Existence conditions for mobility–potential factorization in non-SGD learning dynamics

Determine necessary and sufficient conditions under which a non-conservative update force field F(W) in the parameter space of a non–stochastic-gradient-descent learning algorithm admits a factorization F(W) = μ(W) ∇U(W), where μ(W) is a symmetric positive-definite mobility matrix and U(W) is a scalar potential, so that the stationary-distribution analysis analogous to SGD applies.

Background

In discussing general (non-SGD) learning dynamics under minibatch stochasticity, the paper models weight updates by a Langevin equation and highlights that an SGD-like stationary distribution analysis carries over if the update force can be written in the form F(x) = μ(x)∇U(x) with a symmetric mobility matrix. The authors note that when such a factorization is available, the resulting Fokker–Planck analysis closely parallels the SGD case.

However, the paper explicitly flags uncertainty about when this factorization exists for a given (possibly non-conservative) force field arising from a learning rule. Pinning down exact existence criteria would clarify the scope of applicability of the SGD-inspired stationary distribution framework to broader classes of biologically plausible or other non-SGD learning rules.