Assessing the Quality of a Set of Basis Functions for Inverse Optimal Control via Projection onto Global Minimizers

TL;DR

This paper introduces a method to evaluate the quality of basis functions in inverse optimal control by measuring the distance to global optima, addressing the limitations of previous convexity assumptions.

Contribution

It proposes a novel distance-based metric for basis function quality and explores properties of global optima in various settings, including extensions to nonconvex functions.

Findings

Bounds for minimum distance in quadratic settings are derived.

The approach can identify inadequacies in basis function sets.

Extensions include handling nonconvex and polynomial basis functions.

Abstract

Inverse optimization (Inverse optimal control) is the task of imputing a cost function such that given test points (trajectories) are (nearly) optimal with respect to the discovered cost. Prior methods in inverse optimization assume that the true cost is a convex combination of a set of convex basis functions and that this basis is consistent with the test points. However, the consistency assumption is not always justified, as in many applications the principles by which the data is generated are not well understood. This work proposes using the distance between a test point and the set of global optima generated by the convex combinations of the convex basis functions as a measurement for the expressive quality of the basis with respect to the test point. A large minimal distance invalidates the set of basis functions. The concept of a set of global optima is introduced and its properties are explored in unconstrained and constrained settings. Upper and lower bounds for the minimum distance in the convex quadratic setting are implemented by bi-level gradient descent and an enriched linear matrix inequality respectively. Extensions to this framework include max-representable basis functions, nonconvex basis functions (local minima), and applying polynomial optimization techniques.