Optimism in the face of uncertainty

A decision principle that maximizes possible reward jointly over an action and a parameter in a confidence set. On the event that the set contains the true parameter, the reward predicted for this selected pair is at least the true optimal reward. This inequality bounds pseudoregret by the width of the confidence set in the selected action's direction. For an $L$-Lipschitz continuous reward link and a radius-$\beta$ confidence ellipsoid, Cauchy-Schwarz inequality and the triangle inequality bound that width by $2L\beta \sqrt{x^T{V}^{-1}x}$.