Distribution-free two-sample testing with blurred total variation distance

Two-sample testing, where we aim to determine whether two distributions are equal or not equal based on samples from each one, is challenging if we cannot place assumptions on the properties of the two distributions. In particular, certifying equality of distributions, or even providing a tight upper bound on the total variation (TV) distance between the distributions, is impossible to achieve in a distribution-free regime. In this work, we examine the blurred TV distance, a relaxation of TV distance that enables us to perform inference without assumptions on the distributions. We provide theoretical guarantees for distribution-free upper and lower bounds on the blurred TV distance, and examine its properties in high dimensions.

Concentration Inequalities for Exchangeable Tensors and Matrix-valued Data

We study concentration inequalities for structured weighted sums of random data, including (i) tensor inner products and (ii) sequential matrix sums. We are interested in tail bounds and concentration inequalities for those structured weighted sums under exchangeability, extending beyond the classical framework of independent terms. We develop Hoeffding and Bernstein bounds provided with structure-dependent exchangeability. Along the way, we recover known results in weighted sum of exchangeable random variables and i.i.d. sums of random matrices to the optimal constants. Notably, we develop a sharper concentration bound for combinatorial sum of matrix arrays than the results previously derived from Chatterjee's method of exchangeable pairs. For applications, the richer structures provide us with novel analytical tools for estimating the average effect of multi-factor response models and studying fixed-design sketching methods in federated averaging. We apply our results to these problems, and find that our theoretical predictions are corroborated by numerical evidence.

Stability and Accuracy Trade-offs in Statistical Estimation

Algorithmic stability is a central concept in statistics and learning theory that measures how sensitive an algorithm's output is to small changes in the training data. Stability plays a crucial role in understanding generalization, robustness, and replicability, and a variety of stability notions have been proposed in different learning settings. However, while stability entails desirable properties, it is typically not sufficient on its own for statistical learning -- and indeed, it may be at odds with accuracy, since an algorithm that always outputs a constant function is perfectly stable but statistically meaningless. Thus, it is essential to understand the potential statistical cost of stability. In this work, we address this question by adopting a statistical decision-theoretic perspective, treating stability as a constraint in estimation. Focusing on two representative notions-worst-case stability and average-case stability-we first establish general lower bounds on the achievable estimation accuracy under each type of stability constraint. We then develop optimal stable estimators for four canonical estimation problems, including several mean estimation and regression settings. Together, these results characterize the optimal trade-offs between stability and accuracy across these tasks. Our findings formalize the intuition that average-case stability imposes a qualitatively weaker restriction than worst-case stability, and they further reveal that the gap between these two can vary substantially across different estimation problems.

Can a calibration metric be both testable and actionable?

Forecast probabilities often serve as critical inputs for binary decision making. In such settings, calibration$\unicode{x2014}$ensuring forecasted probabilities match empirical frequencies$\unicode{x2014}$is essential. Although the common notion of Expected Calibration Error (ECE) provides actionable insights for decision making, it is not testable: it cannot be empirically estimated in many practical cases. Conversely, the recently proposed Distance from Calibration (dCE) is testable but is not actionable since it lacks decision-theoretic guarantees needed for high-stakes applications. We introduce Cutoff Calibration Error, a calibration measure that bridges this gap by assessing calibration over intervals of forecasted probabilities. We show that Cutoff Calibration Error is both testable and actionable and examine its implications for popular post-hoc calibration methods, such as isotonic regression and Platt scaling.

Are all models wrong? Fundamental limits in distribution-free empirical model falsification

In statistics and machine learning, when we train a fitted model on available data, we typically want to ensure that we are searching within a model class that contains at least one accurate model -- that is, we would like to ensure an upper bound on the model class risk (the lowest possible risk that can be attained by any model in the class). However, it is also of interest to establish lower bounds on the model class risk, for instance so that we can determine whether our fitted model is at least approximately optimal within the class, or, so that we can decide whether the model class is unsuitable for the particular task at hand. Particularly in the setting of interpolation learning where machine learning models are trained to reach zero error on the training data, we might ask if, at the very least, a positive lower bound on the model class risk is possible -- or are we unable to detect that "all models are wrong"? In this work, we answer these questions in a distribution-free setting by establishing a model-agnostic, fundamental hardness result for the problem of constructing a lower bound on the best test error achievable over a model class, and examine its implications on specific model classes such as tree-based methods and linear regression.

