News for August 2022

A eerily quiet month, this August of ’22. We only found a single paper to cover!

Training Overparametrized Neural Networks in Sublinear Time by Hang Hu, Zhao Song, Omri Weinstein, Danyang Zhuo (arXiv). Think of a classification problem where the inputs are in \(\mathbb{R}^d\). We have \(n\) such points (with their true labels, as training data) and wish to train a Neural Network. A two layer Rectified Linear Unit (ReLU) Neural Network (NN) works as follows. The first layer has \(m\) vertices, where each vertex has vector weight \(\vec{w}_i \in \mathbb{R}^d\). The second “hidden layer” has \(m\) vertices, each with a scalar weight \(a_1, a_2, \ldots, a_m\). This network is called overparametrized when \(m \gg n\). The output of this NN on input vector \(\vec{x}\) is (up to scaling) \(\sum_{i \leq m} a_i \phi(\vec{w_i} \cdot \vec{x})\) (where \(\phi\) is a thresholded linear function). Observe that to compute the value on a single input takes \(O(md)\) time, so the total time to compute all values on \(n\) training inputs takes \(O(mnd)\) time. The training is done by gradient descent methods; given a particular setting of weights, we compute the total loss, and then modify the weights along the gradient. Previous work showed how a single iteration can be done in time \(O(mnd + n^3)\). When \(m \gg n^2\), this can be thought of as linear in computing the loss function (which requires evaluating the NN on all the \(n\) points). This paper shows how to implement a single iteration in \(O(m^{1-\alpha}nd + n^3)\) time, for some \(\alpha > 0\). Hence, the time for an iteration is sublinear in the trivial computation. The techniques used are sparse recovery methods and random projections.

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