| import OAI.Probability.MatroidProphet.Information |
| import OAI.Probability.MatroidProphet.FiniteBits |
| import OAI.Probability.MatroidProphet.Pivots.GreedyScan |
|
|
| namespace OAI |
|
|
| open Finset Set |
|
|
| namespace MatroidProphet |
| namespace Candidates |
|
|
| variable {n : ℕ} |
|
|
| def prior (time : Fin n → ℕ) (H : Finset (Fin n)) (e : Fin n) : Set (Fin n) := |
| {f | f ∈ H ∧ time f < time e} |
|
|
| noncomputable def greedy (M : Matroid (Fin n)) (time : Fin n → ℕ) |
| (H : Finset (Fin n)) : Finset (Fin n) := by |
| classical |
| exact H.filter fun e => e ∉ M.closure (prior time H e) |
|
|
| noncomputable def candidates (M : Matroid (Fin n)) (time : Fin n → ℕ) |
| (V H : Finset (Fin n)) : Finset (Fin n) := by |
| classical |
| exact (V \ H).filter fun e => e ∉ M.closure (prior time H e) |
|
|
| lemma prior_insert_self (time : Fin n → ℕ) (H : Finset (Fin n)) (e : Fin n) : |
| prior time (insert e H) e = prior time H e := by |
| ext f |
| simp only [prior, Set.mem_ofPred_eq, Finset.mem_insert] |
| constructor |
| · rintro ⟨rfl | hf, hlt⟩ |
| · exact (lt_irrefl _ hlt).elim |
| · exact ⟨hf, hlt⟩ |
| · rintro ⟨hf, hlt⟩ |
| exact ⟨Or.inr hf, hlt⟩ |
|
|
| lemma prior_mono (time : Fin n → ℕ) {H V : Finset (Fin n)} (h : H ⊆ V) |
| (e : Fin n) : prior time H e ⊆ prior time V e := |
| fun _ hf => ⟨h hf.1, hf.2⟩ |
|
|
| lemma greedy_indep (M : Matroid (Fin n)) (time : Fin n → ℕ) |
| (ht : Function.Injective time) (H : Finset (Fin n)) |
| (hground : ∀ e ∈ H, e ∈ M.E) : M.Indep (greedy M time H : Set (Fin n)) := by |
| classical |
| let label : H → Fin n := Subtype.val |
| let arrival : H → ℕ := fun e => time e.val |
| have htime : Function.Injective arrival := fun _ _ h => Subtype.ext (ht h) |
| have hp (e : H) : label '' {p : H | arrival p < arrival e} = prior time H e.val := by |
| ext f |
| simp [label, arrival, prior, and_comm] |
| have hi := Pivots.keptOccurrences_indep M label arrival (fun e => hground e.val e.property) htime |
| have heq : label '' (Pivots.keptOccurrences M label arrival : Set H) = |
| (greedy M time H : Set (Fin n)) := by |
| ext e |
| simp only [Set.mem_image, Finset.mem_coe, Pivots.keptOccurrences, Finset.mem_filter, |
| Finset.mem_univ, true_and, greedy] |
| constructor |
| · rintro ⟨f, hf, rfl⟩ |
| exact ⟨f.property, by simpa only [hp] using hf⟩ |
| · rintro ⟨he, hp'⟩ |
| refine ⟨⟨e, he⟩, ?_, rfl⟩ |
| simpa only [hp] using hp' |
| rwa [heq] at hi |
|
|
| lemma greedy_sdiff_subset_candidates (M : Matroid (Fin n)) (time : Fin n → ℕ) |
| (V H : Finset (Fin n)) (hHV : H ⊆ V) : |
| greedy M time V \ H ⊆ candidates M time V H := by |
| classical |
| intro e he |
| rcases Finset.mem_sdiff.mp he with ⟨heG, heH⟩ |
| rcases Finset.mem_filter.mp heG with ⟨heV, hecl⟩ |
| apply Finset.mem_filter.mpr |
| refine ⟨Finset.mem_sdiff.mpr ⟨heV, heH⟩, ?_⟩ |
| exact fun h => hecl (M.closure_subset_closure (prior_mono time hHV e) h) |
|
|
| lemma expectation_sum (q : Fin n → ℝ) (V I : Finset (Fin n)) |
| (f : Fin n → Finset (Fin n) → ℝ) : |
| bitsExpectation q V (fun H => ∑ e ∈ I, f e H) = |
| ∑ e ∈ I, bitsExpectation q V (f e) := by |
| simp only [bitsExpectation, Finset.mul_sum] |
| rw [Finset.sum_comm] |
|
|
| lemma expectation_not_mem (q : Fin n → ℝ) (V : Finset (Fin n)) |
| (e : Fin n) (he : e ∈ V) (c : ℝ) : |
| bitsExpectation q V (fun H => if e ∉ H then c else 0) = (1 - q e) * c := by |
