Binarized Neural Networks (BNNs) are receiving increasing attention due to their lightweight architecture and ability to run on low-power devices. The state-of-the-art for training classification BNNs restricted to few-shot learning is based on a Mixed Integer Programming (MIP) approach. This paper proposes the BeMi ensemble, a structured architecture of BNNs based on training a single BNN for each possible pair of classes and applying a majority voting scheme to predict the final output. The training of a single BNN discriminating between two classes is achieved by a MIP model that optimizes a lexicographic multi-objective function according to robustness and simplicity principles. This approach results in training networks whose output is not affected by small perturbations on the input and whose number of active weights is as small as possible, while good accuracy is preserved. We computationally validate our model using the MNIST and Fashion-MNIST datasets using up to 40 training images per class. Our structured ensemble outperforms both BNNs trained by stochastic gradient descent and state-of-the-art MIP-based approaches. While the previous approaches achieve an average accuracy of 51.1% on the MNIST dataset, the BeMi ensemble achieves an average accuracy of 61.7% when trained with 10 images per class and 76.4% when trained with 40 images per class.

In this paper, we propose a novel ensembling technique for deep neural networks, which is able to drastically reduce the required memory compared to alternative approaches. In particular, we propose to extract multiple sub-networks from a single, untrained neural network by solving an end-to-end optimization task combining differentiable scaling over the original architecture, with multiple regularization terms favouring the diversity of the ensemble. Since our proposal aims to detect and extract sub-structures, we call it Structured Ensemble. On a large experimental evaluation, we show that our method can achieve higher or comparable accuracy to competing methods while requiring significantly less storage. In addition, we evaluate our ensembles in terms of predictive calibration and uncertainty, showing they compare favourably with the state-of-the-art. Finally, we draw a link with the continual learning literature, and we propose a modification of our framework to handle continuous streams of tasks with a sub-linear memory cost. We compare with a number of alternative strategies to mitigate catastrophic forgetting, highlighting advantages in terms of average accuracy and memory.