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feat: one sample-complexity reduction for the plug-in GLRT, and Track-and-Stop with D-tracking from it #374

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lmlexplore-automation merged 1 commit into from
Nov 14, 2026
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feat: one sample-complexity reduction for the plug-in GLRT, and Track-and-Stop with D-tracking from it #374

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22 changes: 9 additions & 13 deletions LMLExplore/Identification/AlmostSure.lean
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Expand Up @@ -10,8 +10,9 @@ public import LMLExplore.Identification.TrackAndStop
/-!
# Almost sure sample complexity of the GLRT

The almost sure counterpart [GK16, Proposition 13] of the sample complexity bound in expectation
`IdentTask.eventually_expectedStoppingTime_glrt_le`. Consider the GLRTs with thresholds
The almost sure counterpart [GK16, Proposition 13] of the sample complexity bounds in expectation
`IdentTask.eventually_expectedStoppingTime_glrt_le` and
`IdentTask.eventually_expectedStoppingTime_glrt_plugInCost_le`. Consider the GLRTs with thresholds
`β δ T h ≤ log (1 / δ) + f T`, where `f T ≤ ε T` eventually for every `ε > 0`.

* **Pathwise.** Along one sequence of histories, if the GLR statistic of the recommended answer
Expand Down Expand Up @@ -150,17 +151,12 @@ lemma eventually_mul_le_glr_plugInCost {rec : Θ → ℐ} (hW : IsCompact W)
(h_prop : Tendsto (fun n a ↦ (pullCount X a n ω : ℝ) / n) atTop (𝓝ˢ W)) :
∀ᶠ T in atTop, κ * T ≤ task.glr (task.plugInCost est) T (history O X Y T ω)
(rec (est T (history O X Y T ω))) := by
-- Near `θ`, on the allocations of a thickening of `W`, the cost of the alternatives exceeds `κ`.
obtain ⟨U, hU, η, hη, hUη⟩ := hW.exists_forall_mem_thickening_lt_of_lowerSemicontinuousWithinAt
h_lsc hκ
filter_upwards [h_est.eventually_mem (inter_mem hU hrec_eq),
h_prop.eventually_mem (thickening_mem_nhdsSet W hη), eventually_ne_atTop 0] with T hT hTW hT0
have h_eq : (fun a ↦ (pullCount' T (history O X Y T ω) a : ℝ) / T) =
fun a ↦ (pullCount X a T ω : ℝ) / T := by
simp_rw [pullCount_eq_pullCount' (O := O) (R' := Y)]
rw [← h_eq] at hTW
rw [hT.2, glr_plugInCost_eq_mul task est hT0, mul_comm]
exact mul_le_mul_right (hUη _ hT.1 _ (pullCount'_div_mem_simplex _ hT0) hTW).le _
obtain ⟨U, hU, V, hV, hUV⟩ :=
task.exists_mem_nhds_mul_le_glr_plugInCost est hW h_lsc hκ hrec_eq
filter_upwards [h_est.eventually_mem hU, h_prop.eventually_mem hV, eventually_ne_atTop 0]
with T hT hTV hT0
refine hUV T hT0 _ hT ?_
simpa only [pullCount_eq_pullCount' (O := O) (R' := Y)] using hTV

/-! ### Almost sure sample complexity -/

Expand Down
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