| import Mathlib |
| import OAI.Probability.GaussianRegression.Main |
| import OAI.Probability.MemoryPrecision.KernelEvaluation |
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| namespace OAI |
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| open MeasureTheory ProbabilityTheory Set Filter EuclideanGeometry |
| open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise |
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| noncomputable section |
| universe u v |
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| 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) |
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| 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))) |
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| 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 |
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| 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 |
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| end |
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| end OAI |
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