| import OAI.Probability.GaussianRegression.Transfer |
| import OAI.Probability.GaussianRegression.Rotational |
|
|
| namespace OAI |
|
|
| /-! |
| Uniform-prior success bounds for finite-state estimators. |
| -/ |
|
|
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal NNReal RealInnerProductSpace Topology |
| open InnerProductSpace |
| open scoped Pointwise |
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| noncomputable section |
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| universe u |
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| namespace NoiselessRegression |
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| namespace Learner |
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| lemma frozen_success_bound {d M T : ℕ} {Ω : Type u} (L : Learner d M T Ω) |
| (ω : Ω) (hd : 3 ≤ d) |
| (htrans : ∀ t j, AEMeasurable (fun obs => (L.transition t ω j obs).2) (observationReference d)) |
| {ε : ℝ} (hε : 0 < ε) {a q : ℕ} (ha : 2*a ≤ d-1) (hq : 2 ≤ q) (hqa : 2*q ≤ a) : |
| ((uniformSphere d).prod (sampleLaw d T)) |
| {z | angularError z.1 (L.estimate z.1 ω z.2) ≤ ε} ≤ |
| (T+1 : ℝ≥0∞) * (ENNReal.ofReal ((3 : ℝ)*4^d) * ENNReal.ofReal (ε^a)) * |
| streamFactor (E := Vector d) (J := State M) q ^ (T/q+1) := by |
| classical |
| let : NeZero d := ⟨by omega⟩ |
| let δ : ℕ → State M → (Vector d × ℝ) → State M := fun t j => (htrans t j).mk _ |
| have hδ (t : ℕ) (j : State M) : Measurable (δ t j) := (htrans t j).measurable_mk |
| let μ := (uniformSphere d).prod (sampleLaw d T) |
| have hgood : ∀ᵐ z ∂μ, ∀ t j (i : Fin T), |
| (L.transition t ω j (z.2 i,⟪z.1.val,z.2 i⟫)).2 = δ t j (z.2 i,⟪z.1.val,z.2 i⟫) := by |
| simp only [ae_all_iff] |
| intro t j i |
| exact (sphere_sample_observation_quasi (by simpa using hd) i).ae (htrans t j).ae_eq_mk |
| let S : Fin (T+1) → Set (UnitSphere d × Samples d T) := fun t => |
| {z | dist z.1.val (L.output t ω (streamRoute δ 0 t (L.initialChoice ω).2 |
| ((fun i => z.2 (Fin.castLE (by omega) i)),fun i => ⟪z.1.val,z.2 (Fin.castLE (by omega) i)⟫))).val ≤ ε} |
| have hmono : {z : UnitSphere d × Samples d T | angularError z.1 (L.estimate z.1 ω z.2) ≤ ε} |
| ≤ᵐ[μ] ⋃ t, S t := by |
| filter_upwards [hgood] with z hz |
| intro hs |
| rcases L.terminal_forced_witness z.1 ω z.2 with ⟨t,ht,he⟩ |
| have hf : L.forcedRun z.1 ω z.2 t = streamRoute δ 0 t (L.initialChoice ω).2 |
| ((fun i => z.2 (Fin.castLE ht i)),fun i => ⟪z.1.val,z.2 (Fin.castLE ht i)⟫) := by |
| rw [L.forcedRun_eq_route z.1 ω z.2 ht] |
| exact streamRoute_congr _ _ 0 t _ _ _ (fun n j i => hz n j (Fin.castLE ht i)) |
| have hc : dist z.1.val (L.estimate z.1 ω z.2).val ≤ ε := by |
| apply le_trans _ hs |
| simpa only [angularError, dist_comm] using chord_le_arccos (L.estimate z.1 ω z.2) z.1 |
| refine Set.mem_iUnion.mpr ⟨⟨t,by omega⟩, ?_⟩ |
| change dist z.1.val (L.output t ω _) ≤ ε |
| simpa only [estimate, he, hf] using hc |
| let B : ℝ≥0∞ := (ENNReal.ofReal ((3 : ℝ)*4^d) * ENNReal.ofReal (ε^a)) * |
| streamFactor (E := Vector d) (J := State M) q ^ (T/q+1) |
| have hb (t : Fin (T+1)) : μ (S t) ≤ B := by |
| simpa only [B, μ, S, uniformSphere, sphereLaw, sampleLaw, finrank_euclideanSpace_fin] using |
| prefix_route_cap_bound hδ (fun j => (L.output t ω j).val) hε |
| (by simpa using ha) hq hqa (by omega : (t : ℕ) ≤ T) (L.initialChoice ω).2 |
| calc |
| _ ≤ μ (⋃ t, S t) := measure_mono_ae hmono |
| _ ≤ ∑' t, μ (S t) := measure_iUnion_le S |
| _ ≤ ∑' _ : Fin (T+1), B := ENNReal.tsum_le_tsum hb |
| _ = _ := by simp [B, tsum_fintype, nsmul_eq_mul, mul_assoc] |
|
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| lemma streaming_success_bound {d M T : ℕ} {Ω : Type u} [MeasurableSpace Ω] |
| (L : Learner d M T Ω) (ρ : Measure Ω) [IsProbabilityMeasure ρ] |
| (hL : L.Admissible ρ) (hd : 3 ≤ d) {ε : ℝ} (hε : 0 < ε) |
| {a q : ℕ} (ha : 2*a ≤ d-1) (hq : 2 ≤ q) (hqa : 2*q ≤ a) : |
| L.uniformSuccess ρ ε ≤ |
| (T+1 : ℝ≥0∞) * (ENNReal.ofReal ((3 : ℝ)*4^d) * ENNReal.ofReal (ε^a)) * |
| streamFactor (E := Vector d) (J := State M) q ^ (T/q+1) := by |
| let : NeZero d := ⟨by omega⟩ |
| let : SigmaFinite (observationReference d) := by unfold observationReference; infer_instance |
| have hω : ∀ᵐ ω ∂ρ, ∀ t j, |
| AEMeasurable (fun obs => (L.transition t ω j obs).2) (observationReference d) := by |
| simp only [ae_all_iff] |
| intro t j |
| filter_upwards [ae_fiber_aemeasurable (hL.transition t j).prod_swap] with ω hω |
| exact measurable_snd.comp_aemeasurable hω |
| have hp := ((MeasurePreserving.id (uniformSphere d)).prod |
| (Measure.measurePreserving_swap (μ := sampleLaw d T) (ν := ρ))).comp |
| (measurePreserving_prodAssoc (uniformSphere d) (sampleLaw d T) ρ) |
| let S := {z : UnitSphere d × (Ω × Samples d T) | |
| angularError z.1 (L.estimate z.1 z.2.1 z.2.2) ≤ ε} |
| have hS : NullMeasurableSet S ((uniformSphere d).prod (ρ.prod (sampleLaw d T))) := |
| L.angularSuccess_nullMeasurable ρ ε hL |
| have he := hp.measure_preimage hS |
| have hb := prod_null_set_bound (hS.preimage hp.quasiMeasurePreserving) |
| (c := (T+1 : ℝ≥0∞) * (ENNReal.ofReal ((3 : ℝ)*4^d) * ENNReal.ofReal (ε^a)) * |
| streamFactor (E := Vector d) (J := State M) q ^ (T/q+1)) |
| (by |
| filter_upwards [hω] with ω hω |
| exact L.frozen_success_bound ω hd hω hε ha hq hqa) |
| simpa only [he, measure_univ, mul_one, S, uniformSuccess] using hb |
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| end Learner |
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| end NoiselessRegression |
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| end |
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| end OAI |
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