Tracking the Best Expert
AbstractWe generalize the recent relative loss bounds for on-line algorithms where the additional loss of the algorithm on the whole sequence of examples over the loss of the best expert is bounded. The generalization allows the sequence to be partitioned into segments, and the goal is to bound the additional loss of the algorithm over the sum of the losses of the best experts for each segment. This is to model situations in which the examples change and different experts are best for certain segments of the sequence of examples. In the single segment case, the additional loss is proportional to log n, where n is the number of experts and the constant of proportionality depends on the loss function. Our algorithms do not produce the best partition; however the loss bound shows that our predictions are close to those of the best partition. When the number of segments is k+1 and the sequence is of length &ell, we can bound the additional loss of our algorithm over the best partition by 
$$O\left( {klogn + k\log \left( {{\ell \mathord{\left/ {\vphantom {\ell k}} \right. \kern-\nulldelimiterspace} k}} \right)} \right)$$
. For the case when the loss per trial is bounded by one, we obtain an algorithm whose additional loss over the loss of the best partition is independent of the length of the sequence. The additional loss becomes 
$$O\left( {klogn + k\log \left( {{\ell \mathord{\left/ {\vphantom {\ell k}} \right. \kern-\nulldelimiterspace} k}} \right)} \right)$$
 , where L is the loss of the best partitionwith k+1 segments. Our algorithms for tracking the predictions of the best expert aresimple adaptations of Vovk's original algorithm for the single best expert case. As in the original algorithms, we keep one weight per expert, and spend O(1) time per weight in each trial.