joint_empirical_squared_risk_replacement

Plain-English statement

Compiled generic library interface. Exact hypotheses, actual integrals and remaining global-law boundaries are exposed in the authored lesson. Current public context-bound independent semantic reviews are accepted; serialized stabilization remains pending; no stochastic algorithm or model theorem is claimed.

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Lean statement

theorem joint_empirical_squared_risk_replacement
    {W X : Type*} [MeasurableSpace W] [MeasurableSingletonClass W]
    [MeasurableSpace X] (P : Measure (X × ℝ)) [IsProbabilityMeasure P]
    (K : W → X → ℝ) (b : ℝ) (hb : 0 < b)
    (hfeature : ∀ q, MemLp (K q) 2 (P.map Prod.fst))
    (hbound : ∀ q, ‖(hfeature q).toLp (K q)‖ ≤ b)
    (hlabel : MemLp (fun z : X × ℝ => z.2) 2 P)
    {N : ℕ} (hN : 0 < N) (w : Fin N → W) (i : Fin N) (u : W) :
    let rho : Measure W := (N : ℝ≥0∞)⁻¹ • ∑ j, Measure.dirac (w j)
    let rhoNew : Measure W :=
      (N : ℝ≥0∞)⁻¹ • ∑ j, Measure.dirac (Function.update w i u j)
    let predictor : Measure W → X → ℝ := fun law x => ∫ q, K q x ∂law
    let risk : Measure W → ℝ :=
      fun law => ∫ z, (predictor law z.1 - z.2) ^ 2 ∂P
    let potential : W → ℝ :=
      fun q => 2 * ∫ z, (predictor rho z.1 - z.2) * K q z.1 ∂P
    let pairing : ℝ := (∫ q, potential q ∂rhoNew) - (∫ q, potential q ∂rho)
    (0 < 4 * b ^ 2) ∧ IsProbabilityMeasure rho ∧ IsProbabilityMeasure rhoNew ∧
    Integrable (fun z : X × ℝ => (predictor rho z.1 - z.2) ^ 2) P ∧
    Integrable (fun z : X × ℝ => (predictor rhoNew z.1 - z.2) ^ 2) P ∧
    Integrable potential rho ∧ Integrable potential rhoNew ∧
    (pairing = (N : ℝ)⁻¹ * (potential u - potential (w i))) ∧
    (risk rhoNew - risk rho - pairing =
      ∫ x, (predictor rhoNew x - predictor rho x) ^ 2 ∂(P.map Prod.fst)) ∧
    (0 ≤ risk rhoNew - risk rho - pairing) ∧
    (risk rhoNew - risk rho - pairing ≤ 4 * b ^ 2 / (N : ℝ) ^ 2) := by
  classical
  dsimp only
  let rhoOf : (Fin N → W) → Measure W :=
    fun cfg => (N : ℝ≥0∞)⁻¹ • ∑ j, Measure.dirac (cfg j)
  let cfgNew := Function.update w i u
  let predictor : Measure W → X → ℝ := fun law x => ∫ q, K q x ∂law
  let risk : Measure W → ℝ := fun law => ∫ z, (predictor law z.1 - z.2) ^ 2 ∂P
  let potential : W → ℝ :=
    fun q => 2 * ∫ z, (predictor (rhoOf w) z.1 - z.2) * K q z.1 ∂P
  let pairing : ℝ := (∫ q, potential q ∂rhoOf cfgNew) - (∫ q, potential q ∂rhoOf w)
  change (0 < 4 * b ^ 2) ∧ IsProbabilityMeasure (rhoOf w) ∧
    IsProbabilityMeasure (rhoOf cfgNew) ∧ _
  have hmarg : MeasurePreserving (Prod.fst : X × ℝ → X) P (P.map Prod.fst) :=
    measurable_fst.measurePreserving P
  have hfeatureP (q : W) : MemLp (fun z : X × ℝ => K q z.1) 2 P :=
    (hfeature q).comp_measurePreserving hmarg
  let feature : W → Lp ℝ 2 P :=
    fun q => (hfeatureP q).toLp (fun z : X × ℝ => K q z.1)
  let label : Lp ℝ 2 P := hlabel.toLp (fun z : X × ℝ => z.2)
  let mean : (Fin N → W) → Lp ℝ 2 P :=
    fun cfg => (N : ℝ)⁻¹ • ∑ j, feature (cfg j)
  have hfeatureClass (q : W) : feature q =
      Lp.compMeasurePreserving Prod.fst hmarg ((hfeature q).toLp (K q)) := rfl
