Emmanuel J. Candès’s research while affiliated with Stanford University and other places

We study the coverage properties of full conformal regression in the proportional asymptotic regime where the ratio of the dimension and the sample size converges to a constant. In this setting, existing theory tells us only that full conformal inference is unbiased, in the sense that its average coverage lies at the desired level when marginalized over both the new test point and the training data. Considerably less is known about the behaviour of these methods conditional on the training set. As a result, the exact benefits of full conformal inference over much simpler alternative methods is unclear. This paper investigates the behaviour of full conformal inference and natural uncorrected alternatives for a broad class of $L_2$-regularized linear regression models. We show that in the proportional asymptotic regime the training-conditional coverage of full conformal inference concentrates at the target value. On the other hand, simple alternatives that directly compare test and training residuals realize constant undercoverage bias. While these results demonstrate the necessity of full conformal in correcting for high-dimensional overfitting, we also show that this same methodology is redundant for the related task of tuning the regularization level. In particular, we show that full conformal inference still yields asymptotically valid coverage when the regularization level is selected using only the training set, without consideration of the test point. Simulations show that our asymptotic approximations are accurate in finite samples and can be readily extended to other popular full conformal variants, such as full conformal quantile regression and the LASSO, that do not directly meet our assumptions.

Hypotheses are central to information acquisition, decision-making, and discovery. However, many real-world hypotheses are abstract, high-level statements that are difficult to validate directly. This challenge is further intensified by the rise of hypothesis generation from Large Language Models (LLMs), which are prone to hallucination and produce hypotheses in volumes that make manual validation impractical. Here we propose Popper, an agentic framework for rigorous automated validation of free-form hypotheses. Guided by Karl Popper's principle of falsification, Popper validates a hypothesis using LLM agents that design and execute falsification experiments targeting its measurable implications. A novel sequential testing framework ensures strict Type-I error control while actively gathering evidence from diverse observations, whether drawn from existing data or newly conducted procedures. We demonstrate Popper on six domains including biology, economics, and sociology. Popper delivers robust error control, high power, and scalability. Furthermore, compared to human scientists, Popper achieved comparable performance in validating complex biological hypotheses while reducing time by 10 folds, providing a scalable, rigorous solution for hypothesis validation.

The practice of deep learning has shown that neural networks generalize remarkably well even with an extreme number of learned parameters. This appears to contradict traditional statistical wisdom, in which a trade-off between model complexity and fit to the data is essential. We aim to address this discrepancy by adopting a convex optimization and sparse recovery perspective. We consider the training and generalization properties of two-layer ReLU networks with standard weight decay regularization. Under certain regularity assumptions on the data, we show that ReLU networks with an arbitrary number of parameters learn only simple models that explain the data. This is analogous to the recovery of the sparsest linear model in compressed sensing. For ReLU networks and their variants with skip connections or normalization layers, we present isometry conditions that ensure the exact recovery of planted neurons. For randomly generated data, we show the existence of a phase transition in recovering planted neural network models, which is easy to describe: whenever the ratio between the number of samples and the dimension exceeds a numerical threshold, the recovery succeeds with high probability; otherwise, it fails with high probability. Surprisingly, ReLU networks learn simple and sparse models that generalize well even when the labels are noisy. The phase transition phenomenon is confirmed through numerical experiments.

Motivation Conditional testing via the knockoff framework allows one to identify—among large number of possible explanatory variables—those that carry unique information about an outcome of interest, and also provides a false discovery rate guarantee on the selection. This approach is particularly well suited to the analysis of genome wide association studies (GWAS), which have the goal of identifying genetic variants which influence traits of medical relevance. Results While conditional testing can be both more powerful and precise than traditional GWAS analysis methods, its vanilla implementation encounters a difficulty common to all multivariate analysis methods: it is challenging to distinguish among multiple, highly correlated regressors. This impasse can be overcome by shifting the object of inference from single variables to groups of correlated variables. To achieve this, it is necessary to construct ‘‘group knockoffs." While successful examples are already documented in the literature, this paper substantially expands the set of algorithms and software for group knockoffs. We focus in particular on second-order knockoffs, for which we describe correlation matrix approximations that are appropriate for GWAS data and that result in considerable computational savings. We illustrate the effectiveness of the proposed methods with simulations and with the analysis of albuminuria data from the UK Biobank. Availability The described algorithms are implemented in an open-source Julia package Knockoffs.jl. R and Python wrappers are available as knockoffsr and knockoffspy packages. Supplementary information Supplementary data are available from Bioinformatics online.

