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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 | import OAI.Probability.Localization.Termination
namespace OAI
open MeasureTheory ProbabilityTheory Set
open scoped BigOperators ENNReal
namespace RepeatedLocalization
namespace FiniteEntropy
noncomputable def terminalKL {n : ℕ} (P : Measure (Coordinate n)) (Q : Cell n) (z : Coordinate n) : ℝ≥0∞ :=
InformationTheory.klDiv (terminalLaw P Q (terminalLog P Q z))
(uniformCell (terminalCell P Q (terminalLog P Q z)))
lemma measurable_terminalKL {n : ℕ} (hn : 0 < n) (P : Measure (Coordinate n)) (Q : Cell n) :
Measurable (terminalKL P Q) :=
(measurable_of_countable (fun h : Log n => InformationTheory.klDiv (terminalLaw P Q h)
(uniformCell (terminalCell P Q h)))).comp (measurable_terminalLog hn P Q)
lemma measurable_depth_real {n : ℕ} (hn : 0 < n) (P : Measure (Coordinate n)) (Q : Cell n) :
Measurable (fun z => (depth P Q z:ℝ)) :=
(measurable_of_countable (fun h : Log n => (((terminalCell P Q h).1-Q.1:ℕ):ℝ))).comp
(measurable_terminalLog hn P Q)
lemma terminal_historyKL {n : ℕ} (hn : 0 < n) (P : Measure (Coordinate n)) (Q : Cell n)
(hroot : ∀ᵐ z ∂P,z ∈ cellSet Q) {z : Coordinate n} (hz : z ∈ cellSet Q) (ht : Terminates P Q z) :
terminalKL P Q z = historyKL P Q (terminalLog P Q z) := by
unfold terminalKL historyKL terminalLaw terminalCell
rw [cond_congr_ae P (terminal_fiber_ae_event hn P Q hroot hz ht)]
lemma terminal_cost_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) ≠ ∞) :
Integrable (fun z => (terminalKL P Q z).toReal + entropySlope n/2*(depth P Q z:ℝ)) P ∧
(∫ z,(terminalKL P Q z).toReal + entropySlope n/2*(depth P Q z:ℝ) ∂P) ≤
(InformationTheory.klDiv P (uniformCell Q)).toReal + 2/Real.exp 1 := by
have hn0 : 0 < n := by omega
have hs : 0 ≤ entropySlope n/2 := le_of_lt (div_pos (entropySlope_pos hn0) (by norm_num))
apply frozen_nonnegative_bound P
(fun N z => (historyKL P Q (run P Q N z)).toReal + entropySlope n/2*(truncatedDepth P Q N z:ℝ))
_ _ (budget_nonneg P Q)
· intro N
exact (truncated_KL_integrable hn0 P Q hac hK N).add
((truncated_depth_integrable hn0 P Q hac hK N).const_mul _)
· intro N z
exact add_nonneg ENNReal.toReal_nonneg (mul_nonneg hs (Nat.cast_nonneg _))
· exact ((measurable_terminalKL hn0 P Q).ennreal_toReal.add
((measurable_depth_real hn0 P Q).const_mul _)).aestronglyMeasurable
· intro z
exact add_nonneg ENNReal.toReal_nonneg (mul_nonneg hs (Nat.cast_nonneg _))
· have hroot := ae_mem_cell_of_ac hac
filter_upwards [hroot,almost_sure_termination hn P Q hac hK] with z hz ht
refine ⟨attempts P Q z,fun M hM => ?_⟩
have he := run_eventually_terminal P Q ht M hM
simp only [truncatedDepth,he,← terminal_historyKL hn0 P Q hroot hz ht,depth,terminalCell]
· exact truncated_final_budget hn P Q hac hK
lemma terminal_depth_integrable {n : ℕ} (hn : 32 ≤ n) (P : Measure (Coordinate n)) [IsProbabilityMeasure P]
(Q : Cell n) (hac : P ≪ uniformCell Q) (hK : InformationTheory.klDiv P (uniformCell Q) ≠ ∞) :
Integrable (fun z => (depth P Q z:ℝ)) P := by
have hn0 : 0 < n := by omega
have hc := (terminal_cost_budget hn P Q hac hK).1
apply Integrable.mono' (hc.div_const (entropySlope n/2)) (measurable_depth_real hn0 P Q).aestronglyMeasurable
filter_upwards [] with z
rw [Real.norm_eq_abs,abs_of_nonneg (Nat.cast_nonneg _),le_div_iff₀ (div_pos (entropySlope_pos hn0) (by norm_num))]
