非单调凸岭赌博机的汤普森采样：多项式遗憾不需要单调性

中文摘要 AI 辅助

Bakhtiari、Lattimore和Szepesvári（COLT 2025）证明了对于具有凸单调岭损失$f(x)=\ell(\ip{x}{\theta})$的赌博机凸优化，汤普森采样（TS）具有贝叶斯遗憾$\tilde O(d^{5/2}\sqrt n)$，并询问链接的单调性是否是必要的。我们给出了一个定性的否定答案。对于取值于$[0,1]$的$1$-Lipschitz凸岭损失的任意先验，且链接为任意凸的、可能非单调的链接，以及任何固定的可测最小化器选择，精确后验TS具有贝叶斯遗憾$O\big((d+1)^4\sqrt{dn}\\,\log(e+nd\max\{1,\diam K\})\big)=\tilde O(d^{9/2}\sqrt n)$。单调情形的证明依赖于单次移除的John椭球二分法；我们通过一个明确的十二点配置表明该二分法对非单调链接失效，并用“无信息”配置的$O(d^2)$基数界替代它。该界使用布尔舍入论证：一个在最大范数下与秩为$r$的矩阵距离在$1/(4r)$以内的$0$-$1$矩阵的秩至多为$2r-1$。我们构造了$d(d+1)$个无信息损失，表明在大直径与间隙之比区域中该基数界在常数意义下是紧的，并给出了一个自包含的信息比到遗憾的转换，该转换对固定的可测选择是一致的。单调情形中$d^{5/2}$的依赖是否能够保留仍然是一个开放问题。

英文摘要

Bakhtiari, Lattimore and Szepesvári (COLT 2025) proved that Thompson sampling (TS) has Bayesian regret $\tilde O(d^{5/2}\sqrt n)$ for bandit convex optimisation with convex \emph{monotone} ridge losses $f(x)=\ell(\ip{x}θ)$, and asked whether monotonicity of the link is necessary. We give a qualitative negative answer. For every prior on $[0,1]$-valued, $1$-Lipschitz convex ridge losses with an arbitrary convex, possibly non-monotone, link, and for any fixed measurable selection of minimisers, exact-posterior TS has Bayesian regret $O\big((d+1)^4\sqrt{dn}\,\log(e+nd\max\{1,\diam K\})\big)=\tilde O(d^{9/2}\sqrt n)$. The monotone proof relies on a single-removal John-ellipsoid dichotomy; we show by an explicit twelve-point configuration that this dichotomy fails for non-monotone links, and replace it by an $O(d^2)$ cardinality bound for ``uninformative'' configurations. The bound uses a Boolean rounding argument: a $0$-$1$ matrix within $1/(4r)$ in max-norm of a rank-$r$ matrix has rank at most $2r-1$. We construct $d(d+1)$ uninformative losses, showing that the cardinality bound is tight up to constants in the large-diameter-to-gap regime, and give a self-contained information-ratio-to-regret transfer that is uniform over fixed measurable selections. Whether the $d^{5/2}$ dependence of the monotone case can be retained remains open.

发表机构

• University of New South Wales（新南威尔士大学）

机构由 AI 辅助整理，请以论文原文为准。