Tracking the best regressor

Abstract

In most of the on-line learning research the total on-line loss of the algorithm is compared to the total loss of the best off-line predictor u from a comparison class of predictors. We call such bounds static bounds. The interesting feature of these bounds is that they hold for an arbitrary sequence of examples. Recently some work has been done where the comparison vector ut at each trial t is allowed to change with time, and the total online loss of the algorithm is compared to the sum of the losses of ut at each trial plus the total "cost" for shifting to successive comparison vectors. This is to model situations in which the examples change over time and different predictors from the comparison class are best for different segments of the sequence of examples. We call such bounds shifting bounds. Shifting bounds still hold for arbitrary sequences of examples and also for arbitrary partitions. The algorithm does not know the offline partition and the sequence of predictors that its performance is compared against. Naturally shifting bounds are much harder to prove. The only known bounds are for the case when the comparison class consists of a finite sets of experts or boolean disjunctions. In this paper we develop the methodology for lifting known static bounds to the shifting case. In particular we obtain bounds when the comparison class consists of linear neurons (linear combinations of experts). Our essential technique consists of the following. At the end of each trial we project the hypothesis of the static algorithm into a suitably chosen convex region. This keeps the hypothesis of the algorithm well-behaved and the static bounds can be converted to shifting bounds so that the cost for shifting remains reasonable.

On the equivalence of weak learnability and linear separability: new relaxations and efficient boosting algorithms

“…(12). A similar problem was tackled by Herbster and Warmuth [HW01], who provided O(m log(m)) and O(m) algorithms for performing entropic projections. For completeness, in the rest of this section we outline the simpler O(m log(m)) algorithm.…”

Learning Additive Models Online with Fast Evaluating Kernels

“…Local bounds are thus appropriate when the best predictor for the example sequence is changing over time. There have been a number of papers [28,20,3,43,6,21] which prove loss bounds in terms of a measure of the amount of change of the best predictor over time. These bounds have been called shifting or switching bounds.…”

“…These bounds have been called shifting or switching bounds. The local bounds of this section are direct simplifications of the shifting bounds in [21]. Here we give local bounds rather than shifting bounds, however, since less introductory machinery is required, the bounds are easier to interpret, and weaker assumptions on the example sequence are possible in the theorem statements.…”

“…We give a simple total loss bound for this algorithm with the -insensitive square loss. Finally, in Part 2 we show that the recent results of [21] may be applied and implemented efficiently for online learning with kernels; this allows us to prove local loss bounds. These bounds differ from total loss bounds in that they hold for any segment of the sequence.…”

Untitled

“…Again we aim to bound the total loss of the on-line algorithm minus the total loss of the best comparator of this form. Such bound have been obtained by Herbster and Warmuth (1998) for the case of linear regression using first-order algorithms. We would like to know whether there is a simple second-order algorithm for linear regression that requires O(n 2 ) update time per trial and for which the additional loss grows with the sums of the logs of the section lengths.…”