| import OAI.Probability.Localization.TruncatedBounds |
| import OAI.Probability.Localization.EntropyChain |
|
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| namespace OAI |
|
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| open MeasureTheory ProbabilityTheory Set |
| open scoped BigOperators ENNReal |
|
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| namespace RepeatedLocalization |
|
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| namespace FiniteEntropy |
|
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| noncomputable def entropySlope (n : ℕ) : ℝ := (n:ℝ)*Real.log 2/2 |
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| noncomputable def historyKL {n : ℕ} (P : Measure (Coordinate n)) (Q : Cell n) (h : Log n) : ℝ≥0∞ := |
| InformationTheory.klDiv (ProbabilityTheory.cond P (state P Q h).event) (uniformCell (state P Q h).cell) |
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| noncomputable def historyCost {n : ℕ} (P : Measure (Coordinate n)) (Q : Cell n) (h : Log n) : ℝ := |
| (historyKL P Q h).toReal + entropySlope n * ((state P Q h).cell.1-Q.1 : ℕ) |
|
|
| lemma entropySlope_pos {n : ℕ} (hn : 0 < n) : 0 < entropySlope n := by |
| unfold entropySlope |
| exact div_pos (mul_pos (Nat.cast_pos.mpr hn) (Real.log_pos (by norm_num))) (by norm_num) |
|
|
| lemma historyCost_nonneg {n : ℕ} (P : Measure (Coordinate n)) (Q : Cell n) (h : Log n) : |
| 0 ≤ historyCost P Q h := by |
| unfold historyCost entropySlope |
| exact add_nonneg ENNReal.toReal_nonneg (mul_nonneg (div_nonneg (mul_nonneg (Nat.cast_nonneg _) (Real.log_nonneg (by norm_num))) (by norm_num)) (Nat.cast_nonneg _)) |
|
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| lemma historyKL_ne_top_budget {n : ℕ} (hn : 0 < n) (P : Measure (Coordinate n)) [IsProbabilityMeasure P] |
| (Q : Cell n) (hac : P ≪ uniformCell Q) (hK : InformationTheory.klDiv P (uniformCell Q) ≠ ∞) |
| (h : Log n) (hp : P (state P Q h).event ≠ 0) : |
| historyKL P Q h ≠ ∞ ∧ |
| (P (state P Q h).event).toReal * historyCost P Q h ≤ |
| (∫ z in (state P Q h).event,llr P (uniformCell Q) z ∂P) + |
| (P (state P Q h).event).toReal * rewardLog P Q (complementReward P) h := by |
| let := uniformCell_probability hn Q |
| have hd := positive_event_descendant P Q h hp |
| have hk := klDiv_cond_cond P (uniformCell Q) hac hK (state_event_measurable P Q h) |
| (measurableSet_cellSet _) (state_event_subset_cell P Q h) hp (uniformCell_descendant_pos hn hd) |
| rw [uniformCell_cond hd,uniformCell_log_mass hn hd] at hk |
| refine ⟨hk.1,?_⟩ |
| have hroot : P (cellSet Q) = 1 := (mem_ae_iff_prob_eq_one (measurableSet_cellSet Q)).mp (ae_mem_cell_of_ac hac) |
| have hs := mul_le_mul_of_nonneg_left (path_label_surprisal P Q hroot h hp) (ENNReal.toReal_nonneg (a := P (state P Q h).event)) |
| have he := hk.2 |
| change (P (state P Q h).event).toReal * ((historyKL P Q h).toReal + entropySlope n * _) ≤ _ |
| change (P (state P Q h).event).toReal * (historyKL P Q h).toReal = _ at he |
| unfold entropySlope |
| nlinarith |
|
|
| lemma truncated_cost_bound {n : ℕ} (hn : 0 < n) (P : Measure (Coordinate n)) [IsProbabilityMeasure P] |
| (Q : Cell n) (hac : P ≪ uniformCell Q) (hK : InformationTheory.klDiv P (uniformCell Q) ≠ ∞) (N : ℕ) : |
