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 a pplied to the problem of on-line linear prediction in arbitrary inner product spaces, We prove worst-case bounds on the sum of the squared p rediction errors under various assumptions concerning the amount of a priori information about the sequence to predict, The algorithms we us e are variants and extensions of on-line GD, Whereas our algorithms al ways 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, A ll the upper bounds are tight to within constants, Matching lower boun ds are provided in some cases. Finally, we apply our results to the pr oblem of on-line prediction for classes of smooth functions.