Training neural networks (NNs) using combinatorial optimization solvers has gained attention in recent years. In low-data settings, the use of state-of-the-art mixed integer linear programming solvers, for instance, has the potential to exactly train an NN while avoiding computing-intensive training and hyperparameter tuning and simultaneously training and sparsifying the network. We study the case of few-bit discrete-valued neural networks, both binarized neural networks (BNNs) whose values are restricted to 61 and integer-valued neural networks (INNs) whose values lie in the range {―P, ::: , P}. Few-bit NNs receive increasing recognition because of their lightweight architecture and ability to run on low-power devices: for example, being implemented using Boolean operations. This paper proposes new methods to improve the training of BNNs and INNs. Our contribution is a multiobjective ensemble approach based on training a single NN for each possible pair of classes and applying a majority voting scheme to predict the final output. Our approach results in the training of robust sparsified networks whose output is not affected by small perturbations on the input and whose number of active weights is as small as possible. We empirically compare this BeMi approach with the current state of the art in solver-based NN training and with traditional gradient-based training, focusing on BNN learning in few-shot contexts. We compare the benefits and drawbacks of INNs versus BNNs, bringing new light to the distribution of weights over the {―P, ::: , P} interval. Finally, we compare multiobjective versus single-objective training of INNs, showing that robustness and network simplicity can be acquired simultaneously, thus obtaining better test performances. Although the previous state-of-the-art approaches achieve an average accuracy of 51:1% on the Modified National Institute of Standards and Technology data set, the BeMi ensemble approach achieves an average accuracy of 68.4% when trained with 10 images per class and 81.8% when trained with 40 images per class while having up to 75.3% NN links removed.

Early detection of the emergence of a new variant of concern (VoC) is essential to develop strategies that contain epidemic outbreaks. For example, knowing in which region a VoC starts spreading enables prompt actions to circumscribe the geographical area where the new variant can spread, by containing it locally. This paper presents ‘funnel plots’ as a statistical process control method that, unlike tools whose purpose is to identify rises of the reproduction number (R_t), detects when a regional R_t departs from the national average and thus represents an anomaly. The name of the method refers to the funnel-like shape of the scatter plot that the data take on. Control limits with prescribed false alarm rate are derived from the observation that regional R_t's are normally distributed with variance inversely proportional to the number of infectious cases. The method is validated on public COVID-19 data demonstrating its efficacy in the early detection of SARS-CoV-2 variants in India, South Africa, England, and Italy, as well as of a malfunctioning episode of the diagnostic infrastructure in England, during which the Immensa lab in Wolverhampton gave 43,000 incorrect negative tests relative to South West and West Midlands territories.