The structure of the majority of modern deep neural networks is characterized by uni- directional feed-forward connectivity across a very large number of layers. By contrast, the architecture of the cortex of vertebrates contains fewer hierarchical levels but many recurrent and feedback connections. Here we show that a small, few-layer artificial neural network that employs feedback will reach top level performance on a standard benchmark task, otherwise only obtained by large feed-forward structures. To achieve this we use feed-forward transfer entropy between neurons to structure feedback connectivity. Transfer entropy can here intuitively be understood as a measure for the relevance of certain pathways in the network, which are then amplified by feedback. Feedback may therefore be key for high network performance in small brain-like architectures.