Reward Maximization for Pure Exploration: Minimax Optimal Good Arm Identification for Nonparametric Multi-Armed Bandits

In multi-armed bandits, the tasks of reward maximization and pure exploration are often at odds with each other. The former focuses on exploiting arms with the highest means, while the latter may require constant exploration across all arms. In this work, we focus on good arm identification (GAI), a practical bandit inference objective that aims to label arms with means above a threshold as quickly as possible. We show that GAI can be efficiently solved by combining a reward-maximizing sampling algorithm with a novel nonparametric anytime-valid sequential test for labeling arm means. We first establish that our sequential test maintains error control under highly nonparametric assumptions and asymptotically achieves the minimax optimal e-power, a notion of power for anytime-valid tests. Next, by pairing regret-minimizing sampling schemes with our sequential test, we provide an approach that achieves minimax optimal stopping times for labeling arms with means above a threshold, under an error probability constraint. Our empirical results validate our approach beyond the minimax setting, reducing the expected number of samples for all stopping times by at least 50% across both synthetic and real-world settings.

LoSAM: Local Search in Additive Noise Models with Unmeasured Confounders, a Top-Down Global Discovery Approach

We address the challenge of causal discovery in structural equation models with additive noise without imposing additional assumptions on the underlying data-generating process. We introduce local search in additive noise model (LoSAM), which generalizes an existing nonlinear method that leverages local causal substructures to the general additive noise setting, allowing for both linear and nonlinear causal mechanisms. We show that LoSAM achieves polynomial runtime, and improves runtime and efficiency by exploiting new substructures to minimize the conditioning set at each step. Further, we introduce a variant of LoSAM, LoSAM-UC, that is robust to unmeasured confounding among roots, a property that is often not satisfied by functional-causal-model-based methods. We numerically demonstrate the utility of LoSAM, showing that it outperforms existing benchmarks.

CSPI-MT: Calibrated Safe Policy Improvement with Multiple Testing for Threshold Policies

When modifying existing policies in high-risk settings, it is often necessary to ensure with high certainty that the newly proposed policy improves upon a baseline, such as the status quo. In this work, we consider the problem of safe policy improvement, where one only adopts a new policy if it is deemed to be better than the specified baseline with at least pre-specified probability. We focus on threshold policies, a ubiquitous class of policies with applications in economics, healthcare, and digital advertising. Existing methods rely on potentially underpowered safety checks and limit the opportunities for finding safe improvements, so too often they must revert to the baseline to maintain safety. We overcome these issues by leveraging the most powerful safety test in the asymptotic regime and allowing for multiple candidates to be tested for improvement over the baseline. We show that in adversarial settings, our approach controls the rate of adopting a policy worse than the baseline to the pre-specified error level, even in moderate sample sizes. We present CSPI and CSPI-MT, two novel heuristics for selecting cutoff(s) to maximize the policy improvement from baseline. We demonstrate through both synthetic and external datasets that our approaches improve both the detection rates of safe policies and the realized improvement, particularly under stringent safety requirements and low signal-to-noise conditions.

Local Causal Discovery for Structural Evidence of Direct Discrimination

Fairness is a critical objective in policy design and algorithmic decision-making. Identifying the causal pathways of unfairness requires knowledge of the underlying structural causal model, which may be incomplete or unavailable. This limits the practicality of causal fairness analysis in complex or low-knowledge domains. To mitigate this practicality gap, we advocate for developing efficient causal discovery methods for fairness applications. To this end, we introduce local discovery for direct discrimination (LD3): a polynomial-time algorithm that recovers structural evidence of direct discrimination. LD3 performs a linear number of conditional independence tests with respect to variable set size. Moreover, we propose a graphical criterion for identifying the weighted controlled direct effect (CDE), a qualitative measure of direct discrimination. We prove that this criterion is satisfied by the knowledge returned by LD3, increasing the accessibility of the weighted CDE as a causal fairness measure. Taking liver transplant allocation as a case study, we highlight the potential impact of LD3 for modeling fairness in complex decision systems. Results on real-world data demonstrate more plausible causal relations than baselines, which took 197x to 5870x longer to execute.

Hybrid Global Causal Discovery with Local Search

Learning the unique directed acyclic graph corresponding to an unknown causal model is a challenging task. Methods based on functional causal models can identify a unique graph, but either suffer from the curse of dimensionality or impose strong parametric assumptions. To address these challenges, we propose a novel hybrid approach for global causal discovery in observational data that leverages local causal substructures. We first present a topological sorting algorithm that leverages ancestral relationships in linear structural equation models to establish a compact top-down hierarchical ordering, encoding more causal information than linear orderings produced by existing methods. We demonstrate that this approach generalizes to nonlinear settings with arbitrary noise. We then introduce a nonparametric constraint-based algorithm that prunes spurious edges by searching for local conditioning sets, achieving greater accuracy than current methods. We provide theoretical guarantees for correctness and worst-case polynomial time complexities, with empirical validation on synthetic data.

