Why do Activation Functions have to be Monotonic?

Answer: Activation functions need to be monotonic to ensure stable and predictable gradient descent during training.

Activation functions in neural networks are typically required to be monotonic to ensure that the network can learn effectively through backpropagation. When an activation function is monotonic, it means that its output increases or decreases in a consistent direction as its input increases. This property is crucial for several reasons:

1. Gradient Sign Preservation: Monotonic activation functions preserve the sign of the gradient during backpropagation. This property ensures that the gradient always points in a consistent direction, allowing the network to update its parameters in a stable and predictable manner.
2. Stable Gradient Descent: Monotonic activation functions help stabilize the gradient descent optimization process. With monotonic functions, the loss function decreases consistently as the network parameters are updated, leading to smoother convergence and better training stability.
3. Avoiding Vanishing or Exploding Gradients: Non-monotonic activation functions can lead to vanishing or exploding gradients, where the gradients become extremely small or large during backpropagation. Monotonic activation functions help mitigate this issue by ensuring that the gradients remain within a reasonable range, facilitating more stable and efficient learning.
4. Interpretability and Predictability: Monotonic activation functions provide more interpretable and predictable behavior, making it easier to understand how changes in input values affect the network’s output. This property is especially important in applications where model interpretability is crucial, such as in healthcare or finance.

Overall, the monotonicity of activation functions is essential for ensuring stable and efficient learning in neural networks, facilitating gradient-based optimization methods like backpropagation, and improving the overall performance and reliability of the network.