We consider the optimal decision-making problem in a primary sample of interest with multiple auxiliary sources available. The outcome of interest is limited in the sense that it is only observed in the primary sample. In reality, such multiple data sources may belong to heterogeneous studies and thus cannot be combined directly. This paper proposes a new framework to handle heterogeneous studies and address the limited outcome simultaneously through a novel calibrated optimal decision making (CODA) method, by leveraging the common intermediate outcomes in multiple data sources. Specifically, CODA allows the baseline covariates across different samples to have either homogeneous or heterogeneous distributions. Under a mild and testable assumption that the conditional means of intermediate outcomes in different samples are equal given baseline covariates and the treatment information, we show that the proposed CODA estimator of the conditional mean outcome is asymptotically normal and more efficient than using the primary sample solely. In addition, the variance of the CODA estimator can be easily obtained using the simple plug-in method due to the rate double robustness. Extensive experiments on simulated datasets demonstrate empirical validity and improved efficiency using CODA, followed by a real application to a MIMIC-III dataset as the primary sample with the auxiliary data from eICU.
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Survival outcomes are common in comparative effectiveness studies and require unique handling because they are usually incompletely observed due to right-censoring. A ``once for all'' approach for causal inference with survival outcomes constructs pseudo-observations and allows standard methods such as propensity score weighting to proceed as if the outcomes are completely observed. For a general class of model-free causal estimands with survival outcomes on user-specified target populations, we develop corresponding propensity score weighting estimators based on the pseudo-observations and establish their asymptotic properties. In particular, utilizing the functional delta-method and the von Mises expansion, we derive a new closed-form variance of the weighting estimator that takes into account the uncertainty due to both pseudo-observation calculation and propensity score estimation. This allows valid and computationally efficient inference without resampling. We also prove the optimal efficiency property of the overlap weights within the class of balancing weights for survival outcomes. The proposed methods are applicable to both binary and multiple treatments. Extensive simulations are conducted to explore the operating characteristics of the proposed method versus other commonly used alternatives. We apply the proposed method to compare the causal effects of three popular treatment approaches for prostate cancer patients.
We consider the conditional treatment effect for competing risks data in observational studies. While it is described as a constant difference between the hazard functions given the covariates, we do not assume specific functional forms for the covariates. We derive the efficient score for the treatment effect using modern semiparametric theory, as well as two doubly robust scores with respect to 1) the assumed propensity score for treatment and the censoring model, and 2) the outcome models for the competing risks. An important asymptotic result regarding the estimators is rate double robustness, in addition to the classical model double robustness. Rate double robustness enables the use of machine learning and nonparametric methods in order to estimate the nuisance parameters, while preserving the root-$n$ asymptotic normality of the estimators for inferential purposes. We study the performance of the estimators using simulation. The estimators are applied to the data from a cohort of Japanese men in Hawaii followed since 1960s in order to study the effect of mid-life drinking behavior on late life cognitive outcomes.
Gaussian processes (GPs) are an important tool in machine learning and statistics with applications ranging from social and natural science through engineering. They constitute a powerful kernelized non-parametric method with well-calibrated uncertainty estimates, however, off-the-shelf GP inference procedures are limited to datasets with several thousand data points because of their cubic computational complexity. For this reason, many sparse GPs techniques have been developed over the past years. In this paper, we focus on GP regression tasks and propose a new approach based on aggregating predictions from several local and correlated experts. Thereby, the degree of correlation between the experts can vary between independent up to fully correlated experts. The individual predictions of the experts are aggregated taking into account their correlation resulting in consistent uncertainty estimates. Our method recovers independent Product of Experts, sparse GP and full GP in the limiting cases. The presented framework can deal with a general kernel function and multiple variables, and has a time and space complexity which is linear in the number of experts and data samples, which makes our approach highly scalable. We demonstrate superior performance, in a time vs. accuracy sense, of our proposed method against state-of-the-art GP approximation methods for synthetic as well as several real-world datasets with deterministic and stochastic optimization.
Federated Learning (FL) makes a large amount of edge computing devices (e.g., mobile phones) jointly learn a global model without data sharing. In FL, data are generated in a decentralized manner with high heterogeneity. This paper studies how to perform statistical estimation and inference in the federated setting. We analyze the so-called Local SGD, a multi-round estimation procedure that uses intermittent communication to improve communication efficiency. We first establish a {\it functional central limit theorem} that shows the averaged iterates of Local SGD weakly converge to a rescaled Brownian motion. We next provide two iterative inference methods: the {\it plug-in} and the {\it random scaling}. Random scaling constructs an asymptotically pivotal statistic for inference by using the information along the whole Local SGD path. Both the methods are communication efficient and applicable to online data. Our theoretical and empirical results show that Local SGD simultaneously achieves both statistical efficiency and communication efficiency.
