Differentiable Combinatorial Losses through Generalized Gradients of Linear Programs

When samples have internal structure, we often see a mismatch between the objective optimized during training and the model's goal during inference. For example, in sequence-to-sequence modeling we are interested in high-quality translated sentences, but training typically uses maximum likelihood at the word level. The natural training-time loss would involve a combinatorial problem -- dynamic programming-based global sequence alignment -- but solutions to combinatorial problems are not differentiable with respect to their input parameters, so surrogate, differentiable losses are used instead. Here, we show how to perform gradient descent over combinatorial optimization algorithms that involve continuous parameters, for example edge weights, and can be efficiently expressed as linear programs. We demonstrate usefulness of gradient descent over combinatorial optimization in sequence-to-sequence modeling using differentiable encoder-decoder architecture with softmax or Gumbel-softmax, and in image classification in a weakly supervised setting where instead of the correct class for each photo, only groups of photos labeled with correct but unordered set of classes are available during training.