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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
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