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Register a new userA Max-Sum algorithm for training discrete neural networks
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We present an efficient learning algorithm for the problem of training neural networks with discrete synapses, a well-known hard (NP-complete) discrete optimization problem. The algorithm is a variant of the so-called Max-Sum (MS) algorithm. In particular, we show how, for bounded integer weights with $q$ distinct states and independent concave a priori distribution (e.g. $l_{1}$ regularization), the algorithms time complexity can be made to scale as $Oleft(Nlog Nright)$ per node update, thus putting it on par with alternative schemes, such as Belief Propagation (BP), without resorting to approximations. Two special cases are of particular interest: binary synapses $Win{-1,1}$ and ternary synapses $Win{-1,0,1}$ with $l_{0}$ regularization. The algorithm we present performs as well as BP on binary perceptron learning problems, and may be better suited to address the problem on fully-connected two-layer networks, since inherent symmetries in two layer networks are naturally broken using the MS approach.
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Stochasticity and limited precision of synaptic weights in neural network models are key aspects of both biological and hardware modeling of learning processes. Here we show that a neural network model with stochastic binary weights naturally gives prominence to exponentially rare dense regions of solutions with a number of desirable properties such as robustness and good generalization performance, while typical solutions are isolated and hard to find. Binary solutions of the standard perceptron problem are obtained from a simple gradient descent procedure on a set of real values parametrizing a probability distribution over the binary synapses. Both analytical and numerical results are presented. An algorithmic extension aimed at training discrete deep neural networks is also investigated.
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