Mo Liu

Beichen Wan, Mo Liu, Paul Grigas et al.

We consider the sequential experimental design problem in the predict-then-optimize paradigm. In this paradigm, the outputs of the prediction model are used as coefficient vectors in a downstream linear optimization problem. Traditional sequential experimental design aims to control the input variables (features) so that the improvement in prediction accuracy from each experimental outcome (label) is maximized. However, in the predict-then-optimize setting, performance is ultimately evaluated based on the decision loss induced by the downstream optimization, rather than by prediction error. This mismatch between prediction accuracy and decision loss renders traditional decision-blind designs inefficient. To address this issue, we propose a directional-based metric to quantify predictive uncertainty. This metric does not require solving an optimization oracle and is therefore computationally tractable. We show that the resulting sequential design criterion enjoys strong consistency and convergence guarantees. Under a broad class of distributions, we demonstrate that our directional uncertainty-based design attains an earlier stopping time than decision-blind designs. This advantage is further supported by real-world experiments on an LLM job allocation problem.

Mo Liu, Paul Grigas, Heyuan Liu et al.

We develop the first active learning method for contextual linear optimization. Specifically, we introduce a label acquisition algorithm that sequentially decides whether to request the ``labels'' of feature samples from an unlabeled data stream, where the labels correspond to the coefficients of the objective in the linear optimization. Our method is the first to be directly informed by the decision loss induced by the predicted coefficients, referred to as the Smart Predict-then-Optimize (SPO) loss. Motivated by the structure of the SPO loss, our algorithm adopts a margin-based criterion utilizing the concept of distance to degeneracy. In particular, we design an efficient active learning algorithm with theoretical excess risk (i.e., generalization) guarantees. We derive upper bounds on the label complexity, defined as the number of samples whose labels are acquired to achieve a desired small level of SPO risk. These bounds show that our algorithm has a much smaller label complexity than the naive supervised learning approach that labels all samples, particularly when the SPO loss is minimized directly on the collected data. To address the discontinuity and nonconvexity of the SPO loss, we derive label complexity bounds under tractable surrogate loss functions. Under natural margin conditions, these bounds also outperform naive supervised learning. Using the SPO+ loss, a specialized surrogate of the SPO loss, we establish even tighter bounds under separability conditions. Finally, we present numerical evidence showing the practical value of our algorithms in settings such as personalized pricing and the shortest path problem.