The problem of neural network association is to retrieve a previously memorized pattern from its noisy version using a network of neurons. An ideal neural network should include three components simultaneously: a learning algorithm, a large pattern retrieval capacity and resilience against noise. Prior works in this area usually improve one or two aspects at the cost of the third. Our work takes a step forward in closing this gap. More specifically, we show that by forcing natural constraints on the set of learning patterns, we can drastically improve the retrieval capacity of our neural network. Moreover, we devise a learning algorithm whose role is to learn those patterns satisfying the above mentioned constraints. Finally we show that our neural network can cope with a fair amount of noise.
In addition to neural networks capable of learning patterns gradually, in @cite_18 , the authors calculate the weight matrix offline (as opposed to gradual learning) using the pseudo-inverse rule @cite_15 which in return help them improve the capacity of a Hopfield network to @math with the ability of error correction.
@cite_18 @cite_15
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Rumelhart, Hinton and Williams [ 86] describe a learning procedure for layered networks of deterministic, neuron-like units. This paper describes further research on the learning procedure. We start by describing the units, the way they are connected, the learning procedure, and the extension to iterative nets. We then give an example in which a network learns a set of filters that enable it to discriminate formant-like patterns in the presence of noise. The speed of learning is strongly dependent on the shape of the surface formed by the error measure in "weight space." We give examples of the shape of the error surface for a typical task and illustrate how an acceleration method speeds up descent in weight space. The main drawback of the learning procedure is the way it scales as the size of the task and the network increases. We give some preliminary results on scaling and show how the magnitude of the optimal weight changes depends on the fan-in of the units. Additional results illustrate the effects on learning speed of the amount of interaction between the weights. A variation of the learning procedure that back-propagates desired state information rather than error gradients is developed and compared with the standard procedure. Finally, we discuss the relationship between our iterative networks and the "analog" networks described by Hopfield and Tank [Hopfield and Tank 85]. The learning procedure can discover appropriate weights in their kind of network, as well as determine an optimal schedule for varying the nonlinearity of the units during a search. Multi-layer neural networks have lead to remarkable performance on many kinds of benchmark tasks in text, speech and image processing. Nonlinear parameter estimation in hierarchical models is known to be subject to overfitting and misspecification. One approach to these estimation and related problems (local minima, colinearity, feature discovery etc.) is called Dropout (Hinton, et al 2012, 2016). The Dropout algorithm removes hidden units according to a Bernoulli random variable with probability @math prior to each update, creating random "shocks" to the network that are averaged over updates. In this paper we will show that Dropout is a special case of a more general model published originally in 1990 called the Stochastic Delta Rule, or SDR (Hanson, 1990). SDR redefines each weight in the network as a random variable with mean @math and standard deviation @math . Each weight random variable is sampled on each forward activation, consequently creating an exponential number of potential networks with shared weights. Both parameters are updated according to prediction error, thus resulting in weight noise injections that reflect a local history of prediction error and local model averaging. SDR therefore implements a more sensitive local gradient-dependent simulated annealing per weight converging in the limit to a Bayes optimal network. Tests on standard benchmarks (CIFAR) using a modified version of DenseNet shows the SDR outperforms standard Dropout in test error by approx. @math with DenseNet-BC 250 on CIFAR-100 and approx. @math in smaller networks. We also show that SDR reaches the same accuracy that Dropout attains in 100 epochs in as few as 35 epochs. Weight modifications in traditional neural nets are computed by hard-wired algorithms. Without exception, all previous weight change algorithms have many specific limitations. Is it (in principle) possible to overcome limitations of hard-wired algorithms by allowing neural nets to run and improve their own weight change algorithms? This paper constructively demonstrates that the answer (in principle) is ‘yes’. I derive an initial gradientbased sequence learning algorithm for a ‘self-referential’ recurrent network that can ‘speak’ about its own weight matrix in terms of activations. It uses some of its input and output units for observing its own errors and for explicitly analyzing and modifying its own weight matrix, including those parts of the weight matrix responsible for analyzing and modifying the weight matrix. The result is the first ‘introspective’ neural net with explicit potential control over all of its own adaptive parameters. A disadvantage of the algorithm is its high computational complexity per time step which is independent of the sequence length and equals O(nconnlognconn), where riconn is the number of connections. Another disadvantage is the high number of local minima of the unusually complex error surface. The purpose of this paper, however, is not to come up with the most efficient ‘introspective’ or ‘self-referential’ weight change algorithm, but to show that such algorithms are possible at all. We investigate the use of information from all second order derivatives of the error function to perform network pruning (i.e., removing unimportant weights from a trained network) in order to improve generalization, simplify networks, reduce hardware or storage requirements, increase the speed of further training, and in some cases enable rule extraction. Our method, Optimal Brain Surgeon (OBS), is Significantly better than magnitude-based methods and Optimal Brain Damage [Le Cun, Denker and Solla, 1990], which often remove the wrong weights. OBS permits the pruning of more weights than other methods (for the same error on the training set), and thus yields better generalization on test data. Crucial to OBS is a recursion relation for calculating the inverse Hessian matrix H-1 from training data and structural information of the net. OBS permits a 90 , a 76 , and a 62 reduction in weights over backpropagation with weight decay on three benchmark MONK's problems [, 1991]. Of OBS, Optimal Brain Damage, and magnitude-based methods, only OBS deletes the correct weights from a trained XOR network in every case. Finally, whereas Sejnowski and Rosenberg [1987] used 18,000 weights in their NETtalk network, we used OBS to prune a network to just 1560 weights, yielding better generalization.