| import OAI.Probability.ProjectionMoments.Realization |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal NNReal RealInnerProductSpace |
| namespace NoiselessRegression |
|
|
| section |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] [Nontrivial E] |
|
|
| lemma capWeight_mass_mixed (y : E) {ε : ℝ} (hε : 0 < ε) (hε1 : ε ≤ 1) |
| (a : ℕ) {b : ℝ} (hb : 0 ≤ b) (hab : (a:ℝ)+b ≤ (Module.finrank ℝ E-1:ℕ)) : |
| PolynomialMass ((ambientSphereLaw (E := E)).withDensity (capWeight y ε)) a |
| (ENNReal.ofReal ((3 : ℝ)*4^Module.finrank ℝ E) * ENNReal.ofReal (ε^b)) := by |
| change PolynomialMass ((ambientSphereLaw (E := E)).withDensity ((Metric.closedBall y ε).indicator 1)) a _ |
| rw [withDensity_indicator_one Metric.isClosed_closedBall.measurableSet] |
| intro z r hr |
| rw [Measure.restrict_apply Metric.isClosed_closedBall.measurableSet] |
| by_cases h : r ≤ ε |
| · apply (measure_mono Set.inter_subset_left).trans |
| apply (ambientSphereLaw_closedBall z hr (Module.finrank ℝ E-1) le_rfl).trans |
| rw [mul_assoc, ← ENNReal.ofReal_mul (Real.rpow_nonneg hε.le _)] |
| apply mul_le_mul' le_rfl |
| apply ENNReal.ofReal_le_ofReal |
| have hpow : r^(Module.finrank ℝ E-1) ≤ r^((a:ℝ)+b) := by |
| rw [← Real.rpow_natCast] |
| exact Real.rpow_le_rpow_of_exponent_ge hr (h.trans hε1) hab |
| apply hpow.trans |
| rw [Real.rpow_add hr,Real.rpow_natCast,mul_comm] |
| exact mul_le_mul_of_nonneg_right (Real.rpow_le_rpow hr.le h hb) (pow_nonneg hr.le _) |
| · apply (measure_mono Set.inter_subset_right).trans |
| apply (ambientSphereLaw_closedBall y hε (Module.finrank ℝ E-1) le_rfl).trans |
| rw [mul_assoc, ← ENNReal.ofReal_mul (Real.rpow_nonneg hε.le _)] |
| apply mul_le_mul' le_rfl |
| apply ENNReal.ofReal_le_ofReal |
| have hpow : ε^(Module.finrank ℝ E-1) ≤ ε^((a:ℝ)+b) := by |
| rw [← Real.rpow_natCast] |
| exact Real.rpow_le_rpow_of_exponent_ge hε hε1 hab |
| apply hpow.trans |
| rw [Real.rpow_add hε,Real.rpow_natCast,mul_comm] |
| exact mul_le_mul_of_nonneg_left (pow_le_pow_left₀ hε.le (le_of_not_ge h) a) (Real.rpow_nonneg hε.le _) |
|
|
| variable {J : Type*} [Fintype J] [MeasurableSpace J] [MeasurableSingletonClass J] [Nonempty J] |
|
|
| lemma continuation_mixed_integral_bound {δ : ℕ → J → (E × ℝ) → J} |
| (hδ : ∀ n j, Measurable (δ n j)) (y : ℕ → J → E) |
| {ε : ℝ} (hε : 0 < ε) (hε1 : ε ≤ 1) {a q : ℕ} {b : ℝ} |
| (hb : 0 ≤ b) (hab : (a:ℝ)+b ≤ (Module.finrank ℝ E-1:ℕ)) |
| (hq : 2 ≤ q) (hqa : 2*q ≤ a) (n k : ℕ) (j : J) : |
| (∫⁻ s, continuation δ (fun n j => capWeight (y n j) ε) n k j s ∂ambientSphereLaw) ≤ |
