Relative loss bounds for single neurons

We analyze and compare the well-known gradient descent algorithm and the more recent exponentiated gradient algorithm for training a single neuron with an arbitrary transfer function. Both algorithms are easily generalized to larger neural networks, and the generalization of gradient descent is the standard backpropagation algorithm. We prove worst-case loss bounds for both algorithms in the single neuron case. Since local minima make it difficult to prove worst case bounds for gradient-based algorithms, we must use a loss function that prevents the formation of spurious local minima. We define such a matching loss function for any strictly increasing differentiable transfer function and prove worst-case loss bounds for any such transfer function and its corresponding matching loss. The different forms of the two algorithms´ bounds indicates that exponentiated gradient outperforms gradient descent when the inputs contain a large number of irrelevant components. Simulations on synthetic data confirm these analytical results