Proximal Gradient Descent: Optimization Techniques in Machine Learning

This lecture covers the principles of proximal gradient descent, a key optimization technique in machine learning. It begins with an introduction to composite optimization problems, where objective functions are expressed as the sum of a 'nice' function and a 'simple' term that may not be differentiable. The instructor explains the convergence properties of proximal gradient descent, highlighting its efficiency in handling non-differentiable functions. The lecture details the algorithm's iteration process, defining the proximal mapping and its role in optimization. The concept of subgradients is introduced, providing a framework for optimization when functions are not differentiable. The instructor discusses the characterization of convexity through subgradients and the differentiability of convex functions. The lecture concludes with a discussion on the optimality of first-order methods and the implications of strong convexity on convergence rates, emphasizing the importance of bounded subgradients for achieving efficient optimization. Overall, this lecture provides a comprehensive overview of proximal gradient descent and its applications in machine learning optimization.