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Learning with differentiable perturbed optimizers

By Quentin Berthet

Machine learning pipelines often rely on optimization procedures to make discrete decisions (e.g. sorting, picking closest neighbors, finding shortest paths or optimal matchings). Although these discrete decisions are easily computed in a forward manner, they cannot be used to modify model parameters using first-order optimization techniques because they break the back-propagation of computational graphs. In order to expand the scope of learning problems that can be solved in an end-to-end fashion, we propose a systematic method to transform a block that outputs an optimal discrete decision into a differentiable operation. Our approach relies on stochastic perturbations of these parameters, and can be used readily within existing solvers without the need for ad hoc regularization or smoothing. These perturbed optimizers yield solutions that are differentiable and never locally constant. The amount of smoothness can be tuned via the chosen noise amplitude, whose impact we analyze. The derivatives of these perturbed solvers can be evaluated eciently. We also show how this framework can be connected to a family of losses developed in structured prediction, and describe how these can be used in unsupervised and supervised learning, with theoretical guarantees. We demonstrate the performance of our approach on several machine learning tasks in experiments on synthetic and real data.

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Citation data

  • DOI 10.24350/CIRM.V.19622903
  • Cite this video BERTHET Quentin (09/03/2020). Learning with differentiable perturbed optimizers. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19622903
  • URL https://dx.doi.org/10.24350/CIRM.V.19622903


  • PAPANDREOU, George et YUILLE, Alan L. Perturb-and-map random fields: Using discrete optimization to learn and sample from energy models. In : 2011 International Conference on Computer Vision. IEEE, 2011. p. 193-200. - https://doi.org/10.1109/ICCV.2011.6126242
  • KALAI, Adam et VEMPALA, Santosh. Efficient algorithms for online decision problems. In : Learning Theory and Kernel Machines. Springer, Berlin, Heidelberg, 2003. p. 26-40. - http://dx.doi.org/10.1007/978-3-540-45167-9_4
  • BERTHET, Quentin, BLONDEL, Mathieu, TEBOUL, Olivier, et al. Learning with Differentiable Perturbed Optimizers. arXiv preprint arXiv:2002.08676, 2020. - https://arxiv.org/abs/2002.08676

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