We consider the problem of training input-output recurrent neural networks (RNN) for sequence labeling tasks. We propose a novel spectral approach for learning the network parameters. It is based on decomposition of the cross-moment tensor between the output and a non-linear transformation of the input, based on score functions. We guarantee consistent learning with polynomial sample and computational complexity under transparent conditions such as non-degeneracy of model parameters, polynomial activations for the neurons, and a Markovian evolution of the input sequence. We also extend our results to Bidirectional RNN which uses both previous and future information to output the label at each time point, and is employed in many NLP tasks such as POS tagging.

Local search heuristics for non-convex optimizations are popular in applied machine learning. However, in general it is hard to guarantee that such algorithms even converge to a local minimum, due to the existence of complicated saddle point structures in high dimensions. Many functions have degenerate saddle points such that the first and second order derivatives cannot distinguish them with local optima. In this paper we use higher order derivatives to escape these saddle points: we design the first efficient algorithm guaranteed to converge to a third order local optimum (while existing techniques are at most second order). We also show that it is NP-hard to extend this further to finding fourth order local optima.

We consider the problem of learning mixtures of generalized linear models (GLM) which arise in classification and regression problems. Typical learning approaches such as expectation maximization (EM) or variational Bayes can get stuck in spurious local optima. In contrast, we present a tensor decomposition method which is guaranteed to correctly recover the parameters. The key insight is to employ certain feature transformations of the input, which depend on the input generative model. Specifically, we employ score function tensors of the input and compute their cross-correlation with the response variable. We establish that the decomposition of this tensor consistently recovers the parameters, under mild non-degeneracy conditions. We demonstrate that the computational and sample complexity of our method is a low order polynomial of the input and the latent dimensions.

Training neural networks is a challenging non-convex optimization problem, and backpropagation or gradient descent can get stuck in spurious local optima. We propose a novel algorithm based on tensor decomposition for guaranteed training of two-layer neural networks. We provide risk bounds for our proposed method, with a polynomial sample complexity in the relevant parameters, such as input dimension and number of neurons. While learning arbitrary target functions is NP-hard, we provide transparent conditions on the function and the input for learnability. Our training method is based on tensor decomposition, which provably converges to the global optimum, under a set of mild non-degeneracy conditions. It consists of simple embarrassingly parallel linear and multi-linear operations, and is competitive with standard stochastic gradient descent (SGD), in terms of computational complexity. Thus, we propose a computationally efficient method with guaranteed risk bounds for training neural networks with one hidden layer.

Tensor CANDECOMP/PARAFAC (CP) decomposition has wide applications in statistical learning of latent variable models and in data mining. In this paper, we propose fast and randomized tensor CP decomposition algorithms based on sketching. We build on the idea of count sketches, but introduce many novel ideas which are unique to tensors. We develop novel methods for randomized com- putation of tensor contractions via FFTs, without explicitly forming the tensors. Such tensor contractions are encountered in decomposition methods such as ten- sor power iterations and alternating least squares. We also design novel colliding hashes for symmetric tensors to further save time in computing the sketches. We then combine these sketching ideas with existing whitening and tensor power iter- ative techniques to obtain the fastest algorithm on both sparse and dense tensors. The quality of approximation under our method does not depend on properties such as sparsity, uniformity of elements, etc. We apply the method for topic mod- eling and obtain competitive results.

We present a novel analysis of the dynamics of tensor power iterations in the overcomplete regime where the tensor CP rank is larger than the input dimension. Finding the CP decomposition of an overcomplete tensor is NP-hard in general. We consider the case where the tensor components are randomly drawn, and show that the simple power iteration recovers the components with bounded error under mild initialization conditions. We apply our analysis to unsupervised learning of latent variable models, such as multi-view mixture models and spherical Gaussian mixtures. Given the third order moment tensor, we learn the parameters using tensor power iterations. We prove it can correctly learn the model parameters when the number of hidden components $k$ is much larger than the data dimension $d$, up to $k = o(d^{1.5})$. We initialize the power iterations with data samples and prove its success under mild conditions on the signal-to-noise ratio of the samples. Our analysis significantly expands the class of latent variable models where spectral methods are applicable. Our analysis also deals with noise in the input tensor leading to sample complexity result in the application to learning latent variable models.

We propose an efficient ADMM method with guarantees for high-dimensional problems. We provide explicit bounds for the sparse optimization problem and the noisy matrix decomposition problem. For sparse optimization, we establish that the modified ADMM method has an optimal convergence rate of $\mathcal{O}(s\log d/T)$, where $s$ is the sparsity level, $d$ is the data dimension and $T$ is the number of steps. This matches with the minimax lower bounds for sparse estimation. For matrix decomposition into sparse and low rank components, we provide the first guarantees for any online method, and prove a convergence rate of $\tilde{\mathcal{O}}((s+r)\beta^2(p) /T) + \mathcal{O}(1/p)$ for a $p\times p$ matrix, where $s$ is the sparsity level, $r$ is the rank and $\Theta(\sqrt{p})\leq \beta(p)\leq \Theta(p)$. Our guarantees match the minimax lower bound with respect to $s,r$ and $T$. In addition, we match the minimax lower bound with respect to the matrix dimension $p$, i.e. $\beta(p)=\Theta(\sqrt{p})$, for many important statistical models including the independent noise model, the linear Bayesian network and the latent Gaussian graphical model under some conditions. Our ADMM method is based on epoch-based annealing and consists of inexpensive steps which involve projections on to simple norm balls. Experiments show that for both sparse optimization and matrix decomposition problems, our algorithm outperforms the state-of-the-art methods. In particular, we reach higher accuracy with same time complexity.

Corrupting the input and hidden layers of deep neural networks (DNNs) with multiplicative noise, often drawn from the Bernoulli distribution (or 'dropout'), provides regularization that has significantly contributed to deep learning's success. However, understanding how multiplicative corruptions prevent overfitting has been difficult due to the complexity of a DNN's functional form. In this paper, we show that when a Gaussian prior is placed on a DNN's weights, applying multiplicative noise induces a Gaussian scale mixture, which can be reparameterized to circumvent the problematic likelihood function. Analysis can then proceed by using a type-II maximum likelihood procedure to derive a closed-form expression revealing how regularization evolves as a function of the network's weights. Results show that multiplicative noise forces weights to become either sparse or invariant to rescaling. We find our analysis has implications for model compression as it naturally reveals a weight pruning rule that starkly contrasts with the commonly used signal-to-noise ratio (SNR). While the SNR prunes weights with large variances, seeing them as noisy, our approach recognizes their robustness and retains them. We empirically demonstrate our approach has a strong advantage over the SNR heuristic and is competitive to retraining with soft targets produced from a teacher model.

We provide novel guaranteed approaches for training feedforward neural networks with sparse connectivity. We leverage on the techniques developed previously for learning linear networks and show that they can also be effectively adopted to learn non-linear networks. We operate on the moments involving label and the score function of the input, and show that their factorization provably yields the weight matrix of the first layer of a deep network under mild conditions. In practice, the output of our method can be employed as effective initializers for gradient descent.

Community detection in graphs has been extensively studied both in theory and in applications. However, detecting communities in hypergraphs is more challenging. In this paper, we propose a tensor decomposition approach for guaranteed learning of communities in a special class of hypergraphs modeling social tagging systems or folksonomies. A folksonomy is a tripartite 3-uniform hypergraph consisting of (user, tag, resource) hyperedges. We posit a probabilistic mixed membership community model, and prove that the tensor method consistently learns the communities under efficient sample complexity and separation requirements.