Gradient
Clarificaton
导数,derivate
偏微分,partial derivate
梯度,gradient
\[\begin{aligned}
&\bigtriangledown{f} = (\frac{\delta{f}}{\delta{x_1}};\frac{\delta{f}}{\delta{x_2}};...;\frac{\delta{f}}{\delta{x_n}})\\
&z = y^2-x^2\\
&\frac{\delta{z}}{\delta{x}}=-2x\\
&\frac{\delta{z}}{\delta{y}}=2y\\
&\bigtriangledown{f} = (-2x,2y)\\
\end{aligned}
\]
How to search for minima?
\[\begin{aligned}
&\theta_{t+1} = \theta_{t} - \alpha_{t}\bigtriangledown{f(\theta_{t})}\\
&Function:\\
&J(\theta_1,\theta_2) = {\theta_1}^2 + {\theta_2}^2\\
&Objective:\\
&\substack{\min\\{\theta_1,\theta_2}} J(\theta_1,\theta_2)\\
&Update rules:\\
&\theta_1:=\theta_1 - \alpha \frac{d}{d\theta_1}J(\theta_1,\theta_2)\\
&\theta_2:=\theta_2 - \alpha \frac{d}{d\theta_2}J(\theta_1,\theta_2)\\
&Derivatives:\\
&\frac{d}{d\theta_1}J(\theta_1,\theta_2)=\frac{d}{d\theta_1}{\theta_1}^2 + \frac{d}{d\theta_1}{\theta_2}^2=2\theta_1\\
&\frac{d}{d\theta_2}J(\theta_1,\theta_2)=\frac{d}{d\theta_2}{\theta_1}^2 + \frac{d}{d\theta_2}{\theta_2}^2=2\theta_2\\
\end{aligned}
\]
Common Functions
Common Functions | Function | Derivative |
---|---|---|
Constant | $$C$$ | $$0$$ |
Line | $$x$$ | $$1$$ |
$$ax$$ | $$a$$ | |
Square | $$x^2$$ | $$2x$$ |
Square root | $$\sqrt{x}$$ | $$\frac{1}{2}x^{-\frac{1}{2}}$$ |
Exponential | $$e^x$$ | $$e^x$$ |
$$a^x$$ | $$\ln(a)a^x$$ | |
Logarithms | $$\ln(x)$$ | $$\frac{1}{x}$$ |
$$\log_a(x)$$ | $$\frac{1}{x\ln(a)}$$ | |
Trigonometry | $$\sin(x)$$ | $$\cos(x)$$ |
$$\cos(x)$$ | $$-\sin(x)$$ | |
$$\tan(x)$$ | $${\sec(x)}^2$$ | |
eg. |
\[\begin{aligned}
&f=[y-(xw+b)]^2\\
&g=xw+b\\
&\frac{\delta f}{\delta w}=2(y-g)\frac{\delta g}{\delta w}=2(y-(xw+b))*(-x)\\
&\frac{\delta f}{\delta b}=2(y-g)\frac{\delta g}{\delta b}=2(y-(xw+b))*(-1)\\
&\bigtriangledown=(2(y-(xw+b))*(-x),2(y-(xw+b))*(-1))\\
\end{aligned}
\]