Relative Expected Instantaneous Loss Bounds

In the literature a number of relative loss bounds have been shown for on-line learning algorithms. Here the relative loss is the total loss of the on-line algorithm in all trials minus the total loss of the best comparator that is chosen off-line. However, for many applications instantaneous loss bounds are more interesting where the learner first sees a batch of examples and then uses these examples to make a prediction on a new instance. We show relative expected instantaneous loss bounds for the case when the examples are i.i.d. with an unknown distribution. We bound the expected loss of the algorithm on the last example minus the expected loss of best comparator on a random example. In particular, we study linear regression and density estimation problems and show how the leave-oneout loss can be used to prove instantaneous loss bound for these cases. For linear regression we use an algorithm that is similar to a new on-line learning algorithm developed by Vovk. Recently a large number of relative total loss bounds have been shown that have the form O(lnT ), where T is the number of trials/examples. Standard conversions of on-line algorithms to batch algorithms result in relative expected instantaneous loss bounds of the form O( lnT T ). Our methods lead to O( 1 T ) bounds. We also prove lower bounds that show that our upper bound on the relative expected instantaneous loss for Gaussian density estimation is optimal. In the case of linear regression we can show that our bounds are tight within a factor of two.