$(\text{DNN})^2$: Doubly Non-Negative Relaxations for Deep Neural Networks

Abstract

Existing linear program (LP) and semidefinite program (SDP) relaxations for rectified linear unit (ReLU) neural network (NN) verification yield overly-conservative safety guarantees due to significant relaxation gaps. While the completely positive program (CPP) formulation closes this gap, it is NP-hard to solve. Its cheapest tractable relaxation, the doubly non-negative program (DNN), retains critical constraints as an SDP, but one whose size exceeds the reach of interior-point methods at practical scale. While Burer-Monteiro (BM) factorization has been applied to make SDP-based verification scalable, no such result exists for the strictly tighter DNN formulation. A key obstacle is that additional non-negativity constraints in the DNN cause dual multipliers for optimality certification to be non-unique, making standard certification methods inapplicable. We propose a novel eigenvalue maximization procedure that searches the non-unique multiplier space for a valid certificate, i.e. a global optimality guarantee. Experiments demonstrate that our approach $(\text{DNN})^2$ produces bounds consistently tighter than the standard SDP method, often matching the exact solution, and that our certification procedure confirms global optimality when a valid certificate exists. These results are a key step toward providing tight, certifiable, and computationally scalable verification guarantees needed to deploy neural network controllers and perception modules in safety-critical autonomous systems.