Proximal Operator and Gradient Descent

This lecture introduces the concepts of proximal operators and gradient descent. Proximal projection is a powerful tool for minimizing nonsmooth convex functions by transforming the minimization problem into finding fixed points for contractive operators. The lecture covers the definition, properties, and role of proximal operators, highlighting soft-thresholding. It also explains the optimality conditions for differentiable and non-differentiable functions, illustrating the proximal construction and its relation to the notion of projection. The lecture further discusses the proximal point algorithm and the convergence analysis of the proximal gradient algorithm, emphasizing the importance of conditions on the risk function for optimization problems.