import OAI.Combinatorics.EditApproximation.Geometry.ActionVectors import OAI.Combinatorics.EditApproximation.Definitions namespace OAI /-! # Regret of the computed rational band mixtures The optimistic objective is computed in rational arithmetic. The finite optimizer supplies every predicted vector used here; no maximization oracle or unproved entropy-regret hypothesis appears in the conclusion. -/ namespace EditApproximation open Finset theorem rationalOptimisticHistory_cast {ι : Type*} (κ : ℚ) (gain decrement : ℕ → ι → ℚ) (t : ℕ) (i : ι) : (rationalOptimisticHistory κ gain decrement t i : ℝ) = optimisticHistory (κ : ℝ) (fun j i => (gain j i : ℝ)) (fun j i => (decrement j i : ℝ)) t i := by simp [rationalOptimisticHistory, optimisticHistory] def computedOnlineVector {ι : Type*} (coordinates : Finset ι) (vertices : List (ι → ℚ)) (η κ : ℚ) (gain decrement : ℕ → ι → ℚ) (H t : ℕ) : ι → ℚ := finiteRationalOptimizer coordinates vertices (rationalOnlineLinear η κ gain decrement t) (H + coordinates.card + 2) ((H + coordinates.card + 2) ^ 80) theorem rationalOnlineLinear_objective {ι : Type*} (coordinates : Finset ι) (η κ : ℚ) (gain decrement : ℕ → ι → ℚ) (t : ℕ) (p : ι → ℝ) : entropyValue coordinates 1 (fun i => (rationalOnlineLinear η κ gain decrement t i : ℝ)) p = entropyValue coordinates (η : ℝ) (fun i => optimisticHistory (κ : ℝ) (fun j i => (gain j i : ℝ)) (fun j i => (decrement j i : ℝ)) t i + (gain t i : ℝ)) p := by unfold entropyValue apply sum_congr rfl intro i _ simp only [rationalOnlineLinear, Rat.cast_mul, Rat.cast_add, rationalOptimisticHistory_cast, one_mul] /-- Fixed-block regret for the actual rational optimizer, with its prescribed `G^80` iterations and `H^-3` numerical loss. -/ theorem computed_finite_block_entropy_regret {ι β : Type*} [Fintype β] [DecidableEq β] (coordinates : Finset ι) (block : ι → β) (vertices : List (ι → ℚ)) (S H : ℕ) (η κ : ℚ) (rangeBound : ℝ) (gain decrement : ℕ → ι → ℚ) (comparator : ι → ℝ) (hvertices : vertices ≠ []) (hH : 4 ≤ H) (hprobability : ∀ p ∈ vertices, (∀ i ∈ coordinates, (0 : ℝ) ≤ p i) ∧ (∑ i ∈ coordinates, (p i : ℝ)) = 1) (hcovered : ∀ i ∈ coordinates, ∃ p ∈ vertices, (0 : ℝ) < p i) (hcQ : comparator ∈ convexHull ℝ (rationalVertexSet vertices)) (hblocks : ∀ b, (coordinates.filter (fun i => block i = b)).Nonempty) (hblockMass : ∀ p ∈ convexHull ℝ (rationalVertexSet vertices), ∀ q ∈ convexHull ℝ (rationalVertexSet vertices), ∀ b, (∑ i ∈ coordinates with block i = b, p i) = ∑ i ∈ coordinates with block i = b, q i) (hη : 0 < η) (hκ : 0 ≤ κ) (hκhalf : κ ≤ 1 / 2) (hηrange : (η : ℝ) * rangeBound ≤ (κ : ℝ)) (hdecrementNonneg : ∀ t < S, ∀ i ∈ coordinates, (0 : ℝ) ≤ decrement t i) (hrange : ∀ t < S, ∀ i ∈ coordinates, ∀ j ∈ coordinates, block i = block j → |(decrement t i : ℝ) - decrement t j| ≤ rangeBound) (hdecrement : ∀ i ∈ coordinates, ∑ t ∈ range S, (decrement t i : ℝ) ≤ 1) (hgain : ∀ t < S, ∀ i ∈ coordinates, -1 ≤ (gain t i : ℝ) + (1 - (κ : ℝ)) * decrement t i ∧ (gain t i : ℝ) + (1 - (κ : ℝ)) * decrement t i ≤ 0) : (∑ t ∈ range S, ∑ i ∈ coordinates, ((gain t i : ℝ) + decrement t i) * comparator i) - Real.log coordinates.card / (η : ℝ) - 2 * (κ : ℝ) - (S : ℝ) / (H : ℝ) ^ 3 ≤ ∑ t ∈ range S, ∑ i ∈ coordinates, ((gain t i : ℝ) + (1 - (κ : ℝ)) * decrement t i) * (computedOnlineVector coordinates vertices η κ gain decrement H t i : ℝ) := by classical have hfinite : ({p : ι → ℚ | p ∈ vertices} : Set (ι → ℚ)).Finite := by simp have hcompact := (hfinite.image (fun p : ι → ℚ => fun i => (p i : ℝ))).isCompact_convexHull ℝ obtain ⟨witness, hwQ, hw⟩ := rationalVertexHull_positive_witness coordinates vertices hvertices (fun p hp => (hprobability p hp).1) hcovered have hoptimizer (t : ℕ) := finite_entropy_optimizer coordinates vertices (rationalOnlineLinear η κ gain decrement t) H hvertices hH hprobability hcovered choose predicted hpred using hoptimizer have hpredMax t : IsMaxOn (entropyValue coordinates (η : ℝ) (fun i => optimisticHistory (κ : ℝ) (fun j i => (gain j i : ℝ)) (fun j i => (decrement j i : ℝ)) t i + (gain t i : ℝ))) (convexHull ℝ (rationalVertexSet vertices)) (predicted t) := by intro q hq have hbound := (hpred t).2.1 hq change entropyValue coordinates 1 (fun i => (rationalOnlineLinear η κ gain decrement t i : ℝ)) q ≤ entropyValue coordinates 1 (fun i => (rationalOnlineLinear η κ gain decrement t i : ℝ)) (predicted t) at hbound rw [rationalOnlineLinear_objective, rationalOnlineLinear_objective] at hbound exact hbound have hκhalfR : (κ : ℝ) ≤ 1 / 2 := by calc (κ : ℝ) ≤ ((1 / 2 : ℚ) : ℝ) := Rat.cast_le.mpr hκhalf _ = 1 / 2 := by norm_num have hregret := finite_block_entropy_regret coordinates block (convexHull ℝ (rationalVertexSet vertices)) S (η : ℝ) (κ : ℝ) rangeBound (fun j i => (gain j i : ℝ)) (fun j i => (decrement j i : ℝ)) predicted witness comparator hcompact (convex_convexHull ℝ _) hwQ hcQ hw (rationalVertexHull_probability coordinates vertices hprobability) hblocks hblockMass (by exact_mod_cast hη) (by exact_mod_cast hκ) hκhalfR hηrange (fun t _ => (hpred t).1) (fun t _ => hpredMax t) hdecrementNonneg hrange hdecrement have hloss := sum_le_sum fun t (ht : t ∈ range S) => bounded_gain_l1_approximation coordinates (fun i => (gain t i : ℝ) + (1 - (κ : ℝ)) * decrement t i) (predicted t) (fun i => (computedOnlineVector coordinates vertices η κ gain decrement H t i : ℝ)) (1 / (H : ℝ) ^ 3) (hgain t (mem_range.mp ht)) (hpred t).2.2 simp only [sum_sub_distrib, sum_const, card_range, nsmul_eq_mul] at hloss rw [mul_one_div] at hloss linarith /-- Specialization to the actual child/state coordinates and rational fixed child masses of a nonempty band. -/ theorem band_computed_entropy_regret {β σ : Type*} [Fintype β] [Nonempty β] [DecidableEq β] [DecidableEq σ] (band : Finset (β → σ)) (hband : band.Nonempty) (mass : β → ℚ) (hpos : ∀ child, 0 < mass child) (S H : ℕ) (η κ : ℚ) (rangeBound : ℝ) (gain decrement : ℕ → β × σ → ℚ) {comparison : β → σ} (hc : comparison ∈ band) (hH : 4 ≤ H) (hη : 0 < η) (hκ : 0 ≤ κ) (hκhalf : κ ≤ 1 / 2) (hηrange : (η : ℝ) * rangeBound ≤ (κ : ℝ)) (hdecrementNonneg : ∀ t < S, ∀ i ∈ bandCoordinates band, (0 : ℝ) ≤ decrement t i) (hrange : ∀ t < S, ∀ i ∈ bandCoordinates band, ∀ j ∈ bandCoordinates band, i.1 = j.1 → |(decrement t i : ℝ) - decrement t j| ≤ rangeBound) (hdecrement : ∀ i ∈ bandCoordinates band, ∑ t ∈ range S, (decrement t i : ℝ) ≤ 1) (hgain : ∀ t < S, ∀ i ∈ bandCoordinates band, -1 ≤ (gain t i : ℝ) + (1 - (κ : ℝ)) * decrement t i ∧ (gain t i : ℝ) + (1 - (κ : ℝ)) * decrement t i ≤ 0) : (∑ t ∈ range S, ∑ i ∈ bandCoordinates band, ((gain t i : ℝ) + decrement t i) * (actionVector mass comparison i : ℝ)) - Real.log (bandCoordinates band).card / (η : ℝ) - 2 * (κ : ℝ) - (S : ℝ) / (H : ℝ) ^ 3 ≤ ∑ t ∈ range S, ∑ i ∈ bandCoordinates band, ((gain t i : ℝ) + (1 - (κ : ℝ)) * decrement t i) * (computedOnlineVector (bandCoordinates band) (band.toList.map (actionVector mass)) η κ gain decrement H t i : ℝ) := by have hvertices : band.toList.map (actionVector mass) ≠ [] := by intro hnil have hmem : actionVector mass comparison ∈ band.toList.map (actionVector mass) := List.mem_map.mpr ⟨comparison, mem_toList.mpr hc, rfl⟩ simp only [hnil, List.not_mem_nil] at hmem apply computed_finite_block_entropy_regret (bandCoordinates band) Prod.fst (band.toList.map (actionVector mass)) S H η κ rangeBound gain decrement (fun i => (actionVector mass comparison i : ℝ)) hvertices hH · intro p hp obtain ⟨action, ha, rfl⟩ := List.mem_map.mp hp exact actionVector_probability band mass hpos (mem_toList.mp ha) · intro i hi obtain ⟨action, ha, hpositive⟩ := actionVector_coordinate_coverage band mass hpos i hi exact ⟨actionVector mass action, List.mem_map.mpr ⟨action, mem_toList.mpr ha, rfl⟩, hpositive⟩ · exact subset_convexHull ℝ _ ⟨actionVector mass comparison, List.mem_map.mpr ⟨comparison, mem_toList.mpr hc, rfl⟩, rfl⟩ · exact band_block_nonempty band hband · intro p hp q hq b rw [band_hull_block_mass band mass b p hp, band_hull_block_mass band mass b q hq] · exact hη · exact hκ · exact hκhalf · exact hηrange · exact hdecrementNonneg · exact hrange · exact hdecrement · exact hgain end EditApproximation end OAI