Optimal Allocation of Very Limited Search Resources

This paper presents a probabilistic model for studying the question: given n search resources, where in the search tree should they be ezpended? Specifically, a least-cost root-to-leaf path is sought in a random tree. The tree is known to be binary and complete to depth N. Arc costs are independently set either to 1 (with probability p ) or to 0 (with probability l-p ). The cost of a leaf is the sum of the arc costs on the path from the root to that leaf. The searcher (scout) can learn n arc values. How should these scarce resources be dynamically allocated to minimize the average cost of the leaf selected? A natural decision rule for the scout is to allocate resources to arcs that lie above leaves whose current expected cost is minimal. The bad-news theorem says that situations exist for which this rule is nonoptimal, no matter what the value of n . The good-news theorem counters this: for a large class of situations, the aforementioned rule is an optimal decision rule if p 5 0.5 and within a constant of optimal if p > 0.5. This report discusses the lessons provided by these two theorems and presents the proof of the bad-news theorem. I Informal description of the problem Searching the state-space for an acceptable solution is a fundamental activity for many AI programs. Complete search of the state-space is typically infeasible. Instead, one relies on whatever heuristic information is available. Interesting questions then arise as to how much speed-up is obtained and at what price. Many authors have evaluated the complexity of algorithms that invoke heuristic search [I, 3, 6, 7, 9, 10, 111. A typical question asked is: How fast can the algorithm find an optimal (nearly-optimal) (probably nearly-optimal) solution? This paper focuses upon the inverse question: Given n search resources, how good a solution can one obtain? This inverse question is appropriate for real-time processes characterized by an insistence upon an answer (decision) after X seconds have passed. For example, a chess-playing program is limited by the external chess clock. A speech recognizer should maintain pace with the speaker. In these and other processes, search resources are very limited; even linear time may not be fast enough. Heuristics are often said to offer “solutions which are good enough most of the time” [4, page 61. The converse of this phrase implies that heuristics will, by definition, fail some of the time. Worst-case analysis is unilluminating-any algorithm using the heuristic information will, on occasion, perform poorly. One is forced, reluctantly perhaps, to turn to probabilistic, average-case t This report describes work done in the Department of Computer Science at Duke Umverslty It was supported m part by the Au Force Office of Scientific Research, Au Force Systems Command under Grant AF’OSR LB-0205 analysis. Karp and Pearl said it well [lo]: Since the ultimate test for the success of heuristic methods is that they work well “most of the time”, and since probability theory is our principal formalism for quantifying concepts such as “most of the time “, it is only natural that probabilistic models should provide a formal ground for evaluating the performance of heuristic methods quantitatively. In agreement with this philosophy, this paper seeks the algorithm whose average result is best. It must be emphasized from the outset that any conclusions drawn from average-case analysis depend fundamentally on the underlying probability distribution assumed. The concluding section of this paper discusses whether the results of this paper do in fact apply to real-world algorithms.