We perform the first study of the tradeoff space of access methods and replication to support statistical analytics using first-order methods executed in the main memory of a Non-Uniform Memory Access (NUMA) machine. Statistical analytics systems differ from conventional SQL-analytics in the amount and types of memory incoherence they can tolerate. Our goal is to understand tradeoffs in accessing the data in row- or column-order and at what granularity one should share the model and data for a statistical task. We study this new tradeoff space, and discover there are tradeoffs between hardware and statistical efficiency. We argue that our tradeoff study may provide valuable information for designers of analytics engines: for each system we consider, our prototype engine can run at least one popular task at least 100x faster. We conduct our study across five architectures using popular models including SVMs, logistic regression, Gibbs sampling, and neural networks.
* Mathematical Optimization Many statistical analytics tasks are mathematical optimization problems. Recently, the mathematical optimization community has been looking at how to parallelize optimization problems @cite_57 @cite_47 @cite_44 . For example, @cite_57 for SGD and Shotgun @cite_58 for SCD. A lock-free asynchronous variant was recently established by @cite_44 .
@cite_57 @cite_44 @cite_47 @cite_58
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We address the issue of using mini-batches in stochastic optimization of SVMs. We show that the same quantity, the spectral norm of the data, controls the parallelization speedup obtained for both primal stochastic subgradient descent (SGD) and stochastic dual coordinate ascent (SCDA) methods and use it to derive novel variants of mini-batched SDCA. Our guarantees for both methods are expressed in terms of the original nonsmooth primal problem based on the hinge-loss. We consider stochastic strongly convex optimization with a complex inequality constraint. This complex inequality constraint may lead to computationally expensive projections in algorithmic iterations of the stochastic gradient descent (SGD) methods. To reduce the computation costs pertaining to the projections, we propose an Epoch-Projection Stochastic Gradient Descent (Epro-SGD) method. The proposed Epro-SGD method consists of a sequence of epochs; it applies SGD to an augmented objective function at each iteration within the epoch, and then performs a projection at the end of each epoch. Given a strongly convex optimization and for a total number of @math iterations, Epro-SGD requires only @math projections, and meanwhile attains an optimal convergence rate of @math , both in expectation and with a high probability. To exploit the structure of the optimization problem, we propose a proximal variant of Epro-SGD, namely Epro-ORDA, based on the optimal regularized dual averaging method. We apply the proposed methods on real-world applications; the empirical results demonstrate the effectiveness of our methods. We investigate the clustering performances of the relaxed @math means in the setting of sub-Gaussian Mixture Model (sGMM) and Stochastic Block Model (SBM). After identifying the appropriate signal-to-noise ratio (SNR), we prove that the misclassification error decay exponentially fast with respect to this SNR. These partial recovery bounds for the relaxed @math means improve upon results currently known in the sGMM setting. In the SBM setting, applying the relaxed @math means SDP allows to handle general connection probabilities whereas other SDPs investigated in the literature are restricted to the assortative case (where within group probabilities are larger than between group probabilities). Again, this partial recovery bound complements the state-of-the-art results. All together, these results put forward the versatility of the relaxed @math means. We present an algorithm for minimizing a sum of functions that combines the computational efficiency of stochastic gradient descent (SGD) with the second order curvature information leveraged by quasi-Newton methods. We unify these disparate approaches by maintaining an independent Hessian approximation for each contributing function in the sum. We maintain computational tractability and limit memory requirements even for high dimensional optimization problems by storing and manipulating these quadratic approximations in a shared, time evolving, low dimensional subspace. Each update step requires only a single contributing function or minibatch evaluation (as in SGD), and each step is scaled using an approximate inverse Hessian and little to no adjustment of hyperparameters is required (as is typical for quasi-Newton methods). This algorithm contrasts with earlier stochastic second order techniques that treat the Hessian of each contributing function as a noisy approximation to the full Hessian, rather than as a target for direct estimation. We experimentally demonstrate improved convergence on seven diverse optimization problems. The algorithm is released as open source Python and MATLAB packages. We study optimization algorithms based on variance reduction for stochastic gradient descent (SGD). Remarkable recent progress has been made in this direction through development of algorithms like SAG, SVRG, SAGA. These algorithms have been shown to outperform SGD, both theoretically and empirically. However, asynchronous versions of these algorithms—a crucial requirement for modern large-scale applications—have