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b317718 | 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 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 | import OAI.Probability.GaussianRegression.Success
namespace OAI
/-!
Real-valued success bounds and accuracy-uniform precision tradeoffs.
-/
open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal NNReal RealInnerProductSpace Topology
open InnerProductSpace
open scoped Pointwise
noncomputable section
universe u
namespace NoiselessRegression
namespace Learner
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
end Learner
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])
end NoiselessRegression
end
end OAI
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