Peeking with PEAK: Sequential, Nonparametric Composite Hypothesis Tests for Means of Multiple Data Streams

We propose a novel nonparametric sequential test for composite hypotheses for means of multiple data streams. Our proposed method, \emph{peeking with expectation-based averaged capital} (PEAK), builds upon the testing-as-betting framework and provides a non-asymptotic $\alpha$-level test across any stopping time. PEAK is computationally tractable and efficiently rejects hypotheses that are incorrect across all potential distributions that satisfy our nonparametric assumption, enabling joint composite hypothesis testing on multiple streams of data. We numerically validate our theoretical findings under the best arm identification and threshold identification in the bandit setting, illustrating both the competitive performance and the computational efficiency of our method against state-of-the-art testing methods.

Online Uniform Risk Times Sampling: First Approximation Algorithms, Learning Augmentation with Full Confidence Interval Integration

In digital health, the strategy of allocating a limited treatment budget across available risk times is crucial to reduce user fatigue. This strategy, however, encounters a significant obstacle due to the unknown actual number of risk times, a factor not adequately addressed by existing methods lacking theoretical guarantees. This paper introduces, for the first time, the online uniform risk times sampling problem within the approximation algorithm framework. We propose two online approximation algorithms for this problem, one with and one without learning augmentation, and provide rigorous theoretical performance guarantees for them using competitive ratio analysis. We assess the performance of our algorithms using both synthetic experiments and a real-world case study on HeartSteps mobile applications.

Local Discovery by Partitioning: Polynomial-Time Causal Discovery Around Exposure-Outcome Pairs

This work addresses the problem of automated covariate selection under limited prior knowledge. Given an exposure-outcome pair {X,Y} and a variable set Z of unknown causal structure, the Local Discovery by Partitioning (LDP) algorithm partitions Z into subsets defined by their relation to {X,Y}. We enumerate eight exhaustive and mutually exclusive partitions of any arbitrary Z and leverage this taxonomy to differentiate confounders from other variable types. LDP is motivated by valid adjustment set identification, but avoids the pretreatment assumption commonly made by automated covariate selection methods. We provide theoretical guarantees that LDP returns a valid adjustment set for any Z that meets sufficient graphical conditions. Under stronger conditions, we prove that partition labels are asymptotically correct. Total independence tests is worst-case quadratic in |Z|, with sub-quadratic runtimes observed empirically. We numerically validate our theoretical guarantees on synthetic and semi-synthetic graphs. Adjustment sets from LDP yield less biased and more precise average treatment effect estimates than baselines, with LDP outperforming on confounder recall, test count, and runtime for valid adjustment set discovery.

Kernel Debiased Plug-in Estimation

We consider the problem of estimating a scalar target parameter in the presence of nuisance parameters. Replacing the unknown nuisance parameter with a nonparametric estimator, e.g.,a machine learning (ML) model, is convenient but has shown to be inefficient due to large biases. Modern methods, such as the targeted minimum loss-based estimation (TMLE) and double machine learning (DML), achieve optimal performance under flexible assumptions by harnessing ML estimates while mitigating the plug-in bias. To avoid a sub-optimal bias-variance trade-off, these methods perform a debiasing step of the plug-in pre-estimate. Existing debiasing methods require the influence function of the target parameter as input. However, deriving the IF requires specialized expertise and thus obstructs the adaptation of these methods by practitioners. We propose a novel way to debias plug-in estimators which (i) is efficient, (ii) does not require the IF to be implemented, (iii) is computationally tractable, and therefore can be readily adapted to new estimation problems and automated without analytic derivations by the user. We build on the TMLE framework and update a plug-in estimate with a regularized likelihood maximization step over a nonparametric model constructed with a reproducing kernel Hilbert space (RKHS), producing an efficient plug-in estimate for any regular target parameter. Our method, thus, offers the efficiency of competing debiasing techniques without sacrificing the utility of the plug-in approach.

Contextual Bandits with Budgeted Information Reveal

Contextual bandit algorithms are commonly used in digital health to recommend personalized treatments. However, to ensure the effectiveness of the treatments, patients are often requested to take actions that have no immediate benefit to them, which we refer to as pro-treatment actions. In practice, clinicians have a limited budget to encourage patients to take these actions and collect additional information. We introduce a novel optimization and learning algorithm to address this problem. This algorithm effectively combines the strengths of two algorithmic approaches in a seamless manner, including 1) an online primal-dual algorithm for deciding the optimal timing to reach out to patients, and 2) a contextual bandit learning algorithm to deliver personalized treatment to the patient. We prove that this algorithm admits a sub-linear regret bound. We illustrate the usefulness of this algorithm on both synthetic and real-world data.