We present theory of synaptic neural balance and we show experimentally that synaptic neural balance can improve deep learning speed, and accuracy, even in data-scarce environments. Given an additive cost function (regularizer) of the synaptic weights, a neuron is said to be in balance if the total cost of its incoming weights is equal to the total cost of its outgoing weights. For large classes of networks, activation functions, and regularizers, neurons can be balanced fully or partially using scaling operations that do not change their functionality. Furthermore, these balancing operations are associated with a strictly convex optimization problem with a single optimum and can be carried out in any order. In our simulations, we systematically observe that: (1) Fully balancing before training results in better performance as compared to several other training approaches; (2) Interleaving partial (layer-wise) balancing and stochastic gradient descent steps during training results in faster learning convergence and better overall accuracy (with L1 balancing converging faster than L2 balancing); and (3) When given limited training data, neural balanced models outperform plain or regularized models; and this is observed in both feedforward and recurrent networks. In short, the evidence supports that neural balancing operations could be added to the arsenal of methods used to regularize and train neural networks. Furthermore, balancing operations are entirely local and can be carried out asynchronously, making them plausible for biological or neuromorphic systems.

We develop a general theory of synaptic neural balance and how it can emerge or be enforced in neural networks . For a given additive cost function R (regularizer), a neuron is said to be in balance if the total cost of its input weights is equal to the total cost of its output weights. The basic example is provided by feedforward networks of ReLU units trained with L 2 regularizers, which exhibit balance after proper training. The theory explains this phenomenon and extends it in several directions. The first direction is the extension to bilinear and other activation functions . The second direction is the extension to more general regularizers, including all L p ( p > 0 ) regularizers. The third direction is the extension to non-layered architectures, recurrent architectures, convolutional architectures, as well as architectures with mixed activation functions and to different balancing algorithms. Gradient descent on the error function alone does not converge in general to a balanced state, where every neuron is in balance, even when starting from a balanced state. However, gradient descent on the regularized error function ought to converge to a balanced state, and thus network balance can be used to assess learning progress. The theory is based on two local neuronal operations: scaling which is commutative, and balancing which is not commutative. Finally, and most importantly, given any set of weights, when local balancing operations are applied to each neuron in a stochastic manner, global order always emerges through the convergence of the stochastic balancing algorithm to the same unique set of balanced weights. The reason for this convergence is the existence of an underlying strictly convex optimization problem where the relevant variables are constrained to a linear, only architecture-dependent, manifold. Simulations show that balancing neurons prior to learning, or during learning in alternation with gradient descent steps, can improve learning speed and performance thereby expanding the arsenal of available training tools. Scaling and balancing operations are entirely local and thus physically plausible in biological and neuromorphic neural networks .