Besov priors are nonparametric priors that model spatially inhomogeneous functions. They are routinely used in inverse problems and imaging, where they exhibit attractive sparsity-promoting and edge-preserving features. A recent line of work has initiated the study of their asymptotic frequentist convergence properties. In the present paper, we consider the theoretical recovery performance of the posterior distributions associated to Besov-Laplace priors in the density estimation model, under the assumption that the observations are generated by a possibly spatially inhomogeneous true density belonging to a Besov space. We improve on existing results and show that carefully tuned Besov-Laplace priors attain optimal posterior contraction rates. Furthermore, we show that hierarchical procedures involving a hyper-prior on the regularity parameter lead to adaptation to any smoothness level.
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Additive regression models with interactions are widely studied in the literature, using methods such as splines or Gaussian process regression. However, these methods can pose challenges for estimation and model selection, due to the presence of many smoothing parameters and the lack of suitable criteria. We propose to address these challenges by extending the I-prior methodology (Bergsma, 2020) to multiple covariates, which may be multidimensional. The I-prior methodology has some advantages over other methods, such as Gaussian process regression and Tikhonov regularization, both theoretically and practically. In particular, the I-prior is a proper prior, is based on minimal assumptions, yields an admissible posterior mean, and estimation of the scale (or smoothing) parameters can be done using an EM algorithm with simple E and M steps. Moreover, we introduce a parsimonious specification of models with interactions, which has two benefits: (i) it reduces the number of scale parameters and thus facilitates the estimation of models with interactions, and (ii) it enables straightforward model selection (among models with different interactions) based on the marginal likelihood.
This paper addresses the communication issues when estimating hyper-gradients in decentralized federated learning (FL). Hyper-gradients in decentralized FL quantifies how the performance of globally shared optimal model is influenced by the perturbations in clients' hyper-parameters. In prior work, clients trace this influence through the communication of Hessian matrices over a static undirected network, resulting in (i) excessive communication costs and (ii) inability to make use of more efficient and robust networks, namely, time-varying directed networks. To solve these issues, we introduce an alternative optimality condition for FL using an averaging operation on model parameters and gradients. We then employ Push-Sum as the averaging operation, which is a consensus optimization technique for time-varying directed networks. As a result, the hyper-gradient estimator derived from our optimality condition enjoys two desirable properties; (i) it only requires Push-Sum communication of vectors and (ii) it can operate over time-varying directed networks. We confirm the convergence of our estimator to the true hyper-gradient both theoretically and empirically, and we further demonstrate that it enables two novel applications: decentralized influence estimation and personalization over time-varying networks.
We study the problem of model selection in causal inference, specifically for the case of conditional average treatment effect (CATE) estimation under binary treatments. Unlike model selection in machine learning, there is no perfect analogue of cross-validation as we do not observe the counterfactual potential outcome for any data point. Towards this, there have been a variety of proxy metrics proposed in the literature, that depend on auxiliary nuisance models estimated from the observed data (propensity score model, outcome regression model). However, the effectiveness of these metrics has only been studied on synthetic datasets as we can access the counterfactual data for them. We conduct an extensive empirical analysis to judge the performance of these metrics introduced in the literature, and novel ones introduced in this work, where we utilize the latest advances in generative modeling to incorporate multiple realistic datasets. Our analysis suggests novel model selection strategies based on careful hyperparameter tuning of CATE estimators and causal ensembling.
This paper presents a new distance metric to compare two continuous probability density functions. The main advantage of this metric is that, unlike other statistical measurements, it can provide an analytic, closed-form expression for a mixture of Gaussian distributions while satisfying all metric properties. These characteristics enable fast, stable, and efficient calculations, which are highly desirable in real-world signal processing applications. The application in mind is Gaussian Mixture Reduction (GMR), which is widely used in density estimation, recursive tracking, and belief propagation. To address this problem, we developed a novel algorithm dubbed the Optimization-based Greedy GMR (OGGMR), which employs our metric as a criterion to approximate a high-order Gaussian mixture with a lower order. Experimental results show that the OGGMR algorithm is significantly faster and more efficient than state-of-the-art GMR algorithms while retaining the geometric shape of the original mixture.
Conducting valid statistical analyses is challenging in the presence of missing-not-at-random (MNAR) data, where the missingness mechanism is dependent on the missing values themselves even conditioned on the observed data. Here, we consider a MNAR model that generalizes several prior popular MNAR models in two ways: first, it is less restrictive in terms of statistical independence assumptions imposed on the underlying joint data distribution, and second, it allows for all variables in the observed sample to have missing values. This MNAR model corresponds to a so-called criss-cross structure considered in the literature on graphical models of missing data that prevents nonparametric identification of the entire missing data model. Nonetheless, part of the complete-data distribution remains nonparametrically identifiable. By exploiting this fact and considering a rich class of exponential family distributions, we establish sufficient conditions for identification of the complete-data distribution as well as the entire missingness mechanism. We then propose methods for testing the independence restrictions encoded in such models using odds ratio as our parameter of interest. We adopt two semiparametric approaches for estimating the odds ratio parameter and establish the corresponding asymptotic theories: one involves maximizing a conditional likelihood with order statistics and the other uses estimating equations. The utility of our methods is illustrated via simulation studies.
