Local robustness is a property of [[ Neural Network ]]s that is concerned with not changing the network’s classification or prediction around specific inputs.

For formal verification, robustness to [[ adversarial examples ]] in an \(\mathcal{L}_{\infty}\) ball around an input sample \(\mathbf{a}\) with label \(y_a\) can be encoded with a bounded input domain \(\mathcal{C} \triangleq \{\mathbf{x_0} \vert\ \lVert\mathbf{x_0} - \mathbf{a}\rVert_{\infty} \le \epsilon\}\) and a property \(P(\mathbf{\hat{x}_n}) = \{\forall y,\ \hat{x}_{n[y_a]} > \hat{x}_{n[y]}\}\).

In other words, changes in the input image that are smaller than some given value \(\epsilon\) do not lead to a change in the network’s output classification.

Sources

• [[ Branch and Bound for Piecewise Linear Neural Network Verification ]]