Worst-case quadratic loss bounds for prediction using linear functions and gradient descent

In this paper we study the performance of gradient descent (GD) when applied to the problem of on-line linear prediction in arbitrary inner product spaces, We prove worst-case bounds on the sum of the squared prediction errors under various assumptions concerning the amount of a priori information about the sequence to predict, The algorithms we use are variants and extensions of on-line GD, Whereas our algorithms always predict using linear functions as hypotheses, none of our results requires the data to be linearly related. In fact, the bounds proved on the total prediction loss are typically expressed as a function of the total loss of the best fixed linear predictor with bounded norm, All the upper bounds are tight to within constants, Matching lower bounds are provided in some cases. Finally, we apply our results to the problem of on-line prediction for classes of smooth functions.