lean/OAI/Probability/GaussianReplacement/StreamLearner.lean

import OAI.Probability.GaussianReplacement.StreamTransfer

namespace OAI

open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal NNReal RealInnerProductSpace Topology
noncomputable section
universe u
namespace NoiselessRegression
open InnerProductSpace
open scoped Pointwise

namespace Learner

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]


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

end Learner


end NoiselessRegression

namespace CurrentProjection


end CurrentProjection
namespace NoiselessRegression
section ThreeFifthsSphere
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
lemma sphere_projection_large_eighth [Nontrivial E] (U : Submodule ℝ E)
    (hU : 8 * Module.finrank ℝ U ≤ Module.finrank ℝ E) :
    sphereLaw {x : Metric.sphere (0 : E) 1 | (4/5 : ℝ) ≤ ‖U.orthogonalProjectionOnto x.val‖^2} ≤
      ENNReal.ofReal (5/32 : ℝ) := by
  have hi := ofReal_integral_eq_lintegral_ofReal (sphere_projection_sq_integrable U)
    (Filter.Eventually.of_forall fun x => sq_nonneg ‖U.orthogonalProjectionOnto x‖)
  rw [sphere_projection_sq_integral] at hi
  have hmark := meas_ge_le_lintegral_div
    (μ := ambientSphereLaw (E := E)) (f := fun x => ENNReal.ofReal (‖U.orthogonalProjectionOnto x‖^2))
    (by fun_prop) (by norm_num : ENNReal.ofReal (4/5 : ℝ) ≠ 0) (ENNReal.ofReal_ne_top : ENNReal.ofReal (4/5 : ℝ) ≠ ∞)
  rw [← hi] at hmark
  have hset : {x : E | ENNReal.ofReal (4/5 : ℝ) ≤ ENNReal.ofReal (‖U.orthogonalProjectionOnto x‖^2)} =
      {x : E | (4/5 : ℝ) ≤ ‖U.orthogonalProjectionOnto x‖^2} := by
    ext x
    change ENNReal.ofReal (4/5 : ℝ) ≤ _ ↔ _
    exact ENNReal.ofReal_le_ofReal_iff (sq_nonneg _)
  rw [hset, ambientSphereLaw,
    (MeasurableEmbedding.subtype_coe Metric.isClosed_sphere.measurableSet).map_apply] at hmark
  apply hmark.trans
  have hd : (0 : ℝ) < Module.finrank ℝ E := by
    exact_mod_cast Module.finrank_pos (R := ℝ) (M := E)
  have hdiv : (Module.finrank ℝ U : ℝ) / (Module.finrank ℝ E : ℝ) ≤ 1/8 := by
    rw [div_le_iff₀ hd]
    have : (8 : ℝ) * Module.finrank ℝ U ≤ Module.finrank ℝ E := by exact_mod_cast hU
    linarith
  calc
    _ ≤ ENNReal.ofReal (1/8 : ℝ) / ENNReal.ofReal (4/5 : ℝ) := by gcongr
    _ = ENNReal.ofReal (5/32 : ℝ) := by
      rw [← ENNReal.ofReal_div_of_pos (by norm_num : (0 : ℝ) < 4/5)]
      norm_num

