Learning under feature evolution data streams has attracted widespread attention in recent years. Existing methods usually assume that the model predicts and learns from all instances in the data stream. However, when the data stream rate is faster than the model update rate, the model can only learn from some instances. Therefore, this assumption may not always hold in practical scenarios. Additionally, existing methods often update based only on the current instance, ignoring the impact of data stream changes, which further limits their application in practical data streams. This paper proposes a novel learning paradigm to solve this problem: Online Learning under Feature Evolution data streams with A Fast Rate, called OLFE-FR. Specifically, OLFE-FR introduces the concept of relative rate to adaptively determine the prediction mode and update node of the model in the data stream. Additionally, OLFE-FR proposes an adaptive learning rate adjustment strategy based on the upper bound of dynamic regret minimization. This strategy enables the model to find a suitable learning rate based on weights change induced by known data stream variations before using the instance update. Theoretical analysis and experimental results show that OLFE-FR can effectively handle feature evolution data streams with a fast rate.