arXiv subjects

Lucas Janson

Publications and source records attributed to Lucas Janson.

At least 19 recordsLinked to original sources

arXiv2025

The $L$-test: Increasing the Linear Model $F$-test's Power Under Sparsity Without Sacrificing Validity

We introduce a new procedure for testing the significance of a set of regression coefficients in a Gaussian linear model with $n \geq d$. Our method, the $L$-test, provides the same statistical validity guarantee as the classical $F$-test, while attaining higher power when the nuisance coefficients are sparse. Although the $L$-test requires Monte Carlo sampling, each sample's runtime is dominated by simple matrix-vector multiplications so that the overall test remains computationally efficient. Furthermore, we provide a Monte-Carlo-free variant that can be used for particularly large-scale multiple testing applications. We give intuition for the power of our approach, validate its advantages through extensive simulations, and illustrate its practical utility in both single- and multiple-testing contexts with an application to an HIV drug resistance dataset. In the concluding remarks, we also discuss how our methodology can be applied to a more general class of parametric models that admit asymptotically Gaussian estimators.

stat.ME↗

arXiv2025

Bernstein-von Mises for Adaptively Collected Data

Uncertainty quantification (UQ) for adaptively collected data, such as that coming from adaptive experiments, bandits, or reinforcement learning, is necessary for critical elements of data collection such as ensuring safety and conducting after-study inference. The data's adaptivity creates significant challenges for frequentist UQ, yet Bayesian UQ remains the same as if the data were independent and identically distributed (i.i.d.), making it an appealing and commonly used approach. Bayesian UQ requires the (correct) specification of a prior distribution while frequentist UQ does not, but for i.i.d. data the celebrated Bernstein-von Mises theorem shows that as the sample size grows, the prior 'washes out' and Bayesian UQ becomes frequentist-valid, implying that the choice of prior need not be a major impediment to Bayesian UQ as it makes no difference asymptotically. This paper for the first time extends the Bernstein-von Mises theorem to adaptively collected data, proving asymptotic equivalence between Bayesian UQ and Wald-type frequentist UQ in this challenging setting. Our result showing this asymptotic agreement does not require the standard stability condition required by works studying validity of Wald-type frequentist UQ; in cases where stability is satisfied, our results combined with these prior studies of frequentist UQ imply frequentist validity of Bayesian UQ. Counterintuitively however, they also provide a negative result that Bayesian UQ is not asymptotically frequentist valid when stability fails, despite the fact that the prior washes out and Bayesian UQ asymptotically matches standard Wald-type frequentist UQ. We empirically validate our theory (positive and negative) via a range of simulations.

math.ST↗

arXiv2025

Conditioning on posterior samples for flexible frequentist goodness-of-fit testing

Tests of goodness of fit are used in nearly every domain where statistics is applied. One powerful and flexible approach is to sample artificial data sets that are exchangeable with the real data under the null hypothesis (but not under the alternative), as this allows the analyst to conduct a valid test using any test statistic they desire. Such sampling is typically done by conditioning on either an exact or approximate sufficient statistic, but existing methods for doing so have significant limitations, which either preclude their use or substantially reduce their power or computational tractability for many important models. In this paper, we propose to condition on samples from a Bayesian posterior distribution, which constitute a very different type of approximate sufficient statistic than those considered in prior work. Our approach, approximately co-sufficient sampling via Bayes (aCSS-B), considerably expands the scope of this flexible type of goodness-of-fit testing. We prove the approximate validity of the resulting test, and demonstrate its utility on three common null models where no existing methods apply, as well as its outperformance on models where existing methods do apply.

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arXiv2025

Chiseling: Powerful and Valid Subgroup Selection via Interactive Machine Learning

In regression and causal inference, controlled subgroup selection aims to identify, with inferential guarantees, a subgroup (defined as a subset of the covariate space) on which the average response or treatment effect is above a given threshold. E.g., in a clinical trial, it may be of interest to find a subgroup with a positive average treatment effect. However, existing methods either lack inferential guarantees, heavily restrict the search for the subgroup, or sacrifice efficiency by naive data splitting. We propose a novel framework called chiseling that allows the analyst to interactively refine and test a candidate subgroup by iteratively shrinking it. The sole restriction is that the shrinkage direction only depends on the points outside the current subgroup, but otherwise the analyst may leverage any prior information or machine learning algorithm. Despite this flexibility, chiseling controls the probability that the discovered subgroup is null (e.g., has a non-positive average treatment effect) under minimal assumptions: for example, in randomized experiments, this inferential validity guarantee holds under only bounded moment conditions. When applied to a variety of simulated datasets and a real survey experiment, chiseling identifies substantially better subgroups than existing methods with inferential guarantees.