| Integrable (fun z => historyCost P Q (run P Q N z)) P ∧ |
| (∫ z,historyCost P Q (run P Q N z) ∂P) ≤ |
| (InformationTheory.klDiv P (uniformCell Q)).toReal + ∫ z,complementTotal P Q N z ∂P := by |
| let := uniformCell_probability hn Q |
| have hroot := ae_mem_cell_of_ac hac |
| have hi := (InformationTheory.klDiv_ne_top_iff.mp hK).2 |
| have hJ := (truncated_reward_bounds hn P Q hroot N).1 |
| have hb (h : Log n) : (P {z | run P Q N z = h}).toReal * historyCost P Q h ≤ |
| ∫ z in {z | run P Q N z = h},llr P (uniformCell Q) z + complementTotal P Q N z ∂P := by |
| by_cases hp : P {z | run P Q N z = h} = 0 |
| · rw [hp,ENNReal.toReal_zero,zero_mul,Measure.restrict_eq_zero.mpr hp,integral_zero_measure] |
| obtain ⟨z,hz,hzh⟩ := exists_run_fiber_root P Q hroot N h hp |
| have he : {w | run P Q N w = h} =ᵐ[P] (state P Q h).event := by |
| simpa only [hzh] using run_fiber_ae_event hn P Q hroot N hz |
| have hp' : P (state P Q h).event ≠ 0 := by rwa [← measure_congr he] |
| have hb := (historyKL_ne_top_budget hn P Q hac hK h hp').2 |
| rw [integral_add hi.integrableOn hJ.integrableOn] |
| change _ ≤ (∫ z in {z | run P Q N z = h},llr P (uniformCell Q) z ∂P) + |
| ∫ z in {z | run P Q N z = h},(fun h => rewardLog P Q (complementReward P) h) (run P Q N z) ∂P |
| rw [integral_comp_fiber P (run P Q N) (rewardLog P Q (complementReward P)) h |
| ((measurable_run hn P Q N) (measurableSet_singleton h)),Measure.restrict_congr_set he,measure_congr he] |
| exact hb |
| have hb' := countable_cost_bound P (run P Q N) (measurable_run hn P Q N) |
| (historyCost P Q) (historyCost_nonneg P Q) (hi.add hJ) hb |
| simp only [Pi.add_apply] at hb' |
| rw [integral_add hi hJ,← InformationTheory.toReal_klDiv_of_measure_eq hac (by simp)] at hb' |
| exact hb' |
|
|
| lemma truncated_depth_integrable {n : ℕ} (hn : 0 < n) (P : Measure (Coordinate n)) [IsProbabilityMeasure P] |
| (Q : Cell n) (hac : P ≪ uniformCell Q) (hK : InformationTheory.klDiv P (uniformCell Q) ≠ ∞) (N : ℕ) : |
| Integrable (fun z => (truncatedDepth P Q N z : ℝ)) P := by |
| have hc := (truncated_cost_bound hn P Q hac hK N).1 |
| have hm : Measurable (fun z => (truncatedDepth P Q N z : ℝ)) := |
| (measurable_of_countable (fun h : Log n => (((state P Q h).cell.1-Q.1:ℕ):ℝ))).comp (measurable_run hn P Q N) |
| apply Integrable.mono' (hc.div_const (entropySlope n)) hm.aestronglyMeasurable |
| filter_upwards [] with z |
| rw [Real.norm_eq_abs,abs_of_nonneg (Nat.cast_nonneg _),le_div_iff₀ (entropySlope_pos hn)] |
| change _ ≤ (historyKL P Q (run P Q N z)).toReal + entropySlope n * (truncatedDepth P Q N z:ℝ) |
| nlinarith [ENNReal.toReal_nonneg (a := historyKL P Q (run P Q N z))] |
|
|
| lemma truncated_KL_integrable {n : ℕ} (hn : 0 < n) (P : Measure (Coordinate n)) [IsProbabilityMeasure P] |
| (Q : Cell n) (hac : P ≪ uniformCell Q) (hK : InformationTheory.klDiv P (uniformCell Q) ≠ ∞) (N : ℕ) : |
| Integrable (fun z => (historyKL P Q (run P Q N z)).toReal) P := by |
| have hc := (truncated_cost_bound hn P Q hac hK N).1 |
| have hd := truncated_depth_integrable hn P Q hac hK N |
| have he : (fun z => (historyKL P Q (run P Q N z)).toReal) = |
| (fun z => historyCost P Q (run P Q N z)) - (fun z => entropySlope n*(truncatedDepth P Q N z:ℝ)) := by |
