Saddle to Saddle Dynamic

Saddle-to-saddle dynamics describe the trajectory of optimization algorithms, particularly gradient descent, as they navigate the loss landscape of complex models like neural networks. Current research focuses on understanding this phenomenon in various architectures, including two-layer neural networks and diagonal linear networks, often employing techniques like Sharpness-Aware Minimization (SAM) to improve generalization. This research is crucial for improving the efficiency and robustness of training algorithms, impacting fields ranging from deep learning to constrained reinforcement learning and even differential privacy analysis where accurate composition of privacy guarantees is essential.