not been studied. We bridge this gap by presenting a unifying framework for many variance reduction techniques. Subsequently, we propose an asynchronous algorithm grounded in our framework, and prove its fast convergence. An important consequence of our general approach is that it yields asynchronous versions of variance reduction algorithms such as SVRG and SAGA as a byproduct. Our method achieves near linear speedup in sparse settings common to machine learning. We demonstrate the empirical performance of our method through a concrete realization of asynchronous SVRG. We propose Shotgun, a parallel coordinate descent algorithm for minimizing L1-regularized losses. Though coordinate descent seems inherently sequential, we prove convergence bounds for Shotgun which predict linear speedups, up to a problem-dependent limit. We present a comprehensive empirical study of Shotgun for Lasso and sparse logistic regression. Our theoretical predictions on the potential for parallelism closely match behavior on real data. Shotgun outperforms other published solvers on a range of large problems, proving to be one of the most scalable algorithms for L1. S U M M A R Y Non-linear inverse problems in the geosciences often involve probabilistic sampling of multimodal density functions or global optimization and sometimes both. Efficient algorithmic tools for carrying out sampling or optimization in challenging cases are of major interest. Here results are presented of some numerical experiments with a technique, known as Parallel Tempering, which originated in the field of computational statistics but is finding increasing numbers of applications in fields ranging from Chemical Physics to Astronomy. To date, experience in use of Parallel Tempering within earth sciences problems is very limited. In this paper, we describe Parallel Tempering and compare it to related methods of Simulated Annealing and Simulated Tempering for optimization and sampling, respectively. A key feature of Parallel Tempering is that it satisfies the detailed balance condition required for convergence of Markov chain Monte Carlo (McMC) algorithms while improving the efficiency of probabilistic sampling. Numerical results are presented on use of Parallel Tempering for trans-dimensional inversion of synthetic seismic receiver functions and also the simultaneous fitting of multiple receiver functions using global optimization. These suggest that its use can significantly accelerate sampling algorithms and improve exploration of parameter space in optimization. Parallel Tempering is a meta-algorithm which may be used together with many existing McMC sampling and direct search optimization techniques. It’s generality and demonstrated performance suggests that there is significant potential for applications to both sampling and optimization problems in the geosciences. Consider the convex optimization problem min x ƒ (g(x)) where both ƒ and g are unknown but can be estimated through sampling. We consider the stochastic compositional gradient descent method (SCGD) that updates based on random function and subgradient evaluations, which are generated by a conditional sampling oracle. We focus on the case where samples are corrupted with Markov noise. Under certain diminishing stepsize assumptions, we prove that the iterate of SCGD converges almost surely to an optimal solution if such a solution exists. Under specific constant stepsize assumptions, we obtain finite-sample error bounds for the averaged iterates of the algorithm. We illustrate an application to online value evaluation in dynamic programming. We propose a decomposition framework for the parallel optimization of the sum of a differentiable (possibly nonconvex) function and a (block) separable nonsmooth, convex one. The latter term is usually employed to enforce structure in the solution, typically sparsity. Our framework is very flexible and includes both fully parallel Jacobi schemes and Gauss–Seidel (i.e., sequential) ones, as well as virtually all possibilities “in between” with only a subset of variables updated at each iteration. Our theoretical convergence results improve on existing ones, and numerical results on LASSO, logistic regression, and some nonconvex quadratic problems show that the new method consistently outperforms existing algorithms. We study optimization algorithms based on variance reduction for stochastic gradient descent (SGD). Remarkable recent progress has been made in this direction through development of algorithms like SAG, SVRG, SAGA. These algorithms have been shown to outperform SGD, both theoretically and empirically. However, asynchronous versions of these algorithms---a crucial requirement for modern large-scale applications---have not been studied. We bridge this gap by presenting a unifying framework for many variance reduction techniques. Subsequently, we propose an asynchronous algorithm grounded in our framework, and prove its fast convergence. An important consequence of our general approach is that it yields asynchronous versions of variance reduction algorithms such as SVRG and SAGA as a byproduct. Our method achieves near linear speedup in sparse settings common to machine learning. We demonstrate the empirical performance of our method through a concrete realization of asynchronous SVRG. We present a convex framework to learn sequential decisions and apply it to the problem of learning under a budget. We consider the structure proposed in [1], where sensor measurements are acquired