$\partial\mathbb{B}$ nets: learning discrete functions by gradient descent

B nets are differentiable neural networks that learn discrete boolean-valued functions by gradient descent. B nets have two semantically equivalent aspects: a differentiable soft-net, with real weights, and a non-differentiable hard-net, with boolean weights. We train the soft-net by backpropagation and then'harden' the learned weights to yield boolean weights that bind with the hard-net. The result is a learned discrete function. 'Hardening' involves no loss of accuracy, unlike existing approaches to neural network binarization. Preliminary experiments demonstrate that B nets achieve comparable performance on standard machine learning problems yet are compact (due to 1-bit weights) and interpretable (due to the logical nature of the learnt functions). Neural networks are differentiable functions with weights represented by machine floats. Networks are trained by gradient descent in weight-space, where the direction of descent minimises loss. The gradients are efficiently calculated by the backpropagation algorithm (Rumelhart et al., 1986). This overall approach has led to tremendous advances in machine learning.