Smart Predict-and-Optimize for Hard Combinatorial Optimization Problems

Abstract

Combinatorial optimization assumes that all parameters of the optimization problem, e.g. the weights in the objective function, are fixed. Often, these weights are mere estimates and increasingly machine learning techniques are used to for their estimation. Recently, Smart Predict and Optimize (SPO) has been proposed for problems with a linear objective function over the predictions, more specifically linear programming problems. It takes the regret of the predictions on the linear problem into account, by repeatedly solving it during learning. We investigate the use of SPO to solve more realistic discrete optimization problems. The main challenge is the repeated solving of the optimization problem. To this end, we investigate ways to relax the problem as well as warm-starting the learning and the solving. Our results show that even for discrete problems it often suffices to train by solving the relaxation in the SPO loss. Furthermore, this approach outperforms the state-of-the-art approach of Wilder, Dilkina, and Tambe. We experiment with weighted knapsack problems as well as complex scheduling problems, and show for the first time that a predict-and-optimize approach can successfully be used on large-scale combinatorial optimization problems.

End-to-end Learning for Fair Ranking Systems

“…While this is inevitably computationally more expensive, solving the LP of Model 1 requires low degree polynomial time in the number of items to rank [18], due to the sparsity of its constraints. Fortunately, this issue can be vastly alleviated with the application of hot-starting schemes [14], since the SPO framework relies on iteratively updating a stored solution to each LP instance for slightly different objective coefficients as model weights are updated. Thus, each instance of Model 1 need not be solved from scratch.…”

“…Fortunately, the Smart Predict-and-Optimize framework is particularly amenable to hot-starting. Since a LP instance for each data sample must be solved at each epoch, a feasible solution to each LP is available from the previous epoch, corresponding to the same constraints and a cost vector which changes based on updates to the DNN model parameters during training [14]. Storing a hot-start solution to each LP instance in a training set requires memory no larger than that of training set, and as the model weights converge, these hot-starts are expected to be very close to the optimal policies for each LP.…”

Predictive machine learning for prescriptive applications: a coupled training-validating approach

“…An extension of the proposed methodology called SPO Trees (SPOTs) for training decision trees under this loss is offered again by Elmachtoub et al in [29]. Another extension to the smart SPO approach is presented by Mandi et al in [30] where the use of SPO to solve a relaxed version of a combinatorial optimization problems is investigated. An application of the smart SPO denoted as "semi SPO" is proposed by [31] for efficient inspection of ships at ports in maritime transportation.…”

Hybrid Classification and Reasoning for Image-Based Constraint Solving

“…The end-to-end model of [30] learns the constraints of a satisfiability problem by considering a differentiable SDP relaxation of the problem. A similar work [14] trains an ML model by considering a convex surrogate of the task-loss.…”