Soft Multivariate Truncated Normal Distribution (soft tMVN)
We propose a new distribution, called the soft tMVN distribution, which provides a smooth approximation to the truncated multivariate normal (tMVN) distribution with linear constraints. An efficient blocked Gibbs sampler is developed to sample from the soft tMVN distribution in high dimensions. We provide theoretical support to the approximation capability of the soft tMVN and provide further empirical evidence thereof. The soft tMVN distribution can be used to approximate simulations from a multivariate truncated normal distribution with linear constraints, or itself as a prior in shape-constrained problems. …

We consider distributed gradient descent in the presence of stragglers. Recent work on \em gradient coding \em and \em approximate gradient coding \em have shown how to add redundancy in distributed gradient descent to guarantee convergence even if some workers are \em stragglers\em—that is, slow or non-responsive. In this work we propose an approximate gradient coding scheme called \em Stochastic Gradient Coding \em (SGC), which works when the stragglers are random. SGC distributes data points redundantly to workers according to a pair-wise balanced design, and then simply ignores the stragglers. We prove that the convergence rate of SGC mirrors that of batched Stochastic Gradient Descent (SGD) for the $\ell_2$ loss function, and show how the convergence rate can improve with the redundancy. We also provide bounds for more general convex loss functions. We show empirically that SGC requires a small amount of redundancy to handle a large number of stragglers and that it can outperform existing approximate gradient codes when the number of stragglers is large. …