| import Mathlib |
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| namespace OAI |
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| open MeasureTheory ProbabilityTheory Set Filter EuclideanGeometry |
| open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise |
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| noncomputable section |
| universe u v |
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| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise |
| namespace MemoryPrecision.Kernel |
| abbrev Vector (d : ℕ) := EuclideanSpace ℝ (Fin d) |
| abbrev Sphere (d : ℕ) := Metric.sphere (0 : Vector d) 1 |
| abbrev State (M : ℕ) := Fin (2^M) |
| abbrev Action (M : ℕ) := Bool × State M |
| abbrev Observations (d k : ℕ) := Fin k → (Vector d × ℝ) |
| abbrev Rows (d k : ℕ) := Fin k → Vector d |
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| def UniformSphere (d : ℕ) : Measure (Sphere d) := |
| let euclideanGroup : NormedAddCommGroup (Vector d) := |
| PiLp.normedAddCommGroup |
| (@OfNat.ofNat ℝ≥0∞ 2 |
| (@instOfNatAtLeastTwo ℝ≥0∞ 2 ENNReal.instAddCommMonoidWithOne.toNatCast inferInstance)) |
| (fun _ : Fin d => ℝ) |
| let euclideanInner := PiLp.innerProductSpace (𝕜 := ℝ) (fun _ : Fin d => ℝ) |
| let σ := @Measure.toSphere (Vector d) euclideanGroup |
| (@InnerProductSpace.toNormedSpace ℝ (Vector d) _ |
| euclideanGroup.toSeminormedAddCommGroup euclideanInner) inferInstance |
| (@volume (Vector d) (@measureSpaceOfInnerProductSpace (Vector d) |
| euclideanGroup euclideanInner inferInstance inferInstance inferInstance)) |
| (σ Set.univ)⁻¹ • σ |
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| def RowLaw (d k : ℕ) : @Measure (Rows d k) |
| (@MeasurableSpace.pi (Fin k) (fun _ => Vector d) |
| (fun _ => @WithLp.measurableSpace |
| (@OfNat.ofNat ℝ≥0∞ 2 |
| (@instOfNatAtLeastTwo ℝ≥0∞ 2 ENNReal.instAddCommMonoidWithOne.toNatCast inferInstance)) |
| (Fin d → ℝ) inferInstance)) := |
| let euclideanMeasurable := @WithLp.measurableSpace |
| (@OfNat.ofNat ℝ≥0∞ 2 |
| (@instOfNatAtLeastTwo ℝ≥0∞ 2 ENNReal.instAddCommMonoidWithOne.toNatCast inferInstance)) |
| (Fin d → ℝ) inferInstance |
| @Measure.pi (Fin k) (fun _ => Vector d) inferInstance (fun _ => euclideanMeasurable) |
| (fun _ => @stdGaussian (Vector d) |
| (PiLp.normedAddCommGroup |
| (@OfNat.ofNat ℝ≥0∞ 2 |
| (@instOfNatAtLeastTwo ℝ≥0∞ 2 ENNReal.instAddCommMonoidWithOne.toNatCast inferInstance)) |
| (fun _ : Fin d => ℝ)) |
| (PiLp.innerProductSpace (𝕜 := ℝ) (fun _ : Fin d => ℝ)) inferInstance euclideanMeasurable) |
| def ObservationReference (d : ℕ) : Measure (Vector d × ℝ) := |
| (stdGaussian (Vector d)).prod volume |
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| structure Policy (d M T : ℕ) where |
| initial : Action M → ℝ≥0∞ |
| initial_sum : ∑ a, initial a = 1 |
| transition : ℕ → State M → (Vector d × ℝ) → Action M → ℝ≥0∞ |
| transition_sum : ∀ t j obs, ∑ a, transition t j obs a = 1 |
| output : ℕ → State M → Measure (Sphere d) |
| output_probability : ∀ t j, IsProbabilityMeasure (output t j) |
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| namespace Policy |
| instance {d M T : ℕ} (K : Policy d M T) (t : ℕ) (j : State M) : |
| IsProbabilityMeasure (K.output t j) := K.output_probability t j |
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| def Completed {d M T : ℕ} (K : Policy d M T) : Prop := |
| ∀ t j a, AEMeasurable (fun obs => K.transition t j obs a) (ObservationReference d) |
| def Borel {d M T : ℕ} (K : Policy d M T) : Prop := |
| ∀ t j a, Measurable (fun obs => K.transition t j obs a) |
