  When an online learning algorithm is used to estimate the unknown parameters
of a model, the signals interacting with the parameter estimates should not
decay too quickly for the optimal values to be discovered correctly. This
requirement is referred to as persistency of excitation, and it arises in
various contexts, such as optimization with stochastic gradient methods,
exploration for multi-armed bandits, and adaptive control of dynamical systems.
While training a neural network, the iterative optimization algorithm involved
also creates an online learning problem, and consequently, correct estimation
of the optimal parameters requires persistent excitation of the network
weights. In this work, we analyze the dynamics of the gradient descent
algorithm while training a two-layer neural network with two different loss
functions, the squared-error loss and the cross-entropy loss; and we obtain
conditions to guarantee persistent excitation of the network weights. We then
show that these conditions are difficult to satisfy when a multi-layer network
is trained for a classification task, for the signals in the intermediate
layers of the network become low-dimensional during training and fail to remain
persistently exciting. To provide a remedy, we delve into the classical
regularization terms used for linear models, reinterpret them as a means to
ensure persistent excitation of the model parameters, and propose an algorithm
for neural networks by building an analogy. The results in this work shed some
light on why adversarial examples have become a challenging problem for neural
networks, why merely augmenting training data sets will not be an effective
approach to address them, and why there may not exist a data-independent
regularization term for neural networks, which involve only the model
parameters but not the training data.
