When Neurons Fail

Neural networks have been traditionally considered robust in the sense that their precision degrades gracefully with the failure of neurons and can be compensated by additional learning phases. Nevertheless, critical applications for which neural networks are now appealing solutions, cannot afford any additional learning at run-time. In this paper, we view a multilayer neural network as a distributed system of which neurons can fail independently, and we evaluate its robustness in the absence of any (recovery) learning phase. We give tight bounds on the number of neurons that can fail without harming the result of a computation. To determine our bounds, we leverage the fact that neural activation functions are Lipschitz-continuous. Our bound is given in the form of quantity, we call the Forward Error Propagation, computing this quantity only requires looking at the topology of the network, while experimentally assessing the robustness of a network requires the costly experiment of looking at all the possible inputs and testing all the possible configurations of the network corresponding to different failure situations, facing a discouraging combinatorial explosion. We distinguish the case of neurons that can fail and stop their activity (crashed neurons) from the case of neurons that can fail by transmitting arbitrary values (Byzantine neurons). In the crash case, our bound involves the number of neurons per layer, the Lipschitz constant of the neural activation function, the number of failing neurons, the synaptic weights and the depth of the layer where the failure occurred. In the case of Byzantine failures, our bound involves, in addition, the synaptic transmission capacity. Interestingly, as we show in the paper, our bound can easily be extended to the case where synapses can fail. We present three applications of our results. The first is a quantification of the effect of memory cost reduction on the accuracy of a neural network. The second is a quantification of the amount of information any neuron needs from its preceding layer, enabling thereby a boosting scheme that prevents neurons from waiting for unnecessary signals. Our third application is a quantification of the trade-off between neural networks robustness and learning cost.