Overall framework of our MPHM. In step 1, the server sends the global model θ^t for this round to each client; in step 2, the benign clients compute the stochastic gradients ∇^t based on the model and upload them to the server. And the malicious clients, after computing the gradients, compute the malicious gradients ∇^t′ for by our MPHM, and then upload them to the server; in step 3, the server aggregates these gradients and computes new global model θ^t + 1 for next round.