File size: 3,377 Bytes
dd14c26 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 | 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
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