arXiv subjects

Mark J. van der Laan

Publications and source records attributed to Mark J. van der Laan.

At least 55 records · Page 3Linked to original sources

arXiv2021

Why Machine Learning Cannot Ignore Maximum Likelihood Estimation

The growth of machine learning as a field has been accelerating with increasing interest and publications across fields, including statistics, but predominantly in computer science. How can we parse this vast literature for developments that exemplify the necessary rigor? How many of these manuscripts incorporate foundational theory to allow for statistical inference? Which advances have the greatest potential for impact in practice? One could posit many answers to these queries. Here, we assert that one essential idea is for machine learning to integrate maximum likelihood for estimation of functional parameters, such as prediction functions and conditional densities.

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arXiv2021

Evaluating the Robustness of Targeted Maximum Likelihood Estimators via Realistic Simulations in Nutrition Intervention Trials

Several recently developed methods have the potential to harness machine learning in the pursuit of target quantities inspired by causal inference, including inverse weighting, doubly robust estimating equations and substitution estimators like targeted maximum likelihood estimation. There are even more recent augmentations of these procedures that can increase robustness, by adding a layer of cross-validation (cross-validated targeted maximum likelihood estimation and double machine learning, as applied to substitution and estimating equation approaches, respectively). While these methods have been evaluated individually on simulated and experimental data sets, a comprehensive analysis of their performance across ``real-world'' simulations have yet to be conducted. In this work, we benchmark multiple widely used methods for estimation of the average treatment effect using ten different nutrition intervention studies data. A realistic set of simulations, based on a novel method, highly adaptive lasso, for estimating the data-generating distribution that guarantees a certain level of complexity (undersmoothing) is used to better mimic the complexity of the true data-generating distribution. We have applied this novel method for estimating the data-generating distribution by individual study and to subsequently use these fits to simulate data and estimate treatment effects parameters as well as their standard errors and resulting confidence intervals. Based on the analytic results, a general recommendation is put forth for use of the cross-validated variants of both substitution and estimating equation estimators. We conclude that the additional layer of cross-validation helps in avoiding unintentional over-fitting of nuisance parameter functionals and leads to more robust inferences.

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arXiv2021

Personalized Online Machine Learning

In this work, we introduce the Personalized Online Super Learner (POSL) -- an online ensembling algorithm for streaming data whose optimization procedure accommodates varying degrees of personalization. Namely, POSL optimizes predictions with respect to baseline covariates, so personalization can vary from completely individualized (i.e., optimization with respect to baseline covariate subject ID) to many individuals (i.e., optimization with respect to common baseline covariates). As an online algorithm, POSL learns in real-time. POSL can leverage a diversity of candidate algorithms, including online algorithms with different training and update times, fixed algorithms that are never updated during the procedure, pooled algorithms that learn from many individuals' time-series, and individualized algorithms that learn from within a single time-series. POSL's ensembling of this hybrid of base learning strategies depends on the amount of data collected, the stationarity of the time-series, and the mutual characteristics of a group of time-series. In essence, POSL decides whether to learn across samples, through time, or both, based on the underlying (unknown) structure in the data. For a wide range of simulations that reflect realistic forecasting scenarios, and in a medical data application, we examine the performance of POSL relative to other current ensembling and online learning methods. We show that POSL is able to provide reliable predictions for time-series data and adjust to changing data-generating environments. We further cultivate POSL's practicality by extending it to settings where time-series enter/exit dynamically over chronological time.

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arXiv2021

One-step TMLE for targeting cause-specific absolute risks and survival curves

This paper considers one-step targeted maximum likelihood estimation method for general competing risks and survival analysis settings where event times take place on the positive real line R+ and are subject to right-censoring. Our interest is overall in the effects of baseline treatment decisions, static, dynamic or stochastic, possibly confounded by pre-treatment covariates. We point out two overall contributions of our work. First, our method can be used to obtain simultaneous inference across all absolute risks in competing risks settings. Second, we present a practical result for achieving inference for the full survival curve, or a full absolute risk curve, across time by targeting over a fine enough grid of points. The one-step procedure is based on a one-dimensional universal least favorable submodel for each cause-specific hazard that can be implemented in recursive steps along a corresponding universal least favorable submodel. We present a theorem for conditions to achieve weak convergence of the estimator for an infinite-dimensional target parameter. Our empirical study demonstrates the use of the methods.

