When Neurons Fail

Abstract

We view a 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 on a quantity, we call the \textit{Forward Error Propagation}, capturing how much error is propagated by a neural network when a given number of components is failing, 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). Interestingly, as we show in the paper, our bound can easily be extended to the case where synapses can fail. We show how our bound can be leveraged to quantify the effect of memory cost reduction on the accuracy of a neural network, to estimate the amount of information any neuron needs from its preceding layer, enabling thereby a boosting scheme that prevents neurons from waiting for unnecessary signals. We finally discuss the trade-off between neural networks robustness and learning cost.Comment: 2017 IEEE International Parallel and Distributed Processing Symposium, Orlando, Florid

On the Robustness of a Neural Network

“…We use the same model as in [34]. We view a neural network as a distributed system comprised of computing nodes, neurons, and communicating channels, synapses:…”

“…The failure of a synapse is also independent from that of other synapses and neurons. Based on [34], synapses failures can be abstracted as mathematically equivalent to some related neurons' failures, our experimental work focuses therefore on neurons failures.…”

“…Given an over-provisioned network implementing F neu and achieving -acccuracy ( ≤ ), we would like to evaluate how many neurons can crash without harming the -approximation of F by F neu . We use the same definitions as in [34]. In the case where the activation function is bounded, worstcase theoretical analysis reveals [36] 9 that the robustness of the network is completely encompassed in the weight distribution and the network topology: a network of L layers with a K-Lipschitz activation function tolerates the crash of f l neurons per layer l if the quantity Erf defined in Equation 4 is smaller than the error margin − enabled by the over-provision.…”

“…As we pointed out in [34], the answer to this question is simply none. Indeed, why would a neural network tolerate failures if it was not specifically devised with that purpose in mind?…”

“…In fact, it has been experimentally observed that over-provisioning [8], [27] leads to robustness. In a recent work [34], [36], we provided a first theoretical guarantee of neural networks robustness, given an over-provision budget. However, we focused on worst case situations that can be too pessimistic compared to the average case of practical ones.…”