One-dimensional system arising in stochastic gradient descent

Abstract We consider stochastic differential equations of the form $dX_t = |f(X_t)|/t^{\gamma} dt+1/t^{\gamma} dB_t$ , where f ( x ) behaves comparably to $|x|^k$ in a neighborhood origin, for $k\in [1,\infty)$ . show that there exists threshold value $ \,{:}\,{\raise-1.5pt{=}}\, \tilde{\gamma}$ $\gamma$ depending on k such if $\gamma \in (1/2, \tilde{\gamma})$ then $\mathbb{P}(X_t\rightarrow 0) 0$ and rest permissible values 0)>0$ These results extend discrete processes satisfy $X_{n+1}-X_n f(X_n)/n^\gamma +Y_n/n^\gamma$ Here, $Y_{n+1}$ are martingale differences almost surely bounded. This result shows function F whose second derivative at degenerate saddle points is polynomial order, it always possible escape via iteration =F'(X_n)/n^\gamma suitable choice