Modern convolutional networks, incorporating rectifiers and max-pooling, are neither smooth nor convex; standard guarantees therefore do not apply. Nevertheless, methods from convex optimization such as gradient descent and Adam are widely used as building blocks for deep learning algorithms. This paper provides the first convergence guarantee applicable to modern convnets, which furthermore matches a lower bound for convex nonsmooth functions. The key technical tool is the neural Taylor approximation – a straightforward application of Taylor expansions to neural networks – and the associated Taylor loss. Experiments on a range of optimizers, layers, and tasks provide evidence that the analysis accurately captures the dynamics of neural optimization. The second half of the paper applies the Taylor approximation to isolate the main difficulty in training rectifier nets – that gradients are shattered – and investigates the hypothesis that, by exploring the space of activation configurations more thoroughly, adaptive optimizers such as RMSProp and Adam are able to converge to better solutions.

Feature selection is an important pre-processing step, which can reduce the dimensionality of a dataset and increase the accuracy and efficiency of a learning/classification algorithm. However, existing feature selection algorithms mainly wrappers and filters have their own advantages and disadvantages. This paper proposes two filter-wrapper hybrid feature selection algorithms based on particle swarm optimisation (PSO), where the first algorithm named FastPSO combined filter and wrapper into the search process of PSO for feature selection with most of the evaluations as filters and a small number of evaluations as wrappers. The second algorithm named RapidPSO further reduced the number of wrapper evaluations. Theoretical analysis on FastPSO and RapidPSO is conducted to investigate their complexity. FastPSO and RapidPSO are compared with a pure wrapper algorithm named WrapperPSO and a pure filter algorithm named FilterPSO on nine benchmark datasets of varying difficulty. The experimental results show that both FastPSO and RapidPSO can successfully reduce the number of features and simultaneously increase the classification performance over using all features. The two proposed algorithms maintain the high classification performance achieved by WrapperPSO and significantly reduce the computational time, although the number of features is larger. At the same time, they increase the classification accuracy of FilterPSO and reduce the number of features, but increased the computational cost. FastPSO outperformed RapidPSO in terms of the classification accuracy and the number of features, but increased the computational time, which shows the trade-off between the efficiency and effectiveness.