skip to main content


Title: Learning Two-Layer Neural Networks with Symmetric Inputs
We give a new algorithm for learning a two-layer neural network under a general class of input distributions. Assuming there is a ground-truth two-layer network y = Aσ(Wx) + ξ, where A,W are weight matrices, ξ represents noise, and the number of neurons in the hidden layer is no larger than the input or output, our algorithm is guaranteed to recover the parameters A,W of the ground-truth network. The only requirement on the input x is that it is symmetric, which still allows highly complicated and structured input. Our algorithm is based on the method-of-moments framework and extends several results in tensor decompositions. We use spectral algorithms to avoid the complicated non-convex optimization in learning neural networks. Experiments show that our algorithm can robustly learn the ground-truth neural network with a small number of samples for many symmetric input distributions.  more » « less
Award ID(s):
1704656
PAR ID:
10090510
Author(s) / Creator(s):
; ; ;
Date Published:
Journal Name:
International Conference on Learning Representations
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. Abstract We consider the problem of estimating the input and hidden variables of a stochastic multi-layer neural network (NN) from an observation of the output. The hidden variables in each layer are represented as matrices with statistical interactions along both rows as well as columns. This problem applies to matrix imputation, signal recovery via deep generative prior models, multi-task and mixed regression, and learning certain classes of two-layer NNs. We extend a recently-developed algorithm—multi-layer vector approximate message passing, for this matrix-valued inference problem. It is shown that the performance of the proposed multi-layer matrix vector approximate message passing algorithm can be exactly predicted in a certain random large-system limit, where the dimensions N × d of the unknown quantities grow as N → ∞ with d fixed. In the two-layer neural-network learning problem, this scaling corresponds to the case where the number of input features as well as training samples grow to infinity but the number of hidden nodes stays fixed. The analysis enables a precise prediction of the parameter and test error of the learning. 
    more » « less
  2. Recent advancements in two-photon calcium imaging have enabled scientists to record the activity of thousands of neurons with cellular resolution. This scope of data collection is crucial to understanding the next generation of neuroscience questions, but analyzing these large recordings requires automated methods for neuron segmentation. Supervised methods for neuron segmentation achieve state of-the-art accuracy and speed but currently require large amounts of manually generated ground truth training labels. We reduced the required number of training labels by designing a semi-supervised pipeline. Our pipeline used neural network ensembling to generate pseudolabels to train a single shallow U-Net. We tested our method on three publicly available datasets and compared our performance to three widely used segmentation methods. Our method outperformed other methods when trained on a small number of ground truth labels and could achieve state-of-the-art accuracy after training on approximately a quarter of the number of ground truth labels as supervised methods. When trained on many ground truth labels, our pipeline attained higher accuracy than that of state-of-the-art methods. Overall, our work will help researchers accurately process large neural recordings while minimizing the time and effort needed to generate manual labels.

     
    more » « less
  3. We prove the first superpolynomial lower bounds for learning one-layer neural networks with respect to the Gaussian distribution using gradient descent. We show that any classifier trained using gradient descent with respect to square-loss will fail to achieve small test error in polynomial time given access to samples labeled by a one-layer neural network. For classification, we give a stronger result, namely that any statistical query (SQ) algorithm (including gradient descent) will fail to achieve small test error in polynomial time. Prior work held only for gradient descent run with small batch sizes, required sharp activations, and applied to specific classes of queries. Our lower bounds hold for broad classes of activations including ReLU and sigmoid. The core of our result relies on a novel construction of a simple family of neural networks that are exactly orthogonal with respect to all spherically symmetric distributions. 
    more » « less
  4. We propose a new computationally efficient method for quantizing the weights of pre- trained neural networks that is general enough to handle both multi-layer perceptrons and convolutional neural networks. Our method deterministically quantizes layers in an iterative fashion with no complicated re-training required. Specifically, we quantize each neuron, or hidden unit, using a greedy path-following algorithm. This simple algorithm is equivalent to running a dynamical system, which we prove is stable for quantizing a single-layer neural network (or, alternatively, for quantizing the first layer of a multi-layer network) when the training data are Gaussian. We show that under these assumptions, the quantization error decays with the width of the layer, i.e., its level of over-parametrization. We provide numerical experiments, on multi-layer networks, to illustrate the performance of our methods on MNIST and CIFAR10 data, as well as for quantizing the VGG16 network using ImageNet data. 
    more » « less
  5. Wasserstein Barycenter is a principled approach to represent the weighted mean of a given set of probability distributions, utilizing the geometry induced by optimal transport. In this work, we present a novel scalable algorithm to approximate the Wasserstein Barycenters aiming at highdimensional applications in machine learning. Our proposed algorithm is based on the Kantorovich dual formulation of the Wasserstein-2 distance as well as a recent neural network architecture, input convex neural network, that is known to parametrize convex functions. The distinguishing features of our method are: i) it only requires samples from the marginal distributions; ii) unlike the existing approaches, it represents the Barycenter with a generative model and can thus generate infinite samples from the barycenter without querying the marginal distributions; iii) it works similar to Generative Adversarial Model in one marginal case. We demonstrate the efficacy of our algorithm by comparing it with the state-of-art methods in multiple experiments. 
    more » « less