| (ENNReal.ofReal ((3 : ℝ)*4^Module.finrank ℝ E) * ENNReal.ofReal (ε^b)) * |
| streamFactor (E := E) (J := J) q ^ ((k+q-1)/q) := by |
| have ha : a ≤ Module.finrank ℝ E := by |
| have : (a:ℝ) ≤ (Module.finrank ℝ E:ℝ) := by |
| have hh : ((Module.finrank ℝ E-1:ℕ):ℝ) ≤ Module.finrank ℝ E := by exact_mod_cast Nat.sub_le (Module.finrank ℝ E) 1 |
| linarith |
| exact_mod_cast this |
| apply ambient_mass_bound |
| apply continuation_mass_bound_ceil _ hδ (fun n j => measurable_capWeight _ _) |
| (fun n j s => capWeight_le_one _ _ _) |
| (mul_ne_zero (by positivity) (by positivity)) (ENNReal.mul_ne_top ENNReal.ofReal_ne_top ENNReal.ofReal_ne_top) |
| (fun n j => capWeight_mass_mixed (y n j) hε hε1 a hb hab) |
| (Module.finrank_pos (R := ℝ) (M := E)) ha hq hqa |
|
|
| lemma route_mixed_bound {δ : ℕ → J → (E × ℝ) → J} |
| (hδ : ∀ n j, Measurable (δ n j)) (y : J → E) |
| {ε : ℝ} (hε : 0 < ε) (hε1 : ε ≤ 1) {a q : ℕ} {b : ℝ} |
| (hb : 0 ≤ b) (hab : (a:ℝ)+b ≤ (Module.finrank ℝ E-1:ℕ)) |
| (hq : 2 ≤ q) (hqa : 2*q ≤ a) (k : ℕ) (j : J) : |
| ((sphereLaw (E := E)).prod (Measure.pi (fun _ : Fin k => stdGaussian E))) |
| {z | dist z.1.val (y (streamRoute δ 0 k j (z.2,fun i => ⟪z.1.val,z.2 i⟫))) ≤ ε} ≤ |
| (ENNReal.ofReal ((3 : ℝ)*4^Module.finrank ℝ E) * ENNReal.ofReal (ε^b)) * |
| streamFactor (E := E) (J := J) q ^ ((k+q-1)/q) := by |
| let w : ℕ → J → E → ℝ≥0∞ := fun _ j => capWeight (y j) ε |
| have hV := measurable_continuation (w := w) hδ (fun _ j => measurable_capWeight (y j) ε) 0 k j |
| have he (s : E) : continuation δ w 0 k j s = |
| ∫⁻ A : Fin k → E, capWeight (y (streamRoute δ 0 k j (A,fun i => ⟪s,A i⟫))) ε s |
| ∂Measure.pi (fun _ => stdGaussian E) := by |
| simpa [w, blockValue, continuation] using continuation_unroll (w := w) hδ 0 k 0 j s |
| let S : Set ((Metric.sphere (0 : E) 1) × (Fin k → E)) := |
| {z | dist z.1.val (y (streamRoute δ 0 k j (z.2,fun i => ⟪z.1.val,z.2 i⟫))) ≤ ε} |
| have hS : MeasurableSet S := by |
| apply measurableSet_le _ measurable_const |
| exact measurable_subtype_coe.comp measurable_fst |>.dist |
| ((measurable_of_finite y).comp ((measurable_streamRoute hδ 0 k j).comp (by fun_prop))) |
| change ((sphereLaw (E := E)).prod _) S ≤ _ |
| rw [← one_mul (((sphereLaw (E := E)).prod _) S), ← lintegral_indicator_const hS 1, |
| lintegral_prod _ (measurable_const.indicator hS).aemeasurable] |
| have hf : ∀ s : Metric.sphere (0 : E) 1, |
| (∫⁻ A, S.indicator (fun _ => (1 : ℝ≥0∞)) (s,A) ∂Measure.pi (fun _ => stdGaussian E)) = |
| continuation δ w 0 k j s.val := by |
| intro s |
| rw [he] |
| apply lintegral_congr |
| intro A |
| classical |
