  Federated learning (FL) is a distributed paradigm that coordinates massive
local clients to collaboratively train a global model via stage-wise local
training processes on the heterogeneous dataset. Previous works have implicitly
studied that FL suffers from the ``client-drift'' problem, which is caused by
the inconsistent optimum across local clients. However, till now it still lacks
solid theoretical analysis to explain the impact of this local inconsistency.
To alleviate the negative impact of the ``client drift'' and explore its
substance in FL, in this paper, we first design an efficient FL algorithm
\textit{FedInit}, which allows employing the personalized relaxed
initialization state at the beginning of each local training stage.
Specifically, \textit{FedInit} initializes the local state by moving away from
the current global state towards the reverse direction of the latest local
state. This relaxed initialization helps to revise the local divergence and
enhance the local consistency level. Moreover, to further understand how
inconsistency disrupts performance in FL, we introduce the excess risk analysis
and study the divergence term to investigate the test error of the proposed
\textit{FedInit} method. Our studies show that optimization error is not
sensitive to this local inconsistency, while it mainly affects the
generalization error bound in \textit{FedInit}. Extensive experiments are
conducted to validate this conclusion. Our proposed \textit{FedInit} could
achieve state-of-the-art~(SOTA) results compared to several advanced benchmarks
without any additional costs. Meanwhile, stage-wise relaxed initialization
could also be incorporated into the current advanced algorithms to achieve
higher performance in the FL paradigm.
