Variational inference is a popular alternative to Markov chain Monte Carlo methods that constructs a Bayesian posterior approximation by minimizing a discrepancy to the true posterior within a pre-specified family. This converts Bayesian inference into an optimization problem, enabling the use of simple and scalable stochastic optimization algorithms. However, a key limitation of variational inference is that the optimal approximation is typically not tractable to compute; even in simple settings the problem is nonconvex. Thus, recently developed statistical guarantees -- which all involve the (data) asymptotic properties of the optimal variational distribution -- are not reliably obtained in practice. In this work, we provide two major contributions: a theoretical analysis of the asymptotic convexity properties of variational inference in the popular setting with a Gaussian family; and consistent stochastic variational inference (CSVI), an algorithm that exploits these properties to find the optimal approximation in the asymptotic regime. CSVI consists of a tractable initialization procedure that finds the local basin of the optimal solution, and a scaled gradient descent algorithm that stays locally confined to that basin. Experiments on nonconvex synthetic and real-data examples show that compared with standard stochastic gradient descent, CSVI improves the likelihood of obtaining the globally optimal posterior approximation.
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We develop a new method of online inference for a vector of parameters estimated by the Polyak-Ruppert averaging procedure of stochastic gradient descent (SGD) algorithms. We leverage insights from time series regression in econometrics and construct asymptotically pivotal statistics via random scaling. Our approach is fully operational with online data and is rigorously underpinned by a functional central limit theorem. Our proposed inference method has a couple of key advantages over the existing methods. First, the test statistic is computed in an online fashion with only SGD iterates and the critical values can be obtained without any resampling methods, thereby allowing for efficient implementation suitable for massive online data. Second, there is no need to estimate the asymptotic variance and our inference method is shown to be robust to changes in the tuning parameters for SGD algorithms in simulation experiments with synthetic data.
While many areas of machine learning have benefited from the increasing availability of large and varied datasets, the benefit to causal inference has been limited given the strong assumptions needed to ensure identifiability of causal effects; these are often not satisfied in real-world datasets. For example, many large observational datasets (e.g., case-control studies in epidemiology, click-through data in recommender systems) suffer from selection bias on the outcome, which makes the average treatment effect (ATE) unidentifiable. We propose a general algorithm to estimate causal effects from \emph{multiple} data sources, where the ATE may be identifiable only in some datasets but not others. The key idea is to construct control variates using the datasets in which the ATE is not identifiable. We show theoretically that this reduces the variance of the ATE estimate. We apply this framework to inference from observational data under outcome selection bias, assuming access to an auxiliary small dataset from which we can obtain a consistent estimate of the ATE. We construct a control variate by taking the difference of the odds ratio estimates from the two datasets. Across simulations and two case studies with real data, we show that this control variate can significantly reduce the variance of the ATE estimate.
In this paper, we characterize the asymptotic and large scale behavior of the eigenvalues of wavelet random matrices in high dimensions. We assume that possibly non-Gaussian, finite-variance $p$-variate measurements are made of a low-dimensional $r$-variate ($r \ll p$) fractional stochastic process with non-canonical scaling coordinates and in the presence of additive high-dimensional noise. The measurements are correlated both time-wise and between rows. We show that the $r$ largest eigenvalues of the wavelet random matrices, when appropriately rescaled, converge to scale invariant functions in the high-dimensional limit. By contrast, the remaining $p-r$ eigenvalues remain bounded. Under additional assumptions, we show that, up to a log transformation, the $r$ largest eigenvalues of wavelet random matrices exhibit asymptotically Gaussian distributions. The results have direct consequences for statistical inference.
We consider the inference problem for high-dimensional linear models, when covariates have an underlying spatial organization reflected in their correlation. A typical example of such a setting is high-resolution imaging, in which neighboring pixels are usually very similar. Accurate point and confidence intervals estimation is not possible in this context with many more covariates than samples, furthermore with high correlation between covariates. This calls for a reformulation of the statistical inference problem, that takes into account the underlying spatial structure: if covariates are locally correlated, it is acceptable to detect them up to a given spatial uncertainty. We thus propose to rely on the $\delta$-FWER, that is the probability of making a false discovery at a distance greater than $\delta$ from any true positive. With this target measure in mind, we study the properties of ensembled clustered inference algorithms which combine three techniques: spatially constrained clustering, statistical inference, and ensembling to aggregate several clustered inference solutions. We show that ensembled clustered inference algorithms control the $\delta$-FWER under standard assumptions for $\delta$ equal to the largest cluster diameter. We complement the theoretical analysis with empirical results, demonstrating accurate $\delta$-FWER control and decent power achieved by such inference algorithms.
