Tedyun: Created page with "== Types of Machine Learning == * Supervised Learning Regression Problem: Continuous valued output. Classification Problem: Discrete valued output. * Unsupervised Learn..."

2016-10-30T21:18:53Z

Created page with "== Types of Machine Learning == * Supervised Learning ** Regression Problem: Continuous valued output. ** Classification Problem: Discrete valued output. * Unsupervised Learn..."

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== Types of Machine Learning ==

* Supervised Learning
** Regression Problem: Continuous valued output.
** Classification Problem: Discrete valued output.
* Unsupervised Learning
** Clustering

== Linear Regression ==

=== Terminologies ===

* $x^{(i)}_j$ feature vectors
* $y^{(i)}$ outcomes
* $h_\theta(x)$ the hypothesis
* $J(\theta)$ the cost function
* $\alpha$ the learning rate

=== Advanced Optimization Algorithms ===

There are advanced algorithms (from numerical computing) to minimize the cost function other than the '''gradient descent'''. For all of the following algorithms all we need to supply to the algorithm is a code to compute the function $J(\theta)$ (the cost function) and the partial derivatives of the cost function $\frac{\partial}{\partial \theta_i} J(\theta)$.

# Conjugate gradient
# BFGS
# L-BFGS

Advantages
* No need to manually pick $\alpha$ (the learning rate in gradient descent)
* Often faster than gradient descent

Disadvantages
* More complex

== Classification Problem ==

=== Logistic Regression ===

* $h_\theta(x) = 1 / (1 + e^{-\theta^T x})$. Note $f(z) = 1 / (1 + e^{-z})$ is called the sigmoid function / logistic function.
* $J(\theta) = - y \cdot \log h_\theta(x) - (1 - y) \cdot \log (1-h_\theta(x))$. This comes from Maximum Likelihood Estimation in Statistics.

=== Multi-class Classification ===

* One-vs-all (one-vs-rest): For $n$-class classification, train a logistic regression classifier $h_\theta^{(i)} (x)$ for each class $i = 1, \ldots, n$ to predict the probability that $y = i$. To make a prediction on a new input $x$, pick the class $i$ that maximizes $h_\theta^{(i)} (x)$.

=== Cocktail Party Problem ===
* Algorithm
** [W, s, v] = svd((repmat(sum(x.*x, 1), size(x, 1), 1).*x)*x');

== Overfitting ==

=== Terminologies ===
* "underfitting" or "high bias": not fitting the training set well
* "overfitting" or "high variance": too many features, fails to generalize to new examples

=== Regularization ===
* Modify the cost function to penalize large parameters. $J(\theta) = \frac{1}{2m} \big[ \sum_{i = 1}^m (h_\theta(x^{(i)}) - y^{(i)})^2 + \lambda \sum_{j = 1}^n \theta_j^2 \big]$. $\lambda$ is the regularization parameter. Note that the index $j$ starts from $1$ which means we don't penalize the constant term (by convention).

=== Regularized Linear Regression ===

==== Gradient Descent ====
For a learning rate $\alpha > 0$ and a regularization parameter $\lambda > 0$,

* $\theta_0 := \theta_0 - \alpha \frac{1}{m} \sum_{i = 1}^m (h_\theta(x^{(i)}) - y^{(i)}) x_0^{(i)}$
* $\theta_j := \theta_j(1 - \alpha \frac{\lambda}{m}) - \alpha \frac{1}{m} \sum_{i = 1}^m (h_\theta(x^{(i)}) - y^{(i)}) x_j^{(i)}$ for $j > 0$

==== Normal Equation ====

* Normal equation: $\theta = (X^T X)^{-1} X^T y$
* Normal equation '''with regularization''': $\theta = (X^T X + \lambda K)^{-1} X^T y$, where $K$ is a diagonal matrix whose first diagonal entry is $0$ and the rest of the diagonal is $1$.

