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a2ee19a | 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 | import Mathlib
import OAI.Probability.GaussianRegression.Main
import OAI.Probability.MemoryPrecision.KernelEvaluation
namespace OAI
open MeasureTheory ProbabilityTheory Set Filter EuclideanGeometry
open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise
noncomputable section
universe u v
open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise
namespace MemoryPrecision.Kernel
namespace Policy
variable {d M T : ℕ} (K : Policy d M T)
variable {X : Type*} [MeasurableSpace X]
variable (w : ℕ → State M → X → ℝ≥0∞) (ι : X → Vector d)
lemma measurable_records_at (s : X) (k : ℕ) :
Measurable (fun A : Rows d k => ((s, (fun i => (A i,⟪ι s,A i⟫))) : X × Observations d k)) := by
exact measurable_const.prodMk
(Measurable.of_eval (fun index => (measurable_pi_apply index).prodMk (by fun_prop)))
lemma cont_eq_integral (hK : K.Borel) (hw : ∀ n j, Measurable (w n j))
(n k : ℕ) (j : State M) (s : X) :
K.cont w ι n k j s =
∫⁻ A : Rows d k, K.eval w n k j s (fun i => (A i,⟪ι s,A i⟫)) ∂(RowLaw d k) := by
classical
induction k generalizing n j with
| zero => simp [cont, eval, RowLaw]
| succ k ih =>
have hf : Measurable (fun A : Rows d (k+1) =>
K.eval w n (k+1) j s (fun i => (A i,⟪ι s,A i⟫))) := by
exact (K.measurable_eval w hK hw n (k+1) j).comp (measurable_records_at ι s (k+1))
rw [RowLaw, NoiselessRegression.lintegral_pi_cons _ _ hf]
unfold cont
apply lintegral_congr
intro x
simp only [eval, Fin.cons_zero]
have hm (a : Action M) : Measurable (fun A : Rows d k =>
if a.1 then w (n+1) a.2 s
else K.eval w (n+1) k a.2 s (fun i => (A i,⟪ι s,A i⟫))) := by
rcases a with ⟨b,a⟩
cases b
· exact (K.measurable_eval w hK hw (n+1) k a).comp (measurable_records_at ι s k)
· exact measurable_const
change (∑ a : Action M, K.transition (n+1) j (x,⟪ι s,x⟫) a *
if a.1 then w (n+1) a.2 s else K.cont w ι (n+1) k a.2 s) =
∫⁻ A : Rows d k, ∑ a : Action M, K.transition (n+1) j (x,⟪ι s,x⟫) a *
if a.1 then w (n+1) a.2 s
else K.eval w (n+1) k a.2 s (fun i => (A i,⟪ι s,A i⟫))
∂Measure.pi (fun _ => stdGaussian (Vector d))
rw [lintegral_finsetSum _ (fun a _ => (hm a).const_mul _)]
apply Finset.sum_congr rfl
intro a _
rw [lintegral_const_mul _ (hm a)]
congr 1
rcases a with ⟨b,a⟩
cases b
· exact ih (n+1) a
· simp
lemma pathValue_integral_le (hK : K.Borel) (hw : ∀ n j, Measurable (w n j))
(s : X) :
(∫⁻ A : Rows d T, K.pathValue w s (fun i => (A i,⟪ι s,A i⟫)) ∂(RowLaw d T)) ≤
∑ a : Action M, K.initial a *
(∑ r ∈ Finset.range (T+1), MemoryPrecision.stochCont K.marginal w ι 0 r a.2 s) := by
classical
have hm (a : Action M) : Measurable (fun A : Rows d T =>
if a.1 then w 0 a.2 s else K.eval w 0 T a.2 s (fun i => (A i,⟪ι s,A i⟫))) := by
rcases a with ⟨b,a⟩
cases b
· exact (K.measurable_eval w hK hw 0 T a).comp (measurable_records_at ι s T)
· exact measurable_const
unfold pathValue
rw [lintegral_finsetSum _ (fun a _ => (hm a).const_mul _)]
apply Finset.sum_le_sum
intro a _
