EECS 545 - Machine Learning

Lecture 3: Convex Optimization

Date: January 13, 2016

Instructor: Jacob Abernethy

Some important notes

  • HW1 is out! Due January 25th at 11pm
  • Homework will be submitted via Gradescope. Please see Piazza for precise instructions. Do it soon, not at the last minute!!
  • There is an optional tutorial this evening, 5pm, in Dow 1013. Come see Daniel go over some tough problems.
  • No class on Monday January 18, MLK day!

Python: We recommend Anacdona

Drawing

  • Anaconda is standalone Python distribution that includes all the most important scientific packages: numpy, scipy, matplotlib, sympy, sklearn, etc.
  • Easy to install, available for OS X, Windows, Linux.
  • Small warning: it's kind of large (250MB)

Some notes on using Python

  • HW1 has only a very simple programming exercise, just as a warmup. We don't expect you to submit code this time
  • This is a good time to start learning Python basics
  • There are a ton of good places on the web to learn python, we'll post some
  • Highly recommended: ipython; it's a much more user friendly terminal interface to python
  • Even better: jupyter notebook, a web based interface. This is how I'm making these slides!

Checking if all is installed, and HelloWorld

If you got everything installed, this should run:

# numpy is crucial for vectors, matrices, etc.
import numpy as np              
# Lots of cool plotting tools with matplotlib
import matplotlib.pyplot as plt
# For later: scipy has a ton of stats tools
import scipy as sp
# For later: sklearn has many standard ML algs
import sklearn
# Here we go!
print("Hello World!")

More on learning python

  • We will have one tutorial devoted to this
  • If you're new to Python, go slow!
    • First learn the basics (lists, dicts, for loops, etc.)
    • Then spend a couple days playing with numpy
    • Then explore matplotlib
    • etc.
  • Piazza = your friend. We have a designated python instructor (IA Ben Bray) who has lots of answers.

Outline for this Lecture

  • Convexity
    • Convex Set
    • Convex Function
  • Introduction to Optimization
  • Introduction to Lagrange Duality

Convexity

Convex Sets

  • $C \subseteq \mathbb{R}^n$ is convex if $$t x + (1-t)y \in C$$ for any $x, y \in C$ and $0 \leq t \leq 1$
  • that is, a set is convex if the line connecting any two points in the set is entirely inside the set
(Left: Convex Set; Right: Non-convex Set)

Convex Functions

  • We say that a function $f$ is convex if, for any distinct pair of points $x,y$ we have $$f(tx_1+(1-t)x_2) \leq tf(x_1)+(1-t)f(x_2) \quad \forall t \in[0,1]$$
  • A function $f$ is said to be concave if $-f$ is convex

Fun Facts About Convex Functions

  • If $f$ is differentiable, then $f$ is convex iff $f$ "lies above its linear approximation", i.e.: $$ f(x + y) \geq f(x) + \nabla_x f(x) \cdot y \quad \forall x,y$$
  • If $f$ is twice-differentiable, then the hessian is always positive semi-definite!
  • This last one you will show on your homeowork :-)

Introduction to Optimization

The Most General Optimization Problem

Assume $f$ is some function, and $C \subset \mathbb{R}^n$ is some set. The following is an optimization problem: $$ \begin{array}{ll} \mbox{minimize} & f(x) \\ \mbox{subject to} & x \in C \end{array} $$

  • How hard is it to find a solution that is (near-) optimal? This is one of the fundamental problems in Computer Science and Operations Research.
  • A huge portion of ML relies on this task

A Rough Optimization Hierarchy

$$ \mbox{minimize } \ f(x) \quad \mbox{subject to } x \in C $$
  • [Really Easy] $C = \mathbb{R}^n$ (i.e. problem is unconstrained), $f$ is convex, $f$ is differentiable, strictly convex, and "slowly-changing" gradients
  • [Easyish] $C = \mathbb{R}^n$, $f$ is convex
  • [Medium] $C$ is a convex set, $f$ is convex
  • [Hard] $C$ is a convex set, $f$ is non-convex
  • [REALLY Hard] $C$ is an arbitrary set, $f$ is non-convex

Optimization Without Constraints

$$ \begin{array}{ll} \mbox{minimize} & f(x) \\ \mbox{subject to} & x \in \mathbb{R}^n \end{array} $$
  • This problem tends to be easier than constrained optimization
  • We just need to find an $x$ such that $\nabla f(x) = \vec{0}$
  • Techniques like gradient descent or Newton's method work in this setting. (More on this later)

Optimization With Constraints

$$ \begin{aligned} & {\text{minimize}} & & f(\mathbf{x})\\ & \text{subject to} & & g_i(\mathbf{x}) \leq 0, \quad i = 1, ..., m\\ & & & h_j(x) = 0, \quad j = 1, ..., n \end{aligned} $$
  • Here $C = \{ x : g_i(x) \leq 0,\ h_j(x) = 0, \ i=1, \ldots, m,\ j = 1, ..., n \}$
  • $C$ is convex as long as all $g_i(x)$ convex and all $h_j(x)$ affine
  • The solution of this optimization may occur in the interior of $C$, in which case the optimal $x$ will have $\nabla f(x) = 0$
  • But what if the solution occurs on the boundary of $C$?

Introduction to Lagrange Duality

  • In some cases original (primal) optimization problem can hard to solve, solving a proxy problem sometimes can be easier
  • The proxy problem could be dual problem which is transformed from primal problem
  • Here is how to transform from primal to dual. For primal problem $$ \begin{aligned} & {\text{minimize}} & & f(\mathbf{x})\\ & \text{subject to} & & g_i(\mathbf{x}) \leq 0, \quad i = 1, ..., m\\ & & & h_j(x) = 0, \quad j = 1, ..., n \end{aligned} $$ Its Lagrangian is $$L(x,\boldsymbol{\lambda}, \boldsymbol{\nu}) := f(x) + \sum_{i=1}^m \lambda_i g_i(x) + \sum_{j=1}^n \nu_j h_j(x)$$ of which $\boldsymbol{\lambda} \in \mathbb{R}^m$, $\boldsymbol{\nu} \in \mathbb{R}^n$ are dual variables

  • The Langragian dual function is $$L_D(\boldsymbol{\lambda}, \boldsymbol{\nu}) \triangleq \underset{x}{\inf}L(x,\boldsymbol{\lambda}, \boldsymbol{\nu}) = \underset{x}{\inf} \ \left[ f(x) + \sum_{i=1}^m \lambda_i g_i(x) + \sum_{j=1}^n \nu_j h_j(x) \right] $$ The minization is usually done by finding the stable point of $L(x,\boldsymbol{\lambda}, \boldsymbol{\nu})$ with respect to $x$

The Lagrange Dual Problem

  • Then the dual problem is $$ \begin{aligned} & {\text{maximize}} & & L_D(\mathbf{\lambda}, \mathbf{\nu})\\ & \text{subject to} & & \lambda_i, \nu_j \geq 0 \quad i = 1, \ldots, m ,\ j = 1, ..., n\\ \end{aligned} $$ Instead of solving primal problem with respect to $x$, we now need to solve dual problem with respect to $\mathbf{\lambda}$ and $\mathbf{\nu}$
  • $ L_D(\mathbf{\lambda}, \mathbf{\nu})$ is concave even if primal problem is not convex
  • Let the $p^*$ and $d^*$ denote the optimal values of primal problem and dual problem, we always have weak duality: $p^* \geq d^*$
  • Under nice conditions, we get strong duality: $p^* = d^*$
  • Many details are ommited here and they will come when we study support vector machine (SVM)
  • Free online!
  • Chapter 5 covers duality