Differentiable Combinatorial Losses through Generalized Gradients of Linear Programs

Xi Gao, Han Zhang, Aliakbar Panahi, Tom Arodz

Keywords: combinatorial optimization, linear programs, generalized gradient

Abstract: Combinatorial problems with linear objective function play a central role in many computer science applications, and efficient algorithms for solving them are well known. However, the solutions to these problems are not differentiable with respect to the parameters specifying the problem instance – for example, shortest distance between two nodes in a graph is not a differentiable function of graph edge weights. Recently, attempts to integrate combinatorial and, more broadly, convex optimization solvers into gradient-trained models resulted in several approaches for differentiating over the solution vector to the optimization problem. However, in many cases, the interest is in differentiating over only the objective value, not the solution vector, and using existing approaches introduces unnecessary overhead. Here, we show how to perform gradient descent directly over the objective value of the solution to combinatorial problems. We demonstrate advantage of the approach in examples involving sequence-to-sequence modeling using differentiable encoder-decoder architecture with softmax or Gumbel-softmax, and in weakly supervised learning involving a convolutional, residual feed-forward network for image classification.

One-sentence Summary: We show how to differentiate over the objective value of the optimal solution to a combinatorial problem, using a single run to a black-box combinatorial solver.

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