Theoretical Foundations of Conformal Prediction

This book is about conformal prediction and related inferential techniques that build on permutation tests and exchangeability. These techniques are useful in a diverse array of tasks, including hypothesis testing and providing uncertainty quantification guarantees for machine learning systems. Much of the current interest in conformal prediction is due to its ability to integrate into complex machine learning workflows, solving the problem of forming prediction sets without any assumptions on the form of the data generating distribution. Since contemporary machine learning algorithms have generally proven difficult to analyze directly, conformal prediction's main appeal is its ability to provide formal, finite-sample guarantees when paired with such methods. The goal of this book is to teach the reader about the fundamental technical arguments that arise when researching conformal prediction and related questions in distribution-free inference. Many of these proof strategies, especially the more recent ones, are scattered among research papers, making it difficult for researchers to understand where to look, which results are important, and how exactly the proofs work. We hope to bridge this gap by curating what we believe to be some of the most important results in the literature and presenting their proofs in a unified language, with illustrations, and with an eye towards pedagogy.

Boosted Conformal Prediction Intervals

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.

Is Algorithmic Stability Testable? A Unified Framework under Computational Constraints

Algorithmic stability is a central notion in learning theory that quantifies the sensitivity of an algorithm to small changes in the training data. If a learning algorithm satisfies certain stability properties, this leads to many important downstream implications, such as generalization, robustness, and reliable predictive inference. Verifying that stability holds for a particular algorithm is therefore an important and practical question. However, recent results establish that testing the stability of a black-box algorithm is impossible, given limited data from an unknown distribution, in settings where the data lies in an uncountably infinite space (such as real-valued data). In this work, we extend this question to examine a far broader range of settings, where the data may lie in any space -- for example, categorical data. We develop a unified framework for quantifying the hardness of testing algorithmic stability, which establishes that across all settings, if the available data is limited then exhaustive search is essentially the only universally valid mechanism for certifying algorithmic stability. Since in practice, any test of stability would naturally be subject to computational constraints, exhaustive search is impossible and so this implies fundamental limits on our ability to test the stability property for a black-box algorithm.

Building a stable classifier with the inflated argmax

We propose a new framework for algorithmic stability in the context of multiclass classification. In practice, classification algorithms often operate by first assigning a continuous score (for instance, an estimated probability) to each possible label, then taking the maximizer -- i.e., selecting the class that has the highest score. A drawback of this type of approach is that it is inherently unstable, meaning that it is very sensitive to slight perturbations of the training data, since taking the maximizer is discontinuous. Motivated by this challenge, we propose a pipeline for constructing stable classifiers from data, using bagging (i.e., resampling and averaging) to produce stable continuous scores, and then using a stable relaxation of argmax, which we call the "inflated argmax," to convert these scores to a set of candidate labels. The resulting stability guarantee places no distributional assumptions on the data, does not depend on the number of classes or dimensionality of the covariates, and holds for any base classifier. Using a common benchmark data set, we demonstrate that the inflated argmax provides necessary protection against unstable classifiers, without loss of accuracy.

The Limits of Assumption-free Tests for Algorithm Performance

Algorithm evaluation and comparison are fundamental questions in machine learning and statistics -- how well does an algorithm perform at a given modeling task, and which algorithm performs best? Many methods have been developed to assess algorithm performance, often based around cross-validation type strategies, retraining the algorithm of interest on different subsets of the data and assessing its performance on the held-out data points. Despite the broad use of such procedures, the theoretical properties of these methods are not yet fully understood. In this work, we explore some fundamental limits for answering these questions with limited amounts of data. In particular, we make a distinction between two questions: how good is an algorithm $A$ at the problem of learning from a training set of size $n$, versus, how good is a particular fitted model produced by running $A$ on a particular training data set of size $n$? Our main results prove that, for any test that treats the algorithm $A$ as a ``black box'' (i.e., we can only study the behavior of $A$ empirically), there is a fundamental limit on our ability to carry out inference on the performance of $A$, unless the number of available data points $N$ is many times larger than the sample size $n$ of interest. (On the other hand, evaluating the performance of a particular fitted model is easy as long as a holdout data set is available -- that is, as long as $N-n$ is not too small.) We also ask whether an assumption of algorithmic stability might be sufficient to circumvent this hardness result. Surprisingly, we find that this is not the case: the same hardness result still holds for the problem of evaluating the performance of $A$, aside from a high-stability regime where fitted models are essentially nonrandom. Finally, we also establish similar hardness results for the problem of comparing multiple algorithms.