Coordinate Descent: Efficient Optimization Techniques

This lecture focuses on coordinate descent, a method for optimizing functions by updating one coordinate at a time while keeping others fixed. The instructor begins by discussing the limitations of Newton's method, particularly its computational cost due to the need to compute and invert the Hessian matrix. The lecture introduces the secant method as a derivative-free alternative, which approximates gradients using finite differences. The instructor then explains the concept of coordinate descent, detailing its goal to minimize a function by modifying one coordinate at a time. Variants of coordinate descent, including gradient-based step sizes and exact coordinate minimization, are explored. The lecture also covers convergence analysis, emphasizing the importance of coordinate-wise smoothness and strong convexity. The Polyak-Lojasiewicz condition is introduced, providing insights into linear convergence rates. The instructor concludes by discussing applications of coordinate descent in machine learning, highlighting its efficiency in training generalized linear models and its relevance in modern optimization problems.