Relative Loss Bounds for On-line Density Estimation with the Exponential Family of Distributions

We consider on-line density estimation with a parameterized density from the exponential family. The on-line algorithm receives one example at a time and maintains a parameter that is essentially an average of the past examples. After receiving an example the algorithm incurs a loss which is the negative log-likelihood of the example w.r.t. the past parameter of the algorithm. An off-line algorithm can choose the best parameter based on all the examples. We prove bounds on the additional total loss of the on-line algorithm over the total loss of the off-line algorithm. These relative loss bounds hold for an arbitrary sequence of examples. The goal is to design algorithms with the best possible relative loss bounds. We use a certain divergence to derive and analyze the algorithms. This divergence is a relative entropy between two exponential distributions.