| import OAI.Probability.GaussianRegression.Success |
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| namespace OAI |
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| /-! |
| Real-valued success bounds and accuracy-uniform precision tradeoffs. |
| -/ |
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| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal NNReal RealInnerProductSpace Topology |
| open InnerProductSpace |
| open scoped Pointwise |
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| noncomputable section |
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| universe u |
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| namespace NoiselessRegression |
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| namespace Learner |
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| lemma streaming_success_bound_real {d M T : ℕ} {Ω : Type u} [MeasurableSpace Ω] |
| (L : Learner d M T Ω) (ρ : Measure Ω) [IsProbabilityMeasure ρ] |
| (hL : L.Admissible ρ) (hd : 3 ≤ d) {ε : ℝ} (hε : 0 < ε) |
| {a q : ℕ} (ha : 2*a ≤ d-1) (hq : 2 ≤ q) (hqa : 2*q ≤ a) |
| (hs : (2/3 : ℝ≥0∞) ≤ L.uniformSuccess ρ ε) : |
| (2/3 : ℝ) ≤ ((T : ℝ)+1) * ((3 : ℝ)*4^d * ε^a) * |
| ((32768 : ℝ)^d * ((2 : ℝ)^M)^(1/(q : ℝ)))^(T/q+1) := by |
| have hb := hs.trans (L.streaming_success_bound ρ hL hd hε ha hq hqa) |
| have ht : (T+1 : ℝ≥0∞) * |
| (ENNReal.ofReal ((3 : ℝ)*4^d) * ENNReal.ofReal (ε^a)) * |
| streamFactor (E := Vector d) (J := State M) q ^ (T/q+1) ≠ ∞ := by |
| apply ENNReal.mul_ne_top |
| · apply ENNReal.mul_ne_top (by simp) |
| exact ENNReal.mul_ne_top ENNReal.ofReal_ne_top ENNReal.ofReal_ne_top |
| · exact ENNReal.pow_ne_top (streamFactor_ne_top q) |
| have hr := ENNReal.toReal_mono ht hb |
| simpa [ENNReal.toReal_div, ENNReal.toReal_ofNat, ENNReal.toReal_mul, |
| ENNReal.toReal_add, ENNReal.toReal_natCast, ENNReal.toReal_one, |
| ENNReal.toReal_ofReal (by positivity : (0 : ℝ) ≤ 3*4^d), |
| ENNReal.toReal_ofReal (by positivity : 0 ≤ ε^a), streamFactor, |
| ENNReal.toReal_pow, ← ENNReal.toReal_rpow, finrank_euclideanSpace_fin, |
| Fintype.card_fin, State, Nat.cast_pow, Nat.cast_ofNat, Vector] using hr |
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| end Learner |
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| lemma precision_from_stream_bound {d M T a q : ℕ} {ε A : ℝ} |
| (hd : 32 ≤ d) (hT : d ≤ 4*T) (hq : 0 < q) (hdq : d ≤ 8*q) |
| (ha : d ≤ 4*a) (hM : (M : ℝ) ≤ A * (d : ℝ)^2) (hA : 0 < A) |
| (hε : 0 < ε) (hε1 : ε ≤ 1) |
| (hbound : (2/3 : ℝ) ≤ ((T : ℝ)+1) * ((3 : ℝ)*4^d * ε^a) * |
| ((32768 : ℝ)^d * ((2 : ℝ)^M)^(1/(q : ℝ)))^(T/q+1)) : |
| (1 / (4*(400000+100*A))) * (d : ℝ) * Real.log (1/ε) ≤ (T : ℝ) := by |
| have hd0 : (0 : ℝ) < d := by exact_mod_cast (by omega : 0 < d) |
| have hq0 : (0 : ℝ) < q := by exact_mod_cast hq |
| have hT0 : (0 : ℝ) ≤ T := Nat.cast_nonneg T |
| have hdp : (d : ℝ) ≤ 4*T := by exact_mod_cast hT |
| have hdqp : (d : ℝ) ≤ 8*q := by exact_mod_cast hdq |
| have hap : (d : ℝ) ≤ 4*a := by exact_mod_cast ha |
| have hB : ((T/q+1 : ℕ) : ℝ) * d ≤ 12*T := by |
| have hbd : (T/q+1)*d ≤ 12*T := by |
| have hmul := Nat.div_mul_le_self T q |
| calc |
| (T/q+1)*d = (T/q)*d+d := by ring |