| funext z |
| simp [historyCost,truncatedDepth] |
| rw [he] |
| exact hc.sub (hd.const_mul _) |
|
|
| lemma truncated_attempts_integral {n : ℕ} (hn : 0 < n) (P : Measure (Coordinate n)) [IsProbabilityMeasure P] |
| (Q : Cell n) (hac : P ≪ uniformCell Q) (hK : InformationTheory.klDiv P (uniformCell Q) ≠ ∞) (N : ℕ) : |
| (∫ z,(truncatedAttempts P Q N z:ℝ) ∂P) ≤ 2*(1+∫ z,(truncatedDepth P Q N z:ℝ) ∂P) := by |
| have hroot := ae_mem_cell_of_ac hac |
| have hC := (truncated_reward_bounds hn P Q hroot N).2 |
| have hD := truncated_depth_integrable hn P Q hac hK N |
| have hp : ∀ᵐ z ∂P,(truncatedAttempts P Q N z:ℝ) ≤ |
| (truncatedDepth P Q N z:ℝ) + continuingTotal P Q N z + 1 := by |
| filter_upwards [hroot] with z hz |
| have hb := truncated_attempts_path hn P Q N hz |
| have hs : (if (state P Q (run P Q N z)).stopped then (1:ℝ) else 0) ≤ 1 := by split <;> norm_num |
| linarith |
| have hi := integral_mono_ae (integrable_truncatedAttempts hn P Q N) |
| ((hD.add hC.1).add (integrable_const 1)) hp |
| have had := integral_add (hD.add hC.1) (integrable_const (1:ℝ)) |
| simp only [Pi.add_apply] at hi had |
| rw [had,integral_add hD hC.1] at hi |
| simp only [integral_const,Measure.real,measure_univ,ENNReal.toReal_one,one_smul] at hi |
| linarith [hC.2.2] |
|
|
| lemma slope_absorption {n : ℕ} (hn : 32 ≤ n) : 2/Real.exp 1 ≤ entropySlope n/2 := by |
| have hn' : (32:ℝ) ≤ n := by exact_mod_cast hn |
| have hl : (1/2:ℝ) ≤ Real.log 2 := by linarith [Real.log_two_gt_d9] |
| have he : (2:ℝ)/Real.exp 1 ≤ 2 := (div_le_self (by norm_num) (Real.one_le_exp (by norm_num))) |
| unfold entropySlope |
| nlinarith |
|
|
| lemma truncated_final_budget {n : ℕ} (hn : 32 ≤ n) (P : Measure (Coordinate n)) [IsProbabilityMeasure P] |
| (Q : Cell n) (hac : P ≪ uniformCell Q) (hK : InformationTheory.klDiv P (uniformCell Q) ≠ ∞) (N : ℕ) : |
| (∫ z,(historyKL P Q (run P Q N z)).toReal + entropySlope n/2*(truncatedDepth P Q N z:ℝ) ∂P) ≤ |
| (InformationTheory.klDiv P (uniformCell Q)).toReal + 2/Real.exp 1 := by |
| have hn0 : 0 < n := by omega |
| have hcost := (truncated_cost_bound hn0 P Q hac hK N).2 |
| have hJ := (truncated_reward_bounds hn0 P Q (ae_mem_cell_of_ac hac) N).2.2.1 |
| have hA := truncated_attempts_integral hn0 P Q hac hK N |
| have hD := truncated_depth_integrable hn0 P Q hac hK N |
| have hKI := truncated_KL_integrable hn0 P Q hac hK N |
| change (∫ z,(historyKL P Q (run P Q N z)).toReal + entropySlope n*(truncatedDepth P Q N z:ℝ) ∂P) ≤ _ at hcost |
| rw [integral_add hKI (hD.const_mul (entropySlope n)),integral_const_mul] at hcost |
| rw [integral_add hKI (hD.const_mul (entropySlope n / 2)),integral_const_mul] |
| have hD0 : 0 ≤ ∫ z,(truncatedDepth P Q N z:ℝ) ∂P := integral_nonneg (fun _ => Nat.cast_nonneg _) |
| have habs := mul_le_mul_of_nonneg_right (slope_absorption hn) hD0 |
| have hJA := mul_le_mul_of_nonneg_left hA (by positivity : 0 ≤ (1:ℝ)/Real.exp 1) |
| simp only [div_eq_mul_inv,one_mul] at hJ hJA habs ⊢ |
| nlinarith only [hcost,hJ,habs,hJA] |
|
|
| end FiniteEntropy |
|
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| end RepeatedLocalization |
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| end OAI |
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