| change (if dist s.val (y (streamRoute δ 0 k j (A,fun i => ⟪s.val,A i⟫))) ≤ ε then 1 else 0) = _ |
| unfold capWeight |
| rw [Set.indicator_apply] |
| rfl |
| simp_rw [hf] |
| rw [← lintegral_map hV measurable_subtype_coe] |
| exact continuation_mixed_integral_bound hδ (fun _ => y) hε hε1 hb hab hq hqa 0 k j |
|
|
| end |
|
|
| namespace Learner |
|
|
| section |
| variable {d M : ℕ} |
|
|
| def haltShift {k : ℕ} : TaggedState M k → TaggedState M (k+1) |
| | .inl j => .inl j |
| | .inr (t,j) => .inr (t.succ,j) |
|
|
| def haltRoute (τ : ℕ → State M → (Vector d × ℝ) → Bool × State M) (n : ℕ) : |
| (k : ℕ) → State M → ((Fin k → Vector d) × (Fin k → ℝ)) → TaggedState M k |
| | 0, j, _ => .inl j |
| | k+1, j, X => |
| let a := τ (n+1) j (X.1 0,X.2 0) |
| if a.1 then .inr (⟨1,by omega⟩,a.2) |
| else haltShift (haltRoute τ (n+1) k a.2 (Fin.tail X.1,Fin.tail X.2)) |
|
|
| lemma measurable_haltRoute {τ : ℕ → State M → (Vector d × ℝ) → Bool × State M} |
| (hτ : ∀ t j, Measurable (τ t j)) (n k : ℕ) (j : State M) : |
| Measurable (haltRoute τ n k j) := by |
| induction k generalizing n j with |
| | zero => exact measurable_const |
| | succ k ih => |
| let h : (Bool × State M) → ((Fin (k+1) → Vector d) × (Fin (k+1) → ℝ)) → TaggedState M (k+1) := |
| fun a X => if a.1 then .inr (⟨1,by omega⟩,a.2) |
| else haltShift (haltRoute τ (n+1) k a.2 (Fin.tail X.1,Fin.tail X.2)) |
| change Measurable (fun X => h (τ (n+1) j (X.1 0,X.2 0)) X) |
| apply measurable_select (J := Bool × State M) _ ((hτ (n+1) j).comp (by fun_prop)) |
| intro a |
| dsimp [h] |
| split_ifs |
| · exact measurable_const |
| · exact (measurable_of_finite haltShift).comp ((ih (n+1) a.2).comp (by |
| change Measurable (fun X : ((Fin (k+1) → Vector d) × (Fin (k+1) → ℝ)) => |
| ((fun i : Fin k => X.1 i.succ),(fun i : Fin k => X.2 i.succ))) |
| fun_prop)) |
|
|
| def haltChild (w : ℕ → State M → Vector d → ℝ≥0∞) |
| (v : State M → Vector d → ℝ≥0∞) (n : ℕ) {k : ℕ} : TaggedState M k → Vector d → ℝ≥0∞ |
| | .inl j => v j |
| | .inr (t,j) => w (n+t) j |
|
|
| def stoppedContinuation (τ : ℕ → State M → (Vector d × ℝ) → Bool × State M) |
| (w : ℕ → State M → Vector d → ℝ≥0∞) (n : ℕ) : ℕ → State M → Vector d → ℝ≥0∞ |
| | 0, j, s => w n j s |
| | k+1, j, s => ∫⁻ x, let a := τ (n+1) j (x,⟪s,x⟫) |
| if a.1 then w (n+1) a.2 s else stoppedContinuation τ w (n+1) k a.2 s ∂stdGaussian (Vector d) |
|
|
| lemma measurable_stoppedContinuation {τ : ℕ → State M → (Vector d × ℝ) → Bool × State M} |
| {w : ℕ → State M → Vector d → ℝ≥0∞} |
| (hτ : ∀ t j, Measurable (τ t j)) (hw : ∀ n j, Measurable (w n j)) (n k : ℕ) (j : State M) : |