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| def eval {d M T : ℕ} {X : Type*} (K : Policy d M T) |
| (w : ℕ → State M → X → ℝ≥0∞) (n : ℕ) : |
| (k : ℕ) → State M → X → Observations d k → ℝ≥0∞ |
| | 0, j, s, _ => w n j s |
| | k+1, j, s, obs => ∑ a : Action M, K.transition (n+1) j (obs 0) a * |
| if a.1 then w (n+1) a.2 s else K.eval w (n+1) k a.2 s (Fin.tail obs) |
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| def pathValue {d M T : ℕ} {X : Type*} (K : Policy d M T) |
| (w : ℕ → State M → X → ℝ≥0∞) (s : X) (obs : Observations d T) : ℝ≥0∞ := |
| ∑ a : Action M, K.initial a * if a.1 then w 0 a.2 s else K.eval w 0 T a.2 s obs |
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| def records {d T : ℕ} (s : Sphere d) (A : Rows d T) : Observations d T := |
| fun index => (A index, @inner ℝ (EuclideanSpace ℝ (Fin d)) |
| (@InnerProductSpace.toInner ℝ (EuclideanSpace ℝ (Fin d)) _ |
| (PiLp.seminormedAddCommGroup |
| (@OfNat.ofNat ℝ≥0∞ 2 |
| (@instOfNatAtLeastTwo ℝ≥0∞ 2 ENNReal.instAddCommMonoidWithOne.toNatCast inferInstance)) |
| (fun _ : Fin d => ℝ)) |
| (PiLp.innerProductSpace (𝕜 := ℝ) (fun _ : Fin d => ℝ))) s.val (A index)) |
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| def outputTest {d M T : ℕ} (K : Policy d M T) (f : Sphere d × Sphere d → ℝ≥0∞) |
| (t : ℕ) (j : State M) (s : Sphere d) : ℝ≥0∞ := |
| ∫⁻ v, f (s,v) ∂K.output t j |
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| def angleTest {d : ℕ} (ε : ℝ) |
| (point : MemoryPrecision.Kernel.Sphere d × MemoryPrecision.Kernel.Sphere d) : ℝ≥0∞ := |
| @ite ℝ≥0∞ (Real.arccos (@inner ℝ (EuclideanSpace ℝ (Fin d)) |
| (@InnerProductSpace.toInner ℝ (EuclideanSpace ℝ (Fin d)) _ |
| (PiLp.seminormedAddCommGroup |
| (@OfNat.ofNat ℝ≥0∞ 2 |
| (@instOfNatAtLeastTwo ℝ≥0∞ 2 ENNReal.instAddCommMonoidWithOne.toNatCast inferInstance)) |
| (fun _ : Fin d => ℝ)) |
| (PiLp.innerProductSpace (𝕜 := ℝ) (fun _ : Fin d => ℝ))) |
| point.2.val point.1.val) ≤ ε) (Classical.propDecidable _) 1 0 |
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| def chordTest {d : ℕ} (ε : ℝ) (x : Sphere d × Sphere d) : ℝ≥0∞ := |
| {x : Sphere d × Sphere d | dist x.2.val x.1.val ≤ ε}.indicator (fun _ => 1) x |
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| def experiment {d M T : ℕ} {Ω : Type u} (K : Ω → Policy d M T) |
| (f : Sphere d × Sphere d → ℝ≥0∞) (z : Sphere d × (Ω × Rows d T)) : ℝ≥0∞ := |
| (K z.2.1).pathValue ((K z.2.1).outputTest f) z.1 (records z.1 z.2.2) |
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| structure Admissible {d M T : ℕ} {Ω : Type u} [MeasurableSpace Ω] |
| (K : Ω → Policy d M T) (ρ : Measure Ω) (π : Measure (Sphere d)) : Prop where |
| completed : ∀ᵐ ω ∂ρ, (K ω).Completed |
| experiment : ∀ f : Sphere d × Sphere d → ℝ≥0∞, Measurable f → |
| (∀ x, f x ≤ 1) → AEMeasurable (experiment K f) (π.prod (ρ.prod (RowLaw d T))) |
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| def Success {d M T : ℕ} {Ω : Type u} [MeasurableSpace Ω] |
| (K : Ω → Policy d M T) (ρ : Measure Ω) (π : Measure (Sphere d)) (ε : ℝ) : ℝ≥0∞ := |
| ∫⁻ z, experiment K (angleTest ε) z ∂π.prod (ρ.prod (RowLaw d T)) |
| def ChordSuccess {d M T : ℕ} {Ω : Type u} [MeasurableSpace Ω] |
| (K : Ω → Policy d M T) (ρ : Measure Ω) (π : Measure (Sphere d)) (ε : ℝ) : ℝ≥0∞ := |
| ∫⁻ z, experiment K (chordTest ε) z ∂π.prod (ρ.prod (RowLaw d T)) |
| end Policy |
| end MemoryPrecision.Kernel |
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| end |
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| end OAI |
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