更严格的非指定核化老虎机优化保证
Sharper Guarantees for Misspecified Kernelized Bandit Optimization
AI总结：
本文针对非指定核化老虎机优化问题，提出更高效的保证方法，通过谱局部化和域分割算法，减少非指定误差放大，改进了离线和在线设置下的性能界。
摘要：
现有非指定核化老虎机优化的保证通过核复杂性来处理非指定问题：在通用离线界中，非指定水平ε被乘以√d_eff，其中d_eff是核有效维度，而在在线遗憾界中，相应的惩罚是√γ_n *nε，其中γ_n是n轮交互后的最大信息增益。在本文中，我们证明，对于一大类核，非指定放大可以减少到对数或多项对数增长。在离线设置中，我们首先证明了高概率简单遗憾界，其非指定项由谱Lebesgue常数控制。这导致了一维单调谱的对数放大和多变量傅里叶-对角乘积核的多项对数放大。在在线设置中，我们修改了域分割算法，并在温和的局部特征衰减假设下证明了累积遗憾界为~O(√(γ_n n)+nε)，去除了非指定项中的额外√γ_n因子。共同的原则是局部化：谱局部化控制离线近似算子的Lebesgue常数，而域分割在在线设置中实现了这种机制的空间类比，防止局部非指定误差被全局放大。
英文摘要：
Existing guarantees for misspecified kernelized bandit optimization pay for misspecification through kernel complexity: in generic offline bounds, the misspecification level $\varepsilon$ is multiplied by $\sqrt{d_\mathrm{eff}}$, where $d_\mathrm{eff}$ is the kernel effective dimension, while in online regret bounds, the corresponding penalty is $\sqrt{γ_n}\,n\varepsilon$, where $γ_n$ is the maximum information gain after $n$ rounds of interaction. In this work, we show that, for a large class of kernels, the misspecification amplification can be reduced to logarithmic or polylogarithmic growth. In the offline setting, we first prove high-probability simple-regret bounds whose misspecification term is governed by a spectral Lebesgue constant. This yields logarithmic amplification for one-dimensional monotone spectra and polylogarithmic amplification for multivariate Fourier-diagonal product kernels. In the online setting, we modify a domain-splitting algorithm and prove a cumulative regret bound of $\widetilde{\mathcal O}(\sqrt{γ_n n}+n\varepsilon)$ under mild localized eigendecay assumptions, removing the extra $\sqrt{γ_n}$ factor from the misspecification term. The common principle is localization: spectral localization controls the Lebesgue constant of the offline approximation operator, while domain splitting implements the spatial analogue of this mechanism in the online setting, preventing local misspecification errors from being amplified globally.
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