Theory of Curriculum Learning, with Convex Loss Functions

Curriculum Learning - the idea of teaching by gradually exposing the learner to examples in a meaningful order, from easy to hard, has been investigated in the context of machine learning long ago. Although methods based on this concept have been empirically shown to improve performance of several learning algorithms, no theoretical analysis has been provided even for simple cases. To address this shortfall, we start by formulating an ideal definition of difficulty score - the loss of the optimal hypothesis at a given datapoint. We analyze the possible contribution of curriculum learning based on this score in two convex problems - linear regression, and binary classification by hinge loss minimization. We show that in both cases, the expected convergence rate decreases monotonically with the ideal difficulty score, in accordance with earlier empirical results. We also prove that when the ideal difficulty score is fixed, the convergence rate is monotonically increasing with respect to the loss of the current hypothesis at each point. We discuss how these results bring to term two apparently contradicting heuristics: curriculum learning on the one hand, and hard data mining on the other.

Curriculum Learning by Transfer Learning: Theory and Experiments with Deep Networks

We provide theoretical investigation of curriculum learning in the context of stochastic gradient descent when optimizing the convex linear regression loss. We prove that the rate of convergence of an ideal curriculum learning method is monotonically increasing with the difficulty of the examples. Moreover, among all equally difficult points, convergence is faster when using points which incur higher loss with respect to the current hypothesis. We then analyze curriculum learning in the context of training a CNN. We describe a method which infers the curriculum by way of transfer learning from another network, pre-trained on a different task. While this approach can only approximate the ideal curriculum, we observe empirically similar behavior to the one predicted by the theory, namely, a significant boost in convergence speed at the beginning of training. When the task is made more difficult, improvement in generalization performance is also observed. Finally, curriculum learning exhibits robustness against unfavorable conditions such as excessive regularization.