How to assess that a cross entropy based model has converged
Pure, hardcore Stochastic Gradient Descent (when you feed just one observation at a time) is not advisable at all. The descent of the Gradient is so noisy that after a certain minimal loss reduction it will stop learning anything. It will wander around the loss function in unpredictable ways. In this case, you're right: there's no way to assess final convergence. Even if the Gradient hits the global minimum, it will jump away from it at the following iteration.

The most powerful technique is a robust version of SGD: Mini-Batch Gradient Descent. In this case, you feed a batch of data at each training iteration (usually of size between 30 to 250, but it's up to you). Batch size is another hyperparameter, therefore you must find a trade off between robustness to noise on one side, and on the other side to avoid get stuck in local minima and lower computational times.