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arXiv2021

Nonparametric inverse probability weighted estimators based on the highly adaptive lasso

Inverse probability weighted estimators are the oldest and potentially most commonly used class of procedures for the estimation of causal effects. By adjusting for selection biases via a weighting mechanism, these procedures estimate an effect of interest by constructing a pseudo-population in which selection biases are eliminated. Despite their ease of use, these estimators require the correct specification of a model for the weighting mechanism, are known to be inefficient, and suffer from the curse of dimensionality. We propose a class of nonparametric inverse probability weighted estimators in which the weighting mechanism is estimated via undersmoothing of the highly adaptive lasso, a nonparametric regression function proven to converge at $n^{-1/3}$-rate to the true weighting mechanism. We demonstrate that our estimators are asymptotically linear with variance converging to the nonparametric efficiency bound. Unlike doubly robust estimators, our procedures require neither derivation of the efficient influence function nor specification of the conditional outcome model. Our theoretical developments have broad implications for the construction of efficient inverse probability weighted estimators in large statistical models and a variety of problem settings. We assess the practical performance of our estimators in simulation studies and demonstrate use of our proposed methodology with data from a large-scale epidemiologic study.

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arXiv2021

Efficient Estimation of Pathwise Differentiable Target Parameters with the Undersmoothed Highly Adaptive Lasso

We consider estimation of a functional parameter of a realistically modeled data distribution based on observing independent and identically distributed observations. We define an $m$-th order Spline Highly Adaptive Lasso Minimum Loss Estimator (Spline HAL-MLE) of a functional parameter that is defined by minimizing the empirical risk function over an $m$-th order smoothness class of functions. We show that this $m$-th order smoothness class consists of all functions that can be represented as an infinitesimal linear combination of tensor products of $\leq m$-th order spline-basis functions, and involves assuming $m$-derivatives in each coordinate. By selecting $m$ with cross-validation we obtain a Spline-HAL-MLE that is able to adapt to the underlying unknown smoothness of the true function, while guaranteeing a rate of convergence faster than $n^{-1/4}$, as long as the true function is cadlag (right-continuous with left-hand limits) and has finite sectional variation norm. The $m=0$-smoothness class consists of all cadlag functions with finite sectional variation norm and corresponds with the original HAL-MLE defined in van der Laan (2015). In this article we establish that this Spline-HAL-MLE yields an asymptotically efficient estimator of any smooth feature of the functional parameter under an easily verifiable global undersmoothing condition. A sufficient condition for the latter condition is that the minimum of the empirical mean of the selected basis functions is smaller than a constant times $n^{-1/2}$, which is not parameter specific and enforces the selection of the $L_1$-norm in the lasso to be large enough to include sparsely supported basis. We demonstrate our general result for the $m=0$-HAL-MLE of the average treatment effect and of the integral of the square of the data density. We also present simulations for these two examples confirming the theory.

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arXiv2021

Adaptive Sequential Design for a Single Time-Series

The current work is motivated by the need for robust statistical methods for precision medicine; as such, we address the need for statistical methods that provide actionable inference for a single unit at any point in time. We aim to learn an optimal, unknown choice of the controlled components of the design in order to optimize the expected outcome; with that, we adapt the randomization mechanism for future time-point experiments based on the data collected on the individual over time. Our results demonstrate that one can learn the optimal rule based on a single sample, and thereby adjust the design at any point t with valid inference for the mean target parameter. This work provides several contributions to the field of statistical precision medicine. First, we define a general class of averages of conditional causal parameters defined by the current context for the single unit time-series data. We define a nonparametric model for the probability distribution of the time-series under few assumptions, and aim to fully utilize the sequential randomization in the estimation procedure via the double robust structure of the efficient influence curve of the proposed target parameter. We present multiple exploration-exploitation strategies for assigning treatment, and methods for estimating the optimal rule. Lastly, we present the study of the data-adaptive inference on the mean under the optimal treatment rule, where the target parameter adapts over time in response to the observed context of the individual. Our target parameter is pathwise differentiable with an efficient influence function that is doubly robust - which makes it easier to estimate than previously proposed variations. We characterize the limit distribution of our estimator under a Donsker condition expressed in terms of a notion of bracketing entropy adapted to martingale settings.

