Neural Canyon (Ai Generated)
Function Name: NeuralCanyon
Difficulty: Hard
Type: Continuous, Differentiable, Multi-Modal
Description
The Neural Canyon is a simulation of the loss landscapes frequently encountered in Deep Learning, specifically designed to mimic the “Narrow Valley” problem combined with saturated activations.
The function features a long, curving valley that follows a hyperbolic tangent ($\tanh$) path. This creates a specific challenge: as $x$ increases, the manifold flattens out, simulating the “vanishing gradient” problem found in saturated neural network layers. The landscape is further corrupted by “egg-crate” noise, simulating the stochastic variance of mini-batch training or architectural roughness.
Mathematical Composition
The function combines three terms:
- The Manifold (Tanh Valley):
The valley follows the curve $y = \tanh(x)$. This creates a non-linear dependency. The walls are steep ($Wall \times (y - \text{path})^2$), forcing the optimizer to learn the relationship between parameters.
- Regularization:
A weak global quadratic term ($Bias \times (x^2 + y^2)$) keeps the solution bounded, similar to L2 weight decay.
- Interference Noise:
A noise term using $\cos(x)\cos(y)$ creates local minima. Crucially, this noise is essentially subtractive (digging holes) and is deepest near the center, making the final convergence step prone to getting stuck in sub-optimal “nearby” buckets.
Optimization Challenge
- Manifold Navigation: The optimizer must follow the curved $\tanh$ path.
- Vanishing Gradients: Far from the center, the $\tanh$ curve is flat, providing very little gradient information about direction.
- Noise Tolerance: The landscape is “rough.” Algorithms without adaptive learning rates or momentum may settle in local minima surrounding the global solution.