Optimization Techniques: Convexity and Algorithms in Machine Learning

This lecture introduces the fundamental concepts of optimization in machine learning, focusing on convexity and its significance. The instructor outlines the course structure, emphasizing the importance of optimization algorithms, including gradient descent and its variants. The discussion covers the mathematical modeling of optimization problems, highlighting the distinction between theoretical and practical aspects. The lecture delves into the properties of convex functions and sets, explaining why convexity is crucial for ensuring that local minima are also global minima. The instructor presents various optimization methods, including coordinate descent and stochastic gradient descent, and discusses their historical development and applications in machine learning. The importance of understanding the convergence rates of these algorithms is also emphasized, as well as the computational trade-offs involved in choosing the right optimization technique. The session concludes with an overview of constrained minimization and the conditions under which global minima can be guaranteed, setting the stage for further exploration of optimization techniques in subsequent lectures.