| import OAI.Probability.PosteriorReplicas.Model |
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| namespace OAI |
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| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal NNReal RealInnerProductSpace |
| noncomputable section |
| universe u |
| namespace PosteriorRegression |
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| structure Learner (d M T : ℕ) (Ω : Type u) where |
| initialChoice : Ω → Bool × State M |
| transition : ℕ → Ω → State M → (Vector d × ℝ) → Bool × State M |
| output : ℕ → Ω → State M → UnitSphere d |
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| namespace Learner |
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| def start {d M T : ℕ} {Ω : Type u} (L : Learner d M T Ω) (ω : Ω) : |
| RunStatus M := |
| let a := L.initialChoice ω |
| if a.1 then .inr (0, a.2) else .inl a.2 |
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| def run {d M T : ℕ} {Ω : Type u} (L : Learner d M T Ω) |
| (s : UnitSphere d) (ω : Ω) (x : Samples d T) : ℕ → RunStatus M |
| | 0 => L.start ω |
| | n + 1 => |
| match L.run s ω x n with |
| | .inr done => .inr done |
| | .inl state => |
| if hn : n < T then |
| let row := x ⟨n, hn⟩ |
| let a := L.transition (n + 1) ω state (row, ⟪row, s.val⟫) |
| if a.1 then .inr (n + 1, a.2) else .inl a.2 |
| else .inl state |
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| def terminal {d M T : ℕ} {Ω : Type u} (L : Learner d M T Ω) |
| (s : UnitSphere d) (ω : Ω) (x : Samples d T) : ℕ × State M := |
| match L.run s ω x T with |
| | .inl state => (T, state) |
| | .inr done => done |
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| def estimate {d M T : ℕ} {Ω : Type u} (L : Learner d M T Ω) |
| (s : UnitSphere d) (ω : Ω) (x : Samples d T) : UnitSphere d := |
| let a := L.terminal s ω x |
| L.output a.1 ω a.2 |
|
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| structure Admissible {d M T : ℕ} {Ω : Type u} [MeasurableSpace Ω] |
| (L : Learner d M T Ω) (ρ : Measure Ω) : Prop where |
| initialChoice : AEMeasurable L.initialChoice ρ |
| transition : ∀ t a, AEMeasurable (fun z : Ω × (Vector d × ℝ) => |
| L.transition t z.1 a z.2) (ρ.prod (observationReference d)) |
| output : ∀ t a, AEMeasurable (fun ω => L.output t ω a) ρ |
| fixed_experiment : ∀ s, AEMeasurable (fun z : Ω × Samples d T => |
| L.estimate s z.1 z.2) (ρ.prod (sampleLaw d T)) |
| uniform_experiment : AEMeasurable |
| (fun z : UnitSphere d × (Ω × Samples d T) => L.estimate z.1 z.2.1 z.2.2) |
| ((uniformSphere d).prod (ρ.prod (sampleLaw d T))) |
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| def success {d M T : ℕ} {Ω : Type u} [MeasurableSpace Ω] |
| (L : Learner d M T Ω) (ρ : Measure Ω) (ε : ℝ) (s : UnitSphere d) : ℝ≥0∞ := |
| (ρ.prod (sampleLaw d T)) |
| {z | angularError s (L.estimate s z.1 z.2) ≤ ε} |
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| def uniformSuccess {d M T : ℕ} {Ω : Type u} [MeasurableSpace Ω] |
| (L : Learner d M T Ω) (ρ : Measure Ω) (ε : ℝ) : ℝ≥0∞ := |
| ((uniformSphere d).prod (ρ.prod (sampleLaw d T))) |
| {z | angularError z.1 (L.estimate z.1 z.2.1 z.2.2) ≤ ε} |
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| end Learner |
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| def SubquadraticMemory : Prop := |
| ∃ c : ℝ, 0 < c ∧ ∀ M : ℕ → ℕ, |
| Asymptotics.IsLittleO atTop (fun d => (M d : ℝ)) (fun d => (d : ℝ) ^ 2) → |
| ∃ d₀ : ℕ, ∀ d : ℕ, d₀ ≤ d → ∀ T : ℕ, ∀ ε : ℝ, |
| 0 < ε → ε ≤ 1 / 10 → |
| ∀ (Ω : Type u) [MeasurableSpace Ω] (ρ : Measure Ω), IsProbabilityMeasure ρ → |
| ∀ L : Learner d (M d) T Ω, L.Admissible ρ → |
| ((2 / 3 : ℝ≥0∞) ≤ L.uniformSuccess ρ ε ∨ |
| ∀ s, (2 / 3 : ℝ≥0∞) ≤ L.success ρ ε s) → |
| c * (d : ℝ) * Real.log (1 / ε) ≤ (T : ℝ) |
|
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| def FixedQuadraticMemory : Prop := |
| ∀ A₀ : ℝ, 0 < A₀ → ∃ c : ℝ, 0 < c ∧ ∃ d₀ : ℕ, |
| ∀ d : ℕ, d₀ ≤ d → ∀ M T : ℕ, (M : ℝ) ≤ A₀ * (d : ℝ) ^ 2 → |
| ∀ ε : ℝ, 0 < ε → ε ≤ 1 / 10 → |
| ∀ (Ω : Type u) [MeasurableSpace Ω] (ρ : Measure Ω), IsProbabilityMeasure ρ → |
| ∀ L : Learner d M T Ω, L.Admissible ρ → |
| (2 / 3 : ℝ≥0∞) ≤ L.uniformSuccess ρ ε → |
| c * (d : ℝ) * Real.log (1 / ε) ≤ (T : ℝ) |
|
|
| open scoped Topology |
|
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| end PosteriorRegression |
| end |
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| end OAI |
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|
