Stacked intelligent metasurfaces (SIM) represents an advanced signal processing paradigm that enables over-the-air processing of electromagnetic waves at the speed of light. Its multi-layer structure exhibits customizable increased computational capability compared to conventional single-layer reconfigurable intelligent surfaces and metasurface lenses. In this paper, we deploy SIM to improve the performance of multi-user multiple-input single-output (MISO) wireless systems with low complexity transmit radio frequency (RF) chains. In particular, an optimization formulation for the joint design of the SIM phase shifts and the transmit power allocation is presented, which is efficiently solved via a customized deep reinforcement learning (DRL) approach that continuously observes pre-designed states of the SIM-parametrized smart wireless environment. The presented performance evaluation results showcase the proposed method's capability to effectively learn from the wireless environment while outperforming conventional precoding schemes under low transmit power conditions. Finally, a whitening process is presented to further augment the robustness of the proposed scheme.