Mathieu Lacroix

Francesco Demelas, Joseph Le Roux, Mathieu Lacroix et al.

Lagrangian relaxation stands among the most efficient approaches for solving a Mixed Integer Linear Programs (MILP) with difficult constraints. Given any duals for these constraints, called Lagrangian Multipliers (LMs), it returns a bound on the optimal value of the MILP, and Lagrangian methods seek the LMs giving the best such bound. But these methods generally rely on iterative algorithms resembling gradient descent to maximize the concave piecewise linear dual function: the computational burden grows quickly with the number of relaxed constraints. We introduce a deep learning approach that bypasses the descent, effectively amortizing the local, per instance, optimization. A probabilistic encoder based on a graph convolutional network computes high-dimensional representations of relaxed constraints in MILP instances. A decoder then turns these representations into LMs. We train the encoder and decoder jointly by directly optimizing the bound obtained from the predicted multipliers. Numerical experiments show that our approach closes up to 85~\% of the gap between the continuous relaxation and the best Lagrangian bound, and provides a high quality warm-start for descent based Lagrangian methods.

Caio Corro, Mathieu Lacroix, Joseph Le Roux

We propose a novel discriminative model for sequence labeling called Bregman conditional random fields (BCRF). Contrary to standard linear-chain conditional random fields, BCRF allows fast parallelizable inference algorithms based on iterative Bregman projections. We show how such models can be learned using Fenchel-Young losses, including extension for learning from partial labels. Experimentally, our approach delivers comparable results to CRF while being faster, and achieves better results in highly constrained settings compared to mean field, another parallelizable alternative.