Ebook: Linear Algebra And Optimization With Applications To Machine Learning - Volume II: Fundamentals of Optimization Theory with Applications to Machine Learning

Volume 2 applies the linear algebra concepts presented in Volume 1 to optimization problems which frequently occur throughout Machine Learning. This book blends theory with practice by not only carefully discussing the mathematical under pinnings of each optimization technique but by applying these techniques to linear programming, support vector machines (SVM), principal component analysis (PCA), and ridge regression. Volume 2 begins by discussing preliminary concepts of optimization theory such as metric spaces, derivatives, and the Lagrange multiplier technique for finding extrema of real valued functions. The focus then shifts to the special case of optimizing a linear function over a region determined by affine constraints, namely linear programming. Highlights include careful derivations and applications of the simplex algorithm, the dual-simplex algorithm, and the primal-dual algorithm. The theoretical heart of this book is the mathematically rigorous presentation of various nonlinear optimization methods, including but not limited to gradient decent, the Karush-Kuhn-Tucker (KKT) conditions, Lagrangian duality, alternating direction method of multipliers (ADMM), and the kernel method. These methods are carefully applied to hard margin SVM, soft margin SVM, kernel PCA, ridge regression, lasso regression, and elastic-net regression. Matlab programs implementing these methods are included.

Many books on machine learning struggle with the above problem. How can one understand what are the dual variables of a ridge regression problem if one doesn’t know about the Lagrangian duality framework? Similarly, how is it possible to discuss the dual formulation of SVM without a firm understanding of the Lagrangian framework? The easy way out is to sweep these difficulties under the rug. If one is just a consumer of the techniques we mentioned above, the cookbook recipe approach is probably adequate. But this approach doesn’t work for someone who really wants to do serious research and make significant contributions. To do so, we believe that one must have a solid background in linear algebra and optimization theory.