| import OAI.Probability.MatroidSecretary.Labels.FiniteLabelingModel |
| import OAI.Probability.MatroidProphet.Main |
|
|
| namespace OAI |
|
|
| /-! |
| Finite-label equivalence supporting the scope statement "every finite known |
| labeled matroid". No positivity, nonempty-ground, or nonloop hypothesis occurs |
| in the transport laws. The proof uses mathlib's actual matroid map, not a |
| replacement independence predicate. |
| -/ |
|
|
| namespace MatroidProphet.Labeling |
|
|
| open Finset |
|
|
| variable {E F : Type*} [Fintype E] [Fintype F] |
|
|
| theorem finiteOptimum_nonneg (M : Matroid E) (w : E → ℝ) : |
| 0 ≤ finiteOptimum M w := by |
| classical |
| unfold finiteOptimum |
| apply Finset.le_sup'_of_le _ (Finset.mem_univ ∅) |
| simp |
|
|
| theorem sum_le_finiteOptimum (M : Matroid E) (w : E → ℝ) |
| (I : Finset E) (hI : M.Indep (I : Set E)) : |
| (∑ e ∈ I, w e) ≤ finiteOptimum M w := by |
| classical |
| unfold finiteOptimum |
| apply Finset.le_sup'_of_le _ (Finset.mem_univ I) |
| simp [hI] |
|
|
| omit [Fintype E] [Fintype F] in |
| theorem mapEquiv_finset_indep (M : Matroid E) (e : E ≃ F) (I : Finset E) : |
| (M.mapEquiv e).Indep (I.map e.toEmbedding : Set F) ↔ M.Indep (I : Set E) := by |
| simp |
|
|
| theorem finiteOptimum_mapEquiv (M : Matroid E) (e : E ≃ F) (w : E → ℝ) : |
| finiteOptimum (M.mapEquiv e) (fun f => w (e.symm f)) = finiteOptimum M w := by |
| classical |
| apply le_antisymm |
| · apply Finset.sup'_le |
| intro I _ |
| split_ifs with hI |
| · have hi : M.Indep (I.map e.symm.toEmbedding : Set E) := by |
| simpa using hI |
| simpa using sum_le_finiteOptimum M w (I.map e.symm.toEmbedding) hi |
| · exact finiteOptimum_nonneg M w |
| · apply Finset.sup'_le |
| intro I _ |
| split_ifs with hI |
| · have hi := (mapEquiv_finset_indep M e I).2 hI |
| simpa using sum_le_finiteOptimum (M.mapEquiv e) (fun f => w (e.symm f)) |
| (I.map e.toEmbedding) hi |
| · exact finiteOptimum_nonneg (M.mapEquiv e) _ |
|
|
| theorem finMatroid_ground (M : Matroid E) (hE : M.E = Set.univ) : |
| (finMatroid M).E = Set.univ := by |
| simp [finMatroid, hE] |
|
|
| theorem finWeights_nonnegative_iff (w : E → ℝ) : |
| (∀ i, 0 ≤ finWeights w i) ↔ ∀ e, 0 ≤ w e := by |
| constructor |
| · intro h e |
| simpa [finWeights] using h ((Fintype.equivFin E) e) |
| · intro h i |
| exact h _ |
|
|
| theorem optimum_finMatroid (M : Matroid E) (w : E → ℝ) : |
| MatroidProphet.optimum (finMatroid M) (finWeights w) = finiteOptimum M w := |
| finiteOptimum_mapEquiv M (Fintype.equivFin E) w |
|
|
| section Online |
|
|
| variable [MeasurableSpace E] [MeasurableSingletonClass E] {bits : ℕ} |
|
|
| /-- Conjugate an actual online rule by a fixed label enumeration. In particular, |
| the translated rule does not receive a future order or a future value. -/ |
| noncomputable def relabelRule (A : OnlineRule (Fintype.card E) bits) : |
| LabeledOnlineRule E bits where |
| decide k r s h := A.decide k r (finWeights s) |
| (encodeHistory (Fintype.equivFin E) k h) |
| measurable_decide k := by |
| have hm : Measurable (fun x : Seed bits × |
| ((E → ℝ) × (Fin (k.val + 1) → E × ℝ)) => |
| (x.1, (finWeights x.2.1, encodeHistory (Fintype.equivFin E) k x.2.2))) := by |
| apply Measurable.prodMk measurable_fst |
