Averaged stationarity bound for stochastic gradient descent ($k^{-1}{\sum }_{j<k}\mathbb{E}\Vert \nabla f(X^j){\Vert }^2\le M+2L{\Delta }_0/k$)

Suppose a fixed-step size stochastic gradient descent iteration satisfiesand $f$ has a global minimizer $x^{\ast }$. Taking expectations and forming a telescoping sum proves the displayed bound with ${\Delta }_0=f(X^0)-f(x^{\ast })$ and deterministic $X^0$. The Cauchy-Schwarz inequality then bounds the averaged, or the best, expected gradient Euclidean norm by $\sqrt{M+2L{\Delta }_0/k}$. This is a guarantee of stationarity in the indicated senses, rather than a last-iterate convergence assertion. The additive $M$ is a stochastic gradient noise floor.

Target: real-analysis.bigb, under Strongly convex function.