Training Mixed-Monotone Neural Networks for Verification Without Set Propagation

The increasing integration of deep neural networks in critical systems has spawned a theoretical and practical interest in formally guaranteeing safety properties about their behavior. To achieve this, contemporary verification algorithms rely on computing linear relaxations for a network's non-linear activation functions. Existing approaches for linear relaxations typically fall into one of two categories: single-neuron relaxation, in which each activation neuron is bounded in terms of its sources; and multi-neuron relaxation, in which linear bounds involving multiple activation neurons and their sources are calculated. However, existing methods might fail to balance tightness and scalability, as single-neuron bounds might not derive sufficiently tight bounds necessary for verification to complete, whereas generating multi-neuron relaxation for all activation neurons is computationally expensive. In this paper, we present a middle-ground approach featuring partial multi-neuron relaxation, in which we generate multi-neuron bounds for only a small, heuristically selected subset of neurons. To achieve this, we build upon existing branching heuristics for selecting neurons and for optimizing bounding hyper-planes for multi-neuron bounds. We integrated our proposed method within the Marabou verifier, and obtained favorable results in comparison to existing bound tightening methods. Our experiments showcase the potential of our technique for neural network verification.

Many neural network (NN) verification systems represent the network's input-output relation as a constraint program. Sound and complete, representations involve integer constraints, for simulating the activations. Recent works convexly relax the integer constraints, improving performance, at the cost of soundness. Convex relaxations consider outputs that are unreachable by the original network. We study the worst case divergence between the original network and its convex relaxations; both qualitatively and quantitatively. The relaxations' space forms a lattice, where the top element corresponds to a full relaxation, with every neuron linearized. The bottom element corresponds to the original network. We provide analytical upper and lower bounds for the $\ell_\infty$-distance between the fully relaxed and original outputs. This distance grows exponentially, w.r.t. the network's depth, and linearly w.r.t. the input's radius. The misclassification probability exhibits a step-like behavior, w.r.t. input radius. Our results are supported by experiments on MNIST, Fashion MNIST and random networks.

Branch and Bound (BaB) aims to achieve complete verification of neural networks by adaptively partitioning the problem and applying off-the-shelf verifiers to subproblems. Its problem-splitting history can be represented as a tree, where each subproblem corresponds to a child node. A key problem of BaB lies in searching for the verdict boundaries across all the paths that divide the verified and unverified subproblems. We observe that the existing BaB approach tackles this problem by solving each expensive subproblem sequentially along the tree path as its depth increases, requiring costly bounds propagation at every visited BaB tree node (i.e., subproblem), which is inefficient. To address this issue, we propose effective search approaches that leverage the monotonicity of each path to efficiently and precisely locate the verdict boundary by simultaneously splitting multiple activation functions (e.g., ReLU), rather than processing them one at a time as in the classical approach. Our approach performs an effective exponential search along each path, allowing us to skip many boundary-unrelated subproblems when identifying the verdict boundary. The enhanced version further improves this process by estimating the boundary's position using quantitative information obtained from subproblem solving. We perform experimental evaluation on commonly-used benchmarks to assess our proposed techniques, and compare them with recent BaB-based approaches.