Predict+Optimise with Ranking Objectives: Exhaustively Learning Linear Functions

Abstract

We study the predict+optimise problem, where machine learning and combinatorial optimisation must interact to achieve a common goal. These problems are important when optimisation needs to be performed on input parameters that are not fully observed but must instead be estimated using machine learning. Our contributions are two-fold: 1) we provide theoretical insight into the properties and computational complexity of predict+optimise problems in general, and 2) develop a novel framework that, in contrast to related work, guarantees to compute the optimal parameters for a linear learning function given any ranking optimisation problem. We illustrate the applicability of our framework for the particular case of the unit-weighted knapsack predict+optimise problem and evaluate on benchmarks from the literature.

MIPaaL: Mixed Integer Program as a Layer

“…We want to maximize the revenue we get but we need to predict half-hour prices. We use real-world energy-aware scheduling data from (Demirovic et al 2019) which consist of historical energy data and prices from the ICON energy-aware scheduling competition. The time period weights are drawn uniformly between 0 and 1, and the budget is 10% of the total weight.…”

“…The time period weights are drawn uniformly between 0 and 1, and the budget is 10% of the total weight. In (Demirovic et al 2019), the authors describe and analyze an algorithm for exhaustively searching linear models which yield good knapsack performance; however, the results are for linear predictive components feeding to knapsack instances, and do not directly extend to neural network models.…”

“…MIPs naturally arise in so many settings largely due to their flexibility, computational complexity (ability to capture NPhard problems), and interpretability. In many practical situations it is often necessary to predict some component (e.g., the objective) of the MIP based on historical data, such as estimated demand (O'Mahony and Shmoys 2015), price forecasts (Demirovic et al 2019), or patient readmission rate (Chan et al 2012). Alternatively, practitioners may use a MIP to enforce that the outputs of the predictions meet semantically meaningful objectives such as ensuring predictions result in making fair decisions downstream (Benabbou et al 2018;Trilling, Guinet, and Le Magny 2006;Warner 1976).…”

End-to-End Game-Focused Learning of Adversary Behavior in Security Games

“…We find that typical datasets only contain a set of sampled attacker responses to a particular historical defender mixed strategy or a small set of mixed strategies. Previous work on end-to-end learning for decision problems (Bengio 1997;Donti, Amos, and Kolter 2017;Wilder, Dilkina, and Tambe 2019;Demirovic et al 2019) assumes that the historical data specifies the utility of any possible decision, but this assumption does not hold in SSGs because they are interactions between strategic agents.…”

“…This analytic approach is extended to stochastic optimization by Donti et al (2017) and to submodular optimization by Wilder et al (2019). Demirovic et al (2019) provide a theoretically optimal framework for ranking problems with linear objectives.…”