lemma sphere_reflection_success_bound_eighth [Nontrivial E] (U : Submodule ℝ E)
    (hU : 8 * Module.finrank ℝ U ≤ Module.finrank ℝ E)
    (S : Set (Metric.sphere (0 : E) 1)) (hS : NullMeasurableSet S sphereLaw)
    (hsep : ∀ s ∈ S, sphereIsometry U.reflection s ∈ S →
      dist s.val (U.reflection s.val) ≤ 1/5) :
    sphereLaw S ≤ ENNReal.ofReal (37/64 : ℝ) := by
  let f := sphereIsometry U.reflection
  have hp := sphereIsometry_preserving U.reflection
  have hi : S ∩ f ⁻¹' S ⊆
      {x : Metric.sphere (0 : E) 1 | (4/5 : ℝ) ≤ ‖U.orthogonalProjectionOnto x.val‖^2} := by
    intro s hs
    have hd := hsep s hs.1 hs.2
    rw [reflection_dist] at hd
    have hn := U.norm_sq_eq_add_norm_sq_starProjection s.val
    rw [mem_sphere_zero_iff_norm.mp s.property] at hn
    change (4/5 : ℝ) ≤ ‖U.orthogonalProjectionOnto s.val‖^2
    have he : ‖U.starProjection s.val‖ = ‖U.orthogonalProjectionOnto s.val‖ := rfl
    rw [he] at hn
    nlinarith [norm_nonneg (Uᗮ.starProjection s.val)]
  have hinter := (measure_mono hi).trans (sphere_projection_large_eighth U hU)
  have heq := measure_union_add_inter₀ S (hS.preimage hp.quasiMeasurePreserving)
  rw [hp.measure_preimage hS] at heq
  calc
    sphereLaw S = ENNReal.ofReal (1/2 : ℝ) * (sphereLaw S + sphereLaw S) := by
      rw [← two_mul, ← mul_assoc]
      have hc : ENNReal.ofReal (1/2 : ℝ) * 2 = 1 := by
        rw [← ENNReal.ofReal_ofNat, ← ENNReal.ofReal_mul (by norm_num : (0 : ℝ) ≤ 1/2)]
        norm_num
      rw [hc, one_mul]
    _ = ENNReal.ofReal (1/2 : ℝ) *
        (sphereLaw (S ∪ f ⁻¹' S) + sphereLaw (S ∩ f ⁻¹' S)) := by rw [heq]
    _ ≤ ENNReal.ofReal (1/2 : ℝ) * (1 + ENNReal.ofReal (5/32 : ℝ)) := by
      exact mul_le_mul' le_rfl (add_le_add prob_le_one hinter)
    _ = ENNReal.ofReal (37/64 : ℝ) := by
      rw [← ENNReal.ofReal_one, ← ENNReal.ofReal_add (by norm_num : (0 : ℝ) ≤ 1)
        (by norm_num : (0 : ℝ) ≤ 5/32),
        ← ENNReal.ofReal_mul (by norm_num : (0 : ℝ) ≤ 1/2)]
      norm_num


end ThreeFifthsSphere
namespace Learner
lemma linear_success_bound_three_fifths {d M T : ℕ} {Ω : Type u} [MeasurableSpace Ω]
    (L : Learner d M T Ω) (ρ : Measure Ω) [IsProbabilityMeasure ρ]
    (hd : 0 < d) (hT : 8*T ≤ d) {ε : ℝ} (hε : ε ≤ 1/10) (hL : L.Admissible ρ) :
    L.uniformSuccess ρ ε ≤ ENNReal.ofReal (37/64 : ℝ) := by
  classical
  let : NeZero d := ⟨hd.ne'⟩
  have hmeas := ae_fiber_aemeasurable hL.uniform_experiment
  unfold uniformSuccess
  refine (prod_null_set_bound (L.angularSuccess_nullMeasurable ρ ε hL)
    (c := ENNReal.ofReal (37/64 : ℝ)) ?_).trans_eq (by simp)
  filter_upwards [hmeas] with z hz
  let U : Submodule ℝ (Vector d) := Submodule.span ℝ (Set.range z.2)
  have hdim : Module.finrank ℝ U ≤ T := by
    have h := finrank_span_le_card (R := ℝ) (Set.range z.2)
    apply h.trans
    rw [Set.toFinset_range]
    exact (Finset.card_image_le).trans_eq (by simp)
  have hU : 8 * Module.finrank ℝ U ≤ Module.finrank ℝ (Vector d) := by
    simpa only [Vector, finrank_euclideanSpace_fin] using (Nat.mul_le_mul_left 8 hdim).trans hT
  change sphereLaw {s : UnitSphere d | angularError s (L.estimate s z.1 z.2) ≤ ε} ≤ _
  apply sphere_reflection_success_bound_eighth U hU
  · exact nullMeasurableSet_le
      (Real.continuous_arccos.measurable.comp_aemeasurable
        ((measurable_subtype_coe.comp_aemeasurable hz).inner measurable_subtype_coe.aemeasurable))
      aemeasurable_const
  · intro s hs ht
    let t := sphereIsometry U.reflection s
    have he : L.estimate s z.1 z.2 = L.estimate t z.1 z.2 := by
      apply L.estimate_congr_labels
      intro i
      have hi : z.2 i ∈ U := Submodule.subset_span (Set.mem_range_self i)
      have he := U.reflection.inner_map_map (z.2 i) s.val
      rw [(Submodule.reflection_eq_self_iff (z.2 i)).2 hi] at he
      exact he.symm
    have hs' : dist s.val (L.estimate s z.1 z.2).val ≤ ε := by
      apply (chord_le_arccos s (L.estimate s z.1 z.2)).trans
      simpa only [Set.mem_ofPred_eq, angularError, real_inner_comm] using hs
    have ht' : dist t.val (L.estimate t z.1 z.2).val ≤ ε := by
      apply (chord_le_arccos t (L.estimate t z.1 z.2)).trans
      simpa only [Set.mem_ofPred_eq, angularError, real_inner_comm] using ht
    calc
      _ = dist s.val t.val := rfl
      _ ≤ dist s.val (L.estimate s z.1 z.2).val + dist (L.estimate s z.1 z.2).val t.val := dist_triangle _ _ _
      _ ≤ ε + ε := by rw [he, dist_comm (L.estimate t z.1 z.2).val t.val] at *; exact add_le_add hs' ht'
      _ ≤ 1/5 := by linarith