| Measurable (stoppedContinuation τ w n k j) := by |
| induction k generalizing n j with |
| | zero => exact hw n j |
| | succ k ih => |
| apply Measurable.lintegral_prod_left |
| change Measurable (fun z : Vector d × Vector d => |
| if (τ (n+1) j (z.1,⟪z.2,z.1⟫)).1 then w (n+1) (τ (n+1) j (z.1,⟪z.2,z.1⟫)).2 z.2 |
| else stoppedContinuation τ w (n+1) k (τ (n+1) j (z.1,⟪z.2,z.1⟫)).2 z.2) |
| apply measurable_select (J := Bool × State M) (h := fun a (z : Vector d × Vector d) => |
| if a.1 then w (n+1) a.2 z.2 else stoppedContinuation τ w (n+1) k a.2 z.2) _ |
| ((hτ (n+1) j).comp (by fun_prop)) |
| intro a |
| split_ifs |
| · exact (hw (n+1) a.2).comp measurable_snd |
| · exact (ih (n+1) a.2).comp measurable_snd |
|
|
| lemma stoppedContinuation_le_one |
| {τ : ℕ → State M → (Vector d × ℝ) → Bool × State M} |
| {w : ℕ → State M → Vector d → ℝ≥0∞} |
| (hw : ∀ n j s, w n j s ≤ 1) (n k : ℕ) (j : State M) (s : Vector d) : |
| stoppedContinuation τ w n k j s ≤ 1 := by |
| induction k generalizing n j with |
| | zero => exact hw n j s |
| | succ k ih => |
| change (∫⁻ x, (if (τ (n+1) j (x,⟪s,x⟫)).1 then w (n+1) (τ (n+1) j (x,⟪s,x⟫)).2 s else stoppedContinuation τ w (n+1) k (τ (n+1) j (x,⟪s,x⟫)).2 s) ∂stdGaussian (Vector d)) ≤ 1 |
| apply (lintegral_mono (g := fun _ => (1:ℝ≥0∞)) (fun x => ?_)).trans_eq (by simp) |
| split_ifs |
| · exact hw _ _ _ |
| · exact ih _ _ |
|
|
| lemma stoppedContinuation_unroll |
| {τ : ℕ → State M → (Vector d × ℝ) → Bool × State M} |
| {w : ℕ → State M → Vector d → ℝ≥0∞} |
| (hτ : ∀ t j, Measurable (τ t j)) (n m k : ℕ) (j : State M) (s : Vector d) : |
| stoppedContinuation τ w n (m+k) j s = |
| blockValue (haltChild w (stoppedContinuation τ w (n+m) k) n) (haltRoute τ n m j) s := by |
| induction m generalizing n j with |
| | zero => simp [blockValue,haltRoute,haltChild] |
| | succ m ih => |
| rw [Nat.succ_add,stoppedContinuation] |
| unfold blockValue |
| have hf : Measurable (fun X : Fin (m+1) → Vector d => |
| haltChild w (stoppedContinuation τ w (n+(m+1)) k) n |
| (haltRoute τ n (m+1) j (X,fun i => ⟪s,X i⟫)) s) := |
| (measurable_of_finite (fun a => haltChild w (stoppedContinuation τ w (n+(m+1)) k) n a s)).comp |
| ((measurable_haltRoute hτ n (m+1) j).comp (by fun_prop)) |
| rw [lintegral_pi_cons _ _ hf] |
| apply lintegral_congr |
| intro x |
| have htail (X : Fin m → Vector d) : Fin.tail (fun i : Fin (m+1) => ⟪s,(Fin.cons x X : Fin (m+1) → Vector d) i⟫) = (fun i => ⟪s,X i⟫) := by funext i; rfl |
| simp only [haltRoute,Fin.cons_zero,Fin.tail_cons,htail] |
| split_ifs with h |
| · simp [haltChild] |
| · rw [ih] |
