Gradient descent for linear regression
线性回归的梯度下降

Gradient descent can be used to solve many things. Today we used it to build a linear regression model.
梯度下降法可以用来解决很多问题。今天，我们用它来建立一个线性回归模型。

Our objective function is:
我们的目标函数是

$$f\big(\beta_0, \beta_1\big)=\frac{1}{2}\sum_{i=1}^N(\beta_0+\beta_1x_i-y_i)^2$$

Our gradients are: 我们的梯度是

$$\nabla _{\beta_0} f=\sum_{i=1}^N(\beta_0+\beta_1x_i-y_i)\hspace{1.5cm}\nabla _{\beta_1} f =\sum_{i=1}^Nx_i(\beta_0+\beta_1x_i-y_i)$$

Hence, our final update equations for $\beta_0$ and $\beta_1$ are:
因此， $\beta_0$ 和 $\beta_1$ 的最终更新方程为

$$\beta_0^{i+1}=\beta_0^i-\alpha_0\nabla_{\beta_0}f\hspace{3cm}\beta_1^{i+1}=\beta_1^i-\alpha_1\nabla_{\beta_1}f$$

Loss curves 损耗曲线

Loss curves are used to observe a model's training process. In general, you want to see that at each iteration the error is reducing.
损失曲线用于观察模型的训练过程。一般来说，您希望看到每次迭代时误差都在减少。

Here are some examples of loss curves that you might see.
下面是一些您可能会看到的损失曲线示例。

Mini-batch gradient descent
小批量梯度下降

Mini-batch gradient descent is a variant of gradient descent where at each iteration, the gradient is calculated for only a subset of the full training data.
迷你批量梯度下降法是梯度下降法的一种变体，每次迭代时，只对全部训练数据的一个子集计算梯度。

Stochastic gradient descent is a special case of mini-batch gradient descent where the batch size is 1.
随机梯度下降法是迷你批次梯度下降法的一种特例，其批次大小为 1。