  We demonstrate that the update of weight matrices in learning algorithms can
be described in the framework of Dyson Brownian motion, thereby inheriting many
features of random matrix theory. We relate the level of stochasticity to the
ratio of the learning rate and the mini-batch size, providing more robust
evidence to a previously conjectured scaling relationship. We discuss universal
and non-universal features in the resulting Coulomb gas distribution and
identify the Wigner surmise and Wigner semicircle explicitly in a
teacher-student model and in the (near-)solvable case of the Gaussian
restricted Boltzmann machine.
