When the linear measurements of an instance of low-rank matrix recovery satisfy a restricted isometry property (RIP) --- i.e. they are approximately norm-preserving --- the problem is known to contain no spurious local minima, so exact recovery is guaranteed. In this paper, we show that moderate RIP is not enough to eliminate spurious local minima, so existing results can only hold for near-perfect RIP. In fact, counterexamples are ubiquitous: every $x$ is the spurious local minimum of a rank-1 instance of matrix recovery that satisfies RIP. One specific counterexample has RIP constant $\delta=1/2$, but causes randomly initialized stochastic gradient descent (SGD) to fail 12\% of the time. SGD is frequently able to avoid and escape spurious local minima, but this empirical result shows that it can occasionally be defeated by their existence. Hence, while exact recovery guarantees will likely require a proof of no spurious local minima, arguments based solely on norm preservation will only be applicable to a narrow set of nearly-isotropic instances.

Neural networks are increasingly deployed in real-world safety-critical domains such as autonomous driving, aircraft collision avoidance, and malware detection. However, these networks have been shown to often mispredict on inputs with minor adversarial or even accidental perturbations. Consequences of such errors can be disastrous and even potentially fatal as shown by the recent Tesla autopilot crash. Thus, there is an urgent need for formal analysis systems that can rigorously check neural networks for violations of different safety properties such as robustness against adversarial perturbations within a certain L-norm of a given image. An effective safety analysis system for a neural network must be able to either ensure that a safety property is satisfied by the network or find a counterexample, i.e., an input for which the network will violate the property. Unfortunately, most existing techniques for performing such analysis struggle to scale beyond very small networks and the ones that can scale to larger networks suffer from high false positives and cannot produce concrete counterexamples in case of a property violation. In this paper, we present a new efficient approach for rigorously checking different safety properties of neural networks that significantly outperforms existing approaches by multiple orders of magnitude. Our approach can check different safety properties and find concrete counterexamples for networks that are 10x larger than the ones supported by existing analysis techniques. We believe that our approach to estimating tight output bounds of a network for a given input range can also help improve the explainability of neural networks and guide the training process of more robust neural networks.

Neural Control Barrier Functions (NCBFs) have shown significant promise in enforcing safety constraints on nonlinear autonomous systems.

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/