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arXiv2021

Continuous-time targeted minimum loss-based estimation of intervention-specific mean outcomes

This paper studies the generalization of the targeted minimum loss-based estimation (TMLE) framework to estimation of effects of time-varying interventions in settings where both interventions, covariates, and outcome can happen at subject-specific time-points on an arbitrarily fine time-scale. TMLE is a general template for constructing asymptotically linear substitution estimators for smooth low-dimensional parameters in infinite-dimensional models. Existing longitudinal TMLE methods are developed for data where observations are made on a discrete time-grid. We consider a continuous-time counting process model where intensity measures track the monitoring of subjects, and focus on a low-dimensional target parameter defined as the intervention-specific mean outcome at the end of follow-up. To construct our TMLE algorithm for the given statistical estimation problem we derive an expression for the efficient influence curve and represent the target parameter as a functional of intensities and conditional expectations. The high-dimensional nuisance parameters of our model are estimated and updated in an iterative manner according to separate targeting steps for the involved intensities and conditional expectations. The resulting estimator solves the efficient influence curve equation. We state a general efficiency theorem and describe a highly adaptive lasso estimator for nuisance parameters that allows us to establish asymptotic linearity and efficiency of our estimator under minimal conditions on the underlying statistical model.

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arXiv2020

Sufficient and insufficient conditions for the stochastic convergence of Cesàro means

We study the stochastic convergence of the Cesàro mean of a sequence of random variables. These arise naturally in statistical problems that have a sequential component, where the sequence of random variables is typically derived from a sequence of estimators computed on data. We show that establishing a rate of convergence in probability for a sequence is not sufficient in general to establish a rate in probability for its Cesàro mean. We also present several sets of conditions on the sequence of random variables that are sufficient to guarantee a rate of convergence for its Cesàro mean. We identify common settings in which these sets of conditions hold.

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arXiv2020

tmleCommunity: A R Package Implementing Target Maximum Likelihood Estimation for Community-level Data

Over the past years, many applications aim to assess the causal effect of treatments assigned at the community level, while data are still collected at the individual level among individuals of the community. In many cases, one wants to evaluate the effect of a stochastic intervention on the community, where all communities in the target population receive probabilistically assigned treatments based on a known specified mechanism (e.g., implementing a community-level intervention policy that target stochastic changes in the behavior of a target population of communities). The tmleCommunity package is recently developed to implement targeted minimum loss-based estimation (TMLE) of the effect of community-level intervention(s) at a single time point on an individual-based outcome of interest, including the average causal effect. Implementations of the inverse-probability-of-treatment-weighting (IPTW) and the G-computation formula (GCOMP) are also available. The package supports multivariate arbitrary (i.e., static, dynamic or stochastic) interventions with a binary or continuous outcome. Besides, it allows user-specified data-adaptive machine learning algorithms through SuperLearner, sl3 and h2oEnsemble packages. The usage of the tmleCommunity package, along with a few examples, will be described in this paper.

stat.AP↗

arXiv2020

Targeted Maximum Likelihood Estimation of Community-based Causal Effect of Community-Level Stochastic Interventions

