We present a data-driven framework for incorporating side information in dynamic optimization under uncertainty. Specifically, our approach uses predictive machine learning methods (such as k-nearest neighbors, kernel regression, and random forests) to weight the relative importance of various data-driven uncertainty sets in a robust optimization formulation. Through a novel measure concentration result for local machine learning methods, we prove that the proposed framework is asymptotically optimal for stochastic dynamic optimization with covariates. We also describe a general-purpose approximation for the proposed framework, based on overlapping linear decision rules, which is computationally tractable and produces high-quality solutions for dynamic problems with many stages. Across a variety of examples in shipment planning, inventory management, and finance, our method achieves improvements of up to 15 over alternatives and requires less than one minute of computation time on problems with twelve stages.
As discussed previously, the methodology in this paper also follows recent work on incorporating covariates in optimization under uncertainty using local predictive methods (such as @math -nearest neighbor regression, kernel regression, and random forests). In particular, the asymptotic optimality justification of @cite_11 in single-stage settings relies on the strong universal consistency for local predictive models (, @cite_6 ). Our proof of asymptotic optimality instead relies on convergence guarantees rooted in distributionally robust optimization. The reason we use a different approach is that the arguments for the convergence for local predictive models from @cite_11 require finite dimensional decision variables. In contrast, the convergence guarantees in this paper apply for dynamic optimization over general spaces of policies.
@cite_6 @cite_11
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We consider a Bayesian approach to variable selection in the presence of high dimensional covariates based on a hierarchical model that places prior distributions on the regression coefficients as well as on the model space. We adopt the well-known spike and slab Gaussian priors with a distinct feature, that is, the prior variances depend on the sample size through which appropriate shrinkage can be achieved. We show the strong selection consistency of the proposed method in the sense that the posterior probability of the true model converges to one even when the number of covariates grows nearly exponentially with the sample size. This is arguably the strongest selection consistency result that has been available in the Bayesian variable selection literature; yet the proposed method can be carried out through posterior sampling with a simple Gibbs sampler. Furthermore, we argue that the proposed method is asymptotically similar to model selection with the @math penalty. We also demonstrate through empirical work the fine performance of the proposed approach relative to some state of the art alternatives. In this paper we show how to accelerate randomized coordinate descent methods and achieve faster convergence rates without paying per-iteration costs in asymptotic running time. In particular, we show how to generalize and efficiently implement a method proposed by Nesterov, giving faster asymptotic running times for various algorithms that use standard coordinate descent as a black box. In addition to providing a proof of convergence for this new general method, we show that it is numerically stable, efficiently implementable, and in certain regimes, asymptotically optimal. To highlight the power of this algorithm, we show how it can used to create faster linear system solvers in several regimes: - We show how this method achieves a faster asymptotic runtime than conjugate gradient for solving a broad class of symmetric positive definite systems of equations. - We improve the convergence guarantees for Kaczmarz methods, a popular technique for image reconstruction and solving over determined systems of equations, by accelerating an algorithm of Strohmer and Vershynin. - We achieve the best known running time for solving Symmetric Diagonally Dominant (SDD) system of equations in the unit-cost RAM model, obtaining a running time of O(m log3 2n (log log n)1 2 log((log n) eps)) by accelerating a recent solver by Beyond the independent interest of these solvers, we believe they highlight the versatility of the approach of this paper and we hope that they will open the door for further algorithmic improvements in the future. In this paper we show how to accelerate randomized coordinate descent methods and achieve faster convergence rates without paying per-iteration costs in asymptotic running time. In particular, we show how to generalize and efficiently implement a method proposed by Nesterov, giving faster asymptotic running times for various algorithms that use standard coordinate descent as a black box. In addition to providing a proof of convergence for this new general method, we show that it is numerically stable, efficiently implementable, and in certain regimes, asymptotically optimal. To highlight the computational power of this algorithm, we show how it can used to create faster linear system solvers in several regimes: - We show how this method achieves a faster asymptotic runtime than conjugate gradient for solving a broad class of symmetric positive definite systems of equations. - We improve the best known asymptotic convergence guarantees for Kaczmarz methods, a popular technique for image reconstruction and solving overdetermined systems of equations, by accelerating a randomized algorithm of Strohmer and Vershynin. - We achieve the best known running time for solving Symmetric Diagonally Dominant (SDD) system of equations in the unit-cost RAM model, obtaining an O(m log^ 3 2 n (log log n)^ 1 2 log (log n eps)) asymptotic running time by accelerating