openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /JointExtraction /CompatibilityScaling.lean
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.CompatibilityEntropyBound | |
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.AmbientRateLimit | |
| namespace OAI | |
| /-! Joint tensor extraction, compatibility and entropy estimates. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.JointCompatibilityScaling | |
| open MatrixMultiplication.Foundation JointTypeCounts JointCompatibilityIncidence | |
| open JointCompatibilityEntropyBound Filter | |
| open scoped BigOperators Topology | |
| attribute [local instance] Classical.propDecidable | |
| variable {C : Type*} [Fintype C] [DecidableEq C] | |
| variable {Shape : C → Type*} | |
| [∀ c, Fintype (Shape c)] [∀ c, DecidableEq (Shape c)] | |
| def baseSize (base : ∀ c, Shape c → ℕ) (c : C) : ℕ := ∑ a, base c a | |
| def baseTotal (base : ∀ c, Shape c → ℕ) : ℕ := ∑ c, baseSize base c | |
| def baseEntropy (base : ∀ c, Shape c → ℕ) : ℝ := | |
| ∑ c, (baseSize base c : ℝ) * | |
| finiteEntropy (fun a => (base c a : ℝ) / baseSize base c) | |
| def baseGroupSize (base : ∀ c, Shape c → ℕ) (d : ∀ c, Shape c → Prop) | |
| (g : Σ c, Option (Shape c)) : ℕ := | |
| ∑ a : {a : Shape g.1 // designatedGroup (d g.1) a = g.2}, base g.1 a.val | |
| def baseGroupedEntropy (base : ∀ c, Shape c → ℕ) (d : ∀ c, Shape c → Prop) | |
| (H : (Σ c, Option (Shape c)) → ℝ) : ℝ := | |
| ∑ g, (baseGroupSize base d g : ℝ) * H g | |
| omit [DecidableEq C] [∀ c, DecidableEq (Shape c)] in | |
| theorem baseSize_le_baseTotal (base : ∀ c, Shape c → ℕ) (c : C) : | |
| baseSize base c ≤ baseTotal base := | |
| Finset.single_le_sum (fun _ _ => Nat.zero_le _) (Finset.mem_univ c) | |
| omit [DecidableEq C] in | |
| theorem sum_baseGroupSize (base : ∀ c, Shape c → ℕ) | |
| (d : ∀ c, Shape c → Prop) : | |
| (∑ g, baseGroupSize base d g) = baseTotal base := by | |
| simp only [baseGroupSize, baseTotal, Fintype.sum_sigma] | |
| apply Finset.sum_congr rfl | |
| intro c _ | |
| exact Fintype.sum_fiberwise (designatedGroup (d c)) (base c) | |
| section Repeated | |
| variable (Pos : C → Type*) | |
| [∀ c, Fintype (Pos c)] [∀ c, DecidableEq (Pos c)] | |
| omit [DecidableEq C] [∀ c, DecidableEq (Shape c)] [∀ c, DecidableEq (Pos c)] in | |
| theorem classEntropy_repeated (base : ∀ c, Shape c → ℕ) (t : ℕ) (ht : 0 < t) | |
| (hsize : ∀ c, Fintype.card (Pos c) = t * baseSize base c) : | |
| classEntropy Pos Shape (fun c a => t * base c a) = (t : ℝ) * baseEntropy base := by | |
| simp only [classEntropy, baseEntropy, hsize, Nat.cast_mul, Finset.mul_sum] | |
| apply Finset.sum_congr rfl | |
| intro c _ | |
| rw [show (fun a => (t : ℝ) * base c a / ((t : ℝ) * baseSize base c)) = | |
