{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Optimization - 5 Optimization Optimization plays an...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
5 Optimization Optimization plays an increasingly important role in machine learning. For instance, many machine learning algorithms minimize a regularized risk functional: min f J ( f ) := λ Ω( f ) + R emp ( f ) (5.1) with the empirical risk R emp ( f ) := 1 m m i =1 l ( f ( x i ) � y i ) . (5.2) Here x i are the training instances and y i are the corresponding labels. l the loss function measures the discrepancy between y and the predictions f ( x i ). Finding the optimal f involves solving an optimization problem. This chapter provides a self-contained overview of some basic concepts and tools from optimization, especially geared towards solving machine learning problems. In terms of concepts, we will cover topics related to convexity, duality, and Lagrange multipliers. In terms of tools, we will cover a variety of optimization algorithms including gradient descent, stochastic gradient descent, Newton’s method, and Quasi-Newton methods. We will also look at some specialized algorithms tailored towards solving Linear Programming and Quadratic Programming problems which often arise in machine learning problems. 5.1 Preliminaries Minimizing an arbitrary function is, in general, very difficult, but if the ob- jective function to be minimized is convex then things become considerably simpler. As we will see shortly, the key advantage of dealing with convex functions is that a local optima is also the global optima. Therefore, well developed tools exist to find the global minima of a convex function. Conse- quently, many machine learning algorithms are now formulated in terms of convex optimization problems. We briefly review the concept of convex sets and functions in this section. 129
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
130 5 Optimization 5.1.1 Convex Sets Definition 5.1 �Convex Set) A subset C of R n is said to be convex if (1 λ ) x + λy C whenever x C� y C and 0 < λ < 1 . Intuitively, what this means is that the line joining any two points x and y from the set C lies inside C (see Figure 5.1 ). It is easy to see (Exercise 5.1 ) that intersections of convex sets are also convex. Fig. 5.1. The convex set (left) contains the line joining any two points that belong to the set. A non-convex set (right) does not satisfy this property. A vector sum i λ i x i is called a convex combination if λ i 0 and i λ i = 1. Convex combinations are helpful in defining a convex hull: Definition 5.2 �Convex Hull) The convex hull, conv( X ) , of a finite sub- set X = { x 1 � . . . � x n } of R n consists of all convex combinations of x 1 � . . . � x n . 5.1.2 Convex Functions Let f be a real valued function defined on a set X R n . The set { ( x� µ ) : x X� µ R � µ f ( x ) } (5.3) is called the epigraph of f . The function f is defined to be a convex function if its epigraph is a convex set in R n +1 . An equivalent, and more commonly used, definition (Exercise 5.5 ) is as follows (see Figure 5.2 for geometric intuition): Definition 5.3 �Convex Function) A function f defined on a set X is called convex if, for any x� x X and any 0 < λ < 1 such that λx + (1 λ ) x X , we have f ( λx + (1 λ ) x )
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern