Property Testing of Computational Networks

In this paper we initiate the study of \emph{property testing of weighted computational networks viewed as computational devices}. Our goal is to design property testing algorithms that for a given computational network with oracle access to the weights of the network, accept (with probability at least $\frac23$) any network that computes a certain function (or a function with a certain property) and reject (with probability at least $\frac23$) any network that is \emph{far} from computing the function (or any function with the given property). We parameterize the notion of being far and want to reject networks that are \emph{$(ε,δ)$-far}, which means that one needs to change an $ε$-fraction of the description of the network to obtain a network that computes a function that differs in at most a $δ$-fraction of inputs from the desired function (or any function with a given property). To exemplify our framework, we present a case study involving simple neural Boolean networks with ReLU activation function. As a highlight, we demonstrate that for such networks, any near constant function is testable in query complexity independent of the network's size. We also show that a similar result cannot be achieved in a natural generalization of the distribution-free model to our setting, and also in a related vanilla testing model.