| apply Measurable.prodMk |
| · apply Measurable.of_eval |
| intro i |
| exact (measurable_pi_apply ((Fintype.equivFin E).symm i)).comp |
| (measurable_fst.comp measurable_snd) |
| · apply Measurable.of_eval |
| intro j |
| have hj : Measurable (fun x : Seed bits × |
| ((E → ℝ) × (Fin (k.val + 1) → E × ℝ)) => x.2.2 j) := |
| (measurable_pi_apply j).comp (measurable_snd.comp measurable_snd) |
| exact ((measurable_of_countable (Fintype.equivFin E)).comp |
| (measurable_fst.comp hj)).prodMk (measurable_snd.comp hj) |
| exact (A.measurable_decide k).comp hm |
|
|
| omit [MeasurableSpace E] [MeasurableSingletonClass E] in |
| theorem encode_labeledHistory (v : E → ℝ) (π : LabeledOrder E) |
| (k : Fin (Fintype.card E)) : |
| encodeHistory (Fintype.equivFin E) k (labeledHistory v π k) = |
| history (finWeights v) (π.trans (Fintype.equivFin E)) k := by |
| funext j |
| simp [encodeHistory, labeledHistory, history, finWeights] |
|
|
| theorem relabelRule_acceptedThrough (A : OnlineRule (Fintype.card E) bits) |
| (r : Seed bits) (s v : E → ℝ) (π : LabeledOrder E) (t : ℕ) : |
| (labeledAcceptedThrough (relabelRule A) r s v π t).map |
| (Fintype.equivFin E).toEmbedding = |
| acceptedThrough A r (finWeights s) (finWeights v) |
| (π.trans (Fintype.equivFin E)) t := by |
| classical |
| ext index |
| rw [Finset.mem_map_equiv] |
| unfold labeledAcceptedThrough acceptedThrough |
| rw [Finset.mem_filter, Finset.mem_filter] |
| simp only [Finset.mem_univ, true_and, decisionAt, relabelRule, |
| encode_labeledHistory] |
| rfl |
|
|
| theorem relabelRule_feasible (M : Matroid E) |
| (A : OnlineRule (Fintype.card E) bits) (hA : Feasible (finMatroid M) A) |
| (r : Seed bits) (s v : E → ℝ) (π : LabeledOrder E) (t : ℕ) |
| (hs : ∀ e, 0 ≤ s e) (hv : ∀ e, 0 ≤ v e) : |
| M.Indep (labeledAcceptedThrough (relabelRule A) r s v π t : Set E) := by |
| have hi := hA r (finWeights s) (finWeights v) |
| (π.trans (Fintype.equivFin E)) t |
| ((finWeights_nonnegative_iff s).2 hs) ((finWeights_nonnegative_iff v).2 hv) |
| rw [← relabelRule_acceptedThrough A r s v π t] at hi |
| exact (mapEquiv_finset_indep M (Fintype.equivFin E) _).1 hi |
|
|
| |
|
|
| theorem relabelRule_reward (A : OnlineRule (Fintype.card E) bits) |
| (r : Seed bits) (s v : E → ℝ) (π : LabeledOrder E) : |
| (∑ e ∈ labeledAcceptedThrough (relabelRule A) r s v π (Fintype.card E), v e) = |
| reward A r (finWeights s) (finWeights v) (π.trans (Fintype.equivFin E)) := by |
| classical |
| rw [reward, accepted, ← relabelRule_acceptedThrough] |
| simp [finWeights] |
|
|
| /-- Any pointwise guarantee expressed in the actual `Fin n` model is unchanged |
| by transporting both the selected reward and the offline benchmark. -/ |
| theorem relabelRule_ratio_iff (M : Matroid E) |
| (A : OnlineRule (Fintype.card E) bits) (c : ℝ) |
| (r : Seed bits) (s v : E → ℝ) (π : LabeledOrder E) : |
| (c * finiteOptimum M v ≤ |
| ∑ e ∈ labeledAcceptedThrough (relabelRule A) r s v π (Fintype.card E), v e) ↔ |
| c * MatroidProphet.optimum (finMatroid M) (finWeights v) ≤ |
| reward A r (finWeights s) (finWeights v) (π.trans (Fintype.equivFin E)) := by |
| rw [relabelRule_reward, optimum_finMatroid] |
|
|
| end Online |
|
|
| end MatroidProphet.Labeling |
|
|
| end OAI |
|
|
