File size: 3,258 Bytes
dd14c26 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 | import OAI.Probability.MatroidProphet.Algorithm.WeightedPayoff
import OAI.Probability.MatroidProphet.CompletePayoff
import OAI.Probability.MatroidProphet.Reduction
namespace OAI
namespace MatroidProphet
open MeasureTheory Finset MainAlgorithm
lemma main_branch_guarantee_small {n : ℕ} (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(w : Weights n)
(hm : maximumRounded M w ≤ optimum M (fun e => roundedWeight weightBase (w e)) /
(8 * densityThreshold)) :
thinningRate * retainedFraction / (16 * densityThreshold) *
optimum M (fun e => roundedWeight weightBase (w e)) ≤ mainPayoff M hE w := by
apply main_branch_from_filter Finset.univ
(analyzedComparisonMass M (fun e => roundedLevel weightBase (w e)))
(comparisonMass M (fun e => roundedLevel weightBase (w e)))
(fun H => ∑ e ∈ candidateLabels M (fun e => roundedLevel weightBase (w e)) H,
roundedWeight weightBase (w e))
(fun H => mainMean ⟨H, ∅, ∅, ∅, false⟩
(fun d => mainWorstRounded M hE d (fun e => roundedLevel weightBase (w e))))
(optimum M (fun e => roundedWeight weightBase (w e))) (maximumRounded M w)
(optimum_nonneg M _) hm
· exact expectation_positiveGreedy_survivors M hE w (by norm_num [weightBase])
· exact expectation_candidateLabels_le_optimum M hE w weightBase
· intro H _
exact small_comparison_mass M hE w H
· intro H _
exact conditional_main_payoff M hE ⟨H, ∅, ∅, ∅, false⟩ _
theorem complete_hidden_guarantee {n : ℕ} (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(w : Weights n) (hw : ∀ e, 0 ≤ w e) :
((2 : ℝ) ^ 293)⁻¹ * optimum M w ≤
∫ r, hiddenWorstReward (completeHiddenRule M hE) w r ∂sourceSeedLaw n := by
let W := optimum M (fun e => roundedWeight weightBase (w e))
have hb := branch_mixture_lower W (maximumRounded M w) (mainPayoff M hE w) (maximumPayoff M w)
(optimum_nonneg M _) (maximumRounded_nonneg M w) (mainPayoff_nonneg M hE w)
(maximum_branch_guarantee M w hw) (main_branch_guarantee_small M hE w)
have hi := integrated_branch_lower M hE w hw
have hr := optimum_rounding_bound M weightBase (by norm_num [weightBase]) w hw
have hc0 : 0 ≤ thinningRate * retainedFraction / (32 * densityThreshold * weightBase) :=
div_nonneg (mul_nonneg constants_positive.2.2.1.le constants_positive.2.2.2.2.le)
(mul_nonneg (mul_nonneg (by norm_num) constants_positive.2.1.le) constants_positive.1.le)
have hscaled := mul_le_mul_of_nonneg_left hr hc0
have hcancel : thinningRate * retainedFraction / (32 * densityThreshold * weightBase) *
(weightBase * W) = thinningRate * retainedFraction / (32 * densityThreshold) * W := by
field_simp [ne_of_gt constants_positive.1, ne_of_gt constants_positive.2.1]
rw [hcancel] at hscaled
rw [hidden_fraction_exact] at hscaled
exact hscaled.trans (hb.trans hi)
theorem hidden_vector : HiddenVectorChallenge := by
intro n M hE
refine ⟨mainSeedBits n, sourceSeedLaw n, inferInstance, completeHiddenRule M hE, ?_, ?_⟩
· exact completeHiddenRule_feasible M hE
· exact complete_hidden_guarantee M hE
theorem one_sample.{u} : OneSampleChallenge.{u} :=
hiddenVectorChallenge_implies_oneSampleChallenge hidden_vector
end MatroidProphet
end OAI
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