Learning in Inverse Optimization: Incenter Cost, Augmented Suboptimality Loss, and Algorithms
This work addresses the challenge of robust cost function learning in IO for applications like decision-making systems, though it appears incremental by building on recent circumcenter methods.
The paper tackles the problem of learning an expert's cost function in Inverse Optimization (IO) by introducing the incenter concept and Augmented Suboptimality Loss (ASL), resulting in tractable convex reformulations and a provably efficient algorithm for high-cardinality discrete sets.
In Inverse Optimization (IO), an expert agent solves an optimization problem parametric in an exogenous signal. From a learning perspective, the goal is to learn the expert's cost function given a dataset of signals and corresponding optimal actions. Motivated by the geometry of the IO set of consistent cost vectors, we introduce the "incenter" concept, a new notion akin to circumcenter recently proposed by Besbes et al. (2023). Discussing the geometric and robustness interpretation of the incenter cost vector, we develop corresponding tractable convex reformulations, which are in contrast with the circumcenter, which we show is equivalent to an intractable optimization program. We further propose a novel loss function called Augmented Suboptimality Loss (ASL), a relaxation of the incenter concept for problems with inconsistent data. Exploiting the structure of the ASL, we propose a novel first-order algorithm, which we name Stochastic Approximate Mirror Descent. This algorithm combines stochastic and approximate subgradient evaluations, together with mirror descent update steps, which is provably efficient for the IO problems with discrete feasible sets with high cardinality. We implement the IO approaches developed in this paper as a Python package called InvOpt. Our numerical experiments are reproducible, and the underlying source code is available as examples in the InvOpt package.