  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.