  have hnorm (q : W) : ‖feature q‖ ≤ b := by
    rw [hfeatureClass, Lp.norm_compMeasurePreserving]
    exact hbound q
  have hinv : (N : ℝ≥0∞)⁻¹ ≠ ∞ := by
    apply ENNReal.inv_ne_top.mpr
    exact_mod_cast hN.ne'
  have hInt (cfg : Fin N → W) (g : W → ℝ) :
      Integrable g (rhoOf cfg) := by
    apply Integrable.smul_measure (c := (N : ℝ≥0∞)⁻¹) _ hinv
    exact integrable_finsetSum_measure.mpr (fun j _ => integrable_dirac (by simp))
  have hEval (cfg : Fin N → W) (g : W → ℝ) :
      (∫ q, g q ∂rhoOf cfg) = (N : ℝ)⁻¹ * ∑ j, g (cfg j) := by
    dsimp only [rhoOf]
    rw [integral_smul_measure,
      integral_finsetSum_measure (fun j _ => integrable_dirac (by simp))]
    simp [smul_eq_mul]
  have hPred (cfg : Fin N → W) (x : X) :
      predictor (rhoOf cfg) x = (N : ℝ)⁻¹ * ∑ j, K (cfg j) x :=
    hEval cfg (fun q => K q x)
  have hBase (cfg : Fin N → W) :=
    empirical_law_squared_risk_eq_L2_norm P hN cfg
      (fun q z => K q z.1) (fun z => z.2) (fun j => hfeatureP (cfg j)) hlabel
  have hRisk (cfg : Fin N → W) : risk (rhoOf cfg) = ‖mean cfg - label‖ ^ 2 := by
    simpa only [risk, predictor, rhoOf, mean, feature, label] using (hBase cfg).2.2
  have hMean (cfg : Fin N → W) :
      (fun z => mean cfg z) =ᵐ[P] (fun z : X × ℝ => predictor (rhoOf cfg) z.1) := by
    filter_upwards [Lp.coeFn_smul (N : ℝ)⁻¹ (∑ j, feature (cfg j)),
      Lp.coeFn_fun_finsetSum Finset.univ (fun j => feature (cfg j)),
      ae_all_iff.2 (fun j => (hfeatureP (cfg j)).coeFn_toLp)]
      with z hsmul hsum hf
    change (((N : ℝ)⁻¹ • ∑ j, feature (cfg j)) z) = _
    simp only [Pi.smul_apply, smul_eq_mul] at hsmul
    rw [hsmul, hsum, hPred]
    congr 1
    apply Finset.sum_congr rfl
    intro j _
    exact hf j
  have hResidual : (fun z => (mean w - label) z) =ᵐ[P]
      (fun z : X × ℝ => predictor (rhoOf w) z.1 - z.2) := by
    filter_upwards [Lp.coeFn_sub (mean w) label, hMean w, hlabel.coeFn_toLp]
      with z hsub hm hy
    simp only [Pi.sub_apply] at hsub
    rw [hsub, hm, hy]
  have hPotential (q : W) : potential q = 2 * ⟪mean w - label, feature q⟫_ℝ := by
    dsimp only [potential]
    rw [L2.inner_def]
    congr 1
    apply integral_congr_ae
    filter_upwards [hResidual, (hfeatureP q).coeFn_toLp] with z hr hq
    change feature q z = K q z.1 at hq
    simp only [Real.inner_apply, hr, hq]
  have hPotMean (cfg : Fin N → W) :
      (∫ q, potential q ∂rhoOf cfg) = 2 * ⟪mean w - label, mean cfg⟫_ℝ := by
    rw [hEval]
    simp_rw [hPotential]
    dsimp only [mean]
    rw [real_inner_smul_right, inner_sum, ← Finset.mul_sum]
    ring
  have hPair : pairing = 2 * ⟪mean w - label, mean cfgNew - mean w⟫_ℝ := by
    dsimp only [pairing]
    rw [hPotMean, hPotMean, inner_sub_right]
    ring
  have hUpdate : (fun j => feature (cfgNew j)) =
      Function.update (fun j => feature (w j)) i (feature u) := by
    funext j
    by_cases hji : j = i
    · subst j
      simp [cfgNew]
    · simp [cfgNew, Function.update_of_ne hji]
  have hBudget := empirical_squared_loss_replacement hN
    (fun j => feature (w j)) i (feature u) label b (hnorm (w i)) (hnorm u)