a recent solver by Beyond the independent interest of these solvers, we believe they highlight the versatility of the approach of this paper and we hope that they will open the door for further algorithmic improvements in the future. In this paper, we analyze the convergence of the alternating direction method of multipliers (ADMM) for minimizing a nonconvex and possibly nonsmooth objective function, ( (x_0, ,x_p,y) ), subject to coupled linear equality constraints. Our ADMM updates each of the primal variables (x_0, ,x_p,y ), followed by updating the dual variable. We separate the variable y from (x_i )’s as it has a special role in our analysis. The developed convergence guarantee covers a variety of nonconvex functions such as piecewise linear functions, ( _q ) quasi-norm, Schatten-q quasi-norm ( (0<q<1 )), minimax concave penalty (MCP), and smoothly clipped absolute deviation penalty. It also allows nonconvex constraints such as compact manifolds (e.g., spherical, Stiefel, and Grassman manifolds) and linear complementarity constraints. Also, the (x_0 )-block can be almost any lower semi-continuous function. By applying our analysis, we show, for the first time, that several ADMM algorithms applied to solve nonconvex models in statistical learning, optimization on manifold, and matrix decomposition are guaranteed to converge. Our results provide sufficient conditions for ADMM to converge on (convex or nonconvex) monotropic programs with three or more blocks, as they are special cases of our model. ADMM has been regarded as a variant to the augmented Lagrangian method (ALM). We present a simple example to illustrate how ADMM converges but ALM diverges with bounded penalty parameter ( ). Indicated by this example and other analysis in this paper, ADMM might be a better choice than ALM for some nonconvex nonsmooth problems, because ADMM is not only easier to implement, it is also more likely to converge for the concerned scenarios. In this paper, a novel algorithm to blindly separate an instantaneous linear underdetermined mixture of nonstationary sources is proposed. It means that the number of sources exceeds the number of channels of the available data. The separation is based on the working assumption that the sources are piecewise stationary with a different variance in each block. It proceeds in two steps: 1) estimating the mixing matrix, and 2) computing the optimum beamformer in each block to maximize the signal-to-interference ratio of each separated signal with respect to the remaining signals. Estimating the mixing matrix is accomplished through a specialized tensor decomposition of the set of sample covariance matrices of the received mixture in each block. It utilizes optimum weighting, which allows statistically efficient (CRB attaining) estimation provided that the data obey the assumed Gaussian piecewise stationary model. In simulations, performance of the algorithm is successfully tested on blind separation of 16 speech signals from nine linear instantaneous mixtures of these signals. We investigate a local reparameterizaton technique for greatly reducing the variance of stochastic gradients for variational Bayesian inference (SGVB) of a posterior over model parameters, while retaining parallelizability. This local reparameterization translates uncertainty about global parameters into local noise that is independent across datapoints in the minibatch. Such parameterizations can be trivially parallelized and have variance that is inversely proportional to the mini-batch size, generally leading to much faster convergence. Additionally, we explore a connection with dropout: Gaussian dropout objectives correspond to SGVB with local reparameterization, a scale-invariant prior and proportionally fixed posterior variance. Our method allows inference of more flexibly parameterized posteriors; specifically, we propose variational dropout, a generalization of Gaussian dropout where the dropout rates are learned, often leading to better models. The method is demonstrated through several experiments. In a recent paper, we introduced a trust-region method with variable norms for unconstrained minimization, we proved standard asymptotic convergence results, and we discussed the impact of this method in global optimization. Here we will show that, with a simple modification with respect to the sufficient descent condition and replacing the trust-region approach with a suitable cubic regularization, the complexity of this method for finding approximate first-order stationary points is @math O(ź-3 2). We also prove a complexity result with respect to second-order stationarity. Some numerical experiments are also presented to illustrate the effect of the modification on practical performance. The paper studies distributed static parameter (vector) estimation in sensor networks with nonlinear observation models and noisy intersensor communication. It introduces separably estimable observation models that generalize the observability condition in linear centralized estimation to nonlinear distributed estimation. It studies two distributed estimation algorithms in separably estimable models, the NU (with its linear counterpart LU) and the NLU. Their update rule combines a consensus step (where each sensor updates the state by weight averaging it with its neighbors' states) and an innovation step (where each sensor processes its local current observation). This makes the three algorithms of the consensus + innovations type, very different from traditional consensus. This paper proves consistency (all sensors reach consensus almost surely and converge to the true parameter value), efficiency, and asymptotic unbiasedness. For LU and NU, it proves asymptotic normality and provides convergence rate guarantees. The three algorithms are characterized by appropriately chosen decaying weight sequences. Algorithms LU and NU are analyzed in the framework of stochastic approximation