We consider the stabilization of a linear control system with multiplicative observation noise. Linear strategies have been proven to be ineffective in this system, and non-linear strategies can unboundedly outperform the best linear strategies. The current best-known control strategy is hand-crafted and far from optimal. In this paper, we use neural-network-based controllers to narrow the gap between the achievability and the converse for this problem. We propose a periodic controller structure and a greedy training procedure. This enables us to train our controller on a finite horizon problem but learn strategies that outperform the best-known hand crafted strategy and also generalize, i.e. provide stabilizing control at times beyond the training horizon. Further, we show that for our periodic approach, the learned strategies display a piecewise linear structure and are well approximated by interpretable functions.