From d7de396169885aff0bafe984730b37778516fa0a Mon Sep 17 00:00:00 2001 From: Remy Degenne Date: Sun, 8 Nov 2026 18:19:30 +0100 Subject: [PATCH] =?UTF-8?q?Identification:=20the=20constant=20of=20[GK16,?= =?UTF-8?q?=20Proposition=2012]=20depends=20on=20=CE=B4?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Layer 4.2 asks for C(α, K, δ) with log C = O(log log(1/δ)): no constant independent of δ works, and the paper's proof drops the δ in log² (C t^α / δ) when it chooses C. Layer 4.4 allows thresholds log(1/δ) + g(δ) + f(t) with g(δ) = o(log(1/δ)), so that the thresholds of Layer 4.2 are cases of it. Co-Authored-By: Claude Opus 5.5 Claude-Session: https://claude.ai/code/session_0196ftMToRcTS8JUy4ChhSPh --- LMLRoadmaps/Identification/README.md | 16 ++++++++++------ 1 file changed, 10 insertions(+), 6 deletions(-) diff --git a/LMLRoadmaps/Identification/README.md b/LMLRoadmaps/Identification/README.md index b0db7bc..98d3e04 100644 --- a/LMLRoadmaps/Identification/README.md +++ b/LMLRoadmaps/Identification/README.md @@ -737,8 +737,11 @@ alternatives, and Gaussian costs with arm-dependent variances in design-operator \big(\log((K-1)/\delta)/2\big)$, of order $\log(K/\delta)+O(\log\log t)$, for the covers of item 3. - Prove the union-bound thresholds of [GK16, Proposition 12] ($\log(Ct^\alpha/\delta)$, $\alpha>1$, - with an explicit $C(\alpha,K)$, from the two-sided bound of Layer 1.2, for arms dominated by - any exponential families), of [GK16, Theorem 10] (Bernoulli arms only, + with an explicit $C=C(\alpha,K,\delta)$ such that + $\log C(\alpha,K,\delta)=O(\log\log(1/\delta))$ as $\delta\to0$, from the two-sided bound of + Layer 1.2, for arms dominated by any exponential families; no $C$ independent of $\delta$ + works, and the proof of [GK16, Proposition 12] drops the $\delta$ in + $\log^2(Ct^\alpha/\delta)$ when it chooses $C$), of [GK16, Theorem 10] (Bernoulli arms only, $\log(2t(K-1)/\delta)$, through the Krichevsky–Trofimov mixture of [GK16, Lemma 11]) and of [LS, Lemmas 33.7–33.8, Proposition 33.9], and the uniform deviation bound of [MCP14] used by [DK19, Lemma 32]. LMLExplore's thresholds for Gaussian arms (`unionBoundThreshold`, @@ -783,10 +786,11 @@ alternatives, and Gaussian costs with arm-dependent variances in design-operator $h(T)\le t\le T$). Prove that $\mathbb E[\tau_\delta]\le c\log(1/\delta)$ for all small $\delta$, for every $c>(\inf_{w\in W}D(\theta,w,\neg i))^{-1}$, for every family of thresholds with - $\beta_t(h,\delta)\le\log(1/\delta)+f(t)$ for all histories and $f(t)=o(t)$ - [GK16, proof of Theorem 14]. The bound $\beta_t(h,\delta)\le\log(Ct^\alpha/\delta)$ is a - case. LMLExplore's `IdentTask.eventually_expectedStoppingTime_glrt_le` (linear growth of the - statistic on good events) is the step after the lower semicontinuity. + $\beta_t(h,\delta)\le\log(1/\delta)+g(\delta)+f(t)$ for all histories, with + $g(\delta)=o(\log(1/\delta))$ as $\delta\to0$ and $f(t)=o(t)$ [GK16, proof of Theorem 14]. The + thresholds of item 2 are cases, and so are $\log(r(t)/\delta)$ with $r(t)=O(t^\alpha)$. + LMLExplore's `IdentTask.eventually_expectedStoppingTime_glrt_le` (linear growth of the + statistic on good events), the case $g=0$, is the step after the lower semicontinuity. - Almost sure: with the same statistic, recommendation and thresholds, if almost surely $\hat\theta_t\to\theta$ and the distance from $N(t)/t$ to $W$ tends to $0$, prove that almost surely, for every $c>(\inf_{w\in W}D(\theta,w,\neg i))^{-1}$,