Pros:
 - Smooth gradient, preventing “jumps” in output values.
 - Output values bound between 0 and 1, normalizing the output of each neuron.
 - Clear predictions—For X above 2 or below -2, tends to bring the Y value (the prediction) to the edge of the curve, very close to 1 or 0. This enables clear predictions.

Cons:
 - Vanishing gradient—for very high or very low values of X, there is almost no change to the prediction, causing a vanishing gradient problem. This can result in the network refusing to learn further, or being too slow to reach an accurate prediction.
 - Outputs not zero centered.
 - Computationally expensive


Pros and Cons of Sigmoid/Logistic Function

$$g(x) = \frac{1}{1 + e^{-x}} \Rightarrow g'(x) = \frac{d}{dx}g(x) = \frac{1}{1 + e^{-x}} (1 - \frac{1}{1 + e^{-x}}) = g(x) (1 - g(x))$$ 

Derivative of Sigmoid/Logistic Function

- tanh works almost always better than Sigmoid function, but Sigmoid should be used for the output layer of binary classification models.
- tanh is in range (-1, 1), but Sigmoid is in range (0, 1).
- Using tanh activation functions in the hidden layers optimizes training because it centers the data around 0 rather tahn 0.5

Sigmoid/Logistic vs. TanH/Hyperbolic Tangent functions

The numerical output. ofa sigmoid node in a neural netork

Question about sigmoid node

$S(x) = \frac{1}{1 +e^{-x}} = \frac{e^x}{e^x + 1}$ 


University of Michigan - Ann Arbor

Modern neural network models use non-linear activation functions. They allow the model to create complex mappings between the network’s inputs and outputs, which are essential for learning and modeling complex data, such as images, video, audio, and data sets which are non-linear or have high dimensionality.

There are 8 common non-linear activation functions:
 - Sigmoid / Logistic
 - TanH / Hyperbolic Tangent
 - ELU (Exponential Linear Unit)
 - ReLU (Rectified Linear Unit)
 - Leaky ReLU
 - Parametric ReLU
 - Softmax
 - Swish

Non-Linear Activation Functions

Wikipedia

A helpful website that introduces neural networks:
https://missinglink.ai/guides/neural-network-concepts/

Neural Network Reference

Non-linear functions address the problems of a linear activation function:

 - They allow backpropagation because they have a derivative function which is related to the inputs.
 - They allow “stacking” of multiple layers of neurons to create a deep neural network. Multiple hidden layers of neurons are needed to learn complex data sets with high levels of accuracy.

Linear vs. Non-Linear Activation Functions

Sigmoid/Logistic Function

$S(x)  = tanh(x) = \frac{e^x-e^{-x}}{e^x + e^{-x}}$


TanH/Hyperbolic Tangent Function

The Swish function, introduced by Ramachandran et al. in 2017, is a mathematical function defined as the product of its input and the sigmoid function applied to a scaled version of the input. For a scalar input $x$, it is expressed as:

$ \operatorname{swish}(x) = x \cdot \operatorname{sigmoid}(\beta x) = \frac{x}{1+e^{-\beta x}}$

where $\beta$ is a constant or a trainable parameter. When applied element-wise to a vector $\mathbf{h}$ in neural networks, the formula is written as:

$\sigma_{\text{swish}}(\mathbf{h}) = \mathbf{h} \odot \text{Sigmoid}(\beta \mathbf{h})$

where $\odot$ denotes the element-wise product.

Swish Function

The Rectified Linear Unit (ReLU) is a common choice for the activation function $$\sigma(\cdot)$$ within the hidden layers of neural networks. It is defined to output the positive portion of its argument. When applied to an input vector $$\mathbf{h}$$, the ReLU function is given by the formula: $$\sigma_{\mathrm{relu}}(\mathbf{h}) = \mathrm{max}(0, \mathbf{h})$$.

ReLU (Rectified Linear Unit)

 If x > 0:
 $f(x)=x$
 If x $<=$  1:
$f(x) = α(e^x - 1)$

ELU (Exponential Linear Unit)

Which activation function is represented by each of these plots?

Which of the following introduces nonlinearity into neural networks?

To convert raw, unnormalized outputs $$\mathbf{o}$$ into valid probabilities, the softmax function applies an exponential function to each component and then normalizes them by their sum. The exponentiation ensures that all probabilities are non-negative, while the division ensures that they sum to $$1$$. Mathematically, the predicted probability distribution $$\hat{\mathbf{y}}$$ is defined as:

$$\hat{\mathbf{y}} = \mathrm{softmax}(\mathbf{o}) \quad 	extrm{where}\quad \hat{y}_i = \frac{\exp(o_i)}{\sum_j \exp(o_j)}$$

This guarantees that $$0 \le \hat{y}_i \le 1$$ and $$\sum_j \hat{y}_j = 1$$. Unlike other normalizations or the probit model, the softmax function preserves order and leads to a more well-behaved optimization problem.

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