Estimating out-of-sample risk for models trained on large high-dimensional datasets is an expensive but essential part of the machine learning process, enabling practitioners to optimally tune hyperparameters. Cross-validation (CV) serves as the de facto standard for risk estimation but poorly trades off high bias (K-fold CV) for computational cost (leave-one-out CV). We propose a randomized approximate leave-one-out (RandALO) risk estimator that is not only a consistent estimator of risk in high dimensions but also less computationally expensive than K-fold CV. We support our claims with extensive simulations on synthetic and real data and provide a user-friendly Python package implementing RandALO available on PyPI as randalo and at https://github.com/cvxgrp/randalo.

Large language models (LLMs) have shown high agreement with human raters across a variety of tasks, demonstrating potential to ease the challenges of human data collection. In computational social science (CSS), researchers are increasingly leveraging LLM annotations to complement slow and expensive human annotations. Still, guidelines for collecting and using LLM annotations, without compromising the validity of downstream conclusions, remain limited. We introduce Confidence-Driven Inference: a method that combines LLM annotations and LLM confidence indicators to strategically select which human annotations should be collected, with the goal of producing accurate statistical estimates and provably valid confidence intervals while reducing the number of human annotations needed. Our approach comes with safeguards against LLM annotations of poor quality, guaranteeing that the conclusions will be both valid and no less accurate than if we only relied on human annotations. We demonstrate the effectiveness of Confidence-Driven Inference over baselines in statistical estimation tasks across three CSS settings--text politeness, stance, and bias--reducing the needed number of human annotations by over 25% in each. Although we use CSS settings for demonstration, Confidence-Driven Inference can be used to estimate most standard quantities across a broad range of NLP problems.

We develop new conformal inference methods for obtaining validity guarantees on the output of large language models (LLMs). Prior work in conformal language modeling identifies a subset of the text that satisfies a high-probability guarantee of correctness. These methods work by filtering claims from the LLM's original response if a scoring function evaluated on the claim fails to exceed a threshold calibrated via split conformal prediction. Existing methods in this area suffer from two deficiencies. First, the guarantee stated is not conditionally valid. The trustworthiness of the filtering step may vary based on the topic of the response. Second, because the scoring function is imperfect, the filtering step can remove many valuable and accurate claims. We address both of these challenges via two new conformal methods. First, we generalize the conditional conformal procedure of Gibbs et al. (2023) in order to adaptively issue weaker guarantees when they are required to preserve the utility of the output. Second, we show how to systematically improve the quality of the scoring function via a novel algorithm for differentiating through the conditional conformal procedure. We demonstrate the efficacy of our approach on both synthetic and real-world datasets.

This paper introduces a boosted conformal procedure designed to tailor conformalized prediction intervals toward specific desired properties, such as enhanced conditional coverage or reduced interval length. We employ machine learning techniques, notably gradient boosting, to systematically improve upon a predefined conformity score function. This process is guided by carefully constructed loss functions that measure the deviation of prediction intervals from the targeted properties. The procedure operates post-training, relying solely on model predictions and without modifying the trained model (e.g., the deep network). Systematic experiments demonstrate that starting from conventional conformal methods, our boosted procedure achieves substantial improvements in reducing interval length and decreasing deviation from target conditional coverage.

While reliable data-driven decision-making hinges on high-quality labeled data, the acquisition of quality labels often involves laborious human annotations or slow and expensive scientific measurements. Machine learning is becoming an appealing alternative as sophisticated predictive techniques are being used to quickly and cheaply produce large amounts of predicted labels; e.g., predicted protein structures are used to supplement experimentally derived structures, predictions of socioeconomic indicators from satellite imagery are used to supplement accurate survey data, and so on. Since predictions are imperfect and potentially biased, this practice brings into question the validity of downstream inferences. We introduce cross-prediction: a method for valid inference powered by machine learning. With a small labeled dataset and a large unlabeled dataset, cross-prediction imputes the missing labels via machine learning and applies a form of debiasing to remedy the prediction inaccuracies. The resulting inferences achieve the desired error probability and are more powerful than those that only leverage the labeled data. Closely related is the recent proposal of prediction-powered inference [A. N. Angelopoulos, S. Bates, C. Fannjiang, M. I. Jordan, T. Zrnic, Science 382 , 669–674 (2023)], which assumes that a good pretrained model is already available. We show that cross-prediction is consistently more powerful than an adaptation of prediction-powered inference in which a fraction of the labeled data is split off and used to train the model. Finally, we observe that cross-prediction gives more stable conclusions than its competitors; its CIs typically have significantly lower variability.