The existence of non-strict saddle points in deep linear networks has significant implications for the stability and robustness of implicit regularization. Non-strict saddle points are characterized by a positive semi-definite Hessian with at least one zero eigenvalue, indicating that the optimization landscape is flat in certain directions. This flatness can lead to prolonged training periods where gradient-based algorithms may get "stuck" in these regions, resulting in long plateaus during optimization.
From the perspective of implicit regularization, non-strict saddle points correspond to global minimizers of rank-constrained linear regression problems. This means that while the optimization process may linger around these saddle points, it can still yield solutions that generalize well to unseen data. However, the presence of these saddle points can also introduce instability in the training dynamics, as the algorithm may oscillate or take longer to converge to a solution.
Moreover, the distinction between strict and non-strict saddle points highlights the need for careful consideration of the optimization strategy employed. Algorithms that are designed to escape strict saddle points may not be as effective in navigating non-strict saddle points, potentially leading to suboptimal convergence behavior. Therefore, understanding the role of non-strict saddle points is crucial for developing robust training methodologies that leverage implicit regularization effectively.