Title: FedSpeed: Larger Local Interval, Less Communication Round, and Higher Generalization Accuracy. Abstract: Federated learning (FL) is an emerging distributed machine learning framework which jointly trains a global model via a large number of local devices with data privacy protections. Its performance suffers from the non-vanishing biases introduced by the local inconsistent optimal and the rugged client-drifts by the local over-fitting. In this paper, we propose a novel and practical method, FedSpeed, to alleviate the negative impacts posed by these problems.  Concretely, FedSpeed applies the prox-correction term on the current local updates to efficiently reduce the biases introduced by the prox-term, a necessary regularizer to maintain the strong local consistency. Furthermore, FedSpeed merges the vanilla stochastic gradient with a perturbation computed from an extra gradient ascent step in the neighborhood, thereby alleviating the issue of local over-fitting. Our theoretical analysis indicates that the convergence rate is related to both the communication rounds $T$ and local intervals $K$ with a tighter upper bound $\mathcal{O}(\frac{1}{T})$ if $K=\mathcal{O}(T)$.  Moreover, we conduct extensive experiments on the real-world dataset to demonstrate the efficiency of our proposed FedSpeed, which converges significantly faster and achieves the state-of-the-art (SOTA) performance on the general FL experimental settings than several baselines including FedAvg, FedProx, FedCM, FedAdam, SCAFFOLD, FedDyn, FedADMM, etc.