In this paper, we analyze the inner product of weight vector and input vector in neural networks from the perspective of vector orthogonal decomposition and prove that the local direction gradient of weight vector decreases as the angle between them gets closer to 0 or @math . We propose the PR Product, a substitute for the inner product, which makes the local direction gradient of weight vector independent of the angle and consistently larger than the one in the conventional inner product while keeping the forward propagation identical. As the basic operation in neural networks, the PR Product can be applied into many existing deep learning modules, so we develop the PR Product version of the fully connected layer, convolutional layer, and LSTM layer. In static image classification, the experiments on CIFAR10 and CIFAR100 datasets demonstrate that the PR Product can robustly enhance the ability of various state-of-the-art classification networks. On the task of image captioning, even without any bells and whistles, our PR Product version of captioning model can compete or outperform the state-of-the-art models on MS COCO dataset.
Several recent investigations have considered variants of standard Backpropagation. In particular, @cite_40 presents a surprisingly simple backpropagation mechanism that assigns blame by multiplying errors signals with random weights, instead of the synaptic weights on each neuron, and further downstream. @cite_8 exhaustively considers many Hebbian learning algorithms. The straight-through estimator proposed in @cite_10 heuristically copies the gradient with respect to the stochastic output directly as an estimator of the gradient with respect to the sigmoid argument. @cite_7 proposes Linear Backprop that backpropagates error terms only linearly. Different from these methods, our proposed PR Product changes the local gradients of weights during backpropagation while maintaining the identical forward propagation.
@cite_40 @cite_10 @cite_7 @cite_8
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Multi-layered neural architectures that implement learning require elaborate mechanisms for symmetric backpropagation of errors that are biologically implausible. Here the authors propose a simple resolution to this problem of blame assignment that works even with feedback using random synaptic weights. Stochastic neurons and hard non-linearities can be useful for a number of reasons in deep learning models, but in many cases they pose a challenging problem: how to estimate the gradient of a loss function with respect to the input of such stochastic or non-smooth neurons? I.e., can we "back-propagate" through these stochastic neurons? We examine this question, existing approaches, and compare four families of solutions, applicable in different settings. One of them is the minimum variance unbiased gradient estimator for stochatic binary neurons (a special case of the REINFORCE algorithm). A second approach, introduced here, decomposes the operation of a binary stochastic neuron into a stochastic binary part and a smooth differentiable part, which approximates the expected effect of the pure stochatic binary neuron to first order. A third approach involves the injection of additive or multiplicative noise in a computational graph that is otherwise differentiable. A fourth approach heuristically copies the gradient with respect to the stochastic output directly as an estimator of the gradient with respect to the sigmoid argument (we call this the straight-through estimator). To explore a context where these estimators are useful, we consider a small-scale version of conditional computation , where sparse stochastic units form a distributed representation of gaters that can turn off in combinatorially many ways large chunks of the computation performed in the rest of the neural network. In this case, it is important that the gating units produce an actual 0 most of the time. The resulting sparsity can be potentially be exploited to greatly reduce the computational cost of large deep networks for which conditional computation would be useful. In a physical neural system, where storage and processing are intimately intertwined, the rules for adjusting the synaptic weights can only depend on variables that are available locally, such as the activity of the pre- and post-synaptic neurons, resulting in local learning rules. A systematic framework for studying the space of local learning rules is obtained by first specifying the nature of the local variables, and then the functional form that ties them together into each learning rule. Such a framework enables also the systematic discovery of new learning rules and exploration of relationships between learning rules and group symmetries. We study polynomial local learning rules stratified by their degree and analyze their behavior and capabilities in both linear and non-linear units and networks. Stacking local learning rules in deep feedforward networks leads to deep local learning. While deep local learning can learn interesting representations, it cannot learn complex input-output functions, even when targets are available for the top layer. Learning complex input-output functions requires local deep learning where target information is communicated to the deep layers through a backward learning channel. The nature of the communicated information about the targets and the structure of the learning channel partition the space of learning algorithms. For any learning algorithm, the capacity of the learning channel can be defined as the number of bits provided about the error gradient per weight, divided by the number of required operations per weight. We estimate the capacity associated with several learning algorithms and show that backpropagation outperforms them by simultaneously maximizing the information rate and minimizing the computational cost. This result is also shown to be true for recurrent networks, by unfolding them in time. The theory clarifies the concept of Hebbian learning, establishes the power and limitations of local learning rules, introduces the learning channel which enables a formal analysis of the optimality of backpropagation, and explains the sparsity of the space of learning rules discovered so far.