Efficiently Learning from Revealed Preference

In this paper, we consider the revealed preferences problem from a learning perspective. Every day, a price vector and a budget is drawn from an unknown distribution, and a rational agent buys his most preferred bundle according to some unknown utility function, subject to the given prices and budget constraint. We wish not only to find a utility function which rationalizes a finite set of observations, but to produce a hypothesis valuation function which accurately predicts the behavior of the agent in the future. We give efficient algorithms with polynomial sample-complexity for agents with linear valuation functions, as well as for agents with linearly separable, concave valuation functions with bounded second derivative.

💡 Research Summary

The paper reframes the classic revealed‑preference problem as a learning task. Instead of merely asking whether a finite set of observed consumer choices can be rationalized by some utility function, the authors ask whether one can predict future choices under new price‑budget scenarios. The setting is as follows: each day a price vector p∈ℝⁿ₊ and a budget b∈ℝ₊ are drawn independently from an unknown distribution D. A rational agent then selects a bundle x* that maximizes his (unknown) utility subject to the budget constraint p·x ≤ b. The learner observes a collection of triples (p,b,x*) and must output a hypothesis utility function ŭ that, with high probability, predicts the agent’s optimal bundle for fresh draws from D.

Two utility classes are considered. First, linear utilities u(x)=w·x with a non‑negative weight vector w. Second, linearly separable concave utilities u(x)=∑_{i=1}^n u_i(x_i) where each univariate component u_i is twice differentiable, concave, and has bounded second derivative |u_i’’(·)| ≤ L. The bounded curvature assumption guarantees smoothness and limits how quickly marginal utilities can change.

The core technical tool is the Karush‑Kuhn‑Tucker (KKT) optimality condition. For a linear utility, KKT reduces to a set of linear inequalities: for every observed (p,b,x*), the true weight vector w must satisfy w·x* ≥ w·x for all feasible x. By rearranging, each observation yields a linear constraint on w. Collecting all constraints yields a feasible region that can be found by solving a linear program (LP). The authors prove that, using standard VC‑dimension arguments, Õ(n/ε²·log(1/δ)) i.i.d. samples suffice to guarantee that any w satisfying the LP will predict the agent’s choice with error at most ε with probability 1−δ. The computational cost is polynomial because LPs are solvable in poly‑time.

For the separable concave case, the authors approximate each component u_i by a degree‑k polynomial p_i(x)=∑{j=0}^k a{ij} x^j. The bounded second derivative translates into linear constraints on the polynomial coefficients, ensuring the approximation respects the curvature limit. Applying KKT to each observation yields a system of linear (or quadratic) inequalities involving the coefficients a_{ij} and the Lagrange multiplier λ. To handle the fact that the true optimal bundle may not be exactly represented by the polynomial, the algorithm introduces slack variables and samples gradient information to bound the deviation between the true marginal utilities and the polynomial derivatives. This yields a convex feasibility problem that can be solved with standard interior‑point methods.

The sample‑complexity analysis shows that Õ(n·k/ε³·log(1/δ)) observations are enough to learn a hypothesis that predicts future choices within ε error with confidence 1−δ. The extra factor of k reflects the richer hypothesis class (higher‑degree polynomials) and the ε³ term arises from the need to control approximation error as well as statistical error.

Key contributions of the work are:

1. A PAC‑style learning framework for revealed preference, providing explicit bounds on the number of observations required for accurate prediction.
2. Efficient algorithms for two important utility families—linear and smooth separable concave—both solvable in polynomial time.
3. A systematic reduction from observed optimality conditions to linear/convex constraints on utility parameters, leveraging the KKT conditions to bridge economics and optimization.
4. Theoretical guarantees that are independent of the underlying price‑budget distribution, relying only on i.i.d. sampling.

The paper does not include empirical experiments; the results are purely theoretical. Nonetheless, the algorithms are directly applicable to domains such as e‑commerce pricing, online ad auctions, and personalized recommendation systems, where a platform repeatedly observes consumer purchases under varying price structures and wishes to anticipate future demand.

Limitations and future directions identified by the authors include extending the approach to non‑separable or highly non‑linear utilities, handling dynamic environments where preferences evolve over time, and validating the theory on real‑world datasets. Overall, the work establishes a solid foundation for integrating revealed‑preference economics with modern learning theory, opening a pathway for both deeper theoretical investigations and practical predictive tools.