This paper takes the position that, while cognitive computing today relies heavily on the cloud, we will soon see a paradigm shift where cognitive computing primarily happens on network edges. The shift toward edge devices is fundamentally propelled both by technological constraints in data centers and wireless network infrastructures, as well as practical considerations such as privacy and safety. The remainder of this paper lays out our view of how these constraints will impact future cognitive computing. Bringing cognitive computing to edge devices opens up several new opportunities and challenges, some of which demand new solutions and some of which require us to revisit entrenched techniques in light of new technologies. We close the paper with a call to action for future research.