theory; algorithm NLU exhibits mixed time-scale behavior and biased perturbations, and its analysis requires a different approach that is developed in this paper. This paper is motivated by two applications, namely i) generalizations of cuckoo hashing, a computationally simple approach to assigning keys to objects, and ii) load balancing in content distribution networks, where one is interested in determining the impact of content replication on performance. These two problems admit a common abstraction: in both scenarios, performance is characterized by the maximum weight of a generalization of a matching in a bipartite graph, featuring node and edge capacities. Our main result is a law of large numbers characterizing the asymptotic maximum weight matching in the limit of large bipartite random graphs, when the graphs admit a local weak limit that is a tree. This result specializes to the two application scenarios, yielding new results in both contexts. In contrast with previous results, the key novelty is the ability to handle edge capacities with arbitrary integer values. An analysis of belief propagation algorithms (BP) with multivariate belief vectors underlies the proof. In particular, we show convergence of the corresponding BP by exploiting monotonicity of the belief vectors with respect to the so-called upshifted likelihood ratio stochastic order. This auxiliary result can be of independent interest, providing a new set of structural conditions which ensure convergence of BP. Consider the problem of joint parameter estimation and prediction in a Markov random field: that is, the model parameters are estimated on the basis of an initial set of data, and then the fitted model is used to perform prediction (e.g., smoothing, denoising, interpolation) on a new noisy observation. Working under the restriction of limited computation, we analyze a joint method in which the same convex variational relaxation is used to construct an M-estimator for fitting parameters, and to perform approximate marginalization for the prediction step. The key result of this paper is that in the computation-limited setting, using an inconsistent parameter estimator (i.e., an estimator that returns the "wrong" model even in the infinite data limit) is provably beneficial, since the resulting errors can partially compensate for errors made by using an approximate prediction technique. En route to this result, we analyze the asymptotic properties of M-estimators based on convex variational relaxations, and establish a Lipschitz stability property that holds for a broad class of convex variational methods. This stability result provides additional incentive, apart from the obvious benefit of unique global optima, for using message-passing methods based on convex variational relaxations. We show that joint estimation prediction based on the reweighted sum-product algorithm substantially outperforms a commonly used heuristic based on ordinary sum-product. We study statistical inference and distributionally robust solution methods for stochastic optimization problems, focusing on confidence intervals for optimal values and solutions that achieve exact coverage asymptotically. We develop a generalized empirical likelihood framework---based on distributional uncertainty sets constructed from nonparametric @math -divergence balls---for Hadamard differentiable functionals, and in particular, stochastic optimization problems. As consequences of this theory, we provide a principled method for choosing the size of distributional uncertainty regions to provide one- and two-sided confidence intervals that achieve exact coverage. We also give an asymptotic expansion for our distributionally robust formulation, showing how robustification regularizes problems by their variance. Finally, we show that optimizers of the distributionally robust formulations we study enjoy (essentially) the same consistency properties as those in classical sample average approximations. Our general approach applies to quickly mixing stationary sequences, including geometrically ergodic Harris recurrent Markov chains. We discuss a new robust convergence analysis of the well-known subspace iteration algorithm for computing the dominant singular vectors of a matrix, also known as simultaneous iteration or power method. The result characterizes the convergence behavior of the algorithm when a large amount noise is introduced after each matrix-vector multiplication. While interesting in its own right, the main motivation comes from the problem of privacy-preserving spectral analysis where noise is added in order to achieve the privacy guarantee known as differential privacy. This result leads to nearly tight worst-case bounds for the problem of computing a differentially private low-rank approximation in the spectral norm. Our results extend to privacy-preserving principal component analysis. We obtain improvements for several variants of differential privacy that have been considered. The running time of our algorithm is nearly linear in the input sparsity leading to strong improvements in running time over previous work. Complementing our worst-case bounds, we show that the error dependence of our algorithm on the matrix dimension can be replaced by a tight dependence on the coherence of the matrix. This parameter is always bounded by the matrix dimension but often much smaller. Indeed, the assumption of low coherence is essential in several machine learning and signal processing applications. type="main" xml:id="rssb12047-abs-0001"> We introduce a new estimator, the simultaneous multiscale change point estimator SMUCE, for the change point problem in exponential family regression. An unknown step function is estimated by minimizing the number of change points over the acceptance region of a multiscale test at a level α. The probability of overestimating the true number of change points K is controlled by the asymptotic null