  Self-paced learning (SPL) is a new methodology that simulates the learning
principle of humans/animals to start learning easier aspects of a learning
task, and then gradually take more complex examples into training. This
new-coming learning regime has been empirically substantiated to be effective
in various computer vision and pattern recognition tasks. Recently, it has been
proved that the SPL regime has a close relationship to a implicit self-paced
objective function. While this implicit objective could provide helpful
interpretations to the effectiveness, especially the robustness, insights under
the SPL paradigms, there are still no theoretical results strictly proved to
verify such relationship. To this issue, in this paper, we provide some
convergence results on this implicit objective of SPL. Specifically, we prove
that the learning process of SPL always converges to critical points of this
implicit objective under some mild conditions. This result verifies the
intrinsic relationship between SPL and this implicit objective, and makes the
previous robustness analysis on SPL complete and theoretically rational.
