1 Linear Regression with One Variable

2013-04-23 | 分类 course | 标签 ml

Machine learning algorithms:

Supervised learning
Unsupervised learning
others:
- Reinforcement learning
- Recommender systems

Practical advice for applying learning algorithms

Supervised learning is the machine learning task of inferring a function from labeled training data. The training data consist of a set of training examples.

supervised learning give the “right answer” for each example in the data.

Regression Problem: predict real-valued output(continuous-value)
Classification Problem: discrete-valued output

Notation:

m = Number of training examles
x’s = “input” variable / features
y’s = “output” variable /”target” variable
(x, y) – one training example
(x^i, y^i) – i^th training example

With the training set and learning algorithm, we get hypothesis, which we use to make predictions, is this linear function.

Hypothesis: h_θ(x) = θ₀ + θ₁x

$J(\theta_0, \theta_1) = \frac{1}{2m}\sum_{i=1}^{m}(h_\theta(x^{(i)}) - y^{(i)})^{2}$

minimize θ₀, θ₁, J(θ₀, θ₁) is cost function(squared error function).

Want $\underset{\theta_{0},\theta_{1}}{min} J(\theta_{0},\theta_{1})$

Gradient descent to minimize some arbitrary function J.

Outline

Start with some θ₀,θ₁
Keep changing θ₀,θ₁ to reduce J(θ₀,θ₁) until we hopefully end up at a minimum

Gradient descent algorithm(梯度下降)

$repeat\ until \ convergence \left \{ \theta_j := \theta_j - \alpha\frac{\partial }{\partial \theta_j} J(\theta_0, \theta_1)\ \ (for\ j = 0\ and\ j = 1) \right \}$

Here α is learning rate,it controls how big a step we take when updating parameter theta J. $\frac{\partial}{\partial\theta_{i}}J(\theta_{0},\theta_{1})$ is the derivative(导数) term.

We must update θ₀ and θ₁ simultaneously.

Correct: Simultaneously update

$temp0 := \theta_0 - \alpha \frac{\partial }{\partial \theta_0} J(\theta_0, \theta_1)\\ temp1 := \theta_1 - \alpha \frac{\partial }{\partial \theta_1} J(\theta_0, \theta_1)\\ \theta_0 := temp0 \\ \theta_1 := temp1$

!Incorrect

$temp0 := \theta_0 - \alpha \frac{\partial }{\partial \theta_0} J(\theta_0, \theta_1)\\ \theta_0 := temp0 \\ temp1 := \theta_1 - \alpha \frac{\partial }{\partial \theta_1} J(\theta_0, \theta_1)\\ \theta_1 := temp1$

Gradient descent alogirthm

repeat until convergence {

$\theta_{0}:=\theta_{0}-\frac{1}{m}\sum^{m}_{i=1}(h_{\theta}(x^{(i)})-y^{(i)})$ $\theta_{1}:=\theta_{1}-\frac{1}{m}\sum^{m}_{i=1}(h_{\theta}(x^{(i)})-y^{(i)})\cdot x^{(i)}$

}

Unsupervised learning refers to the problem of trying to find hidden structure in unlabeled data.