1.5.8 Computational Limits

Sometimes an agent can decide on its best action quickly enough for it to act. Often there are computational resource limits that prevent an agent from carrying out the best action. That is, the agent may not be able to find the best action quickly enough within its memory limitations to act while that action is still the best thing to do. For example, it may not be much use to take 10 minutes to derive what was the best thing to do 10 minutes ago, when the agent has to act now. Often, instead, an agent must trade off how long it takes to get a solution with how good the solution is; it may be better to find a reasonable solution quickly than to find a better solution later because the world will have changed during the computation.

The computational limits dimension determines whether an agent has

  • perfect rationality, where an agent reasons about the best action without taking into account its limited computational resources; or
  • bounded rationality, where an agent decides on the best action that it can find given its computational limitations.

Computational resource limits include computation time, memory, and numerical accuracy caused by computers not representing real numbers exactly. An anytime algorithm is an algorithm whose solution quality improves with time. In particular, it is one that can produce its current best solution at any time, but given more time it could produce even better solutions. We can ensure that the quality doesn't decrease by allowing the agent to store the best solution found so far and return that when asked for a solution. However, waiting to act has a cost; it may be better for an agent to act before it has found what would have been the best solution.

Figure 1.5: Solution quality as a function of time for an anytime algorithm. The agent has to choose an action. As time progresses, the agent can determine better actions. The value to the agent of the best action found so far, if it had been carried out initially, is given by the dashed line. The reduction in value to the agent by waiting to act is given by the dotted line. The net value to the agent, as a function of the time it acts, is given by the solid line.

Example 1.9: Figure 1.5 shows how the computation time of an anytime algorithm can affect the solution quality. The agent has to carry out an action but can do some computation to decide what to do. The absolute solution quality, had the action been carried out at time zero, shown as the dashed line at the top, is improving as the agent takes time to reason. However, there is a penalty associated with taking time to act. In this figure, the penalty, shown as the dotted line at the bottom, is proportional to the time taken before the agent acts. These two values can be added to get the discounted quality, the time-dependent value of computation; this is the solid line in the middle of the graph. For the example of Figure 1.5, an agent should compute for about 2.5 time units, and then act, at which point the discounted quality achieves its maximum value. If the computation lasts for longer than 4.3 time units, the resulting discounted solution quality is worse than if the algorithm just outputs the initial guess it can produce with virtually no computation. It is typical that the solution quality improves in jumps; when the current best solution changes, there is a jump in the quality. However, the penalty associated with waiting is often not as simple as a straight line.

To take into account bounded rationality, an agent must decide whether it should act or think more. This is challenging because an agent typically does not know how much better off it would be if it only spent a little bit more time reasoning. Moreover, the time spent thinking about whether it should reason may detract from actually reasoning about the domain. However, bounded rationality can be the basis for approximate reasoning.