Tracking the best quantizer
An algorithm is presented for online prediction that allows to track the best expert eﬃciently even when the number of experts is exponentially large, provided that the set of experts has a certain additive structure. As an example we work out the case where each expert is represented by a path in a directed graph and the loss of each expert is the sum of the weights over the edges in the path. These results are then used to construct universal limited-delay schemes for lossy coding of individual sequences. In particular, we consider the problem of tracking the best scalar quantizer that is adaptively matched to the source sequence with piecewise diﬀerent behavior. A randomized algorithm is presented which can perform, on any source sequence, asymptotically as well as the best scalar quantization algorithm that is matched to the sequence and is allowed to change the employed quantizer for a given number of times. The complexity of the algorithm is quadratic in the se- quence length, but at the price of some deterioration in performance, the complexity can be made linear. Analogous results are obtained for sequential multiresolution and multiple description scalar quantization of individual sequences.