UC Irvine

We study the dynamics of gradient descent in learning neural networks for classification problems. Unlike in existing works, we consider the linearly non-separable case where the training data of different classes lie in orthogonal subspaces. Deep neural networks (DNNs) are quantized for efficient inference on resource-constrained platforms. However, training deep learning models with low-precision weights and activations involves a demanding optimization task, which calls for minimizing a stage-wise loss function subject to a discrete set-constraint. While numerous training methods have been proposed, existing studies for full quantization of DNNs are mostly empirical. From a theoretical point of view, we study practical techniques for overcoming the combinatorial nature of network quantization. Specifically, we investigate a simple yet powerful projected gradient-like algorithm for quantizing two-layer convolution networks, by repeatedly moving one step at float weights in the negative direction of a heuristic fake gradient of the loss function (so-called coarse gradient) evaluated at quantized weights. For the first time, we prove that under mild conditions, the sequence of quantized weights recurrently visit the global optimum of the discrete minimization problem for training fully quantized network. We also show numerical evidence of the recurrence phenomenon of weight evolution in training quantized deep networks.