i.e. Ot+1 =0t- Therefore we use the following "effective update capacity" (9.5) to account for the actual contribution for the weighted average update. As discussed later on in Section 9.4, FL experiences performance degradation when data rate is heteroge- neously distributed. Therefore, to make sure every user has a quality update, we use a widely accepted Jain's fairness index, which is defined as: (9.6) For the maximum fairness, Ju should be close to 1. In practice, PS selects users with high effective update capacities and ensures Ju to be close to 1. We adopt a similar scheduling algorithm in [3]. 9 Wireless Systems for Distributed Machine Learning 2. Time budget: Another factor is the computation time at each device. Since NOMA is a synchronous system, the PS needs to set a hard time budget for the local computation. All the devices may have heterogeneous capacities; hence, they need to estimate the time spent on the training in each iteration. The scheduling used in this chapter selects those who can not only finish the calculation on time but also complete the most iterations of computing in each round. 9.3.3 Adaptive Transmission For the selected user k in each round, we calculate the maximum throughput Mk under the NOMA scheme. The total bit length of gradient G is known once we have deter- mined the ML structure. Thus the compression rate for quantization ra is calculated = max in The quantization bit length ba is calculated by where |.| takes the floor operation, 32 is bit-length of each parameter representation. Afterwards, every gradient value in user k is represented by bits with a length of by Similarly, under sparsification, compression ratio rk can be calculated by solving the following equation (9.7) Hence rkmin{r(n),r(n)is the numeric solution for (9.7). Once rk is obtained, we set (1 - rk) portion of the smallest gradient values to zero. For both compression methods, it is important to keep the gradient residuals for the next round.¹ - where GS(.) and GQ(.) are sparsification and quantization