Insights from the loss landscape analysis of deep linear networks can be extended to understand the optimization dynamics of nonlinear neural networks by recognizing the similarities in their optimization landscapes. Both types of networks exhibit non-convex loss functions, which can contain various critical points, including local minima, strict saddle points, and non-strict saddle points.
In nonlinear networks, the presence of saddle points—both strict and non-strict—can similarly affect the convergence behavior of gradient-based optimization algorithms. The analysis of the loss landscape in linear networks provides a framework for identifying the conditions under which these critical points occur, which can be applied to nonlinear architectures. For instance, the rank conditions that characterize critical points in linear networks can inform the design of initialization strategies and learning rates in nonlinear networks to avoid getting trapped in non-optimal regions of the loss landscape.
Furthermore, the concept of implicit regularization observed in linear networks can also be relevant for nonlinear networks. The tendency of gradient descent to converge to low-rank solutions in linear networks suggests that similar phenomena may occur in nonlinear settings, where the optimization dynamics could favor simpler models that generalize better. By leveraging the understanding of the loss landscape from linear networks, researchers can develop more effective training algorithms for nonlinear networks that account for the complexities introduced by non-convexity and the presence of saddle points.