Geometric Identification in Predict-Then-Optimize Learning

We propose a scalable method for training prediction (machine learning) models in the predict-then-optimize paradigm, where model outputs serve as coefficients for a subsequent linear optimization task. Directly minimizing the empirical decision regret is intractable for linear programming and combinatorial optimization since the decision mapping is piecewise constant, and the gradients are zero almost everywhere. While existing methods address this by smoothing the differentiation process, they suffer from scalability issues, since a computationally expensive solver call is required for every gradient evaluation. To address this, we propose a decision-focused learning pipeline based on a measure transformation principle, which yields a new surrogate loss that is completely optimization-solver-free during training. We establish theoretical guarantees, including Fisher consistency and excess risk bounds. Empirically, our method achieves decision quality competitive with state-of-the-art methods while reducing training time by orders of magnitude.

In this paper, we propose and study a robust variant of the smart predict-then-optimize approach that accounts for prediction shifts due to disturbance in the covariate feature space. While traditional integrated-learning-and-optimization models assume that side information is perfectly revealed, empirical data-driven features are frequently corrupted or noisy at the time of decision-making, leading to fragile operational policies. To bridge this gap, we integrate principles of robust optimization directly into the predictive-prescriptive pipeline via a smart predict-then-robustly optimize loss and establish a computationally tractable convex surrogate, designed to hedge against worst-case feature perturbations. On the theoretical front, we formalize the structural validity of this surrogate by proving its approximation error probability decays exponentially according to a sub-Gaussian concentration profile. Furthermore, we establish that under mild assumptions, the surrogate is Fisher consistent with high probability. We also prove necessary conditions under which our framework outperforms standard smart predict-then-optimize and maintain its superiority even when the standard method is equipped with regularized upstream predictions. Numerical experiments validate that our robust framework consistently yields significant performance improvements over standard methods, both in out-of-sample terms and in training stability.

Decision-focused learning for linear optimization is complicated by the discontinuity of the optimizer, where small cost errors may leave the decision unchanged or move it to a different vertex. We show that this non-smooth pointwise behavior becomes locally quadratic after averaging over the data distribution, and we derive the curvature in closed form, specifically, a matrix-valued measure supported on the walls of the normal fan. This measure depends only on the feasible set, with the data distribution entering only as a weight. We then offer a tractable approximation for this curvature, computable with just one projection to the feasible set. We prove that the approximation weakly converges to the true population curvature. We offer one application of our findings, a decision-aware scenario generation method for expected-cost linear optimization. Our experiments test the quadratic and weak convergence laws and show a 30.8% regret improvement over uniform allocation on battery arbitrage.