Название: Optimization for Learning and Control
Автор: Martin Andersen, Anders Hansson
Издательство: Wiley
Год: 2023
Страниц: 435
Язык: английский
Формат: pdf (true)
Размер: 10.1 MB

Optimization for Learning and Control Comprehensive resource providing a masters’ level introduction to optimization theory and algorithms for learning and control.

Optimization for Learning and Control describes how optimization is used in these domains, giving a thorough introduction to both Unsupervised Learning, Supervised Learning, and Reinforcement Learning, with an emphasis on optimization methods for large-scale learning and control problems. Several applications areas are also discussed, including signal processing, system identification, optimal control, and Machine Learning. Today, most of the material on the optimization aspects of Deep Learning that is accessible for students at a Masters’ level is focused on surface-level computer programming; deeper knowledge about the optimization methods and the trade-offs that are behind these methods is not provided. The objective of this book is to make this scattered knowledge, currently mainly available in publications in academic journals, accessible for Masters’ students in a coherent way. The focus is on basic algorithmic principles and trade-offs.

We are now going to discuss Unsupervised Learning. This is about finding lower-dimensional descriptions of a set of data {x1, … , xN}. One simple such lower-dimensional description is the mean of the data. Another one could be to find a probability function from which the data are the outcome. We will see that there are many more lower-dimensional descriptions of data. We will start the chapter by defining entropy, and we will see that many of the probability density functions that are of interest in learning can be derived from the so-called “maximum entropy principle.” Specifically, we will derive the categorical distribution, the Ising distribution, and the normal distribution. There is a close relationship between the Lagrange dual function of the maximum entropy problem and maximum likelihood (ML) estimation, which will also be investigated. Other topics that we cover are prediction, graphical models, cross entropy, the expectation maximization algorithm, the Boltzmann machine, principal component analysis, mutual information, and cluster analysis. As a prelude to entropy we will start by discussing the so-called Chebyshev bounds.

The CVX modeling package for MATLAB has pioneered what is referred to as disciplined convex programming. It requires that user inputs a problem in a form that allows the software to verify convexity via a number of known composition rules. The problem is then reformulated as a conic optimization problem and passed to one of several possible solvers. The software packages CVXPY, Convex.jl, and CVXR make similar modeling functionality available in the programming languages Python, Julia, and R, respectively.

Optimization for Learning and Control covers sample topics such as:

Optimization theory and optimization methods, covering classes of optimization problems like least squares problems, quadratic problems, conic optimization problems and rank optimization.
First-order methods, second-order methods, variable metric methods, and methods for nonlinear least squares problems.
Stochastic optimization methods, augmented Lagrangian methods, interior-point methods, and conic optimization methods.
Dynamic programming for solving optimal control problems and its generalization to Reinforcement learning.
How optimization theory is used to develop theory and tools of statistics and learning, e.g., the maximum likelihood method, expectation maximization, k-means clustering, and support vector machines.
How calculus of variations is used in optimal control and for deriving the family of exponential distributions.

Optimization for Learning and Control is an ideal resource on the subject for scientists and engineers learning about which optimization methods are useful for learning and control problems; the text will also appeal to industry professionals using Machine Learning for different practical applications.