A Unified View of SDP-based Neural Network Verification through Completely Positive Programming

Verifying that input-output relationships of a neural network conform to prescribed operational specifications is a key enabler towards deploying these networks in safety-critical applications. Semidefinite programming (SDP)-based approaches to Rectified Linear Unit (ReLU) network verification transcribe this problem into an optimization problem, where the accuracy of any such formulation reflects the level of fidelity in how the neural network computation is represented, as well as the relaxations of intractable constraints. While the literature contains much progress on improving the tightness of SDP formulations while maintaining tractability, comparatively little work has been devoted to the other extreme, i.e., how to most accurately capture the original verification problem before SDP relaxation. In this work, we develop an exact, convex formulation of verification as a completely positive program (CPP), and provide analysis showing that our formulation is minimal -- the removal of any constraint fundamentally misrepresents the neural network computation. We leverage our formulation to provide a unifying view of existing approaches, and give insight into the source of large relaxation gaps observed in some cases.

On Local Computation for Optimization in Multi-Agent Systems

A number of prototypical optimization problems in multi-agent systems (e.g., task allocation and network load-sharing) exhibit a highly local structure: that is, each agent's decision variables are only directly coupled to few other agent's variables through the objective function or the constraints. Nevertheless, existing algorithms for distributed optimization generally do not exploit the locality structure of the problem, requiring all agents to compute or exchange the full set of decision variables. In this paper, we develop a rigorous notion of "locality" that quantifies the degree to which agents can compute their portion of the global solution based solely on information in their local neighborhood. This notion provides a theoretical basis for a rather simple algorithm in which agents individually solve a truncated sub-problem of the global problem, where the size of the sub-problem used depends on the locality of the problem, and the desired accuracy. Numerical results show that the proposed theoretical bounds are remarkably tight for well-conditioned problems.