Abstract: Training certifiably robust neural networks remains a notoriously hard problem. While adversarial training optimizes under-approximations of the worst-case loss, which leads to insufficient regularization for certification, certified training methods, optimize loose over-approximations, leading to over-regularization and poor accuracy. In this work, we propose TAPS, a novel certified training method combining IBP and PGD training to optimize more precise, although not necessarily sound, worst-case loss approximations, reducing over-regularization and increasing certified accuracy. Empirically, TAPS achieves a new state-of-the-art in many settings, e.g., reaching a certified accuracy of 22\% on TinyImageNet for $\ell_\infty$-perturbations with radius $\epsilon=1/255$.