lean/OAI/Probability/LogConcave/Model.lean

import Mathlib

namespace OAI

section

section

noncomputable section

open MeasureTheory Filter
open scoped ENNReal NNReal Topology

namespace LogConcaveSampling

abbrev Point (d : ℕ) := EuclideanSpace ℝ (Fin d)
abbrev Reply (d : ℕ) := ℝ × Point d
abbrev Transcript (d q : ℕ) := Fin q → Reply d

/-- Normalized twice continuously differentiable potentials with Hessian between the identity and twice the identity. -/
structure Admissible {d : ℕ} (V : Point d → ℝ) : Prop where
  smooth : ContDiff ℝ 2 V
  value_zero : V 0 = 0
  gradient_zero : gradient V 0 = 0
  hessian_bounds : ∀ x v : Point d,
    ‖v‖ ^ 2 ≤ inner ℝ v (fderiv ℝ (gradient V) x v) ∧
    inner ℝ v (fderiv ℝ (gradient V) x v) ≤ 2 * ‖v‖ ^ 2

/-- The unnormalized Gibbs density relative to Euclidean volume. -/
def gibbsDensity {d : ℕ} (V : Point d → ℝ) (x : Point d) : ℝ≥0∞ :=
  ENNReal.ofReal (Real.exp (-V x))

def partition {d : ℕ} (V : Point d → ℝ) : ℝ≥0∞ :=
  ∫⁻ x, gibbsDensity V x ∂volume

/-- The Gibbs probability law, normalized by its partition function. -/
def gibbs {d : ℕ} (V : Point d → ℝ) : Measure (Point d) :=
  (partition V)⁻¹ • volume.withDensity (gibbsDensity V)

def TVAtMost {E : Type*} [MeasurableSpace E]
    (μ ν : Measure E) (ε : ℝ) : Prop :=
  ∀ s : Set E, MeasurableSet s → |μ.real s - ν.real s| ≤ ε

def firstOrderReply {d : ℕ} (V : Point d → ℝ) (x : Point d) : Reply d :=
  (V x, gradient V x)

/-- Measurable adaptive queries and output, parametrized by private randomness. -/
structure OracleAlgorithm (Ω : Type*) [MeasurableSpace Ω] (d q : ℕ) where
  query : Fin q → Ω × Transcript d q → Point d
  query_measurable : ∀ i, Measurable (query i)
  output : Ω × Transcript d q → Point d
  output_measurable : Measurable output

def OracleAlgorithm.history {Ω : Type*} [MeasurableSpace Ω] {d q : ℕ}
    (A : OracleAlgorithm Ω d q) (V : Point d → ℝ) (ω : Ω) : ℕ → Transcript d q
  | 0 => 0
  | n + 1 =>
      let h := A.history V ω n
      if hn : n < q then
        Function.update h ⟨n, hn⟩ (firstOrderReply V (A.query ⟨n, hn⟩ (ω, h)))
      else h

def OracleAlgorithm.run {Ω : Type*} [MeasurableSpace Ω] {d q : ℕ}
    (A : OracleAlgorithm Ω d q) (V : Point d → ℝ) (ω : Ω) : Point d :=
  A.output (ω, A.history V ω q)

/-- A uniform sampler with a deterministic query budget on every execution and total-variation error at most one tenth. -/
def CanSample (d q : ℕ) : Prop :=
  ∃ (Ω : Type) (_ : MeasurableSpace Ω) (μ : Measure Ω)
    (_ : IsProbabilityMeasure μ) (A : OracleAlgorithm Ω d q),
    ∀ V : Point d → ℝ, Admissible V →
      Measurable (A.run V) ∧ TVAtMost (μ.map (A.run V)) (gibbs V) (1 / 10)

/-- The smallest feasible integer query budget, or infinity when no budget is feasible. -/
def queryComplexity (d : ℕ) : ℕ∞ :=
  sInf ((fun q : ℕ => (q : ℕ∞)) '' {q | CanSample d q})

/-- The quantified dimension-exponent upper bound with every positive exponent slack. -/
def ExponentBound (γ : ℝ) : Prop :=
  0 ≤ γ ∧ ∀ ε : ℝ, 0 < ε → ∃ d₀ : ℕ, ∀ d : ℕ, d₀ ≤ d →
    (queryComplexity d).toENNReal ≤ ENNReal.ofReal ((d : ℝ) ^ (γ + ε))

def gammaStar : ℝ := sInf {γ : ℝ | ExponentBound γ}

end LogConcaveSampling

end
end
end

end OAI