stat.ME↗

arXiv2025

The Relative Instability of Model Comparison with Cross-validation

Cross-validation (CV) is known to provide asymptotically exact tests and confidence intervals for model improvement but only when the model comparison is relatively stable. Surprisingly, we prove that even simple, individually stable models can generate relatively unstable comparisons, calling into question the validity of CV inference. Specifically, we show that the Lasso and its close cousin, soft-thresholding, generate relatively unstable comparisons and invalid CV inferences, even in the most favorable of learning settings and when both models are individually stable. These findings highlight the importance of verifying relative stability before deploying CV for model comparison.

stat.ML↗

arXiv2025

Foundations of Safe Online Reinforcement Learning in the Linear Quadratic Regulator: $\sqrt{T}$-Regret

Understanding how to efficiently learn while adhering to safety constraints is essential for using online reinforcement learning in practical applications. However, proving rigorous regret bounds for safety-constrained reinforcement learning is difficult due to the complex interaction between safety, exploration, and exploitation. In this work, we seek to establish foundations for safety-constrained reinforcement learning by studying the canonical problem of controlling a one-dimensional linear dynamical system with unknown dynamics. We study the safety-constrained version of this problem, where the state must with high probability stay within a safe region, and we provide the first safe algorithm that achieves regret of $\tilde{O}_T(\sqrt{T})$. Furthermore, the regret is with respect to the baseline of truncated linear controllers, a natural baseline of non-linear controllers that are well-suited for safety-constrained linear systems. In addition to introducing this new baseline, we also prove several desirable continuity properties of the optimal controller in this baseline. In showing our main result, we prove that whenever the constraints impact the optimal controller, the non-linearity of our controller class leads to a faster rate of learning than in the unconstrained setting.

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arXiv2025

Semiparametric Inference for Partially Identifiable Data Fusion Estimands via Double Machine Learning

Many statistical estimands of interest (e.g., in regression or causality) are functions of the joint distribution of multiple random variables. But in some applications, data is not available that measures all random variables on each subject, and instead the only possible approach is one of data fusion, where multiple independent data sets, each measuring a subset of the random variables of interest, are combined for inference. In general, since all random variables are never observed jointly, their joint distribution, and hence also the estimand which is a function of it, is only partially identifiable. Unfortunately, the endpoints of the partially identifiable region depend in general on entire conditional distributions, rendering them hard both operationally and statistically to estimate. To address this, we present a novel outer-bound on the region of partial identifiability (and establish conditions under which it is tight) that depends only on certain conditional first and second moments. This allows us to derive semiparametrically efficient estimators of our endpoint outer-bounds that only require the standard machine learning toolbox which learns conditional means. We prove asymptotic normality and semiparametric efficiency of our estimators and provide consistent estimators of their variances, enabling asymptotically valid confidence interval construction for our original partially identifiable estimand. We demonstrate the utility of our method in simulations and a data fusion problem from economics.

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arXiv2024

Compositional Covariate Importance Testing via Partial Conjunction of Bivariate Hypotheses

Compositional data (i.e., data comprising random variables that sum up to a constant) arises in many applications including microbiome studies, chemical ecology, political science, and experimental designs. Yet when compositional data serve as covariates in a regression, the sum constraint renders every covariate automatically conditionally independent of the response given the other covariates, since each covariate is a deterministic function of the others. Since essentially all covariate importance tests and variable selection methods, including parametric ones, are at their core testing conditional independence, they are all completely powerless on regression problems with compositional covariates. In fact, compositionality causes ambiguity in the very notion of relevant covariates. To address this problem, we identify a natural way to translate the typical notion of relevant covariates to the setting with compositional covariates and establish that it is intuitive, well-defined, and unique. We then develop corresponding hypothesis tests and controlled variable selection procedures via a novel connection with \emph{bivariate} conditional independence testing and partial conjunction hypothesis testing. Finally, we provide theoretical guarantees of the validity of our methods, and through numerical experiments demonstrate that our methods are not only valid but also powerful across a range of data-generating scenarios.

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arXiv2024

Foundations of Safe Online Reinforcement Learning in the Linear Quadratic Regulator: Generalized Baselines

Many practical applications of online reinforcement learning require the satisfaction of safety constraints while learning about the unknown environment. In this work, we establish theoretical foundations for reinforcement learning with safety constraints by studying the canonical problem of Linear Quadratic Regulator learning with unknown dynamics, but with the additional constraint that the position must stay within a safe region for the entire trajectory with high probability. Our primary contribution is a general framework for studying stronger baselines of nonlinear controllers that are better suited for constrained problems than linear controllers. Due to the difficulty of analyzing non-linear controllers in a constrained problem, we focus on 1-dimensional state- and action- spaces, however we also discuss how we expect the high-level takeaways can generalize to higher dimensions. Using our framework, we show that for \emph{any} non-linear baseline satisfying natural assumptions, $\tilde{O}_T(\sqrt{T})$-regret is possible when the noise distribution has sufficiently large support, and $\tilde{O}_T(T^{2/3})$-regret is possible for \emph{any} subgaussian noise distribution. In proving these results, we introduce a new uncertainty estimation bound for nonlinear controls which shows that enforcing safety in the presence of sufficient noise can provide ``free exploration'' that compensates for the added cost of uncertainty in safety-constrained control.

