Graph Oracle Models, Lower Bounds, and Gaps for Parallel Stochastic Optimization

We suggest a general oracle-based framework that captures different parallel stochastic optimization settings described by a dependency graph, and derive generic lower bounds in terms of this graph. We then use the framework and derive lower bounds for several specific parallel optimization settings, including delayed updates and parallel processing with intermittent communication. We highlight gaps between lower and upper bounds on the oracle complexity, and cases where the "natural" algorithms are not known to be optimal.

Distributed Stochastic Multi-Task Learning with Graph Regularization

We propose methods for distributed graph-based multi-task learning that are based on weighted averaging of messages from other machines. Uniform averaging or diminishing stepsize in these methods would yield consensus (single task) learning. We show how simply skewing the averaging weights or controlling the stepsize allows learning different, but related, tasks on the different machines.

Rate Optimal Estimation and Confidence Intervals for High-dimensional Regression with Missing Covariates

Although a majority of the theoretical literature in high-dimensional statistics has focused on settings which involve fully-observed data, settings with missing values and corruptions are common in practice. We consider the problems of estimation and of constructing component-wise confidence intervals in a sparse high-dimensional linear regression model when some covariates of the design matrix are missing completely at random. We analyze a variant of the Dantzig selector [9] for estimating the regression model and we use a de-biasing argument to construct component-wise confidence intervals. Our first main result is to establish upper bounds on the estimation error as a function of the model parameters (the sparsity level s, the expected fraction of observed covariates $\rho_*$, and a measure of the signal strength $\|\beta^*\|_2$). We find that even in an idealized setting where the covariates are assumed to be missing completely at random, somewhat surprisingly and in contrast to the fully-observed setting, there is a dichotomy in the dependence on model parameters and much faster rates are obtained if the covariance matrix of the random design is known. To study this issue further, our second main contribution is to provide lower bounds on the estimation error showing that this discrepancy in rates is unavoidable in a minimax sense. We then consider the problem of high-dimensional inference in the presence of missing data. We construct and analyze confidence intervals using a de-biased estimator. In the presence of missing data, inference is complicated by the fact that the de-biasing matrix is correlated with the pilot estimator and this necessitates the design of a new estimator and a novel analysis. We also complement our mathematical study with extensive simulations on synthetic and semi-synthetic data that show the accuracy of our asymptotic predictions for finite sample sizes.

Parallel Bayesian Global Optimization of Expensive Functions

We consider parallel global optimization of derivative-free expensive-to-evaluate functions, and propose an efficient method based on stochastic approximation for implementing a conceptual Bayesian optimization algorithm proposed by Ginsbourger et al. (2007). To accomplish this, we use infinitessimal perturbation analysis (IPA) to construct a stochastic gradient estimator and show that this estimator is unbiased. We also show that the stochastic gradient ascent algorithm using the constructed gradient estimator converges to a stationary point of the q-EI surface, and therefore, as the number of multiple starts of the gradient ascent algorithm and the number of steps for each start grow large, the one-step Bayes optimal set of points is recovered. We show in numerical experiments that our method for maximizing the q-EI is faster than methods based on closed-form evaluation using high-dimensional integration, when considering many parallel function evaluations, and is comparable in speed when considering few. We also show that the resulting one-step Bayes optimal algorithm for parallel global optimization finds high quality solutions with fewer evaluations that a heuristic based on approximately maximizing the q-EI. A high quality open source implementation of this algorithm is available in the open source Metrics Optimization Engine (MOE).

Gradient Sparsification for Communication-Efficient Distributed Optimization

Modern large scale machine learning applications require stochastic optimization algorithms to be implemented on distributed computational architectures. A key bottleneck is the communication overhead for exchanging information such as stochastic gradients among different workers. In this paper, to reduce the communication cost we propose a convex optimization formulation to minimize the coding length of stochastic gradients. To solve the optimal sparsification efficiently, several simple and fast algorithms are proposed for approximate solution, with theoretical guaranteed for sparseness. Experiments on $\ell_2$ regularized logistic regression, support vector machines, and convolutional neural networks validate our sparsification approaches.

Improved Optimization of Finite Sums with Minibatch Stochastic Variance Reduced Proximal Iterations

We present novel minibatch stochastic optimization methods for empirical risk minimization problems, the methods efficiently leverage variance reduced first-order and sub-sampled higher-order information to accelerate the convergence speed. For quadratic objectives, we prove improved iteration complexity over state-of-the-art under reasonable assumptions. We also provide empirical evidence of the advantages of our method compared to existing approaches in the literature.

A General Distributed Dual Coordinate Optimization Framework for Regularized Loss Minimization

In modern large-scale machine learning applications, the training data are often partitioned and stored on multiple machines. It is customary to employ the "data parallelism" approach, where the aggregated training loss is minimized without moving data across machines. In this paper, we introduce a novel distributed dual formulation for regularized loss minimization problems that can directly handle data parallelism in the distributed setting. This formulation allows us to systematically derive dual coordinate optimization procedures, which we refer to as Distributed Alternating Dual Maximization (DADM). The framework extends earlier studies described in (Boyd et al., 2011; Ma et al., 2015a; Jaggi et al., 2014; Yang, 2013) and has rigorous theoretical analyses. Moreover with the help of the new formulation, we develop the accelerated version of DADM (Acc-DADM) by generalizing the acceleration technique from (Shalev-Shwartz and Zhang, 2014) to the distributed setting. We also provide theoretical results for the proposed accelerated version and the new result improves previous ones (Yang, 2013; Ma et al., 2015a) whose runtimes grow linearly on the condition number. Our empirical studies validate our theory and show that our accelerated approach significantly improves the previous state-of-the-art distributed dual coordinate optimization algorithms.

Memory and Communication Efficient Distributed Stochastic Optimization with Minibatch-Prox

We present and analyze an approach for distributed stochastic optimization which is statistically optimal and achieves near-linear speedups (up to logarithmic factors). Our approach allows a communication-memory tradeoff, with either logarithmic communication but linear memory, or polynomial communication and a corresponding polynomial reduction in required memory. This communication-memory tradeoff is achieved through minibatch-prox iterations (minibatch passive-aggressive updates), where a subproblem on a minibatch is solved at each iteration. We provide a novel analysis for such a minibatch-prox procedure which achieves the statistical optimal rate regardless of minibatch size and smoothness, thus significantly improving on prior work.

Exploiting Strong Convexity from Data with Primal-Dual First-Order Algorithms

We consider empirical risk minimization of linear predictors with convex loss functions. Such problems can be reformulated as convex-concave saddle point problems, and thus are well suitable for primal-dual first-order algorithms. However, primal-dual algorithms often require explicit strongly convex regularization in order to obtain fast linear convergence, and the required dual proximal mapping may not admit closed-form or efficient solution. In this paper, we develop both batch and randomized primal-dual algorithms that can exploit strong convexity from data adaptively and are capable of achieving linear convergence even without regularization. We also present dual-free variants of the adaptive primal-dual algorithms that do not require computing the dual proximal mapping, which are especially suitable for logistic regression.

Efficient coordinate-wise leading eigenvector computation

We develop and analyze efficient "coordinate-wise" methods for finding the leading eigenvector, where each step involves only a vector-vector product. We establish global convergence with overall runtime guarantees that are at least as good as Lanczos's method and dominate it for slowly decaying spectrum. Our methods are based on combining a shift-and-invert approach with coordinate-wise algorithms for linear regression.