理论 · statistics

Performativity of predictions refers to the phenomenon where prediction-informed decisions influence the very targets they aim to predict—a dynamic commonly observed in policy-making, social sciences, and economics. In this paper, we initiate an end-to-end framework of statistical inference under performativity. Our contributions are twofold. First, we establish a central limit theorem for estimation and inference in the performative setting, enabling standard inferential tasks such as constructing confidence intervals and conducting hypothesis tests in policy-making contexts. Second, we leverage this central limit theorem to study prediction-powered inference (PPI) under performativity. This approach yields more precise estimates and tighter confidence regions for the model parameters (i.e., policies) of interest in performative prediction. We validate the effectiveness of our framework through numerical experiments. To the best of our knowledge, this is the first work to establish a complete statistical inference under performativity, introducing new challenges and inference settings that we believe will provide substantial value to policy-making, statistics, and machine learning.

The optimal transport (OT) map is a geometry-driven transformation between high-dimensional probability distributions which underpins a wide range of tasks in statistics, applied probability, and machine learning. However, existing statistical theory for OT map estimation is quite restricted, hinging on Brenier's theorem (quadratic cost, absolutely continuous source) to guarantee existence and uniqueness of a deterministic OT map, on which various additional regularity assumptions are imposed to obtain quantitative error bounds. In many real‐world problems these conditions fail or cannot be certified, in which case optimal transportation is possible only via stochastic maps that can split mass. To broaden the scope of map estimation theory to such settings, this work introduces a novel metric for evaluating the transportation quality of stochastic maps. Under this metric, we develop computationally efficient map estimators with near-optimal finite-sample risk bounds, subject to easy-to-verify minimal assumptions. Our analysis further accommodates common forms of adversarial sample contamination, yielding estimators with robust estimation guarantees. Empirical experiments are provided which validate our theory and demonstrate the utility of the proposed framework in settings where existing theory fails. These contributions constitute the first general-purpose theory for map estimation, compatible with a wide spectrum of real-world applications where optimal transport may be intrinsically stochastic.

We address the problem of causal effect estimation in the presence of hidden confounders, using nonparametric instrumental variable (IV) regression. A leading strategy employs \emph{spectral features} - that is, learned features spanning the top eigensubspaces of the operator linking treatments to instruments. We derive a generalization error bound for a two-stage least squares estimator based on spectral features, and gain insights into the method's performance and failure modes. We show that performance depends on two key factors, leading to a clear taxonomy of outcomes. In a \emph{good} scenario, the approach is optimal. This occurs with strong \emph{spectral alignment}, meaning the structural function is well-represented by the top eigenfunctions of the conditional operator, coupled with this operator's slow eigenvalue decay, indicating a strong instrument. Performance degrades in a \emph{bad} scenario: spectral alignment remains strong, but rapid eigenvalue decay (indicating a weaker instrument) demands significantly more samples for effective feature learning. Finally, in the \emph{ugly} scenario, weak spectral alignment causes the method to fail, regardless of the eigenvalues' characteristics. Our synthetic experiments empirically validate this taxonomy. We further introduce a practical procedure to estimate these spectral properties from data, allowing practitioners to diagnose which regime a given problem falls into. We apply this method to the dSprites dataset, demonstrating its utility.

We consider the problem of recovering the support of a sparse signal using noisy projections. While extensive work has been done on the dense measurement matrix setting, the sparse setting remains less explored. In this work, we establish sufficient conditions on the sample size for successful sparse recovery using sparse measurement matrices. Bringing together our result with previously known necessary conditions, we discover that, in the regime where $ds/p \rightarrow +\infty$, sparse recovery using a sparse design exhibits a phase transition at an information-theoretic threshold of $n_{\text{INF}}^{\text{SP}} = \Theta\left(s\log\left(p/s\right)/\log\left(ds/p\right)\right)$ for the number of measurements, where \(p\) denotes the signal dimension, $s$ the number of non-zero components of the signal, and $d$ the expected number of non-zero components per row of measurement. This expression makes the price of sparsity explicit: restricting each measurement to $d$ non‑zeros inflates the required sample size by a factor of $\log{s}/\log\left(ds/p\right)$, revealing a precise trade‑off between sampling complexity and measurement sparsity. Additionally, we examine the effect of sparsifying an originally dense measurement matrix on sparse signal recovery. We prove in the regime of $s = \alpha p$ and $d = \psi p$ with $\alpha, \psi \in \left(0,1\right)$ and $\psi$ small that a sample of size $n^{\text{Sp-ified}}_{\text{INF}} = \Theta\left(p / \psi^2\right)$ is sufficient for recovery, subject to a certain uniform integrability conjecture, the proof of which is work in progress.