end Learner
namespace Learner
lemma streaming_success_bound_real_three_fifths {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)
    (hs : (3/5 : ℝ≥0∞) ≤ L.uniformSuccess ρ ε) :
    (3/5 : ℝ) ≤ ((T : ℝ)+1) * ((3 : ℝ)*4^d * ε^a) *
      ((32768 : ℝ)^d * ((2 : ℝ)^M)^(1/(q : ℝ)))^(T/q+1) := by
  have hb := hs.trans (L.streaming_success_bound ρ hL hd hε ha hq hqa)
  have ht : (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
    apply ENNReal.mul_ne_top
    · apply ENNReal.mul_ne_top (by simp)
      exact ENNReal.mul_ne_top ENNReal.ofReal_ne_top ENNReal.ofReal_ne_top
    · exact ENNReal.pow_ne_top (streamFactor_ne_top q)
  have hr := ENNReal.toReal_mono ht hb
  simpa [ENNReal.toReal_div, ENNReal.toReal_ofNat, ENNReal.toReal_mul,
    ENNReal.toReal_add, ENNReal.toReal_natCast, ENNReal.toReal_one,
    ENNReal.toReal_ofReal (by positivity : (0 : ℝ) ≤ 3*4^d),
    ENNReal.toReal_ofReal (by positivity : 0 ≤ ε^a), streamFactor,
    ENNReal.toReal_pow, ← ENNReal.toReal_rpow, finrank_euclideanSpace_fin,
    Fintype.card_fin, State, Nat.cast_pow, Nat.cast_ofNat, Vector] using hr

