L-BFGS does not compute the full Hessian, like BFGS does, it approximates it. The Hessian's size is quadratic in the number of parameters so that wouldn't scale up to even modestly large networks.

Also "parameter search" can be confusing. I'd just sum the whole up by saying L-BFGS is a fast training algorithm but it only works in batch mode, then refer to SciPy for details.