Federated learning of causal estimands may greatly improve estimation efficiency by aggregating estimates from multiple study sites, but robustness to extreme estimates is vital for maintaining consistency. We develop a federated adaptive causal estimation (FACE) framework to incorporate heterogeneous data from multiple sites to provide treatment effect estimation and inference for a target population of interest. Our strategy is communication-efficient and privacy-preserving and allows for flexibility in the specification of the target population. Our method accounts for site-level heterogeneity in the distribution of covariates through density ratio weighting. To safely aggregate estimates from all sites and avoid negative transfer, we introduce an adaptive procedure of weighing the estimators constructed using data from the target and source populations through a penalized regression on the influence functions, which achieves 1) consistency and 2) optimal efficiency. We illustrate FACE by conducting a comparative effectiveness study of BNT162b2 (Pfizer) and mRNA-1273 (Moderna) vaccines on COVID-19 outcomes in U.S. veterans using electronic health records from five VA sites.
Parameters of the covariance kernel of a Gaussian process model often need to be estimated from the data generated by an unknown Gaussian process. We consider fixed-domain asymptotics of the maximum likelihood estimator of the scale parameter under smoothness misspecification. If the covariance kernel of the data-generating process has smoothness $\nu_0$ but that of the model has smoothness $\nu \geq \nu_0$, we prove that the expectation of the maximum likelihood estimator is of the order $N^{2(\nu-\nu_0)/d}$ if the $N$ observation points are quasi-uniform in $[0, 1]^d$. This indicates that maximum likelihood estimation of the scale parameter alone is sufficient to guarantee the correct rate of decay of the conditional variance. We also discuss a connection the expected maximum likelihood estimator has to Driscoll's theorem on sample path properties of Gaussian processes. The proofs are based on reproducing kernel Hilbert space techniques and worst-case case rates for approximation in Sobolev spaces.
Over-parameterized deep neural networks are able to achieve excellent training accuracy while maintaining a small generalization error. It has also been found that they are able to fit arbitrary labels, and this behaviour is referred to as the phenomenon of memorization. In this work, we study the phenomenon of memorization with turn-over dropout, an efficient method to estimate influence and memorization, for data with true labels (real data) and data with random labels (random data). Our main findings are: (i) For both real data and random data, the optimization of easy examples (e.g., real data) and difficult examples (e.g., random data) are conducted by the network simultaneously, with easy ones at a higher speed; (ii) For real data, a correct difficult example in the training dataset is more informative than an easy one. By showing the existence of memorization on random data and real data, we highlight the consistency between them regarding optimization and we emphasize the implication of memorization during optimization.
We consider the problem where $n$ clients transmit $d$-dimensional real-valued vectors using $d(1+o(1))$ bits each, in a manner that allows the receiver to approximately reconstruct their mean. Such compression problems naturally arise in distributed and federated learning. We provide novel mathematical results and derive computationally efficient algorithms that are more accurate than previous compression techniques. We evaluate our methods on a collection of distributed and federated learning tasks, using a variety of datasets, and show a consistent improvement over the state of the art.
Multifidelity methods are widely used for estimation of quantities of interest (QoIs) in uncertainty quantification using simulation codes of differing costs and accuracies. Many methods approximate numerical-valued statistics that represent only limited information of the QoIs. In this paper, we generalize the ideas in \cite{xu2021bandit} to develop a multifidelity method that approximates the distribution of scalar-valued QoI. Under a linear model hypothesis, we propose an exploration-exploitation strategy to reconstruct the full distribution, not just statistics, of a scalar-valued QoI using samples from a subset of low-fidelity regressors. We derive an informative asymptotic bound for the mean 1-Wasserstein distance between the estimator and the true distribution, and use it to adaptively allocate computational budget for parametric estimation and non-parametric approximation of the probability distribution. Assuming the linear model is correct, we prove that such a procedure is consistent and converges to the optimal policy (and hence optimal computational budget allocation) under an upper bound criterion as the budget goes to infinity. As a corollary, we obtain convergence of the approximated distribution in the mean 1-Wasserstein metric. The major advantages of our approach are that convergence to the full distribution of the output is attained under appropriate assumptions, and that the procedure and implementation require neither a hierarchical model setup, knowledge of cross-model information or correlation, nor \textit{a priori} known model statistics. Numerical experiments are provided in the end to support our theoretical analysis.
This paper focuses on the expected difference in borrower's repayment when there is a change in the lender's credit decisions. Classical estimators overlook the confounding effects and hence the estimation error can be magnificent. As such, we propose another approach to construct the estimators such that the error can be greatly reduced. The proposed estimators are shown to be unbiased, consistent, and robust through a combination of theoretical analysis and numerical testing. Moreover, we compare the power of estimating the causal quantities between the classical estimators and the proposed estimators. The comparison is tested across a wide range of models, including linear regression models, tree-based models, and neural network-based models, under different simulated datasets that exhibit different levels of causality, different degrees of nonlinearity, and different distributional properties. Most importantly, we apply our approaches to a large observational dataset provided by a global technology firm that operates in both the e-commerce and the lending business. We find that the relative reduction of estimation error is strikingly substantial if the causal effects are accounted for correctly.
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