Unlike the commonly used parametric regression models such as mixed models, that can easily violate the required statistical assumptions and result in invalid statistical inference, target maximum likelihood estimation allows more realistic data-generative models and provides double-robust, semi-parametric and efficient estimators. Target maximum likelihood estimators (TMLEs) for the causal effect of a community-level static exposure were previously proposed by Balzer et al. In this manuscript, we build on this work and present identifiability results and develop two semi-parametric efficient TMLEs for the estimation of the causal effect of the single time-point community-level stochastic intervention whose assignment mechanism can depend on measured and unmeasured environmental factors and its individual-level covariates. The first community-level TMLE is developed under a general hierarchical non-parametric structural equation model, which can incorporate pooled individual-level regressions for estimating the outcome mechanism. The second individual-level TMLE is developed under a restricted hierarchical model in which the additional assumption of no covariate interference within communities holds. The proposed TMLEs have several crucial advantages. First, both TMLEs can make use of individual level data in the hierarchical setting, and potentially reduce finite sample bias and improve estimator efficiency. Second, the stochastic intervention framework provides a natural way for defining and estimating casual effects where the exposure variables are continuous or discrete with multiple levels, or even cannot be directly intervened on. Also, the positivity assumption needed for our proposed causal parameters can be weaker than the version of positivity required for other casual parameters.

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arXiv2020

Targeting Learning: Robust Statistics for Reproducible Research

Targeted Learning is a subfield of statistics that unifies advances in causal inference, machine learning and statistical theory to help answer scientifically impactful questions with statistical confidence. Targeted Learning is driven by complex problems in data science and has been implemented in a diversity of real-world scenarios: observational studies with missing treatments and outcomes, personalized interventions, longitudinal settings with time-varying treatment regimes, survival analysis, adaptive randomized trials, mediation analysis, and networks of connected subjects. In contrast to the (mis)application of restrictive modeling strategies that dominate the current practice of statistics, Targeted Learning establishes a principled standard for statistical estimation and inference (i.e., confidence intervals and p-values). This multiply robust approach is accompanied by a guiding roadmap and a burgeoning software ecosystem, both of which provide guidance on the construction of estimators optimized to best answer the motivating question. The roadmap of Targeted Learning emphasizes tailoring statistical procedures so as to minimize their assumptions, carefully grounding them only in the scientific knowledge available. The end result is a framework that honestly reflects the uncertainty in both the background knowledge and the available data in order to draw reliable conclusions from statistical analyses - ultimately enhancing the reproducibility and rigor of scientific findings.

stat.ME↗

arXiv2020

Rate-adaptive model selection over a collection of black-box contextual bandit algorithms

We consider the model selection task in the stochastic contextual bandit setting. Suppose we are given a collection of base contextual bandit algorithms. We provide a master algorithm that combines them and achieves the same performance, up to constants, as the best base algorithm would, if it had been run on its own. Our approach only requires that each algorithm satisfy a high probability regret bound. Our procedure is very simple and essentially does the following: for a well chosen sequence of probabilities $(p_{t})_{t\geq 1}$, at each round $t$, it either chooses at random which candidate to follow (with probability $p_{t}$) or compares, at the same internal sample size for each candidate, the cumulative reward of each, and selects the one that wins the comparison (with probability $1-p_{t}$). To the best of our knowledge, our proposal is the first one to be rate-adaptive for a collection of general black-box contextual bandit algorithms: it achieves the same regret rate as the best candidate. We demonstrate the effectiveness of our method with simulation studies.

cs.LG↗

arXiv2020

Non-parametric efficient causal mediation with intermediate confounders

Interventional effects for mediation analysis were proposed as a solution to the lack of identifiability of natural (in)direct effects in the presence of a mediator-outcome confounder affected by exposure. We present a theoretical and computational study of the properties of the interventional (in)direct effect estimands based on the efficient influence fucntion (EIF) in the non-parametric statistical model. We use the EIF to develop two asymptotically optimal, non-parametric estimators that leverage data-adaptive regression for estimation of the nuisance parameters: a one-step estimator and a targeted minimum loss estimator. A free and open source \texttt{R} package implementing our proposed estimators is made available on GitHub. We further present results establishing the conditions under which these estimators are consistent, multiply robust, $n^{1/2}$-consistent and efficient. We illustrate the finite-sample performance of the estimators and corroborate our theoretical results in a simulation study. We also demonstrate the use of the estimators in our motivating application to elucidate the mechanisms behind the unintended harmful effects that a housing intervention had on adolescent girls' risk behavior.

stat.ME↗

arXiv2020

Efficient nonparametric inference on the effects of stochastic interventions under two-phase sampling, with applications to vaccine efficacy trials

