Learning, Teaching, Optimization and Approximation

A model of problem solving is given in terms of problem information, problem features, solution sets, solution selection forms, performance measures and performance requirements. Learning is then defined in terms of identifying problem features and selection forms. The selection forms include, for example, mathematical formulas, sets of rules, decision trees, exemplars, and neural networks. These forms have parameters that can be optimized and an additional aspect of learning is this optimization. Several examples are given and teaching is defined in this context. Lessons learned from the theories of optimization and approximation about learning to solve problems are summarized. The potential that supercomputers plus superoptimizers might create magic in learning how to solve problems is analyzed and the difficulties faced are discussed. There are closing comments on automatically computing geometrical features and rule sets.