rw [lintegral_const_mul _ (hm a)]
apply mul_le_mul' le_rfl
rcases a with ⟨b,a⟩
cases b
· change (∫⁻ A : Rows d T, K.eval w 0 T a s (fun i => (A i,⟪ι s,A i⟫)) ∂(RowLaw d T)) ≤ _
rw [← K.cont_eq_integral w ι hK hw 0 T a s]
exact K.cont_le_forced_sum w ι hK 0 T a s
· change (∫⁻ _A : Rows d T, w 0 a s ∂(RowLaw d T)) ≤ _
rw [lintegral_const]
have : IsProbabilityMeasure (RowLaw d T) := by unfold RowLaw; infer_instance
simp only [measure_univ, mul_one]
simpa only [MemoryPrecision.stochCont] using
(Finset.single_le_sum (fun length _ =>
(show (0:ℝ≥0∞) ≤ MemoryPrecision.stochCont K.marginal w ι 0 length a s from zero_le))
(show 0 ∈ Finset.range (T+1) by simp) :
MemoryPrecision.stochCont K.marginal w ι 0 0 a s ≤
∑ r ∈ Finset.range (T+1), MemoryPrecision.stochCont K.marginal w ι 0 r a s)
lemma experiment_fixed_measurable (hK : K.Borel)
{f : Sphere d × Sphere d → ℝ≥0∞} (hf : Measurable f) :
Measurable (fun z : Sphere d × Rows d T => K.pathValue (K.outputTest f) z.1 (records z.1 z.2)) := by
apply Finset.measurable_sum _
intro a _
apply Measurable.const_mul
rcases a with ⟨b,a⟩
cases b
· exact (K.measurable_eval (K.outputTest f) hK (K.outputTest_measurable hf) 0 T a).comp (by
exact measurable_fst.prodMk (Measurable.of_eval (fun index => by
change Measurable (fun z : Sphere d × Rows d T => (z.2 index,⟪z.1.val,z.2 index⟫))
fun_prop)))
· exact (K.outputTest_measurable hf 0 a).comp measurable_fst
lemma integral_pathValue_bound (hK : K.Borel) (hw : ∀ n j, Measurable (w n j))
(hι : Measurable ι) (ν : Measure X) {B : ℝ≥0∞}
(hb : ∀ (r : ℕ), r ≤ T → ∀ j,
(∫⁻ s, MemoryPrecision.stochCont K.marginal w ι 0 r j s ∂ν) ≤ B) :
(∫⁻ s, ∫⁻ A : Rows d T, K.pathValue w s (fun i => (A i,⟪ι s,A i⟫)) ∂(RowLaw d T) ∂ν) ≤
(T+1:ℝ≥0∞) * B := by
classical
have hm (a : Action M) : Measurable (fun s =>
∑ r ∈ Finset.range (T+1), MemoryPrecision.stochCont K.marginal w ι 0 r a.2 s) :=
Finset.measurable_sum _ (fun r _ =>
MemoryPrecision.measurable_stochCont (K.marginal_measurable hK) hw hι 0 r a.2)
calc
_ ≤ ∫⁻ s, ∑ a : Action M, K.initial a *
(∑ r ∈ Finset.range (T+1), MemoryPrecision.stochCont K.marginal w ι 0 r a.2 s) ∂ν :=
lintegral_mono (K.pathValue_integral_le w ι hK hw)
_ = ∑ a : Action M, K.initial a *
(∑ r ∈ Finset.range (T+1), ∫⁻ s,
MemoryPrecision.stochCont K.marginal w ι 0 r a.2 s ∂ν) := by
rw [lintegral_finsetSum _ (fun a _ => (hm a).const_mul _)]
apply Finset.sum_congr rfl
intro a _
rw [lintegral_const_mul _ (hm a), lintegral_finsetSum _ (fun r _ =>
MemoryPrecision.measurable_stochCont (K.marginal_measurable hK) hw hι 0 r a.2)]
_ ≤ ∑ a : Action M, K.initial a * ((T+1:ℝ≥0∞)*B) := by
apply Finset.sum_le_sum
intro a _
apply mul_le_mul' le_rfl
calc
_ ≤ ∑ _r ∈ Finset.range (T+1), B := Finset.sum_le_sum (fun r hr => hb r (by simpa using Nat.le_of_lt_succ (Finset.mem_range.mp hr)) a.2)
_ = _ := by simp [nsmul_eq_mul]
_ = _ := by rw [← Finset.sum_mul, K.initial_sum, one_mul]
end Policy
end MemoryPrecision.Kernel
end
end OAI
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