[Paper Review] Learning from Rational Behavior: Predicting Solutions to Unknown Linear Programs

Shahin Jabbari, Ryan Rogers+2 more|arXiv (Cornell University)|Jan 1, 2016

Advanced Algebra and Logic7 citations

TL;DR

This paper introduces a novel framework for predicting solutions to unknown linear programs (LPs) using partial information about objectives and constraints, generalizing revealed preferences to optimization. It proposes mistake-bound learning algorithms for two settings: (1) known objective with unknown constraints, and (2) unknown, changing objective with known constraints, achieving provable generalization with bounded errors.

ABSTRACT

We define and study the problem of predicting the solution to a linear program (LP) 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, that 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 LP is known to the learner but there is an arbitrary, fixed set of constraints which are unknown. Each example is defined by an additional known constraint and the goal of the learner is to predict the optimal solution of the LP given the union of the known and unknown constraints. This models 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 LP is unknown, and changing in a controlled way. The constraints of the LP 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 LP.

Motivation & Objective

• To model and predict the optimal solution of a linear program when only partial information about its objective function and constraints is available.
• To generalize the revealed preferences framework from consumer choice to optimization problems involving unknown constraints or objectives.
• To develop learning algorithms that make a bounded number of mistakes when predicting optimal solutions under partial information.
• To handle dynamic settings where the objective function changes daily but constraints remain known and fixed.
• To provide theoretical guarantees on prediction accuracy through mistake-bound learning in both known-objective and unknown-objective scenarios.

Proposed method

• In the first setting, the learner observes a sequence of known constraints and must predict the optimal solution of the LP formed by combining these with an unknown fixed set of constraints.
• The learner uses a mistake-bound algorithm that updates its hypothesis based on feedback from the true optimal solution after each constraint is revealed.
• In the second setting, the objective function is unknown but varies in a controlled manner; the learner observes partial objective information and known constraints per example.
• The algorithm maintains a hypothesis space of possible objective functions and prunes inconsistent candidates using duality and optimality conditions.
• It leverages duality theory and optimality conditions to validate or reject candidate solutions, ensuring convergence to a correct prediction within a bounded number of errors.
• The framework ensures generalization by maintaining a consistent set of feasible solutions and refining it iteratively using revealed optimal outcomes.

Experimental results

Research questions

• RQ1 Can we predict the optimal solution of a linear program when only partial information about its objective and constraints is available?
• RQ2 How can we model rational behavior in optimization settings where agents have unknown constraints or objectives?
• RQ3 What learning algorithms can guarantee a bounded number of mistakes when predicting solutions to partially specified LPs?
• RQ4 How does the structure of known constraints influence the learnability of unknown objectives in linear programming?
• RQ5 What theoretical guarantees can be provided for prediction accuracy in dynamic settings with changing objectives?

Key findings

• The paper establishes mistake-bound learning algorithms that predict the optimal solution of a linear program with a finite, bounded number of errors, even when constraints or objectives are partially unknown.
• In the setting with known objective and unknown constraints, the algorithm achieves a mistake bound that depends on the dimensionality and structure of the constraint space.
• For the setting with unknown, changing objectives and known constraints, the algorithm maintains a hypothesis space of possible objectives and refines it using optimality feedback.
• The framework generalizes revealed preferences to optimization problems, enabling prediction of rational behavior in resource-constrained decision-making.
• Theoretical analysis shows that the number of prediction mistakes is polynomial in the input size, providing strong generalization guarantees.
• The approach is robust to changes in the objective function and maintains correctness under partial information, as long as the optimal solution is revealed after each example.