Learning Two Layer Rectified Neural Networks in Polynomial Time

By Ainesh Bakshi, Rajesh Jayaram, David P. Woodruff. 2018

71 citations recorded by the source corpus.

Consider the following fundamental learning problem: given input examples $x \in \mathbb{R}^d$ and their vector-valued labels, as defined by an underlying generative neural network, recover the weight matrices of this network. We consider two-layer networks, mapping $\mathbb{R}^d$ to $\mathbb{R}^m$, with $k$ non-linear activation units $f(\cdot)$, where $f(x) = \max \{x , 0\}$ is the ReLU. Such a network is specified by two weight matrices, $\mathbf{U}^* \in \mathbb{R}^{m \times k}, \mathbf{V}^* \in \mathbb{R}^{k \times d}$, such that the label of an example $x \in \mathbb{R}^{d}$ is given by $\mathbf{U}^* f(\mathbf{V}^* x)$, where $f(\cdot)$ is applied coordinate-wise. Given $n$ samples as a matrix $\mathbf{X} \in \mathbb{R}^{d \times n}$ and the (possibly noisy) labels $\mathbf{U}^* f(\mathbf{V}^* \mathbf{X}) + \mathbf{E}$ of the network on these samples, where $\mathbf{E}$ is a noise matrix, our goal is to recover the weight matrices $\mathbf{U}^*$ and $\mathbf{V}^*$. In this work, we develop algorithms and hardness results under varying assumptions on the input and noise. Although the problem is NP-hard even for $k=2$, by assuming Gaussian marginals over the input $\mathbf{X}$ we are able to develop polynomial time algorithms for the approximate recovery of $\mathbf{U}^*$ and $\mathbf{V}^*$. Perhaps surprisingly, in the noiseless case our algorithms recover $\mathbf{U}^*,\mathbf{V}^*$ exactly, i.e., with no error. To the best of the our knowledge, this is the first algorithm to accomplish exact recovery. For the noisy case, we give the first polynomial time algorithm that approximately recovers the weights in the presence of mean-zero noise $\mathbf{E}$. Our algorithms generalize to a larger class of rectified activation functions, $f(x) = 0$ when $x\leq 0$, and $f(x) > 0$ otherwise.

Subfields: Artificial Intelligence

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Citations in the corpus

Learning and Generalization in Overparameterized Neural Networks, Going Beyond Two Layers (2018, background): “1 What Can Neural Networks Provably Learn? Most existing works analyzing the learnability of neural networks [9, 12, 13, 20, 21, 28, 33, 34, 42, 43, 47, 49, 50, 56] make unrealistic assumptions about the data distribution (such as being random Gaussian), and/or make strong assumptions about the netw”
Backward Feature Correction: How Deep Learning Performs Deep Learning (2020, background): “For such reason, it is not surprising that most existing theoretical works on the efficient learning regime of neural networks either focus on (1) two-layer networks [9, 10, 12, 22, 23, 36, 41, 41, 42, 46, 50, 51, 53, 54, 56, 58, 61] which do not have any hierarchical structure, or (2) a multi-layer”
Learning Over-Parametrized Two-Layer ReLU Neural Networks beyond NTK (2020, methodology): “Apart from the gradient descent algorithm, the method of moments has also been shown to be an effective strategy with provable guarantees (e.g. (Bakshi et al., 2018; Ge et al., 2018)).”