openAI-math-dataset / lean /OAI /Combinatorics /IndependentSets /Repetition /ScalarSampleInfluence.lean
| import Mathlib | |
| import OAI.Combinatorics.IndependentSets.Repetition.ScalarCell | |
| namespace OAI | |
| namespace LargeIndependentSets.BooleanJunta | |
| open MeasureTheory Set | |
| open scoped BigOperators Classical NNReal ENNReal | |
| noncomputable def scalarSampleInfluence {m : ℕ} (f : ℝ → ℝ) (i : Fin m) : ℝ := | |
| mean (fun q : Cube m => ∫ y in (0:ℝ)..1, | |
| |(f (scalarCell q y) - f (scalarCell (flip i q) y))/2|) | |
| lemma scalarSampleInfluence_succ {m : ℕ} (f : ℝ → ℝ) (i : Fin m) : | |
| scalarSampleInfluence (m:=m+1) f i.succ = | |
| (scalarSampleInfluence (m:=m) (f ∘ halfInput false) i + | |
| scalarSampleInfluence (m:=m) (f ∘ halfInput true) i)/2 := by | |
| rw [scalarSampleInfluence, mean_succ] | |
| rfl | |
| lemma scalarCell_difference {m : ℕ} {L : ℝ≥0} {f : ℝ → ℝ} | |
| (hf : LipschitzWith L f) (q r : Cube m) {y : ℝ} (hy : y ∈ Icc (0:ℝ) 1) : | |
| |f (scalarCell q y) - f (scalarCell r y)| ≤ variation f := by | |
| have hq := scalarCell_mem q hy | |
| have hr := scalarCell_mem r hy | |
| rcases le_total (scalarCell q y) (scalarCell r y) with h | h | |
| · rw [abs_sub_comm] | |
| exact CubeGradient.slice_difference_le_variation hf hq.1 h hr.2 | |
| · exact CubeGradient.slice_difference_le_variation hf hr.1 h hq.2 | |
| theorem dyadic_sample_total_influence {m : ℕ} {L : ℝ≥0} {f : ℝ → ℝ} | |
| (hf : LipschitzWith L f) : | |
| (∑ i, scalarSampleInfluence (m:=m) f i) ≤ variation f := by | |
| induction m generalizing L f with | |
| | zero => simpa using variation_nonneg f | |
| | succ m ih => | |
| rw [Fin.sum_univ_succ] | |
| have hhead : scalarSampleInfluence (m:=m+1) f 0 ≤ variation f/2 := by | |
| apply (mean_mono (fun q => ?_)).trans_eq | |
| (show mean (fun _ : Cube (m+1) => variation f/2) = variation f/2 by simp [mean]) | |
| have hc : Continuous (fun y => |(f (scalarCell q y) - f (scalarCell (flip 0 q) y))/2|) := by | |
| exact (((hf.continuous.comp (scalarCell_lipschitz q).continuous).sub | |
| (hf.continuous.comp (scalarCell_lipschitz (flip 0 q)).continuous)).div_const 2).abs | |
| have hh : (∫ y in (0:ℝ)..1, |(f (scalarCell q y)-f (scalarCell (flip 0 q) y))/2|) ≤ | |
| ∫ _y in (0:ℝ)..1, variation f/2 := by | |
| apply intervalIntegral.integral_mono_on (by norm_num) (hc.intervalIntegrable _ _) | |
| (continuous_const.intervalIntegrable _ _) | |
| intro y hy | |
| rw [abs_div, abs_of_pos (by norm_num : (0:ℝ)<2)] | |
| exact div_le_div_of_nonneg_right (scalarCell_difference hf _ _ hy) (by norm_num) | |
| simpa using hh | |
| have ht : (∑ i : Fin m, scalarSampleInfluence (m:=m+1) f i.succ) = | |
| ((∑ i : Fin m, scalarSampleInfluence (m:=m) (f ∘ halfInput false) i) + | |
| (∑ i : Fin m, scalarSampleInfluence (m:=m) (f ∘ halfInput true) i))/2 := by | |
| simp only [scalarSampleInfluence_succ] | |
| rw [← Finset.sum_div, Finset.sum_add_distrib] | |
| rw [ht] | |
| have h0 := ih (hf.comp (halfInput_lipschitz false)) | |
| have h1 := ih (hf.comp (halfInput_lipschitz true)) | |
| have hv := variation_half_sum hf | |
| linarith | |
| end LargeIndependentSets.BooleanJunta | |
| end OAI | |