Since network data commonly consists of observations on a single large network, researchers often partition the network into clusters in order to apply cluster-robust inference methods. All existing such methods require clusters to be asymptotically independent. We prove under mild conditions that, in order for this requirement to hold for network-dependent data, it is necessary and sufficient for clusters to have low conductance, the ratio of edge boundary size to volume. This yields a simple measure of cluster quality. We find in simulations that, when clusters have low conductance, cluster-robust methods outperform HAC estimators in terms of size control. However, for important classes of networks lacking low-conductance clusters, the methods can exhibit substantial size distortion. To assess the existence of low-conductance clusters and construct them, we draw on results in spectral graph theory that connect conductance to the spectrum of the graph Laplacian. Based on these results, we propose to use the spectrum to determine the number of low-conductance clusters and spectral clustering to construct them.
Linear regression is a fundamental modeling tool in statistics and related fields. In this paper, we study an important variant of linear regression in which the predictor-response pairs are partially mismatched. We use an optimization formulation to simultaneously learn the underlying regression coefficients and the permutation corresponding to the mismatches. The combinatorial structure of the problem leads to computational challenges. We propose and study a simple greedy local search algorithm for this optimization problem that enjoys strong theoretical guarantees and appealing computational performance. We prove that under a suitable scaling of the number of mismatched pairs compared to the number of samples and features, and certain assumptions on problem data; our local search algorithm converges to a nearly-optimal solution at a linear rate. In particular, in the noiseless case, our algorithm converges to the global optimal solution with a linear convergence rate. We also propose an approximate local search step that allows us to scale our approach to much larger instances. We conduct numerical experiments to gather further insights into our theoretical results and show promising performance gains compared to existing approaches.
We propose a novel method for sampling and optimization tasks based on a stochastic interacting particle system. We explain how this method can be used for the following two goals: (i) generating approximate samples from a given target distribution; (ii) optimizing a given objective function. The approach is derivative-free and affine invariant, and is therefore well-suited for solving inverse problems defined by complex forward models: (i) allows generation of samples from the Bayesian posterior and (ii) allows determination of the maximum a posteriori estimator. We investigate the properties of the proposed family of methods in terms of various parameter choices, both analytically and by means of numerical simulations. The analysis and numerical simulation establish that the method has potential for general purpose optimization tasks over Euclidean space; contraction properties of the algorithm are established under suitable conditions, and computational experiments demonstrate wide basins of attraction for various specific problems. The analysis and experiments also demonstrate the potential for the sampling methodology in regimes in which the target distribution is unimodal and close to Gaussian; indeed we prove that the method recovers a Laplace approximation to the measure in certain parametric regimes and provide numerical evidence that this Laplace approximation attracts a large set of initial conditions in a number of examples.
We propose a new method of estimation in topic models, that is not a variation on the existing simplex finding algorithms, and that estimates the number of topics K from the observed data. We derive new finite sample minimax lower bounds for the estimation of A, as well as new upper bounds for our proposed estimator. We describe the scenarios where our estimator is minimax adaptive. Our finite sample analysis is valid for any number of documents (n), individual document length (N_i), dictionary size (p) and number of topics (K), and both p and K are allowed to increase with n, a situation not handled well by previous analyses. We complement our theoretical results with a detailed simulation study. We illustrate that the new algorithm is faster and more accurate than the current ones, although we start out with a computational and theoretical disadvantage of not knowing the correct number of topics K, while we provide the competing methods with the correct value in our simulations.
Amortized inference has led to efficient approximate inference for large datasets. The quality of posterior inference is largely determined by two factors: a) the ability of the variational distribution to model the true posterior and b) the capacity of the recognition network to generalize inference over all datapoints. We analyze approximate inference in variational autoencoders in terms of these factors. We find that suboptimal inference is often due to amortizing inference rather than the limited complexity of the approximating distribution. We show that this is due partly to the generator learning to accommodate the choice of approximation. Furthermore, we show that the parameters used to increase the expressiveness of the approximation play a role in generalizing inference rather than simply improving the complexity of the approximation.
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