  We study how to make decisions that minimize Bayesian regret in offline
linear bandits. Prior work suggests that one must take actions with maximum
lower confidence bound (LCB) on their reward. We argue that the reliance on LCB
is inherently flawed in this setting and propose a new algorithm that directly
minimizes upper bounds on the Bayesian regret using efficient conic
optimization solvers. Our bounds build heavily on new connections to monetary
risk measures. Proving a matching lower bound, we show that our upper bounds
are tight, and by minimizing them we are guaranteed to outperform the LCB
approach. Our numerical results on synthetic domains confirm that our approach
is superior to LCB.