stat.ML↗

arXiv2024

SOAP: Improving and Stabilizing Shampoo using Adam

There is growing evidence of the effectiveness of Shampoo, a higher-order preconditioning method, over Adam in deep learning optimization tasks. However, Shampoo's drawbacks include additional hyperparameters and computational overhead when compared to Adam, which only updates running averages of first- and second-moment quantities. This work establishes a formal connection between Shampoo (implemented with the 1/2 power) and Adafactor -- a memory-efficient approximation of Adam -- showing that Shampoo is equivalent to running Adafactor in the eigenbasis of Shampoo's preconditioner. This insight leads to the design of a simpler and computationally efficient algorithm: $\textbf{S}$hampo$\textbf{O}$ with $\textbf{A}$dam in the $\textbf{P}$reconditioner's eigenbasis (SOAP). With regards to improving Shampoo's computational efficiency, the most straightforward approach would be to simply compute Shampoo's eigendecomposition less frequently. Unfortunately, as our empirical results show, this leads to performance degradation that worsens with this frequency. SOAP mitigates this degradation by continually updating the running average of the second moment, just as Adam does, but in the current (slowly changing) coordinate basis. Furthermore, since SOAP is equivalent to running Adam in a rotated space, it introduces only one additional hyperparameter (the preconditioning frequency) compared to Adam. We empirically evaluate SOAP on language model pre-training with 360m and 660m sized models. In the large batch regime, SOAP reduces the number of iterations by over 40% and wall clock time by over 35% compared to AdamW, with approximately 20% improvements in both metrics compared to Shampoo. An implementation of SOAP is available at https://github.com/nikhilvyas/SOAP.

cs.LG↗

arXiv2024

Optimal ablation for interpretability

Interpretability studies often involve tracing the flow of information through machine learning models to identify specific model components that perform relevant computations for tasks of interest. Prior work quantifies the importance of a model component on a particular task by measuring the impact of performing ablation on that component, or simulating model inference with the component disabled. We propose a new method, optimal ablation (OA), and show that OA-based component importance has theoretical and empirical advantages over measuring importance via other ablation methods. We also show that OA-based component importance can benefit several downstream interpretability tasks, including circuit discovery, localization of factual recall, and latent prediction.

cs.LG↗

arXiv2024

The $\ell$-test: leveraging sparsity in the Gaussian linear model for improved inference

We develop novel LASSO-based methods for coefficient testing and confidence interval construction in the Gaussian linear model with $n\ge d$. Our methods' finite-sample validity is identical to that of their ubiquitous ordinary-least-squares-$t$-test-based analogues, yet have substantially higher power when the true coefficient vector is sparse. In particular, under sparsity our coefficient test, which we call the $\ell$-test, performs like the \emph{one-sided} $t$-test (despite not being given any information about the sign), and $\ell$-test-based confidence intervals are correspondingly shorter than the standard $t$-test-based intervals. The nature of the $\ell$-test directly provides a novel exact adjustment conditional on LASSO selection for post-selection inference, allowing for the construction of post-selection $p$-values and confidence intervals. None of our methods require resampling or Monte Carlo estimation. We perform a variety of simulations and a real data analysis on an HIV drug resistance data set to demonstrate the benefits of the $\ell$-test. We additionally show that the $\ell$-test can be applied to a large class of asymptotically Gaussian estimators, dramatically expanding its applicability beyond linear models.

stat.ME↗

arXiv2024

A New Perspective on Shampoo's Preconditioner

Shampoo, a second-order optimization algorithm which uses a Kronecker product preconditioner, has recently garnered increasing attention from the machine learning community. The preconditioner used by Shampoo can be viewed either as an approximation of the Gauss--Newton component of the Hessian or the covariance matrix of the gradients maintained by Adagrad. We provide an explicit and novel connection between the $\textit{optimal}$ Kronecker product approximation of these matrices and the approximation made by Shampoo. Our connection highlights a subtle but common misconception about Shampoo's approximation. In particular, the $\textit{square}$ of the approximation used by the Shampoo optimizer is equivalent to a single step of the power iteration algorithm for computing the aforementioned optimal Kronecker product approximation. Across a variety of datasets and architectures we empirically demonstrate that this is close to the optimal Kronecker product approximation. Additionally, for the Hessian approximation viewpoint, we empirically study the impact of various practical tricks to make Shampoo more computationally efficient (such as using the batch gradient and the empirical Fisher) on the quality of Hessian approximation.

