent fair bits and the completion process depends only on mismatch counts. Uniform pair selection (or a hidden uniformly random pair permutation) makes terminal count TV equal full TV. Deterministic labeled batch assignments should not be mistaken for this randomized comparison.

• Equal: cumulative completions ceil(n*j/(K-1)).
• Geometric heuristic: cumulative completions floor(n**(j/(K-1))). This is a comparison schedule, not an assertion that this rounding instantiates every constant in the paper's upper-bound theorem.
• Best fixed completion grid: every nondecreasing integer prefix in [0,n], followed by completion of all n pairs. For K=3, this is exactly the complete grid of fixed pilot sizes. For K>3, it is the analogous fixed completion schedule class, not a new all-policy optimum.

Within each oracle and dimension, optimal TV must not increase with K, and the LP optimum must not exceed any feasible baseline. The grid is deliberately small. A 250,000 state-action guard and a per-case solver limit of 120 seconds prevent accidental expansion. No large-dimensional all-policy claim follows from these numerical optima.

Invocation and output

python src/e3_policy_lp.py --config configs/experiment_config.json --output results/e3

The JSON configuration section e3 accepts n_values, rounds, tau, w, delta, ladder_lambda, time_limit_seconds, state_action_limit, and literal_n_values. The main output is e3_results.json, incrementally updated, when --output is a directory. Passing a .json filename writes that file directly and stores policies in a sibling <filename-stem>_policies/ directory. The top-level cases list contains each result. --validate-only runs the randomization regression and both oracles at n=2, K=2,3, including independent literal-coordinate checks. Recovered positive-mass policies are retained. Errors are preserved in the result rather than silently dropping failed cases. If any case fails a check, the process exits nonzero and the result status records this.