  I introduce and analyse an anytime version of the Optimally Confident UCB
(OCUCB) algorithm designed for minimising the cumulative regret in finite-armed
stochastic bandits with subgaussian noise. The new algorithm is simple,
intuitive (in hindsight) and comes with the strongest finite-time regret
guarantees for a horizon-free algorithm so far. I also show a finite-time lower
bound that nearly matches the upper bound.
