Optimization Techniques: Gradient Descent and Convex Functions

This lecture covers optimization techniques in machine learning, focusing on gradient descent and the properties of convex functions. It begins with the definition of critical points and their significance as global minima in convex functions. The instructor presents the concept of constrained minimization, explaining how to identify minimizers within a convex set. The existence of global minima is discussed, emphasizing that not all functions guarantee a minimum. The lecture introduces iterative algorithms for approaching optimal solutions, detailing the assumptions required for effective gradient descent. Examples illustrate the application of these concepts, including bounding the difference between function values at critical points. The discussion extends to Lipschitz convex functions, highlighting their implications for convergence rates in optimization. The lecture concludes with an exploration of smooth functions and their role in optimization, providing insights into the relationship between smoothness and convergence in gradient descent methods. Overall, the lecture provides a comprehensive overview of essential optimization principles relevant to machine learning.