| _ ≤ (T/q)*(8*q)+d := Nat.add_le_add_right (Nat.mul_le_mul_left _ hdq) _ |
| _ = 8*((T/q)*q)+d := by ring |
| _ ≤ 8*T+d := Nat.add_le_add_right (Nat.mul_le_mul_left 8 hmul) _ |
| _ ≤ 12*T := by omega |
| exact_mod_cast hbd |
| have hMq : (M : ℝ) / q ≤ 8*A*d := by |
| apply (div_le_iff₀ hq0).2 |
| calc |
| (M : ℝ) ≤ A*d^2 := hM |
| _ ≤ A*d*(8*q) := by nlinarith [mul_le_mul_of_nonneg_left hdqp (by positivity : 0 ≤ A*d)] |
| _ = 8*A*d*q := by ring |
| have hlog := Real.log_le_log (by norm_num : (0 : ℝ) < 2/3) hbound |
| rw [Real.log_mul (by positivity) (by positivity), |
| Real.log_mul (by positivity) (by positivity), |
| Real.log_mul (by positivity) (by positivity), |
| Real.log_mul (by positivity) (by positivity), |
| Real.log_pow, Real.log_pow, Real.log_pow, |
| Real.log_mul (by positivity) (by positivity), Real.log_pow, |
| Real.log_rpow (by positivity), Real.log_pow] at hlog |
| have hlogε : Real.log (1/ε) = -Real.log ε := by simp |
| have hlogε0 : 0 ≤ Real.log (1/ε) := Real.log_nonneg ((le_div_iff₀ hε).2 (by simpa using hε1)) |
| have heq : (1/(q : ℝ)) * ((M : ℝ)*Real.log 2) = ((M : ℝ)/q)*Real.log 2 := by ring |
| rw [heq] at hlog |
| have hlT : Real.log ((T : ℝ)+1) ≤ T := by |
| simpa using Real.log_le_sub_one_of_pos (by positivity : (0 : ℝ) < T+1) |
| have hl3 : Real.log (3 : ℝ) ≤ 3 := Real.log_le_self (by norm_num) |
| have hl4 : Real.log (4 : ℝ) ≤ 4 := Real.log_le_self (by norm_num) |
| have hl32 : Real.log (32768 : ℝ) ≤ 32768 := Real.log_le_self (by norm_num) |
| have hl2 : Real.log (2 : ℝ) ≤ 1 := by |
| convert Real.log_le_sub_one_of_pos (by norm_num : (0 : ℝ) < 2) using 1; norm_num |
| have hl23 : -(1 : ℝ) ≤ Real.log (2/3 : ℝ) := by |
| have h := Real.one_sub_inv_le_log_of_pos (by norm_num : (0 : ℝ) < 2/3) |
| norm_num at h |
| linarith |
| have hinside : (d : ℝ)*Real.log 32768 + ((M : ℝ)/q)*Real.log 2 ≤ |
| (32768+8*A)*d := by |
| have hm0 : (0 : ℝ) ≤ (M : ℝ)/q := by positivity |
| have h1 := mul_le_mul_of_nonneg_left hl32 hd0.le |
| have h2 := mul_le_mul_of_nonneg_left hl2 hm0 |
| nlinarith |
| have hfull : ((T/q+1 : ℕ) : ℝ) * |
| ((d : ℝ)*Real.log 32768 + ((M : ℝ)/q)*Real.log 2) ≤ |
| 12*T*(32768+8*A) := by |
| calc |
| _ ≤ ((T/q+1 : ℕ) : ℝ) * ((32768+8*A)*d) := |
| mul_le_mul_of_nonneg_left hinside (Nat.cast_nonneg _) |
| _ = (32768+8*A)*(((T/q+1 : ℕ) : ℝ)*d) := by ring |
| _ ≤ (32768+8*A)*(12*T) := mul_le_mul_of_nonneg_left hB (by positivity) |
| _ = _ := by ring |
| have hd4 : (4 : ℝ) ≤ d := by exact_mod_cast (by omega : 4 ≤ d) |
| have hlogbound : (a : ℝ)*Real.log (1/ε) ≤ (400000+100*A)*T := by |
| have h4 := mul_le_mul_of_nonneg_left hl4 hd0.le |
| rw [hlogε] |
| nlinarith [mul_nonneg hA.le hT0] |
| have hmain : (d : ℝ)*Real.log (1/ε) ≤ 4*(400000+100*A)*T := by |
| have ham := mul_le_mul_of_nonneg_right hap hlogε0 |
| linarith |
| have hden : (0 : ℝ) < 4*(400000+100*A) := by positivity |
| calc |
| _ = ((d : ℝ)*Real.log (1/ε))/(4*(400000+100*A)) := by ring |
| _ ≤ T := (div_le_iff₀ hden).2 (by nlinarith [hmain]) |
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| end NoiselessRegression |
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| end |
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| end OAI |
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