Title: Exploring The Capacity Mismatch Problem in Knowledge Distillation from the View of Soft Labels. Abstract: Knowledge distillation (KD) has been extensively employed to transfer the knowledge using the soft label from a large teacher model to the smaller students, where the parameters of the teacher are fixed (or partially) during training. Recent studies show that this mode may cause difficulties in knowledge transfer due to the mismatched model capacities. To alleviate the mismatch problem, adjustment of temperature parameters, label smoothing and teacher-student joint training methods (online distillation) to smooth the soft label of a teacher network, have been proposed. But those methods rarely explain the effect of smoothed soft labels to enhance the KD performance. The main contributions of our work are the discovery, analysis, and validation of the effect of the smoothed soft label and a less time-consuming and adaptive transfer of the teacher's knowledge method, namely PESF-KD by adaptive tuning soft labels of the teacher network. Technically, we first mathematically formulate the mismatch as the sharpness gap between teachers' and students' predictive distributions, where we show such a gap can be narrowed with the appropriate smoothness of the soft label. Then, we introduce an adapter module for the teacher and only update the adapter to obtain soft labels with appropriate smoothness. Experiments on various benchmarks show that PESF-KD can significantly reduce the training cost while obtaining competitive results compared to advanced online distillation methods. 