Consider the stochastic iterative updates 
\begin{align}
\theta_{t+1} \leftarrow \theta_t +  \alpha_t \cdot \left [ h(\theta_t) + M_t \right ],
\end{align}
where $\theta_t \in \mathrm{R}^d$, $h \colon \mathrm{R} ^d \rightarrow \mathrm{R}^d$ is Lipschitz continuous, $M_t$ is a martingale difference. Here the stepsize $\alpha_t $ satisfies 
\begin{align}
\sum_{t\geq 0} \alpha_t = \infty, ~~~~\sum_{t \geq 0} \alpha_t^2 < \infty. 
\end{align}
Standard stochastic approximation results suggest that  $\{ \theta_t \}_{t\geq 0}$ converges to the limit point of an ODE
\begin{align}
\dot \theta(t) = h( \theta(t) ).
\end{align}
Moreover, under the assumption that the ODE has an unique global equilibrium $\theta^*$, it can be shown that $\theta_t \rightarrow \theta^*$ as $t$ goes to infinity. 

However, one interesting question is whether $\{ \theta_t\}_{t\geq 0}$ converges if $h$ has multiple zeros. That is, there exists multiple $\bar \theta$'s such that $h( \bar \theta) = 0$. Under what condition can we show that the algorithm converges to any one of these $\bar \theta$'s (equilibria)? Moreover, under what conditions can we show that the algorithm only converges to a locally stable equilibrium?