Abstract: We develop $\beta$-CROWN, a new bound propagation based method that can fully encode neuron split constraints in branch-and-bound (BaB) based complete verification via optimizable parameters $\beta$. When jointly optimized in intermediate layers, $\beta$-CROWN generally produces better bounds than typical LP verifiers with neuron split constraints, while being as efficient and parallelizable as CROWN on GPUs. Applied to complete robustness verification benchmarks, $\beta$-CROWN with BaB is close to three orders of magnitude faster than LP-based BaB methods, and is at least 3 times faster than winners of VNN-COMP 2020 competition while producing lower timeout rates. By terminating BaB early, our method can also be used for efficient incomplete verification. We achieve higher verified accuracy in many settings over powerful incomplete verifiers, including those based on convex barrier breaking techniques. Compared to the typically tightest but very costly semidefinite programming (SDP) based incomplete verifiers, we obtain higher verified accuracy with three orders of magnitudes less verification time, and enable better certification for verification-agnostic (e.g., adversarially trained) networks.