Relative Loss Bounds for Multidimensional Regression Problems

We study online generalized linear regression with multidimensional outputs, i.e., neural networks with multiple output nodes but no hidden nodes. We allow at the final layer transfer functions such as the softmax function that need to consider the linear activations to all the output neurons. We use distance functions of a certain kind in two completely independent roles in deriving and analyzing online learning algorithms for such tasks. We use one distance function to define a matching loss function for the (possibly multidimensional) transfer function, which allows us to generalize earlier results from one-dimensional to multidimensional outputs. We use another distance function as a tool for measuring progress made by the online updates. This shows how previously studied algorithms such as gradient descent and exponentiated gradient fit into a common framework. We evaluate the performance of the algorithms using relative loss bounds that compare the loss of the online algoritm to the best off-line predictor from the relevant model class, thus completely eliminating probabilistic assumptions about the data.