| (fun a => (base c a : ℝ) / baseSize base c) by | |
| funext a | |
| exact mul_div_mul_left _ _ (Nat.cast_ne_zero.mpr (Nat.ne_of_gt ht))] | |
| ring | |
| omit [Fintype C] [DecidableEq C] [∀ c, DecidableEq (Pos c)] in | |
| theorem groupPos_card_repeated (base : ∀ c, Shape c → ℕ) | |
| (d : ∀ c, Shape c → Prop) (t : ℕ) | |
| (w : FixedClassWords Pos Shape (fun c a => t * base c a)) | |
| (g : Σ c, Option (Shape c)) : | |
| Fintype.card (GroupPos (w g.1).val (d g.1) g.2) = | |
| t * baseGroupSize base d g := by | |
| calc | |
| Fintype.card (GroupPos (w g.1).val (d g.1) g.2) = | |
| Fintype.card (Σ a : {a : Shape g.1 // designatedGroup (d g.1) a = g.2}, | |
| {i : Pos g.1 // (w g.1).val i = a.val}) := | |
| Fintype.card_congr (Equiv.sigmaSubtypeFiberEquivSubtype (w g.1).val | |
| (p := fun i => designatedGroup (d g.1) ((w g.1).val i) = g.2) | |
| (q := fun a => designatedGroup (d g.1) a = g.2) (fun _ => Iff.rfl)).symm | |
| _ = t * baseGroupSize base d g := by | |
| simp only [Fintype.card_sigma] | |
| change (∑ a : {a : Shape g.1 // designatedGroup (d g.1) a = g.2}, | |
| wordPopulation (w g.1).val a.val) = _ | |
| simp only [(w g.1).property, baseGroupSize, Finset.mul_sum] | |
| omit [DecidableEq C] [∀ c, DecidableEq (Pos c)] in | |
| theorem groupedEntropy_repeated (base : ∀ c, Shape c → ℕ) | |
| (d : ∀ c, Shape c → Prop) (t : ℕ) | |
| (H : (Σ c, Option (Shape c)) → ℝ) | |
| (w : FixedClassWords Pos Shape (fun c a => t * base c a)) : | |
| groupedEntropy Pos Shape (fun c a => t * base c a) d H w = | |
| (t : ℝ) * baseGroupedEntropy base d H := by | |
| simp only [JointCompatibilityEntropyBound.groupedEntropy, groupPos_card_repeated, Nat.cast_mul, | |
| baseGroupedEntropy, Finset.mul_sum, mul_assoc] | |
| end Repeated | |
| def typeErrorFactor (B t : ℕ) : ℝ := 1 + Real.log (t * B + 1 : ℕ) | |
| theorem typeErrorFactor_nonneg (B t : ℕ) : 0 ≤ typeErrorFactor B t := by | |
| unfold typeErrorFactor | |
| have hlog : 0 ≤ Real.log (t * B + 1 : ℕ) := | |
| Real.log_nonneg (by exact_mod_cast Nat.succ_le_succ (Nat.zero_le (t * B))) | |
| linarith | |
| def classErrorCoefficient (A : C → Type*) [∀ c, Fintype (A c)] : ℝ := | |
| ∑ c, ((Fintype.card (A c) : ℝ) + 1) | |
| def groupErrorCoefficient (Shape A : C → Type*) | |
| [∀ c, Fintype (Shape c)] [∀ c, Fintype (A c)] : ℝ := | |
| ∑ g : Σ c, Option (Shape c), (2 * (Fintype.card (A g.1) : ℝ) + 1) | |
| omit [DecidableEq C] in | |
| theorem classTypeError_le (Pos A : C → Type*) | |
| [∀ c, Fintype (Pos c)] [∀ c, DecidableEq (Pos c)] | |
| [∀ c, Fintype (A c)] [∀ c, DecidableEq (A c)] | |
| (B t : ℕ) (hsize : ∀ c, Fintype.card (Pos c) ≤ t * B) : | |
| classTypeError Pos A ≤ classErrorCoefficient A * typeErrorFactor B t := by | |
