Dimension dependence of Thompson-sampling regret for non-monotone convex ridge losses

Determine whether the monotone-case dimension exponent 5/2 can be attained for exact-posterior Thompson sampling on arbitrary convex, non-monotone, 1-Lipschitz ridge losses, and characterize the correct dimension dependence of Thompson sampling on this class, including whether a lower bound specific to non-monotone links exists.

Background

The paper proves a Bayesian regret bound of order $\tilde O(d^{9/2}\sqrt n)$ for exact-posterior Thompson sampling on convex ridge losses with arbitrary convex links, removing the monotonicity assumption required in the earlier monotone-link analysis. The authors explicitly distinguish this qualitative resolution from the unresolved quantitative question of whether the earlier $\tilde O(d^{5/2}\sqrt n)$ dependence can be preserved.

The paper explains that its information-ratio certificate has order $O(d^8(1+\log(1/\alpha)))$, partly because uninformative configurations can have quadratic cardinality and because the rounding argument uses a tolerance of order $1/d$. Although the quadratic cardinality lower construction is tight up to constants, it does not establish that the resulting regret exponent is optimal. The authors also state that no lower bound tailored specifically to non-monotone links is known.