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Bound propagation based incomplete neural network verifiers such as CROWN arevery efficient and can significantly accelerate branch-and-bound (BaB) basedcomplete verification of neural networks. However, bound propagation cannotfully handle the neuron split constraints introduced by BaB commonly handled byexpensive linear programming (LP) solvers, leading to loose bounds and hurtingverification efficiency. In this work, we develop $\beta$-CROWN, a new boundpropagation based method that can fully encode neuron splits via optimizableparameters $\beta$ constructed from either primal or dual space. When jointlyoptimized in intermediate layers, $\beta$-CROWN generally produces betterbounds than typical LP verifiers with neuron split constraints, while being asefficient and parallelizable as CROWN on GPUs. Applied to complete robustnessverification benchmarks, $\beta$-CROWN with BaB is up to three orders ofmagnitude faster than LP-based BaB methods, and is notably faster than allexisting approaches while producing lower timeout rates. By terminating BaBearly, our method can also be used for efficient incomplete verification. Weconsistently achieve higher verified accuracy in many settings compared topowerful incomplete verifiers, including those based on convex barrier breakingtechniques. Compared to the typically tightest but very costly semidefiniteprogramming (SDP) based incomplete verifiers, we obtain higher verifiedaccuracy with three orders of magnitudes less verification time. Our algorithmempowered the $\alpha,\!\beta$-CROWN (alpha-beta-CROWN) verifier, the winningtool in VNN-COMP 2021. Our code is available at http://PaperCode.cc/BetaCROWN

Beta-CROWN: Efficient Bound Propagation with Per-neuron Split Constraints for Complete and Incomplete Neural Network Robustness Verification