Joint lower bound for tilted-sampling regret

Establish a matching lower bound for the joint dependence of tilted-sampling regret on the time horizon, inverse temperature, and function-class complexity, complementing the upper bounds proved for UCS, TS$^2$, and SWC.

Background

The paper develops upper bounds for sampling regret when learning a Gibbs distribution induced by an unknown reward function under noisy bandit feedback. For UCS and TS$^2$, the bounds exhibit complementary dependence on the horizon $T$, inverse temperature $\beta$, and statistical complexity of the function class; generic SWC bounds additionally contain an $\exp(\beta C)$ factor. The authors do not establish whether these dependencies are jointly optimal. A matching lower bound would determine the fundamental minimax difficulty of tilted sampling across its temporal, temperature, and function-class dimensions.