| simp only [classTypeError, classErrorCoefficient, Finset.sum_mul] | |
| apply Finset.sum_le_sum | |
| intro c _ | |
| apply mul_le_mul_of_nonneg_left _ (by positivity) | |
| apply add_le_add_right | |
| apply Real.log_le_log (by positivity) | |
| exact_mod_cast Nat.succ_le_succ (hsize c) | |
| theorem wordTypeError_le (I A : Type*) [Fintype I] [Fintype A] | |
| (B t : ℕ) (hsize : Fintype.card I ≤ t * B) : | |
| wordTypeError I A ≤ (2 * (Fintype.card A : ℝ) + 1) * typeErrorFactor B t := by | |
| have hlog : Real.log (Fintype.card I + 1 : ℕ) ≤ Real.log (t * B + 1 : ℕ) := | |
| Real.log_le_log (by positivity) (by exact_mod_cast Nat.succ_le_succ hsize) | |
| have h₁ := mul_le_mul_of_nonneg_left (add_le_add_left hlog 1) | |
| (by positivity : (0 : ℝ) ≤ (Fintype.card A : ℝ) + 1) | |
| have h₂ := mul_le_mul_of_nonneg_left hlog (Nat.cast_nonneg (Fintype.card A) : | |
| (0 : ℝ) ≤ Fintype.card A) | |
| unfold wordTypeError typeErrorFactor | |
| nlinarith [show (0 : ℝ) ≤ Fintype.card A from Nat.cast_nonneg _] | |
| omit [DecidableEq C] in | |
| theorem groupedTypeError_le (Pos A : C → Type*) | |
| [∀ c, Fintype (Pos c)] [∀ c, DecidableEq (Pos c)] | |
| [∀ c, Fintype (A c)] [∀ c, DecidableEq (A c)] | |
| (counts : ∀ c, Shape c → ℕ) (d : ∀ c, Shape c → Prop) | |
| (w : FixedClassWords Pos Shape counts) | |
| (B t : ℕ) (hsize : ∀ c, Fintype.card (Pos c) ≤ t * B) : | |
| groupedTypeError Pos Shape A counts d w ≤ | |
| groupErrorCoefficient Shape A * typeErrorFactor B t := by | |
| simp only [groupedTypeError, groupErrorCoefficient, Finset.sum_mul] | |
| apply Finset.sum_le_sum | |
| intro g _ | |
| apply wordTypeError_le | |
| exact (Fintype.card_subtype_le _).trans (hsize g.1) | |
| theorem tendsto_typeErrorFactor_div (B : ℕ) : | |
| Tendsto (fun t : ℕ => typeErrorFactor B t / (t : ℝ)) atTop (𝓝 0) := by | |
| have hratio : Tendsto (fun t : ℕ => ((t * B : ℕ) : ℝ) / (t : ℝ)) | |
| atTop (𝓝 (B : ℝ)) := by | |
| apply tendsto_const_nhds.congr' | |
| filter_upwards [eventually_ne_atTop (0 : ℕ)] with t ht | |
| simp only [Nat.cast_mul] | |
| exact (mul_div_cancel_left₀ (B : ℝ) (Nat.cast_ne_zero.mpr ht)).symm | |
| have hlog := JointAmbientRateLimit.tendsto_log_succ_div_nat_of_ratio hratio | |
| have hinv : Tendsto (fun t : ℕ => (1 : ℝ) / t) atTop (𝓝 0) := | |
| tendsto_const_nhds.div_atTop tendsto_natCast_atTop_atTop | |
| simpa only [typeErrorFactor, add_div, add_zero] using hinv.add hlog | |
| theorem tendsto_mul_typeErrorFactor_div (K : ℝ) (B : ℕ) : | |
| Tendsto (fun t : ℕ => K * typeErrorFactor B t / (t : ℝ)) atTop (𝓝 0) := by | |
| simpa only [mul_div_assoc, mul_zero] using (tendsto_typeErrorFactor_div B).const_mul K | |
| end MatrixMultiplication.JointCompatibilityScaling | |
| end | |
| end OAI | |