There has been a great deal of recent interest in binarized neural networks, especially because of their explainability. At the same time, automatic differentiation algorithms such as backpropagation fail for binarized neural networks, which limits their applicability. By reformulating the problem of training binarized neural networks as a subadditive dual of a mixed-integer program, we show that binarized neural networks admit a tame representation. This, in turn, makes it possible to use the framework of Bolte et al. for implicit differentiation, which offers the possibility for practical implementation of backpropagation in the context of binarized neural networks. This approach could also be used for a broader class of mixed-integer programs, beyond the training of binarized neural networks, as encountered in symbolic approaches to AI and beyond.

翻译：近期，二值化神经网络因其可解释性而备受关注。然而，反向传播等自动微分算法难以应用于二值化神经网络，这限制了其适用性。通过将二值化神经网络的训练问题重新表述为混合整数规划的子可加对偶形式，我们证明了二值化神经网络具备驯服表示。这一发现使得利用Bolte等人的框架进行隐式微分成为可能，从而为二值化神经网络中的反向传播提供了实际实现的可能性。该方法还可推广至更广泛的混合整数规划问题，不仅限于二值化神经网络的训练，还可应用于符号化人工智能及其他领域。