Suppose we had an infinitely powerful mathematician sitting in a room and proving theorems about learning. Could he solve machine learning?
The answer is “no”. This answer is both obvious and sometimes underappreciated.
There are several ways to conclude that some bias is necessary in order to succesfully learn. For example, suppose we are trying to solve classification. At prediction time, we observe some features X and want to make a prediction of either 0 or 1. Bias is what makes us prefer one answer over the other based on past experience. In order to learn we must:
- Have a bias. Always predicting 0 is as likely as 1 is useless.
- Have the “right” bias. Predicting 1 when the answer is 0 is also not helpful.
The implication of “have a bias” is that we can not design effective learning algorithms with “a uniform prior over all possibilities”. The implication of “have the ‘right’ bias” is that our mathematician fails since “right” is defined with respect to the solutions to problems encountered in the real world. The same effect occurs in various sciences such as physics—a mathematician can not solve physics because the “right” answer is defined by the world.
A similar question is “Can an entirely empirical approach solve machine learning?”. The answer to this is “yes”, as long as we accept the evolution of humans and that a “solution” to machine learning is human-level learning ability.
A related question is then “Is mathematics useful in solving machine learning?” I believe the answer is “yes”. Although mathematics can not tell us what the “right” bias is, it can:
- Give us computational shortcuts relevant to machine learning.
- Abstract empirical observations of what an empirically good bias is allowing transference to new domains.
There is a reasonable hope that solving mathematics related to learning implies we can reach a good machine learning system in time shorter than the evolution of a human.
All of these observations imply that the process of solving machine learning must be partially empirical. (What works on real problems?) Anyone hoping to do so must either engage in real-world experiments or listen carefully to people who engage in real-world experiments. A reasonable model here is physics which has benefited from a combined mathematical and empirical study.