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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 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 | import OAI.Probability.Localization.TruncatedBounds
import OAI.Probability.Localization.EntropyChain
namespace OAI
open MeasureTheory ProbabilityTheory Set
open scoped BigOperators ENNReal
namespace RepeatedLocalization
namespace FiniteEntropy
noncomputable def entropySlope (n : ℕ) : ℝ := (n:ℝ)*Real.log 2/2
noncomputable def historyKL {n : ℕ} (P : Measure (Coordinate n)) (Q : Cell n) (h : Log n) : ℝ≥0∞ :=
InformationTheory.klDiv (ProbabilityTheory.cond P (state P Q h).event) (uniformCell (state P Q h).cell)
noncomputable def historyCost {n : ℕ} (P : Measure (Coordinate n)) (Q : Cell n) (h : Log n) : ℝ :=
(historyKL P Q h).toReal + entropySlope n * ((state P Q h).cell.1-Q.1 : ℕ)
lemma entropySlope_pos {n : ℕ} (hn : 0 < n) : 0 < entropySlope n := by
unfold entropySlope
exact div_pos (mul_pos (Nat.cast_pos.mpr hn) (Real.log_pos (by norm_num))) (by norm_num)
lemma historyCost_nonneg {n : ℕ} (P : Measure (Coordinate n)) (Q : Cell n) (h : Log n) :
0 ≤ historyCost P Q h := by
unfold historyCost entropySlope
exact add_nonneg ENNReal.toReal_nonneg (mul_nonneg (div_nonneg (mul_nonneg (Nat.cast_nonneg _) (Real.log_nonneg (by norm_num))) (by norm_num)) (Nat.cast_nonneg _))
lemma historyKL_ne_top_budget {n : ℕ} (hn : 0 < n) (P : Measure (Coordinate n)) [IsProbabilityMeasure P]
(Q : Cell n) (hac : P ≪ uniformCell Q) (hK : InformationTheory.klDiv P (uniformCell Q) ≠ ∞)
(h : Log n) (hp : P (state P Q h).event ≠ 0) :
historyKL P Q h ≠ ∞ ∧
(P (state P Q h).event).toReal * historyCost P Q h ≤
(∫ z in (state P Q h).event,llr P (uniformCell Q) z ∂P) +
(P (state P Q h).event).toReal * rewardLog P Q (complementReward P) h := by
let := uniformCell_probability hn Q
have hd := positive_event_descendant P Q h hp
have hk := klDiv_cond_cond P (uniformCell Q) hac hK (state_event_measurable P Q h)
(measurableSet_cellSet _) (state_event_subset_cell P Q h) hp (uniformCell_descendant_pos hn hd)
rw [uniformCell_cond hd,uniformCell_log_mass hn hd] at hk
refine ⟨hk.1,?_⟩
have hroot : P (cellSet Q) = 1 := (mem_ae_iff_prob_eq_one (measurableSet_cellSet Q)).mp (ae_mem_cell_of_ac hac)
have hs := mul_le_mul_of_nonneg_left (path_label_surprisal P Q hroot h hp) (ENNReal.toReal_nonneg (a := P (state P Q h).event))
have he := hk.2
change (P (state P Q h).event).toReal * ((historyKL P Q h).toReal + entropySlope n * _) ≤ _
change (P (state P Q h).event).toReal * (historyKL P Q h).toReal = _ at he
unfold entropySlope
nlinarith
lemma truncated_cost_bound {n : ℕ} (hn : 0 < n) (P : Measure (Coordinate n)) [IsProbabilityMeasure P]
(Q : Cell n) (hac : P ≪ uniformCell Q) (hK : InformationTheory.klDiv P (uniformCell Q) ≠ ∞) (N : ℕ) :
Integrable (fun z => historyCost P Q (run P Q N z)) P ∧
(∫ z,historyCost P Q (run P Q N z) ∂P) ≤
