  Contextual optimization, also known as predict-then-optimize or prescriptive
analytics, considers an optimization problem with the presence of covariates
(context or side information). The goal is to learn a prediction model (from
the training data) that predicts the objective function from the covariates,
and then in the test phase, solve the optimization problem with the covariates
but without the observation of the objective function. In this paper, we
consider a risk-sensitive version of the problem and propose a generic
algorithm design paradigm called predict-then-calibrate. The idea is to first
develop a prediction model without concern for the downstream risk profile or
robustness guarantee, and then utilize calibration (or recalibration) methods
to quantify the uncertainty of the prediction. While the existing methods
suffer from either a restricted choice of the prediction model or strong
assumptions on the underlying data, we show the disentangling of the prediction
model and the calibration/uncertainty quantification has several advantages.
First, it imposes no restriction on the prediction model and thus fully
unleashes the potential of off-the-shelf machine learning methods. Second, the
derivation of the risk and robustness guarantee can be made independent of the
choice of the prediction model through a data-splitting idea. Third, our
paradigm of predict-then-calibrate applies to both (risk-sensitive) robust and
(risk-neutral) distributionally robust optimization (DRO) formulations.
Theoretically, it gives new generalization bounds for the contextual LP problem
and sheds light on the existing results of DRO for contextual LP. Numerical
experiments further reinforce the advantage of the predict-then-calibrate
paradigm in that an improvement on either the prediction model or the
calibration model will lead to a better final performance.
