Learning representations by back-propagating errors

Demonstrated that back-propagation of error derivatives enables multi-layer neural networks to learn internal representations.

The universal optimization engine powering all gradient-based neural network training across PyTorch, JAX, and TensorFlow.

Assumptions

• Activation functions are continuous and differentiable; gradient chain rule applies across composite layers

Limitations

• Prone to vanishing and exploding gradients in very deep networks without modern activations or residual connections