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Quantized or low-bit neural networks are attractive due to their inference efficiency. However, training deep neural networks with quantized activations involves minimizing a discontinuous and piecewise constant loss function. Such a loss function has zero gradient almost everywhere (a.e.), which makes the conventional gradient-based algorithms inapplicable. To this end, we study a novel class of  biased  first-order oracle, termed coarse gradient, for overcoming the vanished gradient issue. A coarse gradient is generated by replacing the a.e. zero derivative of quantized (i.e., staircase) ReLU activation composited in the chain rule with some heuristic proxy derivative called straight-through estimator (STE). Although having been widely used in training quantized networks empirically, fundamental questions like when and why the ad hoc STE trick works, still lack theoretical understanding. In this paper, we propose a class of STEs with certain monotonicity and consider their applications to the training of a two-linear-layer network with quantized activation functions for nonlinear multi-category classification. We establish performance guarantees for the proposed STEs by showing that the corresponding coarse gradient methods converge to the global minimum, which leads to a perfect classification. Lastly, we present experimental results on synthetic data as well as MNIST dataset to verify our theoretical findings and demonstrate the effectiveness of our proposed STEs.

Learning quantized neural nets by coarse gradient method for nonlinear classification