  For a certain class of McKean--Vlasov processes, we introduce proxy processes
that substitute the mean-field interaction with self-interaction, employing a
weighted occupation measure. Our study encompasses two key achievements. First,
we demonstrate the ergodicity of the self-interacting dynamics, under broad
conditions, by applying the reflection coupling method. Second, in scenarios
where the drifts are negative intrinsic gradients of convex mean-field
potential functionals, we use entropy and functional inequalities to
demonstrate that the stationary measures of the self-interacting processes
approximate the invariant measures of the corresponding McKean--Vlasov
processes. As an application, we show how to learn the optimal weights of a
two-layer neural network by training a single neuron.
