Non-Convex Optimization: Techniques and Applications

This lecture focuses on non-convex optimization, a critical area in machine learning. It begins with an overview of gradient descent applied to smooth functions, discussing the conditions under which it can converge to a global minimum despite the presence of local minima and saddle points. The instructor emphasizes the importance of trajectory analysis, which helps in understanding the behavior of gradient descent from various starting points. The lecture also covers linear models with multiple outputs and the process of minimizing least squares error, illustrating how to compute optimal weight matrices. Furthermore, the discussion extends to deep linear neural networks, highlighting their training dynamics and the challenges posed by non-convexity. The lecture concludes with convergence analysis, detailing the conditions necessary for gradient descent to ensure convergence to a global minimum, and the implications of bounded Hessians along the trajectory. Overall, this lecture provides essential insights into the complexities of non-convex optimization in machine learning contexts.