We report a learning rule for neural networks that computes how much each neuron should contribute to minimize a giving cost function via the estimation of its target value. By theoretical analysis, we show that this learning rule contains backpropagation, Hebian learning, and additional terms. We also give a general technique for weights initialization. Our results are at least as good as those obtained with backpropagation. The neural networks are trained and tested in three problems: MNIST, MNIST-Fashion, and CIFAR-10 datasets. The associated code is available at this https URL
A better approach was reported in @cite_12 , which contains the Xavier initialization as a special case. There the non-linearity of the activation function was considered, so that for each function we will have a different random distribution of parameters. The problem with this approach is that it assumes a specific distribution for the input data.
@cite_12
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A proper initialization of the weights in a neural network is critical to its convergence. Current insights into weight initialization come primarily from linear activation functions. In this paper, I develop a theory for weight initializations with non-linear activations. First, I derive a general weight initialization strategy for any neural network using activation functions differentiable at 0. Next, I derive the weight initialization strategy for the Rectified Linear Unit (RELU), and provide theoretical insights into why the Xavier initialization is a poor choice with RELU activations. My analysis provides a clear demonstration of the role of non-linearities in determining the proper weight initializations. We consider neural networks with a single hidden layer and non-decreasing positively homogeneous activation functions like the rectified linear units. By letting the number of hidden units grow unbounded and using classical non-Euclidean regularization tools on the output weights, they lead to a convex optimization problem and we provide a detailed theoretical analysis of their generalization performance, with a study of both the approximation and the estimation errors. We show in particular that they are adaptive to unknown underlying linear structures, such as the dependence on the projection of the input variables onto a low-dimensional subspace. Moreover, when using sparsity-inducing norms on the input weights, we show that high-dimensional non-linear variable selection may be achieved, without any strong assumption regarding the data and with a total number of variables potentially exponential in the number of observations. However, solving this convex optimization problem in infinite dimensions is only possible if the nonconvex subproblem of addition of a new unit can be solved efficiently. We provide a simple geometric interpretation for our choice of activation functions and describe simple conditions for convex relaxations of the finite-dimensional non-convex subproblem to achieve the same generalization error bounds, even when constant-factor approximations cannot be found. We were not able to find strong enough convex relaxations to obtain provably polynomialtime algorithms and leave open the existence or non-existence of such tractable algorithms with non-exponential sample complexities. A new initialization method for hidden parameters in a neural network is proposed. Derived from the integral representation of the neural network, a nonparametric probability distribution of hidden parameters is introduced. In this proposal, hidden parameters are initialized by samples drawn from this distribution, and output parameters are fitted by ordinary linear regression. Numerical experiments show that backpropagation with proposed initialization converges faster than uniformly random initialization. Also it is shown that the proposed method achieves enough accuracy by itself without backpropagation in some cases. The problem of quantizing the activations of a deep neural network is considered. An examination of the popular binary quantization approach shows that this consists of approximating a classical non-linearity, the hyperbolic tangent, by two functions: a piecewise constant sign function, which is used in feedforward network computations, and a piecewise linear hard tanh function, used in the backpropagation step during network learning. The problem of approximating the ReLU non-linearity, widely used in the recent deep learning literature, is then considered. An half-wave Gaussian quantizer (HWGQ) is proposed for forward approximation and shown to have efficient implementation, by exploiting the statistics of of network activations and batch normalization operations commonly used in the literature. To overcome the problem of gradient mismatch, due to the use of different forward and backward approximations, several piece-wise backward approximators are then investigated. The implementation of the resulting quantized network, denoted as HWGQ-Net, is shown to achieve much closer performance to full precision networks, such as AlexNet, ResNet, GoogLeNet and VGG-Net, than previously available low-precision networks, with 1-bit binary weights and 2-bit quantized activations.