distribution of the multiscale test statistic. Further, we derive exponential bounds for the probability of underestimating K. By balancing these quantities, α will be chosen such that the probability of correctly estimating K is maximized. All results are even non-asymptotic for the normal case. On the basis of these bounds, we construct (asymptotically) honest confidence sets for the unknown step function and its change points. At the same time, we obtain exponential bounds for estimating the change point locations which for example yield the minimax rate O ( n − 1 ) up to a log-term. Finally, the simultaneous multiscale change point estimator achieves the optimal detection rate of vanishing signals as n→∞, even for an unbounded number of change points. We illustrate how dynamic programming techniques can be employed for efficient computation of estimators and confidence regions. The performance of the multiscale approach proposed is illustrated by simulations and in two cutting edge applications from genetic engineering and photoemission spectroscopy. We analyze the convergence of gradient-based optimization algorithms that base their updates on delayed stochastic gradient information. The main application of our results is to the development of gradient-based distributed optimization algorithms where a master node performs parameter updates while worker nodes compute stochastic gradients based on local information in parallel, which may give rise to delays due to asynchrony. We take motivation from statistical problems where the size of the data is so large that it cannot fit on one computer; with the advent of huge datasets in biology, astronomy, and the internet, such problems are now common. Our main contribution is to show that for smooth stochastic problems, the delays are asymptotically negligible and we can achieve order-optimal convergence results. In application to distributed optimization, we develop procedures that overcome communication bottlenecks and synchronization requirements. We show @math -node architectures whose optimization error in stochastic problems---in spite of asynchronous delays---scales asymptotically as @math after @math iterations. This rate is known to be optimal for a distributed system with @math nodes even in the absence of delays. We additionally complement our theoretical results with numerical experiments on a statistical machine learning task. We consider the estimation of an i.i.d. random vector observed through a linear transform followed by a componentwise, probabilistic (possibly nonlinear) measurement channel. A novel algorithm, called generalized approximate message passing (GAMP), is presented that provides computationally efficient approximate implementations of max-sum and sum-problem loopy belief propagation for such problems. The algorithm extends earlier approximate message passing methods to incorporate arbitrary distributions on both the input and output of the transform and can be applied to a wide range of problems in nonlinear compressed sensing and learning. Extending an analysis by Bayati and Montanari, we argue that the asymptotic componentwise behavior of the GAMP method under large, i.i.d. Gaussian transforms is described by a simple set of state evolution (SE) equations. From the SE equations, one can predict the asymptotic value of virtually any componentwise performance metric including mean-squared error or detection accuracy. Moreover, the analysis is valid for arbitrary input and output distributions, even when the corresponding optimization problems are non-convex. The results match predictions by Guo and Wang for relaxed belief propagation on large sparse matrices and, in certain instances, also agree with the optimal performance predicted by the replica method. The GAMP methodology thus provides a computationally efficient methodology, applicable to a large class of non-Gaussian estimation problems with precise asymptotic performance guarantees. Effective training of deep neural networks suffers from two main issues. The first is that the parameter spaces of these models exhibit pathological curvature. Recent methods address this problem by using adaptive preconditioning for Stochastic Gradient Descent (SGD). These methods improve convergence by adapting to the local geometry of parameter space. A second issue is overfitting, which is typically addressed by early stopping. However, recent work has demonstrated that Bayesian model averaging mitigates this problem. The posterior can be sampled by using Stochastic Gradient Langevin Dynamics (SGLD). However, the rapidly changing curvature renders default SGLD methods inefficient. Here, we propose combining adaptive preconditioners with SGLD. In support of this idea, we give theoretical properties on asymptotic convergence and predictive risk. We also provide empirical results for Logistic Regression, Feedforward Neural Nets, and Convolutional Neural Nets, demonstrating that our preconditioned SGLD method gives state-of-the-art performance on these models. We consider the estimation of an independent and identically distributed (i.i.d.) (possibly non-Gaussian) vector x ∈ Rn from measurements y ∈ Rm obtained by a general cascade model consisting of a known linear transform followed by a probabilistic componentwise (possibly nonlinear) measurement channel. A novel method, called adaptive generalized approximate message passing (adaptive GAMP) is presented. It enables the joint learning of the statistics of the prior and measurement channel along with estimation of the unknown vector x. We prove that, for large i.i.d. Gaussian transform matrices, the asymptotic componentwise behavior of the adaptive GAMP is predicted by a simple set of scalar state evolution equations. In addition, we show that the adaptive GAMP yields asymptotically consistent parameter estimates, when a certain maximum-likelihood estimation can be performed in each step. This implies