  In this paper we are concerned with the learnability of nonlocal interaction
kernels for first order systems modeling certain social interactions, from
observations of realizations of their dynamics. This paper is the first of a
series on learnability of nonlocal interaction kernels and presents a
variational approach to the problem. In particular, we assume here that the
kernel to be learned is bounded and locally Lipschitz continuous and that the
initial conditions of the systems are drawn identically and independently at
random according to a given initial probability distribution. Then the
minimization over a rather arbitrary sequence of (finite dimensional) subspaces
of a least square functional measuring the discrepancy from observed
trajectories produces uniform approximations to the kernel on compact sets. The
convergence result is obtained by combining mean-field limits, transport
methods, and a $\Gamma$-convergence argument. A crucial condition for the
learnability is a certain coercivity property of the least square functional,
majoring an $L_2$-norm discrepancy to the kernel with respect to a probability
measure, depending on the given initial probability distribution by suitable
push forwards and transport maps. We illustrate the convergence result by means
of several numerical experiments.
