We study federated machine learning at the wirelessnetwork edge, where limited power wireless devices, each withits own dataset, build a joint model with the help of a remoteparameter server (PS). We consider a bandwidth-limited fadingmultiple access channel (MAC) from the wireless devices to thePS, and propose various techniques to implement distributedstochastic gradient descent (DSGD) over this shared noisywireless channel. We first propose a digital DSGD (D-DSGD)scheme, in which one device is selected opportunistically fortransmission at each iteration based on the channel conditions;the scheduled device quantizes its gradient estimate to a finitenumber of bits imposed by the channel condition, and transmitsthese bits to the PS in a reliable manner. Next, motivated bythe additive nature of the wireless MAC, we propose a novelanalog communication scheme, referred to as thecompressedanalogDSGD (CA-DSGD), where the devices first sparsifytheir gradient estimates while accumulating error from previousiterations, and project the resultant sparse vector into a low-dimensional vector for bandwidth reduction. We also design apower allocation scheme to align the received gradient vectorsat the PS in an efficient manner. Numerical results show thatD-DSGD outperforms other digital approaches in the literature;however, in general the proposed CA-DSGD algorithm convergesfaster than the D-DSGD scheme, and reaches a higher level ofaccuracy. We have observed that the gap between the analogand digital schemes increases when the datasets of devices arenot independent and identically distributed (i.i.d.). Furthermore,the performance of the CA-DSGD scheme is shown to be robustagainst imperfect channel state information (CSI) at the devices.Overall these results show clear advantages for the proposedanalog over-the-air DSGD scheme, which suggests that learningand communication algorithms should be designed jointly toachieve the best end-to-end performance in machine learningapplications at the wireless edge.