Recent works in neural network verification show that cheap incomplete verifiers such as CROWN, based upon bound propagations, can effectively be used in Branch-and-Bound (BaB) methods to accelerate complete verification, achieving significant speedups compared to expensive linear programming (LP) based techniques. However, they cannot fully handle the per-neuron split constraints introduced by BaB like LP verifiers do, leading to looser bounds and hurting their verification efficiency. In this work, we develop $beta$-CROWN, a new bound propagation based method that can fully encode per-neuron splits via optimizable parameters $beta$. When the optimizable parameters are jointly optimized in intermediate layers, $beta$-CROWN has the potential of producing better bounds than typical LP verifiers with neuron split constraints, while being efficiently parallelizable on GPUs. Applied to the complete verification setting, $beta$-CROWN is close to three orders of magnitude faster than LP-based BaB methods for robustness verification, and also over twice faster than state-of-the-art GPU-based complete verifiers with similar timeout rates. By terminating BaB early, our method can also be used for incomplete verification. Compared to the state-of-the-art semidefinite-programming (SDP) based verifier, we show a substantial leap forward by greatly reducing the gap between verified accuracy and empirical adversarial attack accuracy, from 35% (SDP) to 12% on an adversarially trained MNIST network ($epsilon=0.3$), while being 47 times faster. Our code is available at https://github.com/KaidiXu/Beta-CROWN
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