We propose the novel certified training method, SABR, which outperforms existing methods across perturbation magnitudes on MNIST, CIFAR-10, and TinyImageNet, in terms of both standard and certifiable accuracies. The key insight behind SABR is that propagating interval bounds for a small but carefully selected subset of the adversarial input region is sufficient to approximate the worst-case loss over the whole region while significantly reducing approximation errors. SABR does not only establish a new state-of-the-art in all commonly used benchmarks but more importantly, points to a new class of certified training methods promising to overcome the robustness-accuracy trade-off.