in a sequence. The goal after acquiring each new measurement is to make a decision whether to stop and classify or to pay the cost of using the next sensor in the sequence. We introduce a novel formulation of an empirical risk objective for the multi stage sequential decision problem. This objective naturally lends itself to a non-convex multilinear formulation. Nevertheless, we derive a novel perspective that leads to a tight convex objective. This is accomplished by expressing the empirical risk in terms of linear superposition of indicator functions. We then derive an LP formulation by utilizing hinge loss surrogates. Our LP achieves or exceeds the empirical performance of the nonconvex alternating algorithm that requires a large number of random initializations. Consequently, the LP has the advantage of guaranteed convergence, global optimality, repeatability and computation eciency. Many problems in signal processing and statistical inference involve finding sparse solutions to under-determined, or ill-conditioned, linear systems of equations. A standard approach consists in minimizing an objective function which includes a quadratic (squared ) error term combined with a sparseness-inducing regularization term. Basis pursuit, the least absolute shrinkage and selection operator (LASSO), wavelet-based deconvolution, and compressed sensing are a few well-known examples of this approach. This paper proposes gradient projection (GP) algorithms for the bound-constrained quadratic programming (BCQP) formulation of these problems. We test variants of this approach that select the line search parameters in different ways, including techniques based on the Barzilai-Borwein method. Computational experiments show that these GP approaches perform well in a wide range of applications, often being significantly faster (in terms of computation time) than competing methods. Although the performance of GP methods tends to degrade as the regularization term is de-emphasized, we show how they can be embedded in a continuation scheme to recover their efficient practical performance. We consider sequential decision problems under uncertainty, where we seek to optimize an unknown function from noisy samples. This requires balancing exploration (learning about the objective) and exploitation (localizing the maximum), a problem well-studied in the multiarmed bandit literature. In many applications, however, we require that the sampled function values exceed some prespecified "safety" threshold, a requirement that existing algorithms fail to meet. Examples include medical applications where patient comfort must be guaranteed, recommender systems aiming to avoid user dissatisfaction, and robotic control, where one seeks to avoid controls causing physical harm to the platform. We tackle this novel, yet rich, set of problems under the assumption that the unknown function satisfies regularity conditions expressed via a Gaussian process prior. We develop an efficient algorithm called SAFEOPT, and theoretically guarantee its convergence to a natural notion of optimum reachable under safety constraints. We evaluate SAFEOPT on synthetic data, as well as two real applications: movie recommendation, and therapeutic spinal cord stimulation. We propose a novel schedulability analysis for verifying the feasibility of large periodic task sets under the rate monotonic algorithm when the exact test cannot be applied on line due to prohibitively long execution times. The proposed test has the same complexity as the original Liu and Layland (1973) bound, but it is less pessimistic, thus allowing it to accept task sets that would be rejected using the original approach. The performance of the proposed approach is evaluated with respect to the classical Liu and Layland method and theoretical bounds are derived as a function of n (the number of tasks) and for the limit case of n tending to infinity. The analysis is also extended to include aperiodic servers and blocking times due to concurrency control protocols. Extensive simulations on synthetic tasks sets are presented to compare the effectiveness of the proposed test with respect to the Liu and Layland method and the exact response time analysis. Different task models have been proposed to represent the parallel structure of real-time tasks executing on manycore platforms: fork join, synchronous parallel, DAG-based, etc. Despite different schedulability tests and resource augmentation bounds are available for these task systems, we experience difficulties in applying such results to real application scenarios, where the execution flow of parallel tasks is characterized by multiple (and nested) conditional structures. When a conditional branch drives the number and size of sub-jobs to spawn, it is hard to decide which execution path to select for modeling the worst-case scenario. To circumvent this problem, we integrate control flow information in the task model, considering conditional parallel tasks (cp-tasks) represented by DAGs composed of both precedence and conditional edges. For this task model, we identify meaningful parameters that characterize the schedulability of the system, and derive efficient algorithms to compute them. A response time analysis based on these parameters is then presented for different scheduling policies. A set of simulations shows that the proposed approach allows efficiently checking the schedulability of the addressed systems, and that it significantly tightens the schedulability analysis