Note that while $X^T X$ may not be invertible, $X^T X + \lambda K$ is ''always'' invertible for $\lambda > 0$.

=== Regularized Logistic Regression ===

Cost function $J(\theta) = - \frac{1}{m} \sum_{i = 1}^m \big[ y^{(i)} \cdot \log h_\theta(x^{(i)}) + (1 - y^{(i)}) \cdot \log (1-h_\theta(x^{(i)})) \big] + \frac{\lambda}{2m} \sum_{j = 1}^n \theta_j^2$

==== Gradient Descent ====

* $\theta_0 := \theta_0 - \alpha \frac{1}{m} \sum_{i = 1}^m (h_\theta(x^{(i)}) - y^{(i)}) x_0^{(i)}$
* $\theta_j := \theta_j(1 - \alpha \frac{\lambda}{m}) - \alpha \frac{1}{m} \sum_{i = 1}^m (h_\theta(x^{(i)}) - y^{(i)}) x_j^{(i)}$ for $j > 0$

== Neural Network ==

TODO.

== Debugging Learning Algorithm ==

When a learning algorithm makes unacceptably large errors in its predictions (on a new data set), what can you do?

* Get more training examples
* Try smaller sets of features
* Try getting additional features
* Try adding polynomial features
* Try decreasing $\lambda$
* Try increasing $\lambda$

=== Diagnostics ===

Machine Learning Diagnostic: A test that you can run to gain insight what is/isn't working with a learning algorithm, and gain guidance as to how best to improve its performance.

Split the data into training/test sets, use the test set as a diagnostic.

=== Model Selection ===

Suppose we want to choose a degree of polynomial $d$ of a regression model.

Split the data into three sets:

* Training set (e.g. 60%)
* Cross Validation set (e.g. 20%)
* Test set (e.g. 20%)

For each $d$, train the model with the training set, and compute the cross-validation error $J_{cv}(\Theta^{(d)})$ in the cross validation set. Pick $d$ where this cross validation error is the smallest. Finally, report the test set error $J_{test}(\Theta^{(d)})$ as the estimated error rate of the model.

=== Regularizaton and Bias/Variance ===

=== Learning Curves ===

== Machine Learning System Design ==

== Support Vector Machine ==

=== SVM Libraries ===

liblinear, libsvm package

=== Choosing Kernel ===

Not all similarity functions make valid kernels. Need to satisfy "Mercer's Theorem" to make sure SVM packages' optimizations run correctly and do not diverge.

Example: polynomial kernel, string kernel, chi-suqare kernel, histogram intersection kernel, ...

=== Logistic Regression vs SVM ===

$n$ = number of features ($x \in \mathbb{R}^{n+1}$), $m$ = number of training examples

* If $n$ large (relative to $m$): Use logistic regression or SVM without a kernel ("linear kernel").
* If $n$ is small, $m$ is intermediate: Use SVM with Gaussian kernel.
* If $n$ is small, $m$ is large: Create/add more features, then use logistic regression or SVM without a kernel.
* Note: Neural network likely to work well for most of these settings, but may be slower to train.

== Unsupervised Learning ==

=== Clustering ===

K-means, random initialization

=== Dimensionality Reduction ===

Principal Component Analysis (PCA)

=== Anomaly Detection ===

== Gradient Descent with Large Datasets ==

* Batch gradient descent: Use all $m$ examples in each iteration

* Stochastic gradient descent: Use 1 example in each iteration

* Mini-batch gradient descent: Use $b$ examples in each iteration ($b$ is usually a number between 2 and 100)

Machine Learning (Andrew Ng Course) - Revision history

Tedyun: Created page with "== Types of Machine Learning == * Supervised Learning ** Regression Problem: Continuous valued output. ** Classification Problem: Discrete valued output. * Unsupervised Learn..."

Tedyun: Created page with "== Types of Machine Learning == * Supervised Learning Regression Problem: Continuous valued output. Classification Problem: Discrete valued output. * Unsupervised Learn..."