Many machine learning systems make constrained decisions by optimizing factorized objectives, but the context-specific objective is often treated as fixed. We study contextual decision-weight learning: from logged decisions and proxy outputs, learn an optimizer-facing weight vector w(x) over interpretable decision factors z(x,d), rather than a direct policy or generic predictive score. We propose OTSS, an output-targeted soft-segmentation model that deploys the personalized decision-ready weight vector. At the function-class level, the theory highlights a hard-versus-soft distinction. Hard partitions incur an approximation-estimation tradeoff under overlap, while a realizable fixed-K soft class removes the hard-partition approximation floor and attains a parametric rate. We evaluate OTSS in controlled benchmarks with finite evaluation libraries, where the true weight vector and downstream regret can be computed exactly. In the representative overlap setting, OTSS attains the lowest mean regret among the comparators, including EM mixture regression, the strongest soft-mixture baseline in our comparison; it matches EM on coefficient recovery while running about two orders of magnitude faster. In a matched K=5 benchmark, OTSS remains competitive under hard-routed truth and improves as heterogeneity becomes softer and sample size grows. On a fixed Complete Journey retail anchor with real household covariates and action geometry, OTSS again achieves the lowest mean-regret point estimate.
Many real-world combinatorial problems involve uncertain parameters, which can be predicted given contextual features and historical data. These predict-then-optimize' or contextual optimization' problems have gained significant attention: end-to-end training methods can now minimize the downstream task cost rather than the predictive error. However, despite their effectiveness, these decision-focused learning (DFL) approaches often rely on repeated solving of the underlying combinatorial optimization problem during training, making them computationally expensive and difficult to scale. We reframe the learning problem as a cost-sensitive multi-output regression problem: multi-output due to the combinatorial problem having multiple uncertain parameters, and cost-sensitive due to the downstream task cost being the real target. Our technical contribution is the formalization of multiple loss function components that follow from this reframing: cost-insensitive normalization, decision-aware asymmetric penalization of over- and underpredictions, and instance-based costs that mimic the true downstream task-based loss locally. These components require zero or one solve per training data instance, while requiring no further solves during training. Experiments show that the combination of loss components achieves comparable downstream task quality to the state of the art, while being significantly more efficient, enabling scaling to problem sizes that have not been tackled before with DFL.
Many operational decisions rely on predictive models that estimate uncertain outcomes conditional on observable contexts. Training such models, however, often faces a fundamental data asymmetry: labeled outcomes are scarce or costly to obtain, while contextual covariates are abundant. Motivated by this data asymmetry, we develop a decision-aware weak-to-strong (W2S) framework that leverages both labeled and unlabeled data to improve contextual stochastic optimization. Specifically, we first train a weak model using limited labeled data and then use it to generate predicted outcome distributions on unlabeled contexts. These distributions provide soft supervision for training a strong model. We establish a non-asymptotic upper bound on the excess decision risk of W2S and a complementary lower bound for a strong-only benchmark. Their comparison yields explicit sufficient conditions under which W2S improves downstream decision performance. The key quantity is the correlation dimension between the weak and strong feature representations: when it is small, abundant unlabeled data reduce the effect of teacher errors along non-overlapping directions. A synthetic newsvendor experiment and a comment moderation experiment based on real-world data provide empirical evidence consistent with the theory.
Decision-focused learning (DFL) trains predictive models by optimizing downstream decision quality rather than standalone prediction accuracy. For contextual linear optimization, most existing DFL methods assume offline data and full observations of the objective cost vector. We develop an on-policy learning method for sequential contextual linear optimization under partial feedback, generalizing the standard bandit feedback setting. Our method learns a stochastic predict-then-optimize policy that samples a cost-vector prediction from a conditional distribution and solves the resulting downstream linear optimization problem. To update this distributional model, we introduce a two-component hybrid gradient estimator. The first component is a score function estimator, which provides an unbiased but potentially high-variance policy gradient estimate. The second is a decision-focused plug-in component that uses an auxiliary nuisance estimate of the latent cost vector to exploit the downstream optimization structure, becoming more informative as the estimate improves. We prove an O(T−1/2) bound on the average squared policy-gradient norm, matching the standard non-convex SGD rate. Experiments on top-k selection, shortest path, combinatorial pricing, and a real-data energy-scheduling benchmark show that the hybrid gradient approach achieves lower cumulative regret than contextual-bandit-style baselines across all benchmarks, using both Gaussian and richer conditional generative models. Code is available at https://github.com/Joeyetinghan/on-policy-bandit-dfl.
Wyame Benslimane, Tinghan Ye, Pascal Van Hentenryck +1