A Geometric Approach of Gradient Descent Algorithms in Neural Networks

In this paper, we present an original geometric framework to analyze the convergence properties of gradient descent trajectories in the context of linear neural networks. Built upon a key invariance property induced by the network structure, we propose a conjecture called \emph{overfitting conjecture} stating that, for almost every training data, the corresponding gradient descent trajectory converges to a global minimum, for almost every initial condition. This would imply that, for linear neural networks of an arbitrary number of hidden layers, the solution achieved by simple gradient descent algorithm is equivalent to that of least square estimation. Our first result consists in establishing, in the case of linear networks of arbitrary depth, convergence of gradient descent trajectories to critical points of the loss function. Our second result is the proof of the \emph{overfitting conjecture} in the case of single-hidden-layer linear networks with an argument based on the notion of normal hyperbolicity and under a generic property on the training data (i.e., holding for almost every training data).Comment: Preprint. Work in progres

Deep neural network for traffic sign recognition systems: An analysis of spatial transformers and stochastic optimisation methods

This paper presents a Deep Learning approach for traffic sign recognition systems. Several classification experiments are conducted over publicly available traffic sign datasets from Germany and Belgium using a Deep Neural Network which comprises Convolutional layers and Spatial Transformer Networks. Such trials are built to measure the impact of diverse factors with the end goal of designing a Convolutional Neural Network that can improve the state-of-the-art of traffic sign classification task. First, different adaptive and non-adaptive stochastic gradient descent optimisation algorithms such as SGD, SGD-Nesterov, RMSprop and Adam are evaluated. Subsequently, multiple combinations of Spatial Transformer Networks placed at distinct positions within the main neural network are analysed. The recognition rate of the proposed Convolutional Neural Network reports an accuracy of 99.71% in the German Traffic Sign Recognition Benchmark, outperforming previous state-of-the-art methods and also being more efficient in terms of memory requirements.Ministerio de Economía y Competitividad TIN2017-82113-C2-1-RMinisterio de Economía y Competitividad TIN2013-46801-C4-1-

Identifying and attacking the saddle point problem in high-dimensional non-convex optimization

A central challenge to many fields of science and engineering involves minimizing non-convex error functions over continuous, high dimensional spaces. Gradient descent or quasi-Newton methods are almost ubiquitously used to perform such minimizations, and it is often thought that a main source of difficulty for these local methods to find the global minimum is the proliferation of local minima with much higher error than the global minimum. Here we argue, based on results from statistical physics, random matrix theory, neural network theory, and empirical evidence, that a deeper and more profound difficulty originates from the proliferation of saddle points, not local minima, especially in high dimensional problems of practical interest. Such saddle points are surrounded by high error plateaus that can dramatically slow down learning, and give the illusory impression of the existence of a local minimum. Motivated by these arguments, we propose a new approach to second-order optimization, the saddle-free Newton method, that can rapidly escape high dimensional saddle points, unlike gradient descent and quasi-Newton methods. We apply this algorithm to deep or recurrent neural network training, and provide numerical evidence for its superior optimization performance.Comment: The theoretical review and analysis in this article draw heavily from arXiv:1405.4604 [cs.LG