Convergence of the Kiefer-Wolfowitz algorithm under arbitrary disturbances

The Kiefer-Wolfowitz algorithm (1952) for function minimisation under arbitrary deterministic disturbances is studied and necessary and sufficient conditions on the noise sequence are obtained for convergence of the algorithm. We use a notion of persistently disturbing noise sequences, and show that this characterizes convergence of the algorithm under each fixed noise sequence. The results obtained are stronger than previous results and the proof techniques are simpler, involving only basic notions of convergence.