Because the optimistic Gaussian surrogate is only well-behaved on a shrinking ellipsoidal focus region, algorithms that use it are most naturally analysed in the stochastic setting, where it is already a challenge to prove that the optimal action remains in the focus region. In the adversarial setting there is limited hope to ensure the optimal action in hindsight remains in the focus region. The plan is to use a mechanism that detects when the minimiser leaves the focus region and restarts the algorithm. This is combined with an argument that the regret is negative whenever a restart occurs. The formal setting studied in this chapter is characterised by the following assumption: