# Gradient Descent For Multiple Variables

Cost function: $J(\theta) = \dfrac {1}{2m} \displaystyle \sum_{i=1}^m \left (h_\theta (x^{(i)}) – y^{(i)} \right)^2$
$J(\theta) = \dfrac {1}{2m} \displaystyle \sum_{i=1}^m \left (\theta^Tx^{(i)} – y^{(i)} \right)^2$
$J(\theta) = \dfrac {1}{2m} \displaystyle \sum_{i=1}^m \left ( \left( \sum_{j=0}^n \theta_j x_j^{(i)} \right) – y^{(i)} \right)^2$

Gradient descent: \begin{align*} & \text{repeat until convergence:} \; \lbrace \newline \; & \theta_j := \theta_j – \alpha \frac{1}{m} \sum\limits_{i=1}^{m} (h_\theta(x^{(i)}) – y^{(i)}) \cdot x_j^{(i)} \; & \text{for j := 0..n} \newline \rbrace \end{align*}

which breaks down into

\begin{align*} & \text{repeat until convergence:} \; \lbrace \newline \; & \theta_0 := \theta_0 – \alpha \frac{1}{m} \sum\limits_{i=1}^{m} (h_\theta(x^{(i)}) – y^{(i)}) \cdot x_0^{(i)}\newline \; & \theta_1 := \theta_1 – \alpha \frac{1}{m} \sum\limits_{i=1}^{m} (h_\theta(x^{(i)}) – y^{(i)}) \cdot x_1^{(i)} \newline \; & \theta_2 := \theta_2 – \alpha \frac{1}{m} \sum\limits_{i=1}^{m} (h_\theta(x^{(i)}) – y^{(i)}) \cdot x_2^{(i)} \newline & \cdots \newline \rbrace \end{align*}

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