Learning from Rational Behavior: Predicting Solutions to Unknown Linear Programs
Shahin Jabbari, Ryan Rogers, Aaron Roth, Zhiwei Steven Wu
We define and study the problem of predicting the solution to a linear program, given only partial information about its objective and constraints. This generalizes the problem of learning to predict the purchasing behavior of a rational agent who has an unknown objective function, which has been studied under the name "Learning from Revealed Preferences". We give mistake bound learning algorithms in two settings: in the first, the objective of the linear program is known to the learner, but there is an arbitrary, fixed set of constraints which are unknown. Each example given to the learner is defined by an additional, known constraint, and the goal of the learner is to predict the optimal solution of the linear program given the union of the known and unknown constraints. This models, among other things, the problem of predicting the behavior of a rational agent whose goals are known, but whose resources are unknown. In the second setting, the objective of the linear program is unknown, and changing in a controlled way. The constraints of the linear program may also change every day, but are known. An example is given by a set of constraints and partial information about the objective, and the task of the learner is again to predict the optimal solution of the partially known linear program.