Activation Functions¶
| Name | Output \(f(x)\) | Output Type | Range |
|---|---|---|---|
| Identity | \(x\) | Continuous | \([-1, 1]\) |
| Binary Step | \(\begin{cases} 0, &x < 0 \\ 1, & x \ge 0 \end{cases}\) | Binary | \({0, 1}\) |
| Tariff/ Tanh | \(\tanh(x)\) | Discrete | \([-1, 1]\) |
| ArcTan | \(\tan^{-1} (x)\) | Continuous | \((-\pi/2, \pi/2)\) |
| ReLU (Rectified Linear Unit) | \(\begin{cases} 0, &x < 0 \\ x, & x \ge 0 \end{cases}\) | Continuous | \([0, \infty]\) |
| SoftPlus (smooth alt to ReLU) | \(\log(1+e^x)\) | Continuous | \([0, \infty]\) |
| Parametric/ Leaky ReLU | \(\begin{cases} \alpha x, &x < 0 \\ x, & x \ge 0 \end{cases}\) | Continuous | \([-\infty, \infty]\) |
| Exponential Linear Unit | \(\begin{cases} \alpha (e^x-1), &x < 0 \\ x,& x \ge 0 \end{cases}\) | Continuous | \([-\infty, \infty]\) |
| Sigmoid/ Logistic/ Soft Step | \(\dfrac{1}{1+e^{-x}}\) | Binary | \([0, -1]\) |
| Softmax | \(\dfrac{e^{x}}{\sum_{i=1}^k e^{x_i}}\) where \(k=\) no of classes such that \(\dfrac{\sum p_i}{k} = 1\) | Discrete | \([0, 1]\) |
Why use activation function for hidden layers?¶
Else, it would just be regular linear regression/logistic regression, so no point of hidden layers
Not using activation function \(\implies\) using identity activation function
The only place identity activation function is acceptable is for the final output activation function in regression.
Linear Regression¶
flowchart LR
a((x1)) & b((x2)) -->
d((h1)) & e((h2)) -->
y(("ŷ"))\[ \begin{aligned} \hat y &= w_{h_1 \hat y} h_1 + w_{h_2 \hat y} h_2 \\ &= w_{h_1 \hat y} (w_{x_1 h_1} x_1 + w_{x_2 h_1} x_2) + w_{h_2 \hat y} (w_{x_1 h_2} x_1 + w_{x_2 h_2} x_2) \\ &= \cdots \\ &= w_1 x_1 + w_2 x_2 \end{aligned} \]
Logistic Regression¶
flowchart LR
a((x1)) & b((x2)) -->
d((h1)) & e((h2)) -->
s(("σ")) -->
y(("ŷ"))\[ \begin{aligned} \hat y &= \sigma(w_{h_1 \hat y} h_1 + w_{h_2 \hat y} h_2) \\ &= \sigma(w_{h_1 \hat y} (w_{x_1 h_1} x_1 + w_{x_2 h_1} x_2) + w_{h_2 \hat y} (w_{x_1 h_2} x_1 + w_{x_2 h_2} x_2)) \\ &= \cdots \\ &= \sigma(w_1 x_1 + w_2 x_2) \end{aligned} \]