Analysis of two gradient-based algorithms for on-line regression

In this paper we present a new analysis of two algorithms, Gradient Descent and Exponentiated Gradient, for solving regression problems in the on-line framework. Both these algorithms compute a prediction that depends linearly on the current instance, and then update the coefficients of this linear combination according to the gradient of the loss function. However, the two algorithms have distinctive ways of using the gradient information for updating the coefficients. For each algorithm, we show general regression bounds for any convex loss function. Furthermore, we show special bounds for the absolute and the square loss functions, thus extending previous results by Kivinen and Warmuth. In the nonlinear regression case, we show general bounds for pairs of transfer and loss functions satisfying a certain condition. We apply this result to the Hellinger loss and the entropic loss in case of logistic regression (similar results, but only for the entropic loss, were also obtained by Helmbold et al. using a different analysis.) Finally, we describe the connection between our approach and a general family of gradient-based algorithms proposed by Warmuth et nl. in recent works. (C) 1999 Academic Press.