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What drives neural networks to decide as their creators' brains do? Activation functions, the cognitive catalysts within neural networks, are instrumental in shaping the network's decision-making process, mirroring the intricate workings of the human brain. These functions introduce non-linearity, enabling the network to transcend the limitations of linear models and approximate complex patterns.

What is the activation function?

An activation function is a mathematical operation applied to the output of a neuron. It introduces non-linearity to the neural network so that the network can learn complex patterns in data. Essentially, it acts as a decision-maker for its neural network, determining whether a neuron should be activated based on the weighted sum of its inputs.

The choice of activation function significantly impacts network performance. Functions with bounded outputs (e.g., sigmoid or tanh) can stabilize training due to their smoother gradients, but unbounded functions (e.g., ReLU) often offer faster convergence. Yet the latter may face challenges like the vanishing gradient problem – a phenomenon where gradients become smaller as they keep propagating through their network's layers – in deep networks

The fundamentals of activation function

A neural network without nonlinearity is essentially a linear model. Without activation functions, stacking multiple layers would simply result in a single, more complex linear transformation. This limitation severely restricts the network's ability to learn complex patterns and relationships within data.

Activation functions play a pivotal role in enabling gradient-based optimization algorithms like backpropagation to update network parameters effectively. By introducing non-linear transformations, they facilitate the learning process and enhance the model's ability to approximate complex functions.

Common activation functions

The sigmoid function squashes input values into a range between 0 and 1, making it suitable for tasks like binary classification.

While historically popular, the sigmoid function has a tendency to saturate for extreme input values, leading to the vanishing gradient problem during backpropagation. This can hinder the training process of deep neural networks.

Softmax transforms a vector of real numbers into a probability distribution over multiple classes. The output values, which sum to 1, represent the likelihood of each class.

Unlike the max function, which returns the index of the largest value, softmax produces a probability distribution over all classes, providing a more nuanced representation of uncertainty. This property, combined with its differentiability, makes it suitable for gradient-based optimization algorithms employed in training neural networks.

The hyperbolic tangent (tanh) activation function maps input values to a range of -1 to 1. This zero-centered output can contribute to faster convergence during training compared to sigmoid functions, which are bounded between 0 and 1.

But, similar to sigmoid, tanh can suffer from the vanishing gradient problem in deep neural networks, limiting its effectiveness in certain architectures.

ReLU (Rectified Linear Unit) outputs the input directly for positive values and zero for negative inputs.

While ReLU addresses the vanishing gradient problem prevalent in sigmoid and tanh functions, it can suffer from the dying ReLU problem, where neurons become inactive due to negative inputs.

To mitigate the 'dying ReLU' problem mentioned above, where neurons can become permanently inactive, several variations have been introduced:

• Leaky ReLU: introduces a small, constant, and negative slope for negative inputs to prevent the aforementioned problem.

• PReLU: extends Leaky ReLU by making the negative slope a learnable parameter, allowing the network to adapt the slope for each neuron.

Both variants of ReLU aim to improve the performance of neural networks by ensuring that neurons remain active and contribute to the learning process.

The Exponential Linear Unit (ELU) addresses some of the limitations of ReLU. Unlike ReLU, which outputs zero for negative inputs, ELU produces negative values for inputs less than zero.

The property above helps to reduce the 'dying ReLU' problem and pushes the mean activations closer to zero, which can accelerate learning. However, do note that like ReLU, ELU can still suffer from the vanishing gradient problem for extremely negative inputs.

The Maxout function generalizes ReLU and its variants. Instead of applying a fixed function to the weighted sum of inputs, Maxout computes multiple linear functions and selects the maximum output. This approach addresses the dying ReLU problem, adapts to the data by learning the best linear functions, and often yields better results than traditional activation functions in certain scenarios.

However, Maxout also has drawbacks in that doubling the number of parameters compared to standard ReLU can potentially leading to overfitting. Having more parameters additionally requires more computations during training and inference.

Function comparisons: sigmoid vs. tanh vs. softmax

Vanishing gradient problem

The vanishing gradient problem arises from the characteristics of certain activation functions. As gradients propagate backward through the network, they can diminish rapidly, hindering the training of deeper layers. Sigmoid and tanh functions' saturating nature are particularly susceptible to this issue. In the long term, this phenomenon hampers the network's ability to learn from earlier layers, limiting its overall performance.

To address this challenge, activation functions like ReLU and its variants have been introduced, which help mitigate the vanishing gradient problem and enable the training of deeper neural networks.

Selecting the most productive activation function

Activation functions introduce non-linearity into neural networks, enabling them to learn complex patterns. They transform linear outputs into non-linear representations, enhancing the model's capacity to approximate intricate functions. Here are some summarized key considerations when selecting an activation function:

• Output range: bounded (sigmoid or tanh) vs. unbounded outputs (ReLU or Leaky ReLU).

• Gradient behavior: avoiding vanishing or exploding gradients is crucial for training deep networks.

• Computational efficiency: the complexity of the activation function can impact training speed.

• Problem-specific requirements: certain activation functions might be better suited for specific tasks.

  • Sigmoid: suitable for binary classification outputs.

  • Tanh: similar to sigmoid but zero-centered, often used in hidden layers.

  • ReLU: addresses vanishing gradient issues but can suffer from dying ReLU problem.

  • Leaky ReLU, PReLU, ELU: variants of ReLU that mitigate the dying ReLU problem.

  • Softmax: primarily used for multi-class classification.

By carefully selecting activation functions, we can significantly impact the performance and training efficiency of neural networks to our desired settings.

Making activation functions on Python

This exercise will focus on the activation functions ReLU and its derivative. Per the basis of ReLU, any input below or equal to 0 is outputted as 0, while any value above 0 is outputted with no changes. Its deriative instead converts all above-zero inputs into 1s.

As seen in the plot below, ReLU always outputs non-negative values, which can provide specific benefits:

• Introduce Sparsity in the Activations: many neurons may output zero, which can lead to more efficient computations and improved generalization.

• Reduced Vanishing Gradient: non-negative output helps mitigate the vanishing gradient problem, especially in deeper networks.

• Interpretability: ReLU's non-negativity can make it easier to interpret the model's features and decision-making process.

Technically, the derivative is undefined at x = 0 due to the kink in the ReLU function – the kink being the sharp discontinuity where the function abruptly changes from 0 to x. However, in the context of neural network training, we often treat the derivative of ReLU at x = 0 as 0 for practical purposes. This allows the backpropagation algorithm to continue functioning without encountering numerical issues.

While technically incorrect from a strict mathematical standpoint, this convention simplifies the implementation and training of neural networks. It is a common practice that does not significantly impact the overall performance of the model.