| have hno : bitsExpectation q (V.erase e) (fun H => if e ∉ H then c else 0) = c := by |
| calc |
| _ = bitsExpectation q (V.erase e) (fun _ => c) := by |
| apply bitsExpectation_congr |
| intro H hH |
| have heH : e ∉ H := fun hh => (Finset.mem_erase.mp (hH hh)).1 rfl |
| simp [heH] |
| _ = c := bitsExpectation_const q _ c |
| have hyes : bitsExpectation q (V.erase e) (fun H => if e ∉ insert e H then c else 0) = 0 := by |
| simp only [Finset.mem_insert_self, not_true_eq_false, ite_false] |
| exact bitsExpectation_const q _ 0 |
| calc |
| _ = bitsExpectation q (insert e (V.erase e)) (fun H => if e ∉ H then c else 0) := by |
| rw [Finset.insert_erase he] |
| _ = (1 - q e) * c := by |
| rw [bitsExpectation_insert q _ e (by simp), hno, hyes] |
| ring |
|
|
| lemma expectation_sdiff_fair (V I : Finset (Fin n)) (hIV : I ⊆ V) (w : Weights n) : |
| bitsExpectation (fun _ => (1 / 2 : ℝ)) V (fun H => ∑ e ∈ I \ H, w e) = |
| (∑ e ∈ I, w e) / 2 := by |
| classical |
| have hsum (H : Finset (Fin n)) : |
| (∑ e ∈ I \ H, w e) = ∑ e ∈ I, if e ∉ H then w e else 0 := by |
| rw [Finset.sdiff_eq_filter, Finset.sum_filter] |
| simp_rw [hsum] |
| rw [expectation_sum] |
| calc |
| (∑ e ∈ I, bitsExpectation (fun _ => (1 / 2 : ℝ)) V |
| (fun H => if e ∉ H then w e else 0)) = |
| ∑ e ∈ I, (1 - (1 / 2 : ℝ)) * w e := by |
| apply Finset.sum_congr rfl |
| intro e he |
| exact expectation_not_mem _ V e (hIV he) (w e) |
| _ = (∑ e ∈ I, w e) / 2 := by |
| rw [← Finset.mul_sum] |
| ring |
|
|
| lemma greedy_sdiff_indep (M : Matroid (Fin n)) (time : Fin n → ℕ) |
| (ht : Function.Injective time) (V H : Finset (Fin n)) |
| (hground : ∀ e ∈ V, e ∈ M.E) : |
| M.Indep (greedy M time V \ H : Set (Fin n)) := by |
| apply (greedy_indep M time ht V hground).subset |
| intro e he |
| exact he.1 |
|
|
| lemma expectation_greedy_sdiff_fair (M : Matroid (Fin n)) (time : Fin n → ℕ) |
| (V : Finset (Fin n)) (w : Weights n) : |
| bitsExpectation (fun _ => (1 / 2 : ℝ)) V (fun H => ∑ e ∈ greedy M time V \ H, w e) = |
| (∑ e ∈ greedy M time V, w e) / 2 := by |
| classical |
| apply expectation_sdiff_fair |
| exact Finset.filter_subset _ _ |
|
|
| lemma expectation_fair_balance (V : Finset (Fin n)) (e : Fin n) (he : e ∈ V) |
| (p : Finset (Fin n) → Prop) [DecidablePred p] (hp : ∀ H, p (insert e H) ↔ p H) (c : ℝ) : |
| bitsExpectation (fun _ => (1 / 2 : ℝ)) V |
| (fun H => if e ∈ H then 0 else if p H then c else 0) = |
| bitsExpectation (fun _ => (1 / 2 : ℝ)) V |
| (fun H => if e ∈ H then (if p H then c else 0) else 0) := by |
| classical |
| let q : Fin n → ℝ := fun _ => 1 / 2 |
| let f : Finset (Fin n) → ℝ := fun H => if p H then c else 0 |
| have heH : ∀ H ⊆ V.erase e, e ∉ H := by |
| intro H hH he' |
| exact (Finset.mem_erase.mp (hH he')).1 rfl |
| have hn0 : bitsExpectation q (V.erase e) |
| (fun H => if e ∈ H then 0 else if p H then c else 0) = |
| bitsExpectation q (V.erase e) f := by |
| apply bitsExpectation_congr |
| intro H hH |
| simp [heH H hH, f] |
| have hn1 : bitsExpectation q (V.erase e) |
| (fun H => if e ∈ insert e H then 0 else if p (insert e H) then c else 0) = 0 := by |
| simp only [Finset.mem_insert_self, ite_true] |
| exact bitsExpectation_const q _ 0 |
| have hy0 : bitsExpectation q (V.erase e) |
| (fun H => if e ∈ H then (if p H then c else 0) else 0) = 0 := by |
| calc |
| _ = bitsExpectation q (V.erase e) (fun _ => 0) := by |
