Before revision:  Mathematically, SP-BFGS provides a means of incrementally updating the new inverse Hessian approximation with a controlled amount of bias towards the previous inverse Hessian approximation . Practically speaking, this can be used to help replace the overwriting nature of the  BFGS update with an averaging nature that resists the destructive effects of noise . We provide a convergence analysis of SP-BFGS, and present numerical results illustrating the performance of SP-BFGS  in the presence of noisy gradients .
After revision:  Mathematically, SP-BFGS provides a means of incrementally updating the new inverse Hessian approximation with a controlled amount of bias towards the previous inverse Hessian approximation . Practically speaking, this can be used to help replace the overwriting nature of the BFGS update with an averaging nature that resists the destructive effects of noise and can cope with negative curvature measurements. We discuss the theoretical properties of SP-BFGS, and present numerical results illustrating the performance of SP-BFGS in the presence of noisy gradients .