  change (‖(N : ℝ)⁻¹ • ∑ j, Function.update (fun j => feature (w j)) i (feature u) j - label‖ ^ 2 -
      ‖mean w - label‖ ^ 2 -
      2 * ⟪mean w - label,
        (N : ℝ)⁻¹ • ∑ j, Function.update (fun j => feature (w j)) i (feature u) j - mean w⟫_ℝ =
      ‖(N : ℝ)⁻¹ • ∑ j, Function.update (fun j => feature (w j)) i (feature u) j - mean w‖ ^ 2) ∧ _ at hBudget
  rw [← hUpdate] at hBudget
  change (‖mean cfgNew - label‖ ^ 2 - ‖mean w - label‖ ^ 2 -
      2 * ⟪mean w - label, mean cfgNew - mean w⟫_ℝ = ‖mean cfgNew - mean w‖ ^ 2) ∧
    ‖mean cfgNew - mean w‖ ^ 2 ≤ 4 * b ^ 2 / (N : ℝ) ^ 2 at hBudget
  have hBre : risk (rhoOf cfgNew) - risk (rhoOf w) - pairing =
      ‖mean cfgNew - mean w‖ ^ 2 := by
    rw [hRisk, hRisk, hPair]
    exact hBudget.1
  have hMeanDiff : (fun z => (mean cfgNew - mean w) z) =ᵐ[P]
      (fun z : X × ℝ => predictor (rhoOf cfgNew) z.1 - predictor (rhoOf w) z.1) := by
    filter_upwards [Lp.coeFn_sub (mean cfgNew) (mean w), hMean cfgNew, hMean w]
      with z hsub hn hm
    simp only [Pi.sub_apply] at hsub
    rw [hsub, hn, hm]
  have hPredLp (cfg : Fin N → W) : MemLp (predictor (rhoOf cfg)) 2 (P.map Prod.fst) := by
    have havg := (memLp_finsetSum Finset.univ (fun j _ => hfeature (cfg j))).const_smul (N : ℝ)⁻¹
    exact MemLp.ae_eq (Filter.Eventually.of_forall (fun x => (hPred cfg x).symm)) havg
  have hDiffLp := (hPredLp cfgNew).sub (hPredLp w)
  have hNormInt : ‖mean cfgNew - mean w‖ ^ 2 =
      ∫ x, (predictor (rhoOf cfgNew) x - predictor (rhoOf w) x) ^ 2 ∂(P.map Prod.fst) := by
    rw [← real_inner_self_eq_norm_sq, L2.inner_def]
    calc
      (∫ z, ⟪(mean cfgNew - mean w) z, (mean cfgNew - mean w) z⟫_ℝ ∂P) =
          ∫ z : X × ℝ, (predictor (rhoOf cfgNew) z.1 - predictor (rhoOf w) z.1) ^ 2 ∂P := by
        apply integral_congr_ae
        filter_upwards [hMeanDiff] with z hz
        simp only [Real.inner_apply, hz, pow_two]
      _ = _ := (integral_map measurable_fst.aemeasurable hDiffLp.integrable_sq.aestronglyMeasurable).symm
  have hPairUpdate : pairing = (N : ℝ)⁻¹ * (potential u - potential (w i)) := by
    dsimp only [pairing]
    rw [hEval, hEval, ← mul_sub, ← Finset.sum_sub_distrib, Finset.sum_eq_single i]
    · simp [cfgNew]
    · intro j _ hji
      simp [cfgNew, Function.update_of_ne hji]
    · simp
  refine ⟨mul_pos (by norm_num) (sq_pos_of_pos hb), (hBase w).1, (hBase cfgNew).1,
    (hBase w).2.1, (hBase cfgNew).2.1, hInt w potential, hInt cfgNew potential,
    hPairUpdate, ?_, ?_, ?_⟩
  · exact hBre.trans hNormInt
  · rw [hBre]
    exact sq_nonneg _
  · rw [hBre]
    exact hBudget.2

end AutoSamplingTheory.TechnicalLemmas.Analysis.QuadraticFeatureRisk

Open AutoSamplingTheory/TechnicalLemmas/Analysis/QuadraticFeatureRisk.lean:224published source at ed709a7f320d

Proof architecture

Reusable squared-loss, genuine integral and indexed replacement calculations

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Integrability is an input or proved output; a displayed integral alone does not supply it.
Almost-everywhere hypotheses depend on the stated measure and representative.