The advent and subsequent widespread availability of preventive vaccines has altered the course of public health over the past century. Despite this success, effective vaccines to prevent many high-burden diseases, including HIV, have been slow to develop. Vaccine development can be aided by the identification of immune response markers that serve as effective surrogates for clinically significant infection or disease endpoints. However, measuring immune response marker activity is often costly, which has motivated the usage of two-phase sampling for immune response evaluation in clinical trials of preventive vaccines. In such trials, the measurement of immunological markers is performed on a subset of trial participants, where enrollment in this second phase is potentially contingent on the observed study outcome and other participant-level information. We propose nonparametric methodology for efficiently estimating a counterfactual parameter that quantifies the impact of a given immune response marker on the subsequent probability of infection. Along the way, we fill in theoretical gaps pertaining to the asymptotic behavior of nonparametric efficient estimators in the context of two-phase sampling, including a multiple robustness property enjoyed by our estimators. Techniques for constructing confidence intervals and hypothesis tests are presented, and an open source software implementation of the methodology, the txshift R package, is introduced. We illustrate the proposed techniques using data from a recent preventive HIV vaccine efficacy trial.

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arXiv2020

Generalized Policy Elimination: an efficient algorithm for Nonparametric Contextual Bandits

We propose the Generalized Policy Elimination (GPE) algorithm, an oracle-efficient contextual bandit (CB) algorithm inspired by the Policy Elimination algorithm of \cite{dudik2011}. We prove the first regret optimality guarantee theorem for an oracle-efficient CB algorithm competing against a nonparametric class with infinite VC-dimension. Specifically, we show that GPE is regret-optimal (up to logarithmic factors) for policy classes with integrable entropy. For classes with larger entropy, we show that the core techniques used to analyze GPE can be used to design an $\varepsilon$-greedy algorithm with regret bound matching that of the best algorithms to date. We illustrate the applicability of our algorithms and theorems with examples of large nonparametric policy classes, for which the relevant optimization oracles can be efficiently implemented.

cs.LG↗

arXiv2019

More Efficient Off-Policy Evaluation through Regularized Targeted Learning

We study the problem of off-policy evaluation (OPE) in Reinforcement Learning (RL), where the aim is to estimate the performance of a new policy given historical data that may have been generated by a different policy, or policies. In particular, we introduce a novel doubly-robust estimator for the OPE problem in RL, based on the Targeted Maximum Likelihood Estimation principle from the statistical causal inference literature. We also introduce several variance reduction techniques that lead to impressive performance gains in off-policy evaluation. We show empirically that our estimator uniformly wins over existing off-policy evaluation methods across multiple RL environments and various levels of model misspecification. Finally, we further the existing theoretical analysis of estimators for the RL off-policy estimation problem by showing their $O_P(1/\sqrt{n})$ rate of convergence and characterizing their asymptotic distribution.

cs.LG↗

arXiv2019

Fast rates for empirical risk minimization over càdlàg functions with bounded sectional variation norm

Empirical risk minimization over classes functions that are bounded for some version of the variation norm has a long history, starting with Total Variation Denoising (Rudin et al., 1992), and has been considered by several recent articles, in particular Fang et al., 2019 and van der Laan, 2015. In this article, we consider empirical risk minimization over the class $\mathcal{F}_d$ of càdlàg functions over $[0,1]^d$ with bounded sectional variation norm (also called Hardy-Krause variation). We show how a certain representation of functions in $\mathcal{F}_d$ allows to bound the bracketing entropy of sieves of $\mathcal{F}_d$, and therefore derive rates of convergence in nonparametric function estimation. Specifically, for sieves whose growth is controlled by some rate $a_n$, we show that the empirical risk minimizer has rate of convergence $O_P(n^{-1/3} (\log n)^{2(d-1)/3} a_n)$. Remarkably, the dimension only affects the rate in $n$ through the logarithmic factor, making this method especially appropriate for high dimensional problems. In particular, we show that in the case of nonparametric regression over sieves of càdlàg functions with bounded sectional variation norm, this upper bound on the rate of convergence holds for least-squares estimators, under the random design, sub-exponential errors setting.

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