Backpropagation (BP) uses detailed, unit-specific feedback to train deep neural networks (DNNs) with remarkable success. That biological neural circuits appear to perform credit assignment, but cannot implement BP, implies the existence of other powerful learning algorithms. Here, we explore the extent to which a globally broadcast learning signal, coupled with local weight updates, enables training of DNNs. We present both a learning rule, called global error-vector broadcasting (GEVB), and a class of DNNs, called vectorized nonnegative networks (VNNs), in which this learning rule operates. VNNs have vector-valued units and nonnegative weights past the first layer. The GEVB learning rule generalizes three-factor Hebbian learning, updating each weight by an amount proportional to the inner product of the presynaptic activation and a globally broadcast error vector when the postsynaptic unit is active. We prove that these weight updates are matched in sign to the gradient, enabling accurate credit assignment. Moreover, at initialization, these updates are exactly proportional to the gradient in the limit of infinite network width. GEVB matches the performance of BP in VNNs, and in some cases outperforms direct feedback alignment (DFA) applied in conventional networks. Unlike DFA, GEVB successfully trains convolutional layers. Altogether, our theoretical and empirical results point to a surprisingly powerful role for a global learning signal in training DNNs.