Thought as Optimization

Mental process involves a large variety of computational problems. It is not entirely
implausible that the mind deals with each of them in a unique, context-specific way. But, unlike
Minsky and many cognitive scientists, I do not believe this to be the case. Certainly, the mind
contains a huge number of special-purpose procedures. But nearly all the computational
problems associated with mental process can be formulated as optimization problems. And I
propose that, by and large, there is one general methodology according to which these
optimization problems are solved.

Optimization is simply the process of finding that entity which a certain criterion judges to be
"best". Mathematically, a "criterion" is simply a function which maps a set of entities into a set of "values" which has the property that it is possible to say when one value is greater than
another. So the word "optimization" encompasses a very wide range of intellectual and practical
problems.
For instance, virtually all the laws of physics have been expressed as optimization problems,
often with dramatic consequences. Economics, politics, and law all revolve around finding the
"best" solution to various problems. Cognitive science and many forms of therapeutic
psychology depend on finding the model of a person’s internal state which best explains their
behavior. Everyday social activity is based on maximizing the happiness and productivity of
oneself and others. Hearing, seeing, walking, and virtually all other aspects of sensation and
motor control may be viewed as optimization problems. The Traveling Salesman problem is an
optimization problem — it involves finding the shortest path through n cities. And, finally, the methodological principle known as Occam’s razor suggests that the best explanation of a
phenomenon is the simplest one that fits all the facts. In this sense, all inquiry may be an
optimization problem, the criterion being simplicity.

Some of these optimization problems have been formulated mathematically –e.g. in physics
and economics. For others, such as those of politics and psychology, no useful formalization has
yet been found. Nonmathematical optimization problems are usually solved by intuition, or by
the application of extremely simple, rough traditional methods. And, despite a tremendous body
of sophisticated theory, mathematical optimization problems are often solved in a similar
manner.
Although there are dozens and dozens of mathematical optimization techniques, virtually none
of these are applicable beyond a very narrow range of problems. Most of them — steepest
descent, conjugate gradient, dynamic programming, linear programming, etc. etc. (Dixon and
Szego, 1978; Torn et al, 1990) — rely on special properties of particular types of problems. It
seems that most optimization problems are, like the Traveling Salesman problem, very hard to
solve exactly. The best one can hope for is a PAC solutions. And, in the "classical" literature on mathematical optimization, there are essentially only two reasonably general approaches to
finding PAC solutions: the Monte Carlo method, and the Multistart method.

After discussing these methods, and their shortcomings, I will introduce the multilevel
philosophy of optimization, which incorporates both the Monte Carlo and the Multistart methods
in a rigid yet generally applicable framework which applies to virtually any optimization
problem. I will propose that this philosophy of optimization is essential to mentality, not least because of its essential role in the perceptual and motor hierarchies, to be discussed below.

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