nlinarith [ENNReal.toReal_nonneg (a := terminalKL P Q z)]
lemma terminal_KL_integrable {n : ℕ} (hn : 32 ≤ n) (P : Measure (Coordinate n)) [IsProbabilityMeasure P]
(Q : Cell n) (hac : P ≪ uniformCell Q) (hK : InformationTheory.klDiv P (uniformCell Q) ≠ ∞) :
Integrable (fun z => (terminalKL P Q z).toReal) P := by
have hc := (terminal_cost_budget hn P Q hac hK).1
have hd := terminal_depth_integrable hn P Q hac hK
simpa only [add_sub_cancel_right] using
hc.sub' (hd.const_mul (entropySlope n/2))
lemma terminal_attempts_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) ≠ ∞) :
Integrable (fun z => (attempts P Q z:ℝ)) P ∧
(∫ z,(attempts P Q z:ℝ) ∂P) ≤ 2*(1+∫ z,(depth P Q z:ℝ) ∂P) := by
have hn0 : 0 < n := by omega
have hroot := ae_mem_cell_of_ac hac
have hterm := almost_sure_termination hn P Q hac hK
have hd := terminal_depth_integrable hn P Q hac hK
apply frozen_nonnegative_bound P (fun N z => (truncatedAttempts P Q N z:ℝ)) _ _
· positivity
· exact integrable_truncatedAttempts hn0 P Q
· intro N z
exact Nat.cast_nonneg _
· exact ((measurable_of_countable (fun a : ℕ => (a:ℝ))).comp (measurable_attempts hn0 P Q)).aestronglyMeasurable
· intro z
exact Nat.cast_nonneg _
· filter_upwards [hterm] with z ht
refine ⟨attempts P Q z,fun M hM => ?_⟩
simp only [truncatedAttempts,run_eventually_terminal P Q ht M hM,terminalLog_length P Q ht]
· intro N
have he : ∀ᵐ z ∂P,(truncatedDepth P Q N z:ℝ) ≤ (depth P Q z:ℝ) := by
filter_upwards [hroot,hterm] with z hz ht
have hm := truncatedDepth_mono hn0 P Q hz (Nat.le_max_left N (attempts P Q z))
have hf := run_eventually_terminal P Q ht (max N (attempts P Q z)) (Nat.le_max_right _ _)
simp only [truncatedDepth,hf] at hm
exact_mod_cast hm
have hi := integral_mono_ae (truncated_depth_integrable hn0 P Q hac hK N) hd he
exact (truncated_attempts_integral hn0 P Q hac hK N).trans (by linarith)
lemma terminal_ennreal_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) ≠ ∞) :
(∫⁻ z,terminalKL P Q z ∂P) + ENNReal.ofReal ((Real.log 2/4)*n*(∫ z,(depth P Q z:ℝ) ∂P)) ≤
InformationTheory.klDiv P (uniformCell Q) + ENNReal.ofReal (2/Real.exp 1) := by
have hd := terminal_depth_integrable hn P Q hac hK
have hk := terminal_KL_integrable hn P Q hac hK
have hc := (terminal_cost_budget hn P Q hac hK).2
rw [integral_add hk (hd.const_mul _),integral_const_mul] at hc
have hfin : ∀ᵐ z ∂P,terminalKL P Q z ≠ ∞ := by
filter_upwards [ae_fiber_positive P (terminalLog P Q)] with z hz
exact (terminal_regular hn P Q hac hK (terminalLog P Q z) hz).2.2
have he : (∫⁻ z,terminalKL P Q z ∂P) = ENNReal.ofReal (∫ z,(terminalKL P Q z).toReal ∂P) := by
rw [ofReal_integral_eq_lintegral_ofReal hk (ae_of_all _ (fun _ => ENNReal.toReal_nonneg))]
exact lintegral_congr_ae (hfin.mono (fun z hz => (ENNReal.ofReal_toReal hz).symm))
have hs : (Real.log 2/4)*(n:ℝ) = entropySlope n/2 := by unfold entropySlope; ring
rw [he,hs,← ENNReal.ofReal_add (integral_nonneg (fun _ => ENNReal.toReal_nonneg))]
calc
_ ≤ ENNReal.ofReal ((InformationTheory.klDiv P (uniformCell Q)).toReal+2/Real.exp 1) := ENNReal.ofReal_le_ofReal hc
_ = _ := by
rw [ENNReal.ofReal_add ENNReal.toReal_nonneg,ENNReal.ofReal_toReal hK]
positivity
· exact mul_nonneg (le_of_lt (div_pos (entropySlope_pos (by omega)) (by norm_num)))
(integral_nonneg (fun _ => Nat.cast_nonneg _))
end FiniteEntropy
end RepeatedLocalization
end OAI
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