Learning the Switching Rate by Discretising Bernoulli Sources Online

The expert tracking algorithm Fixed-Share depends on a parameter α, called the switching rate. The switching rate can be learned online with regret 1 2 log T + O(1) bits. The current fastest method to achieve this is based on optimal discretisation of the Bernoulli distributions into O( √ T ) bins and runs in O(T √ T ) time. However, the exact locations of these bins have to be determined algorithmically, and the final number of outcomes T must be known in advance. This paper introduces a new discretisation scheme with the same regret bound for known T , that specifies the number and positions of the discretisation points explicitly. The scheme is especially useful, however, when T is not known in advance: a new fully online algorithm is presented, which runs in O(T √ T log T ) time and achieves a regret of 1 2 log 3 log T +O(log log T ) bits.