cs.LG↗

arXiv2024

Evaluating the Effectiveness of Index-Based Treatment Allocation

When resources are scarce, an allocation policy is needed to decide who receives a resource. This problem occurs, for instance, when allocating scarce medical resources and is often solved using modern ML methods. This paper introduces methods to evaluate index-based allocation policies -- that allocate a fixed number of resources to those who need them the most -- by using data from a randomized control trial. Such policies create dependencies between agents, which render the assumptions behind standard statistical tests invalid and limit the effectiveness of estimators. Addressing these challenges, we translate and extend recent ideas from the statistics literature to present an efficient estimator and methods for computing asymptotically correct confidence intervals. This enables us to effectively draw valid statistical conclusions, a critical gap in previous work. Our extensive experiments validate our methodology in practical settings, while also showcasing its statistical power. We conclude by proposing and empirically verifying extensions of our methodology that enable us to reevaluate a past randomized control trial to evaluate different ML allocation policies in the context of a mHealth program, drawing previously invisible conclusions.

cs.LG↗

arXiv2024

Context in Public Health for Underserved Communities: A Bayesian Approach to Online Restless Bandits

Public health programs often provide interventions to encourage program adherence, and effectively allocating interventions is vital for producing the greatest overall health outcomes, especially in underserved communities where resources are limited. Such resource allocation problems are often modeled as restless multi-armed bandits (RMABs) with unknown underlying transition dynamics, hence requiring online reinforcement learning (RL). We present Bayesian Learning for Contextual RMABs (BCoR), an online RL approach for RMABs that novelly combines techniques in Bayesian modeling with Thompson sampling to flexibly model the complex RMAB settings present in public health program adherence problems, namely context and non-stationarity. BCoR's key strength is the ability to leverage shared information within and between arms to learn the unknown RMAB transition dynamics quickly in intervention-scarce settings with relatively short time horizons, which is common in public health applications. Empirically, BCoR achieves substantially higher finite-sample performance over a range of experimental settings, including a setting using real-world adherence data that was developed in collaboration with ARMMAN, an NGO in India which runs a large-scale maternal mHealth program, showcasing BCoR practical utility and potential for real-world deployment.

cs.LG↗

arXiv2023

Total Variation Floodgate for Variable Importance Inference in Classification

Inferring variable importance is the key problem of many scientific studies, where researchers seek to learn the effect of a feature $X$ on the outcome $Y$ in the presence of confounding variables $Z$. Focusing on classification problems, we define the expected total variation (ETV), which is an intuitive and deterministic measure of variable importance that does not rely on any model context. We then introduce algorithms for statistical inference on the ETV under design-based/model-X assumptions. These algorithms build on the floodgate notion for regression problems (Zhang and Janson 2020). The algorithms we introduce can leverage any user-specified regression function and produce asymptotic lower confidence bounds for the ETV. We show the effectiveness of our algorithms with simulations and a case study in conjoint analysis on the US general election.

stat.ME↗

arXiv2023

Estimating the Convex Hull of the Image of a Set with Smooth Boundary: Error Bounds and Applications

We study the problem of estimating the convex hull of the image $f(X)\subset\mathbb{R}^n$ of a compact set $X\subset\mathbb{R}^m$ with smooth boundary through a smooth function $f:\mathbb{R}^m\to\mathbb{R}^n$. Assuming that $f$ is a submersion, we derive a new bound on the Hausdorff distance between the convex hull of $f(X)$ and the convex hull of the images $f(x_i)$ of $M$ sampled inputs $x_i$ on the boundary of $X$. When applied to the problem of geometric inference from a random sample, our results give error bounds that are tighter and more general than in previous work. We present applications to the problems of robust optimization, of reachability analysis of dynamical systems, and of robust trajectory optimization under bounded uncertainty.

math.OC↗

arXiv2023

Randomization Tests for Adaptively Collected Data

Randomization testing is a fundamental method in statistics, enabling inferential tasks such as testing for (conditional) independence of random variables, constructing confidence intervals in semiparametric location models, and constructing (by inverting a permutation test) model-free prediction intervals via conformal inference. Randomization tests are exactly valid for any sample size, but their use is generally confined to exchangeable data. Yet in many applications, data is routinely collected adaptively via, e.g., (contextual) bandit and reinforcement learning algorithms or adaptive experimental designs. In this paper we present a general framework for randomization testing on adaptively collected data (despite its non-exchangeability) that uses a novel weighted randomization test, for which we also present novel computationally tractable resampling algorithms for various popular adaptive assignment algorithms, data-generating environments, and types of inferential tasks. Finally, we demonstrate via a range of simulations the efficacy of our framework for both testing and confidence/prediction interval construction.

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