  Deep neural networks (DNNs), trained with gradient-based optimization and
backpropagation, are currently the primary tool in modern artificial
intelligence, machine learning, and data science. In many applications, DNNs
are trained offline, through supervised learning or reinforcement learning, and
deployed online for inference. However, training DNNs with standard
backpropagation and gradient-based optimization gives no intrinsic performance
guarantees or bounds on the DNN, which is essential for applications such as
controls. Additionally, many offline-training and online-inference problems,
such as sim2real transfer of reinforcement learning policies, experience domain
shift from the training distribution to the real-world distribution. To address
these stability and transfer learning issues, we propose using techniques from
control theory to update DNN parameters online. We formulate the
fully-connected feedforward DNN as a continuous-time dynamical system, and we
propose novel last-layer update laws that guarantee desirable error convergence
under various conditions on the time derivative of the DNN input vector. We
further show that training the DNN under spectral normalization controls the
upper bound of the error trajectories of the online DNN predictions, which is
desirable when numerically differentiated quantities or noisy state
measurements are input to the DNN. The proposed online DNN adaptation laws are
validated in simulation to learn the dynamics of the Van der Pol system under
domain shift, where parameters are varied in inference from the training
dataset. The simulations demonstrate the effectiveness of using
control-theoretic techniques to derive performance improvements and guarantees
in DNN-based learning systems.