(InformationTheory.klDiv P (uniformCell Q)).toReal + ∫ z,complementTotal P Q N z ∂P := by
let := uniformCell_probability hn Q
have hroot := ae_mem_cell_of_ac hac
have hi := (InformationTheory.klDiv_ne_top_iff.mp hK).2
have hJ := (truncated_reward_bounds hn P Q hroot N).1
have hb (h : Log n) : (P {z | run P Q N z = h}).toReal * historyCost P Q h ≤
∫ z in {z | run P Q N z = h},llr P (uniformCell Q) z + complementTotal P Q N z ∂P := by
by_cases hp : P {z | run P Q N z = h} = 0
· rw [hp,ENNReal.toReal_zero,zero_mul,Measure.restrict_eq_zero.mpr hp,integral_zero_measure]
obtain ⟨z,hz,hzh⟩ := exists_run_fiber_root P Q hroot N h hp
have he : {w | run P Q N w = h} =ᵐ[P] (state P Q h).event := by
simpa only [hzh] using run_fiber_ae_event hn P Q hroot N hz
have hp' : P (state P Q h).event ≠ 0 := by rwa [← measure_congr he]
have hb := (historyKL_ne_top_budget hn P Q hac hK h hp').2
rw [integral_add hi.integrableOn hJ.integrableOn]
change _ ≤ (∫ z in {z | run P Q N z = h},llr P (uniformCell Q) z ∂P) +
∫ z in {z | run P Q N z = h},(fun h => rewardLog P Q (complementReward P) h) (run P Q N z) ∂P
rw [integral_comp_fiber P (run P Q N) (rewardLog P Q (complementReward P)) h
((measurable_run hn P Q N) (measurableSet_singleton h)),Measure.restrict_congr_set he,measure_congr he]
exact hb
have hb' := countable_cost_bound P (run P Q N) (measurable_run hn P Q N)
(historyCost P Q) (historyCost_nonneg P Q) (hi.add hJ) hb
simp only [Pi.add_apply] at hb'
rw [integral_add hi hJ,← InformationTheory.toReal_klDiv_of_measure_eq hac (by simp)] at hb'
exact hb'
lemma truncated_depth_integrable {n : ℕ} (hn : 0 < n) (P : Measure (Coordinate n)) [IsProbabilityMeasure P]
(Q : Cell n) (hac : P ≪ uniformCell Q) (hK : InformationTheory.klDiv P (uniformCell Q) ≠ ∞) (N : ℕ) :
Integrable (fun z => (truncatedDepth P Q N z : ℝ)) P := by
have hc := (truncated_cost_bound hn P Q hac hK N).1
have hm : Measurable (fun z => (truncatedDepth P Q N z : ℝ)) :=
(measurable_of_countable (fun h : Log n => (((state P Q h).cell.1-Q.1:ℕ):ℝ))).comp (measurable_run hn P Q N)
apply Integrable.mono' (hc.div_const (entropySlope n)) hm.aestronglyMeasurable
filter_upwards [] with z
rw [Real.norm_eq_abs,abs_of_nonneg (Nat.cast_nonneg _),le_div_iff₀ (entropySlope_pos hn)]
change _ ≤ (historyKL P Q (run P Q N z)).toReal + entropySlope n * (truncatedDepth P Q N z:ℝ)
nlinarith [ENNReal.toReal_nonneg (a := historyKL P Q (run P Q N z))]
lemma truncated_KL_integrable {n : ℕ} (hn : 0 < n) (P : Measure (Coordinate n)) [IsProbabilityMeasure P]
(Q : Cell n) (hac : P ≪ uniformCell Q) (hK : InformationTheory.klDiv P (uniformCell Q) ≠ ∞) (N : ℕ) :
Integrable (fun z => (historyKL P Q (run P Q N z)).toReal) P := by
have hc := (truncated_cost_bound hn P Q hac hK N).1
have hd := truncated_depth_integrable hn P Q hac hK N
have he : (fun z => (historyKL P Q (run P Q N z)).toReal) =
(fun z => historyCost P Q (run P Q N z)) - (fun z => entropySlope n*(truncatedDepth P Q N z:ℝ)) := by