of non-conditional (e.g., Classic DAG) tasks over existing approaches. Sequential quadratic programming (SQP) methods have proved highly effective for solving constrained optimization problems with smooth nonlinear functions in the objective and constraints. Here we consider problems with general inequality constraints (linear and nonlinear). We assume that first derivatives are available and that the constraint gradients are sparse. We discuss an SQP algorithm that uses a smooth augmented Lagrangian merit function and makes explicit provision for infeasibility in the original problem and the QP subproblems. SNOPT is a particular implementation that makes use of a semidefinite QP solver. It is based on a limited-memory quasi-Newton approximation to the Hessian of the Lagrangian and uses a reduced-Hessian algorithm (SQOPT) for solving the QP subproblems. It is designed for problems with many thousands of constraints and variables but a moderate number of degrees of freedom (say, up to 2000). An important application is to trajectory optimization in the aerospace industry. Numerical results are given for most problems in the CUTE and COPS test collections (about 900 examples). We present parallel approximation algorithms for maximization problems expressible by integer linear programs of a restricted syntactic form introduced by [BKT96]. One of our motivations was to show whether the approximation results in the framework of holds in the parallel setting. Our results are a confirmation of this, and thus we have a new common framework for both computational settings. Also, we prove almost tight non-approximability results, thus solving a main open question of We obtain the results through the constraint satisfaction problem over multi-valued domains, for which we show non-approximability results and develop parallel approximation algorithms. Our parallel approximation algorithms are based on linear programming and random rounding; they are better than previously known sequential algorithms. The non-approximability results are based on new recent progress in the fields of Probabilistically Checkable Proofs and Multi-Prover One-Round Proof Systems [Raz95, Has97, AS97, RS97]. The stochastic composition optimization proposed recently by [2014] minimizes the objective with the compositional expectation form: @math It summarizes many important applications in machine learning, statistics, and finance. In this paper, we consider the finite-sum scenario for composition optimization: [ f (x) := 1 n i = 1 ^n F_i ( 1 m j = 1 ^m G_j (x) ). ] We propose two algorithms to solve this problem by combining the stochastic compositional gradient descent (SCGD) and the stochastic variance reduced gradient (SVRG) technique. A constant linear convergence rate is proved for strongly convex optimization, which substantially improves the sublinear rate @math of the best known algorithm. We consider the problem of distributed stochastic optimization, where each of several machines has access to samples from the same source distribution, and the goal is to jointly optimize the expected objective w.r.t. the source distribution, minimizing: (1) overall runtime; (2) communication costs; (3) number of samples used. We study this problem systematically, highlighting fundamental limitations, and differences versus distributed consensus problems where each machine has a different, independent, objective. We show how the best known guarantees are obtained by an accelerated mini-batched SGD approach, and contrast the runtime and sample costs of the approach with those of other distributed optimization algorithms. Given an arc-weighted directed graph G = (V, A, spl lscr ) and a pair of nodes s, t, we seek to find an s-t walk of length at most B that maximizes some given function f of the set of nodes visited by the walk. The simplest case is when we seek to maximize the number of nodes visited: this is called the orienteering problem. Our main result is a quasi-polynomial time algorithm that yields an O(log OPT) approximation for this problem when f is a given submodular set function. We then extend it to the case when a node v is counted as visited only if the walk reaches v in its time window [R(v), D(v)]. We apply the algorithm to obtain several new results. First, we obtain an O(log OPT) approximation for a generalization of the orienteering problem in which the profit for visiting each node may vary arbitrarily with time. This captures the time window problem considered earlier for which, even in undirected graphs, the best approximation ratio known [Bansal, (2004)] is O(log sup 2 OPT). The second application is an O(log sup 2 k) approximation for the k-TSP problem in directed graphs (satisfying asymmetric triangle inequality). This is the first non-trivial approximation algorithm for this problem. The third application is an O(log sup 2 k) approximation (in quasi-poly time) for the group Steiner problem in undirected graphs where k is the number of groups. This improves earlier ratios (Garg, ) by a logarithmic factor and almost matches the inapproximability threshold on trees (Halperin and Krauthgamer, 2003). This connection to group Steiner trees also enables us to prove that the problem we consider is hard to approximate to a ratio better than spl Omega (log sup 1- spl epsi OPT), even in undirected graphs. Even though our algorithm runs in quasi-poly time, we believe that the implications for the approximability of several basic optimization problems are interesting.