end Learner

lemma precision_from_stream_bound_three_fifths {d M T a q : ℕ} {ε A : ℝ}
    (hd : 32 ≤ d) (hT : d ≤ 8*T) (hq : 0 < q) (hdq : d ≤ 8*q)
    (ha : d ≤ 4*a) (hM : (M : ℝ) ≤ A * (d : ℝ)^2) (hA : 0 < A)
    (hε : 0 < ε) (hε1 : ε ≤ 1)
    (hbound : (3/5 : ℝ) ≤ ((T : ℝ)+1) * ((3 : ℝ)*4^d * ε^a) *
      ((32768 : ℝ)^d * ((2 : ℝ)^M)^(1/(q : ℝ)))^(T/q+1)) :
    (1 / (4*(600000+200*A))) * (d : ℝ) * Real.log (1/ε) ≤ (T : ℝ) := by
  have hd0 : (0 : ℝ) < d := by exact_mod_cast (by omega : 0 < d)
  have hq0 : (0 : ℝ) < q := by exact_mod_cast hq
  have hT0 : (0 : ℝ) ≤ T := Nat.cast_nonneg T
  have hdp : (d : ℝ) ≤ 8*T := by exact_mod_cast hT
  have hdqp : (d : ℝ) ≤ 8*q := by exact_mod_cast hdq
  have hap : (d : ℝ) ≤ 4*a := by exact_mod_cast ha
  have hB : ((T/q+1 : ℕ) : ℝ) * d ≤ 16*T := by
    have hbd : (T/q+1)*d ≤ 16*T := by
      have hmul := Nat.div_mul_le_self T q
      calc
        (T/q+1)*d = (T/q)*d+d := by ring
        _ ≤ (T/q)*(8*q)+d := Nat.add_le_add_right (Nat.mul_le_mul_left _ hdq) _
        _ = 8*((T/q)*q)+d := by ring
        _ ≤ 8*T+d := Nat.add_le_add_right (Nat.mul_le_mul_left 8 hmul) _
        _ ≤ 16*T := by omega
    exact_mod_cast hbd
  have hMq : (M : ℝ) / q ≤ 8*A*d := by
    apply (div_le_iff₀ hq0).2
    calc
      (M : ℝ) ≤ A*d^2 := hM
      _ ≤ A*d*(8*q) := by nlinarith [mul_le_mul_of_nonneg_left hdqp (by positivity : 0 ≤ A*d)]
      _ = 8*A*d*q := by ring
  have hlog := Real.log_le_log (by norm_num : (0 : ℝ) < 3/5) hbound
  rw [Real.log_mul (by positivity) (by positivity),
    Real.log_mul (by positivity) (by positivity),
    Real.log_mul (by positivity) (by positivity),
    Real.log_mul (by positivity) (by positivity),
    Real.log_pow, Real.log_pow, Real.log_pow,
    Real.log_mul (by positivity) (by positivity), Real.log_pow,
    Real.log_rpow (by positivity), Real.log_pow] at hlog
  have hlogε : Real.log (1/ε) = -Real.log ε := by simp
  have hlogε0 : 0 ≤ Real.log (1/ε) := Real.log_nonneg ((le_div_iff₀ hε).2 (by simpa using hε1))
  have heq : (1/(q : ℝ)) * ((M : ℝ)*Real.log 2) = ((M : ℝ)/q)*Real.log 2 := by ring
  rw [heq] at hlog
  have hlT : Real.log ((T : ℝ)+1) ≤ T := by
    simpa using Real.log_le_sub_one_of_pos (by positivity : (0 : ℝ) < T+1)
  have hl3 : Real.log (3 : ℝ) ≤ 3 := Real.log_le_self (by norm_num)
  have hl4 : Real.log (4 : ℝ) ≤ 4 := Real.log_le_self (by norm_num)
  have hl32 : Real.log (32768 : ℝ) ≤ 32768 := Real.log_le_self (by norm_num)
  have hl2 : Real.log (2 : ℝ) ≤ 1 := by
    convert Real.log_le_sub_one_of_pos (by norm_num : (0 : ℝ) < 2) using 1; norm_num
  have hl23 : -(1 : ℝ) ≤ Real.log (3/5 : ℝ) := by
    have h := Real.one_sub_inv_le_log_of_pos (by norm_num : (0 : ℝ) < 3/5)
    norm_num at h
    linarith
  have hinside : (d : ℝ)*Real.log 32768 + ((M : ℝ)/q)*Real.log 2 ≤
      (32768+8*A)*d := by
    have hm0 : (0 : ℝ) ≤ (M : ℝ)/q := by positivity
    have h1 := mul_le_mul_of_nonneg_left hl32 hd0.le
    have h2 := mul_le_mul_of_nonneg_left hl2 hm0
    nlinarith
  have hfull : ((T/q+1 : ℕ) : ℝ) *
      ((d : ℝ)*Real.log 32768 + ((M : ℝ)/q)*Real.log 2) ≤
      16*T*(32768+8*A) := by
    calc
      _ ≤ ((T/q+1 : ℕ) : ℝ) * ((32768+8*A)*d) :=
        mul_le_mul_of_nonneg_left hinside (Nat.cast_nonneg _)
      _ = (32768+8*A)*(((T/q+1 : ℕ) : ℝ)*d) := by ring
      _ ≤ (32768+8*A)*(16*T) := mul_le_mul_of_nonneg_left hB (by positivity)
      _ = _ := by ring
  have hd4 : (4 : ℝ) ≤ d := by exact_mod_cast (by omega : 4 ≤ d)
  have hlogbound : (a : ℝ)*Real.log (1/ε) ≤ (600000+200*A)*T := by
    have h4 := mul_le_mul_of_nonneg_left hl4 hd0.le
    rw [hlogε]
    nlinarith [mul_nonneg hA.le hT0]
  have hmain : (d : ℝ)*Real.log (1/ε) ≤ 4*(600000+200*A)*T := by
    have ham := mul_le_mul_of_nonneg_right hap hlogε0
    linarith
  have hden : (0 : ℝ) < 4*(600000+200*A) := by positivity
  calc
    _ = ((d : ℝ)*Real.log (1/ε))/(4*(600000+200*A)) := by ring
    _ ≤ T := (div_le_iff₀ hden).2 (by nlinarith [hmain])


end NoiselessRegression

end

end OAI