Gradient coding is a technique for straggler mitigation in distributed learning. In this paper we design novel gradient codes using tools from classical coding theory, namely, cyclic MDS codes, which compare favorably with existing solutions, both in the applicable range of parameters and in the complexity of the involved algorithms. Second, we introduce an approximate variant of the gradient coding problem, in which we settle for approximate gradient computation instead of the exact one. This approach enables graceful degradation, i.e., the ℓ2error of the approximate gradient is a decreasing function of the number of stragglers. Our main result is that normalized adjacency matrices of expander graphs yield excellent approximate gradient codes, which enable significantly less computation compared to exact gradient coding, and guarantee faster convergence than trivial solutions under standard assumptions. We experimentally test our approach on Amazon EC2, and show that the generalization error of approximate gradient coding is very close to the full gradient while requiring significantly less computation from the workers.

Gradient coding is a technique for straggler mitigation in distributed learning. In this paper we design novel gradient codes using tools from classical coding theory, namely, cyclic MDS codes, which compare favourably with existing solutions, both in the applicable range of parameters and in the complexity of the involved algorithms. Second, we introduce an approximate variant of the gradient coding problem, in which we settle for approximate gradient computation instead of the exact one. This approach enables graceful degradation, i.e., the $\ell_2$ error of the approximate gradient is a decreasing function of the number of stragglers. Our main result is that the normalized adjacency matrix of an expander graph can yield excellent approximate gradient codes, and that this approach allows us to perform significantly less computation compared to exact gradient coding. We experimentally test our approach on Amazon EC2, and show that the generalization error of approximate gradient coding is very close to the full gradient while requiring significantly less computation from the workers.

We propose a novel coding theoretic framework for mitigating stragglers in distributed learning. We show how carefully replicating data blocks and coding across gradients can provide tolerance to failures and stragglers for synchronous Gradient Descent. We implement our schemes in python (using MPI) to run on Amazon EC2, and show how we compare against baseline approaches in running time and generalization error.

We consider the problem of learning sparsely used overcomplete dictionaries, where each observation is a sparse combination of elements from an unknown overcomplete dictionary. We establish exact recovery when the dictionary elements are mutually incoherent. Our method consists of a clustering-based initialization step, which provides an approximate estimate of the true dictionary with guaranteed accuracy. This estimate is then refined via an iterative algorithm with the following alternating steps: 1) estimation of the dictionary coefficients for each observation through \ell_1 minimization, given the dictionary estimate, and 2) estimation of the dictionary elements through least squares, given the coefficient estimates. We establish that, under a set of sufficient conditions, our method converges at a linear rate to the true dictionary as well as the true coefficients for each observation.

We consider the problem of recovering the graph structure of a “hub-networked” Ising model given iid samples, under high-dimensional settings, where number of nodes p could be potentially larger than the number of samples n. By a “hub-networked” graph, we mean a graph with a few “hub nodes” with very large degrees. State of the art estimators for Ising models have a sample complexity that scales polynomially with the maximum node-degree, and are thus ill-suited to recovering such graphs with a few hub nodes. Some recent proposals for specifically recovering hub graphical models do not come with theoretical guarantees, and even empirically provide limited improvements over vanilla Ising model estimators. Here, we show that under such low sample settings, instead of estimating “difficult” components such as hub-neighborhoods, we can use quantitative indicators of our inability to do so, and thereby identify hub-nodes. This simple procedure allows us to recover hub-networked graphs with very strong statistical guarantees even under very low sample settings.

A power-law graph is any graph G = (V, E), whose degree distribution follows a power law i.e. the number of vertices in the graph with degree i, yi, is proportional to i-β: yi∝ i-β. In this paper, we provide information-theoretic lower bounds on the sample complexity of learning such power-law graphical models i.e. graphical models whose Markov graph obeys the power law. In addition, we briefly revisit some existing state of the art estimators, and explicitly derive their sample complexity for power-law graphs.