Gradient Descent: Principles and Applications

This lecture introduces the concept of gradient descent, a fundamental optimization algorithm widely used in machine learning. The instructor begins by discussing critical points and their significance in identifying global minima for convex functions. The lecture covers constrained minimization, defining minimizers within convex sets, and the existence of minimizers. The instructor then presents the gradient descent algorithm, explaining its iterative nature and the importance of the learning rate. An example illustrates how to apply gradient descent to a quadratic function, emphasizing the role of the initial point and step size in convergence. The lecture further explores the convergence rates for smooth convex functions and discusses the implications of Lipschitz conditions on the algorithm's performance. The instructor highlights the practical challenges of selecting appropriate parameters and the theoretical bounds on the distance to the optimal solution after a specified number of iterations. Overall, this lecture provides a comprehensive overview of gradient descent, its theoretical foundations, and practical considerations in optimization for machine learning.