| unfold blockValue |
| apply lintegral_congr |
| intro X |
| have h1 : n+(m+1) = n+1+m := by omega |
| rw [h1] |
| generalize haltRoute τ (n+1) m (τ (n+1) j (x,⟪s,x⟫)).2 (X,fun i => ⟪s,X i⟫) = a |
| cases a with |
| | inl j => rfl |
| | inr done => |
| simp only [haltShift,haltChild,Fin.val_succ] |
| rw [show n+1+done.1.val = n+(done.1.val+1) by omega] |
|
|
| lemma stoppedContinuation_mass_bound {τ : ℕ → State M → (Vector d × ℝ) → Bool × State M} |
| {w : ℕ → State M → Vector d → ℝ≥0∞} |
| (hτ : ∀ t j, Measurable (τ t j)) (hw : ∀ n j, Measurable (w n j)) |
| (hw1 : ∀ n j s, w n j s ≤ 1) |
| {a q : ℕ} {C : ℝ≥0∞} (hC : C ≠ 0) (hCt : C ≠ ∞) |
| (hlocal : ∀ n j, PolynomialMass (ambientSphereLaw.withDensity (w n j)) a C) |
| (hd : 1 ≤ d) (ha : a ≤ d) (hq : 2 ≤ q) (hqa : 2*q ≤ a) |
| (n k : ℕ) (j : State M) : |
| PolynomialMass (ambientSphereLaw.withDensity (stoppedContinuation τ w n k j)) a |
| (C * streamFactor (E := Vector d) (J := TaggedState M q) q ^ ((k+q-1)/q)) := by |
| let : NeZero d := ⟨by omega⟩ |
| let K := streamFactor (E := Vector d) (J := TaggedState M q) q |
| have hK1 : 1 ≤ K := one_le_streamFactor (by omega) |
| have hK0 : K ≠ 0 := ne_of_gt (lt_of_lt_of_le zero_lt_one hK1) |
| have hKt : K ≠ ∞ := streamFactor_ne_top _ |
| have hden : ((q-1 : ℕ) : ℝ)+1 = (q : ℝ) := by rw [Nat.cast_sub (by omega),Nat.cast_one]; ring |
| have hq0 : 0 < q := by omega |
| change PolynomialMass _ a (C*K^_) |
| induction k using Nat.strong_induction_on generalizing n j with |
| | h k ih => |
| by_cases hk0 : k = 0 |
| · subst k |
| simpa only [Nat.zero_add,Nat.div_eq_of_lt (by omega : q-1 < q),pow_zero,mul_one,stoppedContinuation] using hlocal n j |
| let m := min k q |
| have hm0 : 0 < m := by dsimp [m]; omega |
| have hm : m ≤ q := min_le_right _ _ |
| have hmk : m ≤ k := min_le_left _ _ |
| have hki : k-m < k := by omega |
| let C' := C*K^((k-m+q-1)/q) |
| have hC' : C' ≠ 0 := mul_ne_zero hC (pow_ne_zero _ hK0) |
| have hC't : C' ≠ ∞ := ENNReal.mul_ne_top hCt (ENNReal.pow_ne_top hKt) |
| have hCC' : C ≤ C' := le_mul_of_one_le_right' (one_le_pow₀ hK1) |
| let f := haltChild w (stoppedContinuation τ w (n+m) (k-m)) n (k := m) |
| have hf (a : TaggedState M m) : Measurable (f a) := by |
| cases a with |
| | inl j => exact measurable_stoppedContinuation hτ hw _ _ _ |
| | inr done => exact hw _ _ |
| have hf1 (a : TaggedState M m) (s : Vector d) : f a s ≤ 1 := by |
| cases a with |
| | inl j => exact stoppedContinuation_le_one hw1 _ _ _ _ |
| | inr done => exact hw1 _ _ _ |
| have hflocal (a' : TaggedState M m) : PolynomialMass (ambientSphereLaw.withDensity (f a')) a C' := by |
