Interval Bound Propagation

IBP (Interval Bound Propagation) is a neural network verification technique that propagates input intervals through each layer of the network — computing guaranteed lower and upper bounds on output values, enabling certified robustness verification by checking if outputs stay within safe bounds.

How IBP Works

• Input Interval: Define input bounds $[x - epsilon, x + epsilon]$ (the perturbation region).
• Layer-by-Layer: Propagate intervals through each layer: linear layers, activation functions, batch norm.
• Affine: For $y = Wx + b$: $y_{lower} = W^+ x_{lower} + W^- x_{upper} + b$ (using positive/negative weight splitting).
• ReLU: $ReLU([l, u]) = [max(0, l), max(0, u)]$.

Why It Matters

• Fast: IBP is computationally cheap — just forward propagation with intervals.
• Training: IBP bounds can be used as a training objective (IBP-trained networks) for certified robustness.
• Loose Bounds: IBP bounds are often very loose — tighter methods (CROWN, α-CROWN) trade compute for tighter bounds.

IBP is box propagation through the network — a fast method to bound neural network outputs under input perturbations.