Proper Straight-Through Estimator: Breaking symmetry promotes convergence to true minimum

In the quantized network, its gradient shows either vanishing or diverging. The network thus cannot be learned by the standard back-propagation, so that an alternative approach called Straight Through Estimator (STE), which replaces the part of the gradient with a simple differentiable function, is used. While STE is known to work well for learning the quantized network empirically, it has not been established theoretically. A recent study by Yin et. al. (2019) has provided theoretical support for STE. However, its justification is still limited to the model in the one-hidden layer network with the binary activation where Gaussian generates the input data, and the true labels are output from the teacher network with the same binary network architecture. In this paper, we discuss the effectiveness of STEs in more general situations without assuming the shape of the input distribution and the labels. By considering the scale symmetry of the network and specific properties of the STEs, we find that STE with clipped Relu is superior to STEs with identity function and vanilla Relu. The clipped Relu STE, which breaks the scale symmetry, may pick up one of the local minima degenerated in scales, while the identity STE and vanilla Relu STE, which keep the scale symmetry, may not pick it up. To confirm this observation, we further present an analysis of a simple misspecified model as an example. We find that all the stationary points are identical with the vanishing points of the cRelu STE gradient, while some of them are not identical with the vanishing points of the identity and Relu STE.