| cases a' with |
| | inl j => exact ih (k-m) hki (n+m) j |
| | inr done => |
| intro z r hr |
| exact (hlocal (n+done.1) done.2 z r hr).trans (mul_le_mul' hCC' le_rfl) |
| have hb := blockValue_mass_bound ambientSphereLaw hf hf1 hflocal hC' hC't |
| (by simpa using hd) (by simpa using ha) (by simpa using (show m ≤ d by omega)) |
| (by omega : 1 ≤ q-1) (by omega : q-1+m ≤ a) (measurable_haltRoute hτ n m j) |
| have he : stoppedContinuation τ w n k j = blockValue f (haltRoute τ n m j) := by |
| funext s |
| simpa [f,Nat.add_sub_of_le hmk] using stoppedContinuation_unroll hτ n m (k-m) j s |
| rw [he] |
| intro z r hr |
| apply (hb z r hr).trans |
| apply mul_le_mul' _ le_rfl |
| have hdiv : (k+q-1)/q = (k-m+q-1)/q+1 := by |
| by_cases hqk : q ≤ k |
| · have hme : m = q := min_eq_right hqk |
| rw [hme] |
| have he : k+q-1 = q+(k-q+q-1) := by omega |
| rw [he,Nat.add_div_left _ hq0] |
| · have hme : m = k := min_eq_left (by omega) |
| rw [hme,Nat.sub_self,Nat.zero_add,Nat.div_eq_of_lt (by omega : q-1 < q)] |
| have he : k+q-1 = q+(k-1) := by omega |
| rw [he,Nat.add_div_left _ hq0,Nat.div_eq_of_lt (by omega : k-1 < q)] |
| rw [hden] |
| have hcard : (Fintype.card (TaggedState M m):ℝ≥0∞)^(1/(q:ℝ)) ≤ |
| (Fintype.card (TaggedState M q):ℝ≥0∞)^(1/(q:ℝ)) := by |
| apply ENNReal.rpow_le_rpow _ (by positivity) |
| exact_mod_cast (show Fintype.card (TaggedState M m) ≤ Fintype.card (TaggedState M q) by |
| rw [card_taggedState,card_taggedState] |
| exact Nat.mul_le_mul_right _ (Nat.add_le_add_right hm 2)) |
| calc |
| _ ≤ K*C' := mul_le_mul' (mul_le_mul' le_rfl hcard) le_rfl |
| _ = _ := by rw [hdiv,pow_succ]; dsimp [C']; ring |
| |
| end |
| |
| variable {d M T : ℕ} |
|
|
| def taggedWeight (w : ℕ → State M → Vector d → ℝ≥0∞) (n : ℕ) : TaggedState M T → Vector d → ℝ≥0∞ |
| | .inl j => w n j |
| | .inr (t,j) => w t j |
|
|
| lemma measurable_taggedWeight {w : ℕ → State M → Vector d → ℝ≥0∞} |
| (hw : ∀ n j, Measurable (w n j)) (n : ℕ) (j : TaggedState M T) : |
| Measurable (taggedWeight w n j) := by cases j <;> exact hw _ _ |
| |
| lemma continuation_tagged_eq {τ : ℕ → State M → (Vector d × ℝ) → Bool × State M} |
| {w : ℕ → State M → Vector d → ℝ≥0∞} |
| (n k : ℕ) (hnk : n+k ≤ T) (j : TaggedState M T) (s : Vector d) : |
| continuation (taggedStep τ) (taggedWeight w) n k j s = |
| match j with |
| | .inl a => stoppedContinuation τ w n k a s |
| | .inr (t,a) => w t a s := by |
| induction k generalizing n j with |
| | zero => cases j <;> rfl |
| | succ k ih => |
| cases j with |
| | inl j => |
