Reasoning under Varying and Uncertain Resource Constraints

We describe the use of decision-theory to optimize the value of computation under uncertain and varying resource limitations. The research is motivated by the pursuit of formal models of rational decision making for computational agents, centering on the explicit consideration of preferences and resource availability. We focus here on the importance of identifying the multiattribute structure of partial results generated by approximation methods for making control decisions. Work on simple algorithms and on the control of decision-theoretic inference itself is described. 1 Computation Under Uncertainty We are investigating the decision-theoretic control of problem solving under varying constraints in resources required for reasoning, such as time and memory. This work is motivated by the pursuit of formal models of rational decision making under resource constraints and our goal of extending foundational work on normative rationality to computational agents. We describe here a portion of this research that centers on reformulating traditional computational problems into strategies for generating and reasoning about a spectrum of partial results characterized by multiple dimensions of value. After describing work on the solution of classical problems under uncertain and varying resource constraints, we shall return brie y to the larger, motivating problem of computational rationality, focusing on the pursuit of optimal strategies for computing beliefs and actions under resource constraints. A rational agent applies an inference strategy with the intention of performing an analysis that will be of some net bene t. There is usually uncertainty about the best way to solve a problem because of incompleteness in knowledge about (1) the value of alternative computed results This work was supported by a NASA Fellowship under Grant NCC-220-51, by the National Science Foundation under Grant IRI-8703710, and by the National Library of Medicine under Grant RO1LM0429. Computing facilities were provided by the SUMEX-AIM Resource under NIH Grant RR-00785. Published in Proceedings of the Seventh National Conference on Arti cial Intelligence, Minneapolis, MN. August 1988. Morgan Kaufmann, San Mateo, CA. pp. 111-116. in a particular situation, (2) the di culty of generating results from a problem instance, and (3) the costs and availability of resources (such as time) required for reasoning. We have been investigating the use of decision theory for valuating alternative problem-solving strategies under uncertainty. Thus, we de ne components of computational value in terms of expected utility [7]. The use of decision theory to guide the allocation of computational e ort was proposed by Good several decades ago [3]. 1.1 Computational Utility We use the term computational utility, uc, to refer to the net value associated with the commitment to a computational strategy. We decompose uc into two components: the object-level utility, uo, and the inference-related utility, ui. The object-level utility of a strategy is the bene t attributed to acquiring a result, without regard to the costs associated with its computation. For example, the objectlevel utility of a medical expert-system inference strategy is the value associated with the information it generates about the entities in a medical problem, such as alternative treatments and likelihoods of possible outcomes. The inference-related component, ui, is the cost of the reasoning. This includes the disutility of delaying an action while waiting for a reasoner to infer a recommendation. The decomposition of computational utility focuses attention explicitly on the costs and bene ts associated with problem-solving activity. In the general case, we must consider the dependencies between the objectand inferencerelated value. We assume the existence of a function f that relates uc to uo, ui, and additional information about the problem-speci c dependencies that may exist between the two components of value|that is, uc( ; ; ) = f [uo( ; ); ui( ; )] where and represent parameters that in uence respectively the objectand inference-related utilities and represents the parameters that in uence both the objectand the inference-related utilities. 1.2 Multiple Attributes of Utility In real-world applications, the object-level and inferencerelated utilities frequently are functions of multiple attributes. Dimensions of value can be acquired through consultation with computer users. Computational utility may be assessed as numerical quantities for particular outcomes, or may be described by a function that represents the relationships among costs and bene ts associated with alternative outcomes. Such functions assign a single utility measure to computation based on the status of an n-tuple of attributes. Let us assume that we can decompose uc into uo and ui. A set of object-level attributes, vo1 ; : : : ; von , captures dimensions of value in a result, such as accuracy and precision, and de nes an object-level attribute space, Ao: A sequence of computational actions, c, applied to an initial problem instance, I, yields a result, (I), that may be described as a vector ~vo in Ao. Components of the inference-related cost|such as the computation time, memory, and, in some applications, the time required to explain machine reasoning to a human|de ne a resource attribute space, Ar. In this paper, we simplify Ar to r, the scalar quantity of time. If we assume that uo and ui are combined with addition and ui(r) is the cost of delay, we can say that uc(~vo; r) = uo(~vo) ui(r) 2 Toward a Continuum of Value Much of work on the analysis of algorithms has been directed at proving results about the time required for computing a solution de ned by simple goals and termination conditions [1]. Although this perspective imposes useful simpli cation, it has biased synthesis and analysis toward solution policies that are indi erent to variation in the utility of a result or to the costs and availability of resources. We wish to increase the value of computation under limited and varying resources by identifying and characterizing classes of approximate or partial results that can be produced for a fraction of the resources required by the best available methods for generating nal results. Let c refer to a sequence of primitive computational actions. We de ne a subclass of sequences of computational actions, c , that transform a speci c problem instance I (e.g., a randomly mixed le of records) into a nal result, (I) (e.g. a total ordering over records), without assistance from an omniscient oracle, c [I]! (I). We de ne r(c ; I) as the resource required by c to generate (I) from I. A majority of traditional algorithms generate speci c c given I, halting upon reaching a queried (I).