funext z
simp [historyCost,truncatedDepth]
rw [he]
exact hc.sub (hd.const_mul _)
lemma truncated_attempts_integral {n : ℕ} (hn : 0 < n) (P : Measure (Coordinate n)) [IsProbabilityMeasure P]
(Q : Cell n) (hac : P ≪ uniformCell Q) (hK : InformationTheory.klDiv P (uniformCell Q) ≠ ∞) (N : ℕ) :
(∫ z,(truncatedAttempts P Q N z:ℝ) ∂P) ≤ 2*(1+∫ z,(truncatedDepth P Q N z:ℝ) ∂P) := by
have hroot := ae_mem_cell_of_ac hac
have hC := (truncated_reward_bounds hn P Q hroot N).2
have hD := truncated_depth_integrable hn P Q hac hK N
have hp : ∀ᵐ z ∂P,(truncatedAttempts P Q N z:ℝ) ≤
(truncatedDepth P Q N z:ℝ) + continuingTotal P Q N z + 1 := by
filter_upwards [hroot] with z hz
have hb := truncated_attempts_path hn P Q N hz
have hs : (if (state P Q (run P Q N z)).stopped then (1:ℝ) else 0) ≤ 1 := by split <;> norm_num
linarith
have hi := integral_mono_ae (integrable_truncatedAttempts hn P Q N)
((hD.add hC.1).add (integrable_const 1)) hp
have had := integral_add (hD.add hC.1) (integrable_const (1:ℝ))
simp only [Pi.add_apply] at hi had
rw [had,integral_add hD hC.1] at hi
simp only [integral_const,Measure.real,measure_univ,ENNReal.toReal_one,one_smul] at hi
linarith [hC.2.2]
lemma slope_absorption {n : ℕ} (hn : 32 ≤ n) : 2/Real.exp 1 ≤ entropySlope n/2 := by
have hn' : (32:ℝ) ≤ n := by exact_mod_cast hn
have hl : (1/2:ℝ) ≤ Real.log 2 := by linarith [Real.log_two_gt_d9]
have he : (2:ℝ)/Real.exp 1 ≤ 2 := (div_le_self (by norm_num) (Real.one_le_exp (by norm_num)))
unfold entropySlope
nlinarith
lemma truncated_final_budget {n : ℕ} (hn : 32 ≤ n) (P : Measure (Coordinate n)) [IsProbabilityMeasure P]
(Q : Cell n) (hac : P ≪ uniformCell Q) (hK : InformationTheory.klDiv P (uniformCell Q) ≠ ∞) (N : ℕ) :
(∫ z,(historyKL P Q (run P Q N z)).toReal + entropySlope n/2*(truncatedDepth P Q N z:ℝ) ∂P) ≤
(InformationTheory.klDiv P (uniformCell Q)).toReal + 2/Real.exp 1 := by
have hn0 : 0 < n := by omega
have hcost := (truncated_cost_bound hn0 P Q hac hK N).2
have hJ := (truncated_reward_bounds hn0 P Q (ae_mem_cell_of_ac hac) N).2.2.1
have hA := truncated_attempts_integral hn0 P Q hac hK N
have hD := truncated_depth_integrable hn0 P Q hac hK N
have hKI := truncated_KL_integrable hn0 P Q hac hK N
change (∫ z,(historyKL P Q (run P Q N z)).toReal + entropySlope n*(truncatedDepth P Q N z:ℝ) ∂P) ≤ _ at hcost
rw [integral_add hKI (hD.const_mul (entropySlope n)),integral_const_mul] at hcost
rw [integral_add hKI (hD.const_mul (entropySlope n / 2)),integral_const_mul]
have hD0 : 0 ≤ ∫ z,(truncatedDepth P Q N z:ℝ) ∂P := integral_nonneg (fun _ => Nat.cast_nonneg _)
have habs := mul_le_mul_of_nonneg_right (slope_absorption hn) hD0
have hJA := mul_le_mul_of_nonneg_left hA (by positivity : 0 ≤ (1:ℝ)/Real.exp 1)
simp only [div_eq_mul_inv,one_mul] at hJ hJA habs ⊢
nlinarith only [hcost,hJ,habs,hJA]
end FiniteEntropy
end RepeatedLocalization
end OAI
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