Optimization Techniques: Convexity in Machine Learning

This lecture introduces the fundamental concepts of optimization in machine learning, focusing on convexity. It begins with an overview of optimization problems, emphasizing the importance of convex functions and sets. The instructor explains the general optimization problem and the significance of mathematical modeling in defining and measuring machine learning models. Various optimization algorithms are discussed, including gradient descent and coordinate descent, along with their historical development. The lecture highlights the properties of convex functions, such as the Cauchy-Schwarz inequality and the definition of convex sets. The instructor elaborates on the motivation behind convex optimization, explaining that local minima in convex functions are also global minima, which is a crucial property for efficient problem-solving. The lecture concludes with a discussion on the convergence theory of optimization algorithms, reinforcing the importance of convexity in deriving general analysis and guarantees for convergence in optimization problems.