| apply bitsExpectation_congr |
| intro H hH |
| simp [heH H hH] |
| _ = 0 := bitsExpectation_const q _ 0 |
| have hy1 : bitsExpectation q (V.erase e) |
| (fun H => if e ∈ insert e H then (if p (insert e H) then c else 0) else 0) = |
| bitsExpectation q (V.erase e) f := by |
| apply bitsExpectation_congr |
| intro H _ |
| simp [hp H, f] |
| change bitsExpectation q V _ = bitsExpectation q V _ |
| conv_lhs => rw [← Finset.insert_erase he] |
| conv_rhs => rw [← Finset.insert_erase he] |
| rw [bitsExpectation_insert q _ e (by simp), |
| bitsExpectation_insert q _ e (by simp), hn0, hn1, hy0, hy1] |
| dsimp [q] |
| ring |
|
|
| lemma expectation_candidates_eq_greedy (M : Matroid (Fin n)) (time : Fin n → ℕ) |
| (V : Finset (Fin n)) (w : Weights n) : |
| bitsExpectation (fun _ => (1 / 2 : ℝ)) V |
| (fun H => ∑ e ∈ candidates M time V H, w e) = |
| bitsExpectation (fun _ => (1 / 2 : ℝ)) V |
| (fun H => ∑ e ∈ greedy M time H, w e) := by |
| classical |
| let p : Fin n → Finset (Fin n) → Prop := fun e H => e ∉ M.closure (prior time H e) |
| have hc (H : Finset (Fin n)) : |
| (∑ e ∈ candidates M time V H, w e) = |
| ∑ e ∈ V, if e ∈ H then 0 else if p e H then w e else 0 := by |
| simp [candidates, Finset.sdiff_eq_filter, Finset.sum_filter, p] |
| have hg (H : Finset (Fin n)) (hH : H ⊆ V) : |
| (∑ e ∈ greedy M time H, w e) = |
| ∑ e ∈ V, if e ∈ H then (if p e H then w e else 0) else 0 := by |
| have hset : greedy M time H = V.filter (fun e => e ∈ H ∧ p e H) := by |
| ext e |
| simp only [greedy, Finset.mem_filter, p] |
| constructor |
| · intro he |
| exact ⟨hH he.1, he⟩ |
| · intro he |
| exact he.2 |
| rw [hset, Finset.sum_filter] |
| apply Finset.sum_congr rfl |
| intro e _ |
| by_cases he : e ∈ H <;> simp [he] |
| calc |
| _ = bitsExpectation (fun _ => (1 / 2 : ℝ)) V |
| (fun H => ∑ e ∈ V, if e ∈ H then 0 else if p e H then w e else 0) := by |
| apply bitsExpectation_congr |
| intro H _ |
| exact hc H |
| _ = ∑ e ∈ V, bitsExpectation (fun _ => (1 / 2 : ℝ)) V |
| (fun H => if e ∈ H then 0 else if p e H then w e else 0) := expectation_sum _ _ _ _ |
| _ = ∑ e ∈ V, bitsExpectation (fun _ => (1 / 2 : ℝ)) V |
| (fun H => if e ∈ H then (if p e H then w e else 0) else 0) := by |
| apply Finset.sum_congr rfl |
| intro e he |
| apply expectation_fair_balance V e he (p e) _ (w e) |
| intro H |
| simp [p, prior_insert_self] |
| _ = bitsExpectation (fun _ => (1 / 2 : ℝ)) V |
| (fun H => ∑ e ∈ V, if e ∈ H then (if p e H then w e else 0) else 0) := |
| (expectation_sum _ _ _ _).symm |
| _ = _ := by |
| apply bitsExpectation_congr |
| intro H hH |
| exact (hg H hH).symm |
|
|
| theorem expectation_candidates_le_optimum (M : Matroid (Fin n)) (time : Fin n → ℕ) |
| (ht : Function.Injective time) (V : Finset (Fin n)) |
| (hground : ∀ e ∈ V, e ∈ M.E) (w : Weights n) : |
| bitsExpectation (fun _ => (1 / 2 : ℝ)) V |
| (fun H => ∑ e ∈ candidates M time V H, w e) ≤ optimum M w := by |
| rw [expectation_candidates_eq_greedy] |
| calc |
| _ ≤ bitsExpectation (fun _ => (1 / 2 : ℝ)) V (fun _ => optimum M w) := by |
| apply bitsExpectation_mono _ (by intro e; norm_num) (by intro e; norm_num) |
| intro H hH |
| exact sum_le_optimum M w _ (greedy_indep M time ht H (fun e he => hground e (hH he))) |
| _ = _ := bitsExpectation_const _ _ _ |
|
|
| end Candidates |
| end MatroidProphet |
|
|
| end OAI |
|
|
