Data-driven inverse optimization with imperfect information

Abstract

In data-driven inverse optimization an observer aims to learn the preferences of an agent who solves a parametric optimization problem depending on an exogenous signal. Thus, the observer seeks the agent's objective function that best explains a historical sequence of signals and corresponding optimal actions. We focus here on situations where the observer has imperfect information, that is, where the agent's true objective function is not contained in the search space of candidate objectives, where the agent suffers from bounded rationality or implementation errors, or where the observed signal-response pairs are corrupted by measurement noise. We formalize this inverse optimization problem as a distributionally robust program minimizing the worst-case risk that the predicted decision (i.e., the decision implied by a particular candidate objective) differs from the agent's actual response to a random signal. We show that our framework offers rigorous out-of-sample guarantees for different loss functions used to measure prediction errors and that the emerging inverse optimization problems can be exactly reformulated as (or safely approximated by) tractable B Daniel Kuhn

Inverse Optimization: Theory and Applications

“…We present the main theoretical result for convex forward models FOP-CVX(θ) below. Proposition 4.3 (Esfahani et al [68]). Consider FOP-CVX i (θ) instances that are defined by known input parameters ûi .…”

“…Another intuitive and popular loss function to minimize when estimating θ is the degree of suboptimality of the observed decisions xi under the estimated models FOP i (θ) [23,46,47,68,[143][144][145]. We can evaluate sub-optimality with two potential loss functions.…”

“…Generalizing IOP-DD-R(ℓ, P, ρ), Esfahani et al [68] propose a distributionally robust inverse optimization problem where the uncertainty set of the robust problem controls a worst-case data distribution from the empirical P N .…”

“…We refer the reader to Esfahani et al [68] for details on the computation of β. Note that when ρ P (•) = E P [•], the out-of-sample guarantee is similar to a generalization bound (see Theorem 4.4) in that both use the empirical risk to upper bound the true risk.…”

“…Esfahani et al [68] explore the Conditional-Value-at-Risk (CVaR) at level α > 0 as their risk measure for distributionally robust inverse optimization CVaR(ℓ, α, P)…”