Beating the Perils of Non-Convexity: Guaranteed Training of Neural Networks using Tensor Methods

Training neural networks is a challenging non-convex optimization problem, and backpropagation or gradient descent can get stuck in spurious local optima. We propose a novel algorithm based on tensor decomposition for guaranteed training of two-layer neural networks. We provide risk bounds for our proposed method, with a polynomial sample complexity in the relevant parameters, such as input dimension and number of neurons. While learning arbitrary target functions is NP-hard, we provide transparent conditions on the function and the input for learnability. Our training method is based on tensor decomposition, which provably converges to the global optimum, under a set of mild non-degeneracy conditions. It consists of simple embarrassingly parallel linear and multi-linear operations, and is competitive with standard stochastic gradient descent (SGD), in terms of computational complexity. Thus, we propose a computationally efficient method with guaranteed risk bounds for training neural networks with one hidden layer.Comment: The tensor decomposition analysis is expanded, and the analysis of ridge regression is added for recovering the parameters of last layer of neural networ

Hierarchical Composition of Memristive Networks for Real-Time Computing

Advances in materials science have led to physical instantiations of self-assembled networks of memristive devices and demonstrations of their computational capability through reservoir computing. Reservoir computing is an approach that takes advantage of collective system dynamics for real-time computing. A dynamical system, called a reservoir, is excited with a time-varying signal and observations of its states are used to reconstruct a desired output signal. However, such a monolithic assembly limits the computational power due to signal interdependency and the resulting correlated readouts. Here, we introduce an approach that hierarchically composes a set of interconnected memristive networks into a larger reservoir. We use signal amplification and restoration to reduce reservoir state correlation, which improves the feature extraction from the input signals. Using the same number of output signals, such a hierarchical composition of heterogeneous small networks outperforms monolithic memristive networks by at least 20% on waveform generation tasks. On the NARMA-10 task, we reduce the error by up to a factor of 2 compared to homogeneous reservoirs with sigmoidal neurons, whereas single memristive networks are unable to produce the correct result. Hierarchical composition is key for solving more complex tasks with such novel nano-scale hardware

Feature selection using genetic algorithms and probabilistic neural networks

Selection of input variables is a key stage in building predictive models, and an important form of data mining. As exhaustive evaluation of potential input sets using full non-linear models is impractical, it is necessary to use simple fast-evaluating models and heuristic selection strategies. This paper discusses a fast, efficient, and powerful nonlinear input selection procedure using a combination of Probabilistic Neural Networks and repeated bitwise gradient descent. The algorithm is compared with forward elimination, backward elimination and genetic algorithms using a selection of real-world data sets. The algorithm has comparative performance and greatly reduced execution time with respect to these alternative approaches. It is demonstrated empirically that reliable results cannot be gained using any of these approaches without the use of resampling