A lagrangian propagator for artificial neural networks in constraint programming

Abstract

This paper discusses a new method to perform propagation over a (two-layer, feed-forward) Neural Network embedded in a Constraint Programming model. The method is meant to be employed in Empirical Model Learning, a technique designed to enable optimal decision making over systems that cannot be modeled via conventional declarative means. The key step in Empirical Model Learning is to embed a Machine Learning model into a combinatorial model. It has been showed that Neural Networks can be embedded in a Constraint Programming model by simply encoding each neuron as a global constraint, which is then propagated individually. Unfortunately, this decomposition approach may lead to weak bounds. To overcome such limitation, we propose a new network-level propagator based on a non-linear Lagrangian relaxation that is solved with a subgradient algorithm. The method proved capable of dramatically reducing the search tree size on a thermal-aware dispatching problem on multicore CPUs. The overhead for optimizing the Lagrangian multipliers is kept within a reasonable level via a few simple techniques. This paper is an extended version of [27], featuring an improved structure, a new filtering technique for the network inputs, a set of overhead reduction techniques, and a thorough experimentation.

Strong Mixed-Integer Programming Formulations for Trained Neural Networks

“…Beyond MIP and convex relaxations, a number of authors have investigated other algorithmic techniques for modeling trained neural networks in optimization problems, drawing primarily from the satisfiability, constraint programming, and global optimization communities [7,8,33,37,45]. Another intriguing direction studies restrictions to the space of models that may make the optimization problem over the network inputs simpler: for example, the classes of binarized [34] or input convex [1] neural networks.…”

Reward Potentials for Planning with Learned Neural Network Transition Models

“…In this paper, we have focused on the important problem of improving the efficiency of B&B solvers for optimal planning with learned NN transition models in continuous action and state spaces. Parallel to this work, planning and decision making in discrete action and state spaces [12,17,16], verification of learned NNs [9,6,7,14], robustness evaluation of learned NNs [20] and defenses to adversarial attacks for learned NNs [10] have been studied with the focus of solving very similar decision making problems. For example, the verification problem solved by Reluplex [9] 9 is very similar to the planning problem solved by HD-MILP-Plan [18] without the objective function and horizon H = 1.…”

Strong mixed-integer programming formulations for trained neural networks

“…Beyond MIP and convex relaxations, a number of authors have investigated other algorithmic techniques for modeling trained neural networks in optimization problems, drawing primarily from the satisfiability, constraint programming, and global optimization communities [8,9,41,48,59]. Another intriguing direction studies restrictions to the space of models that may make the optimization problem over the network inputs simpler: for example, the classes of binarized [42] or input convex [2] neural networks.…”