The purpose of a learning machine : first discover patterns in the world, and second, exploit those patterns to intervene more effectively in the world. The flip side of optimal learning is optimal control.

Suppose you are walking across a stony plane, perhaps an ancient riverbed. Most of the rocks you encounter are smooth and rounded, of various sizes, a dull collection produced by the power of running water. Then you encounter something surprising: a rock with sharp edges, concave ablations, and an easy fit to your hand. Your surprise reflects the fact that you need to update your expectations about the stony plane. Intuitively, you update your statistical expectations. The distribution of stones has changed. The randomness of stones has been lowered by the unexpected find.

Entropy has two meanings in science. It has a purely mathematical meaning related to statistical uncertainty as the previous paragraph suggests, but that is not how it first entered science. In the early days of the Industrial Revolution, entropy had a very different meaning. It was used as an index to order the kinds of physical changes, natural or engineered, that can occur; heat flows from hot to cold, a drop of ink diffuses uniformly in a glass of water. In both cases the final state has higher entropy than the initial. These processes are spontaneous. The real question is how did the initial state of lower entropy come about in the first place? Why is the distribution of stones not as random as possible.

We can transform a localised system from a high entropy to low entropy by physically intervening on that system. This costs energy and it must be supplied by a controller, for example, a battery or the sun. The energy can be in the form of heat or due to applied forces, for example an electric field in a transistor. However it is done, the price to pay for lowering the entropy of a system is to increase the entropy in the environment by at least as much as the entropy is reduced in the controlled system. Entropy, in this physical sense, imposes constraints on the kinds of interventions that are possible.

The connection between the statistical/information sense of entropy and the physical (thermodynamic) sense can be made in many ways. The most important of these connects physics and learning, and returns us to the stony plane. The sharp-edged stone axe heads, scattered among the water tumbled smooth stones, is evidence of a special kind of controller. Considerable energy was expended in their making for sure, but also a very special kind of controller, a learning machine.

A machine could make stone tools by randomly chipping away at smooth stones, but almost every stone would be useless as a tool and a great deal of energy would be wasted. Almost every attempt would be an ‘error’ , a non-tool. A machine that monitors its own performance and changes its internals settings to reduce the error probability in each trial will learn to make good stone tools efficiently.

This does not come for free as the learning process itself generates waste heat, dissipated energy. Learning itself reduces the thermodynamic entropy of the learning machine and this enables it to more efficiently create good tools with little error. Learning enables a controller that is a pool of low entropy that can be used to efficiently lower the entropy of its surroundings.

Here then is the link between informational entropy and thermodynamic entropy. Learning necessarily lowers the error of control and in so doing enables a powerful thermodynamic resource for reconfiguring the world. Biology discovered this trick a long time ago. We are now learning how to emulate it, thereby creating new paths to wealth and security. After all, a hominid with a hand tool is wealthy relative to hominids without one, and a threat.