The dominant view in neuroscience is that changes in synaptic weights underlie learning. It is unclear, however, how the brain is able to determine which synapses should change, and by how much. This uncertainty stands in sharp contrast to deep learning, where changes in weights are explicitly engineered to optimize performance. However, the main tool for that, backpropagation, has two problems. One is neuro-science related: it is not biologically plausible. The other is inherent: networks trained with this rule tend to forget old tasks when learning new ones. Here we introduce the Dendritic Gated Network (DGN), a variant of the Gated Linear Network, which offers a biologically plausible alternative to backpropagation. DGNs combine dendritic ‘gating’ (whereby interneurons target dendrites to shape neuronal responses) with local learning rules to yield provably efficient performance. They are significantly more data efficient than conventional artificial networks, and are highly resistant to forgetting. Consequently, they perform well on a variety of tasks, in some cases better than backpropagation. Importantly, DGNs have structural and functional similarities to the cerebellum, a link that we strengthen by using in vivo two-photon calcium imaging to show that single interneurons suppress activity in individual dendritic branches of Purkinje cells, a key feature of the model. Thus, DGNs leverage targeted dendritic inhibition and local learning – two features ubiquitous in the brain – to achieve fast and efficient learning.