openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Magnetic /ConstrainedBlockMagneticEstimate.lean
| import OAI.Probability.InvariantIsing.Magnetic.ConstrainedBlockBias | |
| namespace OAI | |
| /-! Finite constrained-block comparison with the actual scalar magnetic | |
| functional. Every error term is independent of the cascade depth. -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory IsingPerceptron | |
| open scoped BigOperators NNReal | |
| namespace InvariantIsing | |
| lemma sum_sites_group {N : ℕ} {A : Type*} [Fintype A] [DecidableEq A] | |
| (group : Fin N → A) (f : A → ℝ) : | |
| (∑ i, f (group i)) = ∑ a, spinGroupSize group a * f a := by | |
| rw [← Finset.sum_fiberwise Finset.univ group] | |
| apply Finset.sum_congr rfl | |
| intro a _ | |
| calc | |
| _ = ∑ _i ∈ Finset.univ.filter (fun i => group i = a), f a := by | |
| apply Finset.sum_congr rfl | |
| intro i hi | |
| rw [(Finset.mem_filter.mp hi).2] | |
| _ = _ := by simp [spinGroupSize] | |
| lemma spinGroupFieldConstant_of_means {N : ℕ} {A : Type*} [Fintype A] [DecidableEq A] | |
| (group : Fin N → A) (k : A → ℕ) (m bias : A → ℝ) | |
| (hc : ∀ a, (k a : ℝ) = spinGroupSize group a * ((1 + m a) / 2)) : | |
| spinGroupFieldConstant group k bias = ∑ i, bias (group i) * m (group i) := by | |
| rw [sum_sites_group group (fun a => bias a * m a)] | |
| unfold spinGroupFieldConstant | |
| apply Finset.sum_congr rfl | |
| intro a _ | |
| rw [hc a] | |
| ring | |
| theorem biasedConstrainedBlockValue_gap {N : ℕ} (hN : 0 < N) | |
| (S : Finset (Spin N)) (hS : S.Nonempty) (h : FieldStep) (b : Fin N → ℝ) : | |
| 0 ≤ biasedConstrainedBlockValue Finset.univ h b - biasedConstrainedBlockValue S h b ∧ | |
| biasedConstrainedBlockValue Finset.univ h b - biasedConstrainedBlockValue S h b ≤ | |
| (N : ℝ)⁻¹ * (∫ y, restrictedFieldTerminal Finset.univ y - restrictedFieldTerminal S y | |
| ∂fieldCanonicalVectorLaw N h b) := by | |
| have hF := restrictedFieldRecursion_biased_root_integrable hN Finset.univ Finset.univ_nonempty h b | |
| have hS' := restrictedFieldRecursion_biased_root_integrable hN S hS h b | |
| have he : biasedConstrainedBlockValue Finset.univ h b - biasedConstrainedBlockValue S h b = | |
| (N : ℝ)⁻¹ * (∫ z, restrictedFieldRecursion Finset.univ h.depth (chainExponent h.cut) | |
| (fieldStepVariance h) z - restrictedFieldRecursion S h.depth (chainExponent h.cut) | |
| (fieldStepVariance h) z | |
| ∂Measure.pi (fun i => gaussianReal (b i) (NNReal.mk (h.height 0) (h.nonneg 0)))) := by | |
| rw [integral_sub hF hS'] | |
| unfold biasedConstrainedBlockValue | |
| ring | |
| rw [he] | |
| constructor | |
| · apply mul_nonneg (by positivity) | |
| apply integral_nonneg | |
| intro z | |
| apply sub_nonneg.mpr | |
| exact restrictedFieldRecursion_mono hN S Finset.univ hS Finset.univ_nonempty | |
| (Finset.subset_univ _) _ _ _ | |
| (fun i hi => ((chainExponent_admissible h.ordered_cut h.first h.last).1 i hi).1) z | |
| · exact mul_le_mul_of_nonneg_left (restrictedFieldRecursion_root_loss hN S hS h b) (by positivity) | |
| lemma magneticBlockGap_eq_biasedGap {N : ℕ} (hN : 0 < N) | |
| {A : Type*} [Fintype A] [DecidableEq A] (group : Fin N → A) (k : A → ℕ) | |
| (hk : ∀ a, k a ≤ spinGroupSize group a) (m : A → ℝ) (hm : ∀ a, |m a| < 1) | |
| (hc : ∀ a, (k a : ℝ) = spinGroupSize group a * ((1 + m a) / 2)) (h : FieldStep) : | |
| (N : ℝ)⁻¹ * (∑ i, constrainedFieldValue h (m (group i))) - | |
| constrainedBlockValue (spinGroupSlice group k) h = | |
| biasedConstrainedBlockValue Finset.univ h (fun i => magneticBias h (m (group i))) - | |
| biasedConstrainedBlockValue (spinGroupSlice group k) h (fun i => magneticBias h (m (group i))) := by | |
| rw [biasedConstrainedBlockValue_univ hN, | |
| biasedConstrainedBlockValue_group hN group k hk (fun a => magneticBias h (m a)), | |
| spinGroupFieldConstant_of_means group k m (fun a => magneticBias h (m a)) hc] | |
| simp_rw [constrainedFieldValue_magneticBias h (hm _)] | |
| rw [Finset.sum_sub_distrib] | |
| ring | |
| theorem magneticBlockGap_estimate {N : ℕ} (hN : 0 < N) | |
| {A : Type*} [Fintype A] [DecidableEq A] (group : Fin N → A) (k : A → ℕ) | |
| (hk : ∀ a, k a ≤ spinGroupSize group a) (m : A → ℝ) (hm : ∀ a, |m a| < 1) | |
| (hc : ∀ a, (k a : ℝ) = spinGroupSize group a * ((1 + m a) / 2)) | |
| (h : FieldStep) {B H ε t : ℝ} (hB : ∀ i, |magneticBias h (m (group i))| ≤ B) | |
| (hH : h.height (Fin.last h.depth) ≤ H) (hε : 0 < ε) (ht : 0 ≤ t) : | |
| 0 ≤ (N : ℝ)⁻¹ * (∑ i, constrainedFieldValue h (m (group i))) - | |
| constrainedBlockValue (spinGroupSlice group k) h ∧ | |
| (N : ℝ)⁻¹ * (∑ i, constrainedFieldValue h (m (group i))) - | |
| constrainedBlockValue (spinGroupSlice group k) h ≤ | |
| (N : ℝ)⁻¹ * (ε * (N * (2 * Real.exp (2 * B + 4 * H))) + | |
| (ε⁻¹ + t) * (Fintype.card A * Real.sqrt N) + N * Real.log (1 + Real.exp (-t))) := by | |
| rw [magneticBlockGap_eq_biasedGap hN group k hk m hm hc] | |
| have hg := biasedConstrainedBlockValue_gap hN (spinGroupSlice group k) | |
| (spinGroupSlice_nonempty group k hk) h (fun i => magneticBias h (m (group i))) | |
| refine ⟨hg.1, hg.2.trans ?_⟩ | |
| apply mul_le_mul_of_nonneg_left _ (by positivity) | |
| have he := magnetic_block_terminal_loss_size group k hk m hm hc h hB hH hε ht | |
| simpa only [restrictedFieldTerminal_univ] using he | |
| end InvariantIsing | |
| end | |
| end OAI | |