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.