| simp only [continuation,stoppedContinuation] |
| apply lintegral_congr |
| intro x |
| rw [ih (n+1) (by omega)] |
| simp only [taggedStep,dite_eq_left (by omega : n+1 ≤ T)] |
| split_ifs <;> rfl |
| | inr done => |
| simp only [continuation,taggedStep] |
| simp only [ih (n+1) (by omega),lintegral_const,measure_univ,mul_one] |
| |
| lemma tagged_continuation_mass {τ : ℕ → State M → (Vector d × ℝ) → Bool × State M} |
| {w : ℕ → State M → Vector d → ℝ≥0∞} |
| (hτ : ∀ t j, Measurable (τ t j)) (hw : ∀ n j, Measurable (w n j)) |
| (hw1 : ∀ n j s, w n j s ≤ 1) |
| {a q : ℕ} {C : ℝ≥0∞} (hC : C ≠ 0) (hCt : C ≠ ∞) |
| (hlocal : ∀ n j, PolynomialMass (ambientSphereLaw.withDensity (w n j)) a C) |
| (hd : 1 ≤ d) (ha : a ≤ d) (hq : 2 ≤ q) (hqa : 2*q ≤ a) |
| (j : TaggedState M T) : |
| PolynomialMass (ambientSphereLaw.withDensity |
| (continuation (taggedStep τ) (taggedWeight w) 0 T j)) a |
| (C * streamFactor (E := Vector d) (J := TaggedState M q) q ^ ((T+q-1)/q)) := by |
| have he := fun s => continuation_tagged_eq (τ := τ) (w := w) 0 T (by omega) j s |
| cases j with |
| | inl j => |
| have he' : continuation (taggedStep (T := T) τ) (taggedWeight w) 0 T (.inl j) = stoppedContinuation τ w 0 T j := |
| funext he |
| rw [he'] |
| exact stoppedContinuation_mass_bound hτ hw hw1 hC hCt hlocal hd ha hq hqa 0 T j |
| | inr done => |
| have he' : continuation (taggedStep (T := T) τ) (taggedWeight w) 0 T (.inr done) = w done.1 done.2 := |
| funext he |
| rw [he'] |
| intro z r hr |
| apply (hlocal _ _ z r hr).trans |
| apply mul_le_mul' _ le_rfl |
| exact le_mul_of_one_le_right' (one_le_pow₀ (one_le_streamFactor (by omega))) |
|
|
| variable [NeZero d] |
|
|
| lemma stopped_route_mixed_bound {τ : ℕ → State M → (Vector d × ℝ) → Bool × State M} |
| (hτ : ∀ t j, Measurable (τ t j)) (y : ℕ → State M → Vector d) |
| {ε : ℝ} (hε : 0 < ε) (hε1 : ε ≤ 1) {a q : ℕ} {b : ℝ} |
| (hb : 0 ≤ b) (hab : (a:ℝ)+b ≤ (d-1:ℕ)) |
| (hq : 2 ≤ q) (hqa : 2*q ≤ a) (j : TaggedState M T) : |
| ((uniformSphere d).prod (sampleLaw d T)) |
| {z | dist z.1.val |
| (match streamRoute (taggedStep (T := T) τ) 0 T j (z.2,fun i => ⟪z.1.val,z.2 i⟫) with |
| | .inl a => y T a |
| | .inr (t,a) => y t a) ≤ ε} ≤ |
| (ENNReal.ofReal ((3:ℝ)*4^d)*ENNReal.ofReal (ε^b)) * |
| streamFactor (E := Vector d) (J := TaggedState M q) q ^ ((T+q-1)/q) := by |
| let w : ℕ → State M → Vector d → ℝ≥0∞ := fun n j => capWeight (y n j) ε |
| let δ := taggedStep (T := T) τ |
| let w' : ℕ → TaggedState M T → Vector d → ℝ≥0∞ := taggedWeight w |
| let yy : TaggedState M T → Vector d := fun a => match a with |
| | .inl a => y T a |
