Stochastic Gradient Hamiltonian Monte Carlo (SGHMC) algorithms have received increasing attention in both theory and practice. In this paper, we propose a Stochastic Recursive Variance-Reduced gradient HMC (SRVR-HMC) algorithm. It makes use of a semi-stochastic gradient estimator that recursively accumulates the gradient information to reduce the variance of the stochastic gradient. We provide a convergence analysis of SRVR-HMC for sampling from a class of non-log-concave distributions and show that SRVR-HMC converges faster than all existing HMC-type algorithms based on underdamped Langevin dynamics. Thorough experiments on synthetic and real-world datasets validate our theory and demonstrate the superiority of SRVR-HMC.

We present a unified framework to analyze the global convergence of Langevin dynamics based algorithms for nonconvex finite-sum optimization with n component functions. At the core of our analysis is a direct analysis of the ergodicity of the numerical approximations to Langevin dynamics, which leads to faster convergence rates.

Distributed learning allows a group of independent data owners to collaboratively learn a model over their data sets without exposing their private data. We present a distributed learning approach that combines differential privacy with secure multiparty computation. We explore two popular methods of differential privacy, output perturbation and gradient perturbation, and advance the state-of-the-art for both methods in the distributed learning setting. In our output perturbation method, the parties combine local models within a secure computation and then add the required differential privacy noise before revealing the model. In our gradient perturbation method, the data owners collaboratively train a global model via an iterative learning algorithm. At each iteration, the parties aggregate their local gradients within a secure computation, adding sufficient noise to ensure privacy before the gradient updates are revealed. For both methods, we show that the noise can be reduced in the multi-party setting by adding the noise inside the secure computation after aggregation, asymptotically improving upon the best previous results. Experiments on real world data sets demonstrate that our methods provide substantial utility gains for typical privacy requirements.

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

In many practical machine learning problems, the acquisition of labeled data is often expensive and/or time consuming. This motivates us to study a problem as follows: given a label budget, how to select data points to label such that the learning performance is optimized. We propose a selective labeling method by analyzing the out-of-sample error of Laplacian regularized Least Squares (LapRLS). In particular, we derive a deterministic out-of-sample error bound for LapRLS trained on subsampled data, and propose to select a subset of data points to label by minimizing this upper bound. Since the minimization is a combinational problem, we relax it into continuous domain and solve it by projected gradient descent. Experiments on benchmark datasets show that the proposed method outperforms the state-of-the-art methods.

We provide a general theory of the expectation-maximization (EM) algorithm for inferring high dimensional latent variable models. In particular, we make two contributions: (i) For parameter estimation, we propose a novel high dimensional EM algorithm which naturally incorporates sparsity structure into parameter estimation.

Stochastic Gradient Hamiltonian Monte Carlo (SGHMC) algorithms have received increasing attention in both theory and practice. In this paper, we propose a Stochastic Recursive Variance-Reduced gradient HMC (SRVR-HMC) algorithm. It makes use of a semi-stochastic gradient estimator that recursively accumulates the gradient information to reduce the variance of the stochastic gradient. We provide a convergence analysis of SRVR-HMC for sampling from a class of non-log-concave distributions and show that SRVR-HMC converges faster than all existing HMC-type algorithms based on underdamped Langevin dynamics. Thorough experiments on synthetic and real-world datasets validate our theory and demonstrate the superiority of SRVR-HMC.

Graph convolutional networks (GCNs) have recently received wide attentions, due to their successful applications in different graph tasks and different domains. Training GCNs for a large graph, however, is still a challenge. Original full-batch GCN training requires calculating the representation of all the nodes in the graph per GCN layer, which brings in high computation and memory costs. To alleviate this issue, several sampling-based methods have been proposed to train GCNs on a subset of nodes. Among them, the node-wise neighbor-sampling method recursively samples a fixed number of neighbor nodes, and thus its computation cost suffers from exponential growing neighbor size; while the layer-wise importance-sampling method discards the neighbor-dependent constraints, and thus the nodes sampled across layer suffer from sparse connection problem.

A recent line of research has shown that gradient-based algorithms with random initialization can converge to the global minima of the training loss for overparameterized (i.e., sufficiently wide) deep neural networks. However, the condition on the width of the neural network to ensure the global convergence is very stringent, which is often a high-degree polynomial in the training sample size n (e.g., O(n