OPTEX is short for optimal experiment and is so named because it is derived from a crude approximation to the optimal value of an experiment (determining the true optimal value is terribly intractible). The OPTEX valuation consists of two different types of cost functions. One is called the "online" cost which is the cost of each experiment. The cost of a set of experiments, then, is the sum of the online costs for each one. The second cost is called the "final" cost. In optimization problems there is criterion being optimized and the value of a set of experiments is just the value of the best result found in the batch. Often, it may be that an optimization problem includeds both online and final costs. The OPTEX experiment valuation consists of three terms which will be discussed separately.
The first and second terms operate on the online cost. The first is E(C(X,f(X))). This is the expected cost of experiment X. Note the difference between this and PMAX, which is the cost of the experiment when the result is the expected result. In order to compute this cost we must evaluate the following integral over the set of all possible Y's:
/
v1 = | C(X,Y) * p(Y|X)
/
where p(Y|X) is the probability of result Y from experiment X. The
implementation of this actually performs only a coarse numerical evaluation
of this integral within a confidence interval.
We define Cstar to be the best of the estimated costs at each of the points we have seen so far. A true optimal experiment valuation would give this experiment credit for all future improvements in cost derived from the result of the experiment. Since computing that is intractible, we make a gross simplification. We assume that after this experiment, our policy will be to choose the best experiment we've seen from then on (we will not do that, of course). Then the future improvement in cost from this experiment is the improvement in its result over the best seen so far times the number of experiments we will perform in the future (the integral is again over all possible Y's:
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v2 = n * | max(0, (C(X,Y) - Cstar)) * p(Y|X)
/
n is the number of experiments remaining to be done. Again, if the object
is minimization, the max operator becomes a min operator. The effect of
multiplying by the number of steps to go is to make the experiments more
aggressive when their is a lot of time left, and more conservative when the
trials are almost done.
The third term is on the final cost function. It looks like the second term without the number of steps factored in. Its purpose is simply to credit the experiment with any improvement it makes upon the best result obtained so far with respect to the final cost function. Assuming an additional subscript to label this as pertaining to final cost rather than online the term is:
/
v3 = | max(0, (C(X,Y) - Cstar) * p(Y|X)
/
Again, the integral is a coarse numerical approximation and the max is a min
in the case of minimization.
The entire experiment value is: V(X) = v1 + v2 + v3 and OPTEX suggests the experiment with the maximum (or minimum) value.