Large and complex datasets are often collected from several, possibly heterogeneous sources. Collaborative learning methods improve efficiency by leveraging commonalities across datasets while accounting for possible differences among them. Here we study collaborative linear regression and contextual bandits, where each instance's associated parameters are equal to a global parameter plus a sparse instance-specific term. We propose a novel two-stage estimator called MOLAR that leverages this structure by first constructing an entry-wise median of the instances' linear regression estimates, and then shrinking the instance-specific estimates towards the median. MOLAR improves the dependence of the estimation error on the data dimension, compared to independent least squares estimates. We then apply MOLAR to develop methods for sparsely heterogeneous collaborative contextual bandits, which lead to improved regret guarantees compared to independent bandit methods. We further show that our methods are minimax optimal by providing a number of lower bounds. Finally, we support the efficiency of our methods by performing experiments on both synthetic data and the PISA dataset on student educational outcomes from heterogeneous countries.
We consider the problem of estimating the causal effect of a treatment on an outcome in linear structural causal models (SCM) with latent confounders when we have access to a single proxy variable. Several methods (such as difference-in-difference (DiD) estimator or negative outcome control) have been proposed in this setting in the literature. However, these approaches require either restrictive assumptions on the data generating model or having access to at least two proxy variables. We propose a method to estimate the causal effect using cross moments between the treatment, the outcome, and the proxy variable. In particular, we show that the causal effect can be identified with simple arithmetic operations on the cross moments if the latent confounder in linear SCM is non-Gaussian. In this setting, DiD estimator provides an unbiased estimate only in the special case where the latent confounder has exactly the same direct causal effects on the outcomes in the pre-treatment and post-treatment phases. This translates to the common trend assumption in DiD, which we effectively relax. Additionally, we provide an impossibility result that shows the causal effect cannot be identified if the observational distribution over the treatment, the outcome, and the proxy is jointly Gaussian. Our experiments on both synthetic and real-world datasets showcase the effectiveness of the proposed approach in estimating the causal effect.
We present a new approach to semiparametric inference using corrected posterior distributions. The method allows us to leverage the adaptivity, regularization and predictive power of nonparametric Bayesian procedures to estimate low-dimensional functionals of interest without being restricted by the holistic Bayesian formalism. Starting from a conventional nonparametric posterior, we target the functional of interest by transforming the entire distribution with a Bayesian bootstrap correction. We provide conditions for the resulting $\textit{one-step posterior}$ to possess calibrated frequentist properties and specialize the results for several canonical examples: the integrated squared density, the mean of a missing-at-random outcome, and the average causal treatment effect on the treated. The procedure is computationally attractive, requiring only a simple, efficient post-processing step that can be attached onto any arbitrary posterior sampling algorithm. Using the ACIC 2016 causal data analysis competition, we illustrate that our approach can outperform the existing state-of-the-art through the propagation of Bayesian uncertainty.
We study causal effect estimation from a mixture of observational and interventional data in a confounded linear regression model with multivariate treatments. We show that the statistical efficiency in terms of expected squared error can be improved by combining estimators arising from both the observational and interventional setting. To this end, we derive methods based on matrix weighted linear estimators and prove that our methods are asymptotically unbiased in the infinite sample limit. This is an important improvement compared to the pooled estimator using the union of interventional and observational data, for which the bias only vanishes if the ratio of observational to interventional data tends to zero. Studies on synthetic data confirm our theoretical findings. In settings where confounding is substantial and the ratio of observational to interventional data is large, our estimators outperform a Stein-type estimator and various other baselines.
We consider the problem of estimating (diagonally dominant) M-matrices as precision matrices in Gaussian graphical models. These models exhibit intriguing properties, such as the existence of the maximum likelihood estimator with merely two observations for M-matrices \citep{lauritzen2019maximum,slawski2015estimation} and even one observation for diagonally dominant M-matrices \citep{truell2021maximum}. We propose an adaptive multiple-stage estimation method that refines the estimate by solving a weighted $\ell_1$-regularized problem at each stage. Furthermore, we develop a unified framework based on the gradient projection method to solve the regularized problem, incorporating distinct projections to handle the constraints of M-matrices and diagonally dominant M-matrices. A theoretical analysis of the estimation error is provided. Our method outperforms state-of-the-art methods in precision matrix estimation and graph edge identification, as evidenced by synthetic and financial time-series data sets.
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