Policy learning utilizing observational data is pivotal across various domains, with the objective of learning the optimal treatment assignment policy while adhering to specific constraints such as fairness, budget, and simplicity. This study introduces a novel positivity-free (stochastic) policy learning framework designed to address the challenges posed by the impracticality of the positivity assumption in real-world scenarios. This framework leverages incremental propensity score policies to adjust propensity score values instead of assigning fixed values to treatments. We characterize these incremental propensity score policies and establish identification conditions, employing semiparametric efficiency theory to propose efficient estimators capable of achieving rapid convergence rates, even when integrated with advanced machine learning algorithms. This paper provides a thorough exploration of the theoretical guarantees associated with policy learning and validates the proposed framework’s finite-sample performance through comprehensive numerical experiments, ensuring the identification of causal effects from observational data is both robust and reliable.

Additive Noise Models (ANMs) are a common model class for causal discovery from observational data. Due to a lack of real-world data for which an underlying ANM is known, ANMs with randomly sampled parameters are commonly used to simulate data for the evaluation of causal discovery algorithms. While some parameters may be fixed by explicit assumptions, fully specifying an ANM requires choosing all parameters. Reisach et al. (2021) show that, for many ANM parameter choices, sorting the variables by increasing variance yields an ordering close to a causal order and introduce ‘var-sortability’ to quantify this alignment. Since increasing variances may be unrealistic and cannot be exploited when data scales are arbitrary, ANM data are often rescaled to unit variance in causal discovery benchmarking. We show that synthetic ANM data are characterized by another pattern that is scale-invariant and thus persists even after standardization: the explainable fraction of a variable’s variance, as captured by the coefficient of determination $R^2$, tends to increase along the causal order. The result is high ‘$R^2$-sortability’, meaning that sorting the variables by increasing $R^2$ yields an ordering close to a causal order. We propose a computationally efficient baseline algorithm termed ‘$R^2$-SortnRegress’ that exploits high $R^2$-sortability and that can match and exceed the performance of established causal discovery algorithms. We show analytically that sufficiently high edge weights lead to a relative decrease of the noise contributions along causal chains, resulting in increasingly deterministic relationships and high $R^2$. We characterize $R^2$-sortability on synthetic data with different simulation parameters and find high values in common settings. Our findings reveal high $R^2$-sortability as an assumption about the data generating process relevant to causal discovery and implicit in many ANM sampling schemes. It should be made explicit, as its prevalence in real-world data is an open question. For causal discovery benchmarking, we provide implementations of $R^2$-sortability, the $R^2$-SortnRegress algorithm, and ANM simulation procedures in our library CausalDisco at https://causaldisco.github.io/CausalDisco/.

Contextual bandit algorithms are increasingly replacing non-adaptive A/B tests in e-commerce, healthcare, and policymaking because they can both improve outcomes for study participants and increase the chance of identifying good or even best policies. To support credible inference on novel interventions at the end of the study, nonetheless, we still want to construct valid confidence intervals on average treatment effects, subgroup effects, or value of new policies. The adaptive nature of the data collected by contextual bandit algorithms, however, makes this difficult: standard estimators are no longer asymptotically normally distributed and classic confidence intervals fail to provide correct coverage. While this has been addressed in non-contextual settings by using stabilized estimators, variance stabilized estimators in the contextual setting pose unique challenges that we tackle for the first time in this paper. We propose the Contextual Adaptive Doubly Robust (CADR) estimator, a novel estimator for policy value that is asymptotically normal under contextual adaptive data collection. The main technical challenge in constructing CADR is designing adaptive and consistent conditional standard deviation estimators for stabilization. Extensive numerical experiments using 57 OpenML datasets demonstrate that confidence intervals based on CADR uniquely provide correct coverage.

Empirical risk minimization (ERM) is the workhorse of machine learning, whether for classification and regression or for off-policy policy learning, but its model-agnostic guarantees can fail when we use adaptively collected data, such as the result of running a contextual bandit algorithm. We study a generic importance sampling weighted ERM algorithm for using adaptively collected data to minimize the average of a loss function over a hypothesis class and provide first-of-their-kind generalization guarantees and fast convergence rates. Our results are based on a new maximal inequality that carefully leverages the importance sampling structure to obtain rates with the good dependence on the exploration rate in the data. For regression, we provide fast rates that leverage the strong convexity of squared-error loss. For policy learning, we provide regret guarantees that close an open gap in the existing literature whenever exploration decays to zero, as is the case for bandit-collected data. An empirical investigation validates our theory.

We propose the Generalized Policy Elimination (GPE) algorithm, an oracle-efficient contextual bandit (CB) algorithm inspired by the Policy Elimination algorithm of Dudik et al. [2011]. 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.

UNLABELLED: We describe the implementation of the method introduced by Chambaz et al. in 2012. We also demonstrate its genome-wide application to the integrative search of new regions with strong association between DNA copy number and gene expression accounting for DNA methylation in breast cancers. AVAILABILITY AND IMPLEMENTATION: An open-source R package tmle.npvi is available from CRAN (http://cran.r-project.org/). CONTACT: [email protected].