Symmetry-invariant deep learning

Symmetry-invariant optimization in deep networks


V. Badrinarayanan, B. Mishra, and R. Cipolla


Recent works have highlighted scale invariance or symmetry that is present in the weight space of a typical deep network and the adverse effect that it has on the Euclidean gradient based stochastic gradient descent optimization. In this work, we show that these and other commonly used deep networks, such as those which use a max-pooling and sub-sampling layer, possess more complex forms of symmetry arising from scaling based reparameterization of the network weights. We then propose two symmetry-invariant gradient based weight updates for stochastic gradient descent based learning. Our empirical evidence based on the MNIST dataset shows that these updates improve the test performance without sacrificing the computational efficiency of the weight updates. We also show the results of training with one of the proposed weight updates on an image segmentation problem.


  • Status: Technical report, 2015. A shorter version has been accepted to 8th NIPS workshop on optimization for machine learning (OPT2015) to be held at Montreal, Canada on December 11, 2015 [arXiv:1511.01029].
  • Paper: [arXiv:1511.01754].
  • Matlab code: The codes are polished enough to read and understand. However, if there are problems, do let us know.
  • Keshar Breen has put together a symmetry-invariant caffe implementation here. It is under development. Feedback is welcome.


The proposed updates for a two layer network with weights W1 and W2 coupled with a soft-max classifier θ. For the SM update, GW1 and others here are defined in the paper [arXiv:1511.01754].