Implicit regularization and momentum algorithms in nonlinear adaptive control and prediction

Stable concurrent learning and control of dynamical systems is the subject of adaptive control. Despite being an established field with many practical applications and a rich theory, much of the development in adaptive control for nonlinear systems revolves around a few key algorithms. By exploiting strong connections between classical adaptive nonlinear control techniques and recent progress in optimization and machine learning, we show that there exists considerable untapped potential in algorithm development for both adaptive nonlinear control and adaptive dynamics prediction. We first introduce first-order adaptation laws inspired by natural gradient descent and mirror descent. We prove that when the system is in the over-parameterized regime typical of modern machine learning or is not persistently excited, such non-Euclidean adaptation laws implicitly regularize the learned model. We apply this result to regularized dynamics predictor and observer design, and consider Hamiltonian systems, Lagrangian systems, and recurrent neural networks as concrete examples. We subsequently develop a variational formalism based on the Bregman Lagrangian to define adaptation laws with momentum applicable to linearly parameterized systems or nonlinearly parameterized systems satisfying monotonicity or convexity requirements. We show that the Euler Lagrange equations for the Bregman Lagrangian lead to natural gradient and mirror descent-like adaptation laws with momentum, and recover their first-order analogues in the infinite friction limit. We illustrate our analysis with simulations using a higher-order algorithm for nonlinearly parameterized systems to learn regularized hidden layer weights in a three-layer feedforward neural network.