No-Regret Linear Bandits under Gap-Adjusted Misspecification
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866916559211986944 |
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| author | Liu, Chong Qiao, Dan Yin, Ming Bogunovic, Ilija Wang, Yu-Xiang |
| author_facet | Liu, Chong Qiao, Dan Yin, Ming Bogunovic, Ilija Wang, Yu-Xiang |
| contents | This work studies linear bandits under a new notion of gap-adjusted misspecification and is an extension of Liu et al. (2023). When the underlying reward function is not linear, existing linear bandits work usually relies on a uniform misspecification parameter $ε$ that measures the sup-norm error of the best linear approximation. This results in an unavoidable linear regret whenever $ε> 0$. We propose a more natural model of misspecification which only requires the approximation error at each input $x$ to be proportional to the suboptimality gap at $x$. It captures the intuition that, for optimization problems, near-optimal regions should matter more and we can tolerate larger approximation errors in suboptimal regions.
Quite surprisingly, we show that the classical LinUCB algorithm -- designed for the realizable case -- is automatically robust against such $ρ$-gap-adjusted misspecification with parameter $ρ$ diminishing at $O(1/(d \sqrt{\log T}))$. It achieves a near-optimal $O(\sqrt{T})$ regret for problems that the best-known regret is almost linear in time horizon $T$. We further advance this frontier by presenting a novel phased elimination-based algorithm whose gap-adjusted misspecification parameter $ρ= O(1/\sqrt{d})$ does not scale with $T$. This algorithm attains optimal $O(\sqrt{T})$ regret and is deployment-efficient, requiring only $\log T$ batches of exploration. It also enjoys an adaptive $O(\log T)$ regret when a constant suboptimality gap exists. Technically, our proof relies on a novel self-bounding argument that bounds the part of the regret due to misspecification by the regret itself, and a new inductive lemma that limits the misspecification error within the suboptimality gap for all valid actions in each batch selected by G-optimal design. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_05361 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | No-Regret Linear Bandits under Gap-Adjusted Misspecification Liu, Chong Qiao, Dan Yin, Ming Bogunovic, Ilija Wang, Yu-Xiang Machine Learning This work studies linear bandits under a new notion of gap-adjusted misspecification and is an extension of Liu et al. (2023). When the underlying reward function is not linear, existing linear bandits work usually relies on a uniform misspecification parameter $ε$ that measures the sup-norm error of the best linear approximation. This results in an unavoidable linear regret whenever $ε> 0$. We propose a more natural model of misspecification which only requires the approximation error at each input $x$ to be proportional to the suboptimality gap at $x$. It captures the intuition that, for optimization problems, near-optimal regions should matter more and we can tolerate larger approximation errors in suboptimal regions. Quite surprisingly, we show that the classical LinUCB algorithm -- designed for the realizable case -- is automatically robust against such $ρ$-gap-adjusted misspecification with parameter $ρ$ diminishing at $O(1/(d \sqrt{\log T}))$. It achieves a near-optimal $O(\sqrt{T})$ regret for problems that the best-known regret is almost linear in time horizon $T$. We further advance this frontier by presenting a novel phased elimination-based algorithm whose gap-adjusted misspecification parameter $ρ= O(1/\sqrt{d})$ does not scale with $T$. This algorithm attains optimal $O(\sqrt{T})$ regret and is deployment-efficient, requiring only $\log T$ batches of exploration. It also enjoys an adaptive $O(\log T)$ regret when a constant suboptimality gap exists. Technically, our proof relies on a novel self-bounding argument that bounds the part of the regret due to misspecification by the regret itself, and a new inductive lemma that limits the misspecification error within the suboptimality gap for all valid actions in each batch selected by G-optimal design. |
| title | No-Regret Linear Bandits under Gap-Adjusted Misspecification |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2501.05361 |