Why is training better when following an easy-to-difficult schedule?

Question

As suggested in the answer to this question, experimental results show that training is most effective when it follows an easy-to-difficult schedule.

What theories and specifically computational models account for that?

In the simple computational models commonly used to demonstrate learning classifications for example, there is no such effect, as far as I know:

• Binary Perceptron - Typical examples (closest to the cluster means, or far apart along the optimal separating vector) are the most informative, leading to a prediction that easy trials would result in faster learning.
  • Some learning algorithms only update the network when an error is made, leading to a prediction that more difficult trials would result in faster learning.
• Support Vector Machinces (SVM) - only the few most difficult examples affect the final result, again leading to a prediction that more difficult trials would result in faster learning.
• Naive Bayes classifier - Typical examples help estimating the parameters with less errors, leading to a prediction that easy trials would result in faster learning.

Answer 1

SVM training is typically done in a batch processing, and thus the order of data presentation doesn't matter. You should consider online learning algorithms, for example, the perceptron learning rule. These algorithms are in general stochastic gradient descent optimization procedures, and easy examples early on with larger learning step would be much more efficient (faster convergence towards the correct answer) than learning with difficult examples first.

Answer 2

Another way of thinking about this is that by progressing easy-to-hard, different intermediate knowledge structures are called into existence in the course of processing. These knowledge structures, built from an agent's encounter with easy problems, can prove useful in its encounter with subsequent and more difficult problems.

This idea has been around for a long time in a variety of cognitive traditions. It's intimately related to developmental theories of, for instance, Piaget, and Mandler (2006); and to the schema literature, which is vast, but of which Minsky (1986) and Schank & Abelson (1977) are exemplars.

This is all very vague, though. The best (or at least, the most precise) way to think about knowledge progression is through hierarchical reinforcement learning. The idea there is that most non-trivial tasks are inherently hierarchical; and so learning how to do a task requires one to learn how to do its constituent sub-tasks. The more tasks you learn how to do, the greater the 'toolbox' you will be able to bring to subsequent tasks.

With regard to the original question, easier examples of some task induce the acquisition of the controllers (knowledge structures) that will aid the performance of later, more complicated, tasks, and the acquisition of later and more complicated knowledge structures. Oudeyer et al. (2007) and Barto (2009) describe this process in detail, the former using a situated robotic agent. (If we replace 'controllers' with 'rules' then the process becomes comparable to the rule search and chunking process used in ACT-R and Soar, as mentioned in Jeromy Anglim's answer)

References

Barto, A. (2009). Skill characterization based on betweenness. In Advances in Neural Information Processing Systems 22.

Mandler, J. M. (2006). The Foundations of Mind: Origins of Conceptual Thought. Oxford University Press, USA.

Minsky, M. (1986). Society of mind. New York, NY: Simon & Shuster.

Oudeyer, P.-Y., Kaplan, F., & Hafner, V. V. (2007). Intrinsic motivation systems for autonomous mental development. Ieee Transactions on Evolutionary Computation, 11(2), 265–286.

Schank, R., & ABELSON, R. (1977). Scripts, plans, goals and understanding: An inquiry into human knowledge structures.

Answer 3

Interesting question. I've written up a discussion of the model-based training literature and how it relates structuring task difficulty with practice. That said, I feel it's only a start and my apologies that it is more pitched at cognitive tasks than perceptual tasks.

A summary of model-based training systems

Fu et al (2006) have a paper on real-time model-based training systems, in which they review some of the work on model-based training systems:

There has been a long history of applying cognitive theory of learning and skill acquisition to model-based training systems (e.g., Anderson et al., 1995; Graesser et al., 2004; Hill and Johnson, 1993; Sleeman and Brown, 1982). The key idea of a model-based training system is that instructions should be given based on a cognitive model of the competence that the trainee is being asked to learn. In other words, the cognitive model should incorporate the underlying skills that allow the model to per- form the task the trainee is expected to perform. Based on the model, the system can monitor actions of the trainee and infer the intentions of the trainee by mapping actions of the trainee to components of the model. In other words, a model of com- petence provides an explanation of actions as trainees interact with the system. Immediate feedback or real-time instructions can then be given to the trainee to facilitate learning.

John R. Anderson and colleagues on cognitive tutors using the computation model ACT-R

Anderson et al (1995) summarise their work on cognitive tutors teaching LISP programming , geometry and algebra. Their system incorporates eight instructional design principles a few of which pertain to task difficulty.

First, let's look at the design principles:

1. represent student competence as a production set
2. communicate the goal structure underlying the problem solving
3. provide instruction in the problem-solving context
4. Promote an abstract understanding of the problem-solving knowledge
5. Minimise the working memory load
6. Provide immediate feedback on errors
7. Adjust the grain size of instruction with learning
8. Facilitate successive approximations to the target skill

I think principles 5 and 8 directly relate to structuring task difficulty with practice, and probably others relate more indirectly. Minimising working memory (i.e., principle 5) requires providing instruction in manageable components. Facilitating successive approximations (i.e., principle 8) involves progressively providing less support to the learning and is an example of increasing task difficulty with practice.

References

• Anderson, J.R., Corbett, A.T., Koedinger, K.R., Pelletier, R., 1995. Cognitive tutors: lessons learned. The Journal of the Learning Sciences 4, 167–207. PDF
• Fu, W.-T., Bothell, D., Douglass, S., Haimson, C., Sohn, M.-H., & Anderson, J. A. (2006), Toward a Real-Time Model-Based Training System. Interacting with Computers, 18(6), 1216-1230. PDF
• Graesser, A.C., Lu, S., Jackson, G.T., Mitchell, H., Ventura, M., Olney, A., Louwerse, M.M., 2004. Auto tutor: a tutor with dialogue in natural language. Behavioral Research Methods, Instruments, and Computers 36, 180–193. PDF
• Hill Jr., R.W., Johnson, W.L., (1993). Designing an intelligent tutoring system based on a reactive model of skill acquisition. Proceedings of the International Conference on AI and Education, Edinburgh, 1993.
• Sleeman, D., Brown, J.S., 1982. Intelligent Tutoring Systems. Academic Press, New York.

Answer 4

The basic effect can be accounted for by connectionist models. See for example Suret & McLaren (2002). To quote the abstract:

This paper details an associative model that is applied to human learning on an artificial dimension. A variety of phenomena, including peak-shift, transfer along a continuum and summation / generalization are considered and simulation results are presented that give a close fit to empirical data.

http://faime.demon.co.uk/~streetm/wcci_2002/IJCNN02/PDFFiles/Papers/1163.pdf