Bounded Linear Activation Function

In neural network design, activation functions introduce non-linearity into the model, enabling it to learn complex patterns. However, some architectures benefit from using bounded activation functions that constrain neuron outputs within a specific range, preventing gradient explosion and ensuring stable training dynamics.

The Bounded Linear Activation (also known as the Clipped Linear Unit) is a piecewise linear function that acts as a hard limiter on input values. It operates by:

1. Clamping values below the minimum threshold to the minimum value
2. Clamping values above the maximum threshold to the maximum value
3. Passing through values within the range unchanged (identity function)

Mathematically, for an input x with bounds ([min_val, max_val]):

$$f(x) = \begin{cases} min_val & \text{if } x < min_val \ x & \text{if } min_val \leq x \leq max_val \ max_val & \text{if } x > max_val \end{cases}$$

This creates a linear region between the bounds with flat regions (zero gradient) outside. With default bounds of -1.0 and 1.0, the function effectively squashes all inputs into the symmetric interval ([-1, 1]).

Key Properties:

• Piecewise linear: Computationally efficient compared to smooth approximations
• Bounded output range: Prevents unbounded activations that could destabilize training
• Gradient behavior: Gradient is 1 within the linear region and 0 outside (saturated regions)
• Memory efficient: Requires no additional learned parameters

Your Task: Write a Python function that implements the bounded linear activation. Given an input value and optional minimum/maximum bounds, return the activated value clipped to the specified range.