Dragon Notes

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  UNDER CONSTRUCTION

Machine Learning


Machine Learning


Subtopics


Basics

Supervised vs. Unsupervised Learning

SupervisedUnsupervised
- Correct output is known
- Goal is to derive relationship between input and output:
Regression: mapping inputs to a continuous output function
Classification: mapping inputs to representative discrete outputs
- Correct output is not known
- Goal is to derive structure from data where the effects of variables aren't necessarily known
- Works by clustering data based on relationships among variables in the data
- No feedback based on prediction results
Examples:
Regression: given a picture of a person, predict their age on the basis of the given picture
Classification: given a patient with a tumor, predict whether the tumor is malignant of benign
Examples:
Clustering: given a collection of 1,000,000 different genes, find a way to automatically group these genes so that they are somehow similar or related by different variables, such as lifespan, location, roles, etc.
Non-clustering: given a mesh of sounds at a cocktail party, identify individual voices and music (finding structure in a chaotic environment)

Cost Function

A measure of accuracy of hypothesis function - the mean squared error:
\(\ds J(\theta_0,\theta_1)=\frac{1}{2m}\sum_{i=1}^{m}(\hat{y}_i-y_i)^2=\frac{1}{2m}\sum_{i=1}^{m}(h_{\theta}(x_i)-y_i)^2\)
\(\hat{y}_i =\) predeicted output
\(y_i =\) true output
\(m =\) # of training samples
\(x_i =\) input
\(h_{\theta}(x_i) =\) hypothesis function
\(\theta_0, \theta_1 =\) model parameters
Gradient Descent Algorithm

A method for determining function optima via step-approximation:
\(\ds \theta_j := \theta_j -\alpha \frac{\partial}{\partial \theta_j}J(\theta_0,\theta_1)\)
  \((\t{for }j=0\t{ and }j=1)\)

\(\alpha =\) learning rate (step-size)
\(\theta_j =\) model parameter

Hypothesis Function

A learned function aiming to map inputs to outputs with least error; multivariate hypothesis function:
\(\ds h_\theta(x) = \theta_0 +\theta_1 x_1 + \theta_2 x_2 + ... + \theta_n x_n\)
\(\ds h_\theta(x) = \bn{\theta}^T \bn{x}\)

\(x^{(i)}_j =\) value of feature \(j\) in the \(i^{\t{th}}\) training example
\(m =\) # of training examples
\(\bn{\theta} =\) model parameter vector, \((n+1)\times 1\)
\(x^{(i)}=\) input (features) of the \(i^{\t{th}}\) training example\(\vph{x^{(i)}_j}\)
\(n =\) # of features
\(\bn{x} =\) input vector, \((n+1)\times 1\), \(x_0 = 1\)

Linear Regression GD

For linear regression involving multiple variables, the GD algorithm is

\(\ds \theta_j := \theta_j -\alpha \frac{1}{m}\sum_{i=1}^{m}(h_{\theta}(x^{(i)})-y^{(i)})\cdot x^{(i)}_j\)
  \((\t{for }j=0,1,...,n)\)
\(\ds \Rightarrow \ \ \begin{align} \theta_0 & := \theta_0 -\alpha \frac{1}{m}\sum_{i=1}^{m}(h_{\theta}(x^{(i)})-y^{(i)})\cdot x^{(i)}_0 \\ \theta_1 & := \theta_1 -\alpha \frac{1}{m}\sum_{i=1}^{m}(h_{\theta}(x^{(i)})-y^{(i)})\cdot x^{(i)}_1 \\ & \vdots \\ \theta_n & := \theta_n -\alpha \frac{1}{m}\sum_{i=1}^{m}(h_{\theta}(x^{(i)})-y^{(i)})\cdot x^{(i)}_n \\ \end{align} \)

\(x^{(i)}_j =\) value of feature \(j\) in the \(i^{\t{th}}\) training example
\(m =\) # of training examples
\(x^{(i)}=\) input (features) of the \(i^{\t{th}}\) training ex.\(\vph{x^{(i)}_j}\)
\(n =\) # of features

Normal Equation

The normal equation minimizes \(J\) explicitly by solving for its parameters' zero derivatives; in matrix form,
\(\ds \bn{\theta} = (\bn{X}^T\bn{X})^{-1}\bn{X}^T\bn{y}\)

Computing intensity grows with \(\sim n^3\)
Vectorization

Optimizing computing efficiency by writing code that utilizes built-in optimized algorithms
Unvectorized implementation
prediction = 0.0; 
for j = 1:n+1, 
  prediction = prediction + 
               theta(j) * x(j) 
end
Vectorized implementation
prediction = theta' * x;
The vectorized implementation uses the IDE's (MATLAB, Octave, etc.) highly-optimized numerical linear algebra routines to compute the vector inner product


Dragon Notes,   Est. 2018     About

By OverLordGoldDragon