| | .inr (t,a) => y t a |
| have hw' (n : ℕ) (a : TaggedState M T) : Measurable (w' n a) := |
| measurable_taggedWeight (fun n j => measurable_capWeight _ _) n a |
| have hδ := measurable_taggedStep (T := T) hτ |
| have hV := measurable_continuation hδ hw' 0 T j |
| have he (s : Vector d) : continuation δ w' 0 T j s = |
| ∫⁻ A : Fin T → Vector d, capWeight (yy (streamRoute δ 0 T j (A,fun i => ⟪s,A i⟫))) ε s |
| ∂Measure.pi (fun _ => stdGaussian (Vector d)) := by |
| have hu := continuation_unroll (w := w') hδ 0 T 0 j s |
| simp only [Nat.add_zero,continuation] at hu |
| rw [hu] |
| unfold blockValue |
| apply lintegral_congr |
| intro A |
| cases streamRoute δ 0 T j (A,fun i => ⟪s,A i⟫) <;> simp [w',w,yy,taggedWeight] |
| let S : Set (UnitSphere d × Samples d T) := |
| {z | dist z.1.val (yy (streamRoute δ 0 T j (z.2,fun i => ⟪z.1.val,z.2 i⟫))) ≤ ε} |
| have hS : MeasurableSet S := by |
| apply measurableSet_le _ measurable_const |
| exact (measurable_subtype_coe.comp measurable_fst).dist |
| ((measurable_of_finite yy).comp ((measurable_streamRoute hδ 0 T j).comp (by fun_prop))) |
| change ((sphereLaw (E := Vector d)).prod _) S ≤ _ |
| rw [← one_mul (((sphereLaw (E := Vector d)).prod _) S), ← lintegral_indicator_const hS 1, |
| lintegral_prod _ (measurable_const.indicator hS).aemeasurable] |
| have hf : ∀ s : UnitSphere d, |
| (∫⁻ A, S.indicator (fun _ => (1:ℝ≥0∞)) (s,A) ∂sampleLaw d T) = continuation δ w' 0 T j s.val := by |
| intro s |
| rw [he] |
| apply lintegral_congr |
| intro A |
| classical |
| change (if dist s.val (yy (streamRoute δ 0 T j (A,fun i => ⟪s.val,A i⟫))) ≤ ε then 1 else 0) = _ |
| unfold capWeight |
| rw [Set.indicator_apply] |
| rfl |
| simp_rw [hf] |
| rw [← lintegral_map hV measurable_subtype_coe] |
| apply ambient_mass_bound (a := a) |
| have ha : a ≤ d := by |
| have hdn : ((d-1:ℕ):ℝ) ≤ (d:ℝ) := by exact_mod_cast Nat.sub_le d 1 |
| have : (a:ℝ) ≤ d := by linarith |
| exact_mod_cast this |
| have hl (n : ℕ) (j : State M) : PolynomialMass (ambientSphereLaw.withDensity (w n j)) a |
| (ENNReal.ofReal ((3:ℝ)*4^d)*ENNReal.ofReal (ε^b)) := by |
| simpa only [finrank_euclideanSpace_fin] using capWeight_mass_mixed (y n j) hε hε1 a hb (by simpa using hab) |
| exact tagged_continuation_mass hτ (fun n j => measurable_capWeight _ _) |
| (fun n j s => capWeight_le_one _ _ _) |
| (mul_ne_zero (by positivity) (by positivity)) |
| (ENNReal.mul_ne_top ENNReal.ofReal_ne_top ENNReal.ofReal_ne_top) |
| hl |
| (Nat.pos_of_ne_zero (NeZero.ne d)) ha hq hqa j |
|
|
| end Learner |
| end NoiselessRegression |
|
|
| end |
|
|
| end OAI |
|
|