that the algorithm achieves a reconstruction quality equivalent to the oracle algorithm that knows the correct parameter values. Remarkably, this result applies to essentially arbitrary parametrizations of the unknown distributions, including nonlinear and non-Gaussian ones. The adaptive GAMP methodology thus provides a systematic, general and computationally efficient method applicable to a large range of linear-nonlinear models with provable guarantees. Nonconvex optimization arises in many areas of computational science and engineering. However, most nonconvex optimization algorithms are only known to have local convergence or subsequence convergence properties. In this paper, we propose an algorithm for nonconvex optimization and establish its global convergence (of the whole sequence) to a critical point. In addition, we give its asymptotic convergence rate and numerically demonstrate its efficiency. In our algorithm, the variables of the underlying problem are either treated as one block or multiple disjoint blocks. It is assumed that each non-differentiable component of the objective function, or each constraint, applies only to one block of variables. The differentiable components of the objective function, however, can involve multiple blocks of variables together. Our algorithm updates one block of variables at a time by minimizing a certain prox-linear surrogate, along with an extrapolation to accelerate its convergence. The order of update can be either deterministically cyclic or randomly shuffled for each cycle. In fact, our convergence analysis only needs that each block be updated at least once in every fixed number of iterations. We show its global convergence (of the whole sequence) to a critical point under fairly loose conditions including, in particular, the Kurdyka–Łojasiewicz condition, which is satisfied by a broad class of nonconvex nonsmooth applications. These results, of course, remain valid when the underlying problem is convex. We apply our convergence results to the coordinate descent iteration for non-convex regularized linear regression, as well as a modified rank-one residue iteration for nonnegative matrix factorization. We show that both applications have global convergence. Numerically, we tested our algorithm on nonnegative matrix and tensor factorization problems, where random shuffling clearly improves the chance to avoid low-quality local solutions. We prove tight upper bounds on the logarithmic derivative of the independence and matching polynomials of d-regular graphs. For independent sets, this theorem is a strengthening of Kahn's result that a disjoint union of copies of Kd;d maximizes the number of independent sets of a bipartite d-regular graph, Galvin and Tet ali's result that the independence polynomial is maximized by the same, and Zhao's extension of both results to all d-regular graphs. For matchings, this shows that the matching polynomial and the total number of matchings of a d-regular graph are maximized by a union of copies of Kd;d. Using this we prove the asymptotic upper matching conjecture of Friedland, Krop, Lundow, and Markstrom. In probabilistic language, our main theorems state that for all d-regular graphs and all �, the occupancy fraction of the hard-core model and the edge occupancy fraction of the monomer-dimer model with fugacity � are maximized by Kd;d. Our method involves constrained optimization problems over distributions of random variables and applies to all d-regular graphs directly, without a reduction to the bipartite case. Using a variant of the method we prove a lower bound on the occupancy fraction of the hard-core model on any d-regular, vertex-transitive, bipartite graph: the occupancy fraction of such a graph is strictly greater than the occupancy fraction of the unique translationinvariant hard-core measure on the infinite d-regular tree These lecture notes highlight the mathematical and computational structure relating to the formulation of, and development of algorithms for, the Bayesian approach to inverse problems in differential equations. This approach is fundamental in the quantification of uncertainty within applications in volving the blending of mathematical models with data. The finite dimensional situation is described first, along with some motivational examples. Then the development of probability measures on separable Banach space is undertaken, using a random series over an infinite set of functions to construct draws; these probability measures are used as priors in the Bayesian approach to inverse problems. Regularity of draws from the priors is studied in the natural Sobolev or Besov spaces implied by the choice of functions in the random series construction, and the Kolmogorov continuity theorem is used to extend regularity considerations to the space of Holder continuous functions. Bayes’ theorem is de rived in this prior setting, and here interpreted as finding conditions under which the posterior is absolutely continuous with respect to the prior, and determining a formula for the Radon-Nikodym derivative in terms of the likelihood of the data. Having established the form of the posterior, we then describe various properties common to it in the infinite dimensional setting. These properties include well-posedness, approximation theory, and the existence of maximum a posteriori estimators. We then describe measure-preserving dynamics, again on the infinite dimensional space, including Markov chain-Monte C arlo and sequential Monte Carlo methods, and measure-preserving reversible stochastic differential equations. By formulating the theory and algorithms on the underlying infinite dimensional space, we obtain a framework suitable for rigorous analysis of the accuracy of reconstructions, of computational complexity, as well as naturally constructing algorithms which perform well under mesh refinement, since they are inherently well-defined in infinite dimensions.