Neurons learn slower than they think

Recent studies revealed complex convergence dynamics in gradient-based methods, which has been little understood so far. Changing the step size to balance between high convergence rate and small generalization error may not be sufficient: maximizing the test accuracy usually requires a larger learning rate than minimizing the training loss. To explore the dynamic bounds of convergence rate, this study introduces \textit{differential capability} into an optimization process, which measures whether the test accuracy increases as fast as a model approaches the decision boundary in a classification problem. The convergence analysis showed that: 1) a higher convergence rate leads to slower capability growth; 2) a lower convergence rate results in faster capability growth and decay; 3) regulating a convergence rate in either direction reduces differential capability.