Lexicographic Lipschitz Bandits:New Algorithms and a Lower Bound
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Bibliographic record
Abstract
This paper studies a multiobjective bandit problem under lexicographic ordering, wherein the learner aims to maximize <i>m </i>objectives, each with different levels of importance. First, we introduce the local trade-off, <i>λ</i><sub>∗</sub>, which depicts the trade-off between different objectives. For the case when an upper bound of <i>λ</i><sub>∗</sub> is known, i.e., <i>λ </i>≥ <i>λ</i><sub>∗</sub>, we develop an algorithm that achieves a general regret bound of <i>Õ</i>(Λ<i><sup>i</sup> </i>(<i>λ</i>)<i>T </i><sup>(<i>d i z</i>+1)/(<i>d i z</i>+2)</sup>) for the <i>i</i>-th objective, where <i>i </i>∈ {1, 2, . . . , <i>m</i>}, Λ<i><sup>i</sup> </i>(<i>λ</i>) = 1 + <i>λ </i>+ · · · + <i>λ</i><sup><i>i</i>−1</sup>, <i>d <sup>i</sup><sub>z</sub> </i>is the zooming dimension for the <i>i</i>-th objective, and <i>T </i>is the time horizon. Next, we provide a matching lower bound for the lexicographic Lipschitz bandit problem, proving that our algorithm is <i>optimal </i>in terms of <i>λ</i><sub>∗</sub> and <i>T</i>. Finally, for the case where <i>m </i>= 2, we remove the dependence on the knowledge about <i>λ</i><sub>∗</sub>, albeit at the cost of increasing the regret bound to <i>Õ</i>(Λ<i><sup>i</sup> </i>(<i>λ</i><sub>∗</sub>)<i>T </i><sup>(3<i>d i z</i>+4)/(3<i>d i z</i>+6)</sup>), which remains optimal in terms of <i>λ</i><sub>∗</sub>. Compared to existing work on lexicographic multiarmed bandits (Hüyük and Tekin, 2021), our approach improves the current regret bound of <i>Õ</i>(<i>T </i><sup>2/3</sup>) and extends the number of arms to infinity. Numerical experiments confirm the effectiveness of our algorithms. ©2025 Bo Xue, Ji Cheng, Fei Liu, Yimu Wang, Lijun Zhang, and Qingfu Zhang.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.006 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.003 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it