Lexicographic Lipschitz Bandits:New Algorithms and a Lower Bound
Bibliographic record
Abstract
This paper studies a multiobjective bandit problem under lexicographic ordering, wherein the learner aims to maximize m objectives, each with different levels of importance. First, we introduce the local trade-off, λ∗, which depicts the trade-off between different objectives. For the case when an upper bound of λ∗ is known, i.e., λ ≥ λ∗, we develop an algorithm that achieves a general regret bound of Õ(Λi (λ)T (d i z+1)/(d i z+2)) for the i-th objective, where i ∈ {1, 2, . . . , m}, Λi (λ) = 1 + λ + · · · + λi−1, d iz is the zooming dimension for the i-th objective, and T is the time horizon. Next, we provide a matching lower bound for the lexicographic Lipschitz bandit problem, proving that our algorithm is optimal in terms of λ∗ and T. Finally, for the case where m = 2, we remove the dependence on the knowledge about λ∗, albeit at the cost of increasing the regret bound to Õ(Λi (λ∗)T (3d i z+4)/(3d i z+6)), which remains optimal in terms of λ∗. Compared to existing work on lexicographic multiarmed bandits (Hüyük and Tekin, 2021), our approach improves the current regret bound of Õ(T 2/3) 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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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.004 | 0.018 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.003 | 0.003 |
| Research integrity | 0.004 | 0.006 |
| Insufficient payload (model declined to judge) | 0.007 | 0.002 |
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".