A classic statistical problem is to study the asymptotic behavior of the order statistics of a large number of independent samples taken from a distribution with finite expectation. This behavior has implications for several core problems in machine learning and economics—including robust learning under adversarial noise, best-arm identification in bandit algorithms, revenue estimation in second- price auctions, and the analysis of tail-sensitive statistics used in out-of-distribution detection. The research question we tackle in this paper is: How large can the expectation of the $\ell$-th maximum of the $n$ samples be? For $\ell=1$, i.e., the maximum, this expectation is known to grow as $o(n)$, which can be shown to be tight. We show that there is a sharp contrast when considering any fixed $\ell>1$. Surprisingly, in this case, the largest possible growth rate for all fixed $\ell>1$ is $O(\frac{n}{\log(n)\log\log(n)})$ and $\Omega(\frac{n}{\log(n)(\log\log(n))^{1.01}})$. Our result is actually finer than the latter and provides a sharp characterization of the largest achievable growth rate for the expectation of the $\ell$-th maximum of $n$ i.i.d. samples. Beyond the theoretical analysis, we support our findings with extensive simulations. These empirical results highlight a notable phenomenon: although the multiplicative gap between the maximum and the second maximum grows quickly with $n$, the ratio remains approximately constant in 99\% of trials. This suggests that while worst-case growth is sharp and meaningful, typical-case behavior may be significantly more stable.

Contrastive learning---a modern approach to extract useful representations from unlabeled data by training models to distinguish similar samples from dissimilar ones---has driven significant progress in foundation models. In this work, we develop a new theoretical framework for analyzing data augmentation-based contrastive learning, with a focus on SimCLR as a representative example. Our approach is based on the concept of \emph{approximate sufficient statistics}, which we extend beyond its original definition in~\cite{oko2025statistical} for contrastive language-image pretraining (CLIP) using KL-divergence. We generalize it to equivalent forms and general $f$-divergences, and show that minimizing SimCLR and other contrastive losses yields encoders that are approximately sufficient. Furthermore, we demonstrate that these near-sufficient encoders can be effectively adapted to downstream regression and classification tasks, with performance depending on their sufficiency and the error induced by data augmentation in contrastive learning. Concrete examples in linear regression and topic classification are provided to illustrate the broad applicability of our results.

Given i.i.d.~samples from an unknown distribution $P$, the goal of distribution learning is to recover the parameters of a distribution that is close to $P$. When $P$ belongs to the class of product distributions on the Boolean hypercube $\{0,1\}^d$, it is known that $\Omega(d/\epsilon^2)$ samples are necessary to learn $P$ within total variation (TV) distance $\epsilon$. We revisit this problem when the learner is also given as advice the parameters of a product distribution $Q$. We show that there is an efficient algorithm to learn $P$ within TV distance $\epsilon$ that has sample complexity $\tilde{O}(d^{1-\eta}/\epsilon^2)$, if $\|\mathbf{p} - \mathbf{q}\|_1<\epsilon d^{0.5 - \Omega(\eta)}$. Here, $\mathbf{p}$ and $\mathbf{q}$ are the mean vectors of $P$ and $Q$ respectively, and no bound on $\|\mathbf{p} - \mathbf{q}\|_1$ is known to the algorithm a priori.

Gradient boosting is widely popular due to its flexibility and predictive accuracy. However, statistical inference and uncertainty quantification for gradient boosting remain challenging and under-explored. We propose a unified framework for statistical inference in gradient boosting regression. Our framework integrates dropout or parallel training with a recently proposed regularization procedure called Boulevard that allows for a central limit theorem (CLT) for boosting. With these enhancements, we surprisingly find that \textit{increasing} the dropout rate and the number of trees grown in parallel at each iteration substantially enhances signal recovery and overall performance. Our resulting algorithms enjoy similar CLTs, which we use to construct built-in confidence intervals, prediction intervals, and rigorous hypothesis tests for assessing variable importance in only $O(nd^2)$ time with the Nyström method. Numerical experiments verify the asymptotic normality and demonstrate that our algorithms perform well, do not require early stopping, interpolate between regularized boosting and random forests, and confirm the validity of their built-in statistical inference procedures.

In this position paper we argue that conclusions drawn about relative system safety or attack method efficacy via AI red teaming are often not supported by evidence provided by attack success rate (ASR) comparisons. We show, through conceptual, theoretical, and empirical contributions, that many conclusions are founded on apples-to-oranges comparisons or low-validity measurements. Our arguments are grounded in asking a simple question: When can attack success rates be meaningfully compared? To answer this question, we draw on ideas from social science measurement theory and inferential statistics, which, taken together, provide a conceptual grounding for understanding when numerical values obtained through the quantification of system attributes can be meaningfully compared. Through this lens, we articulate conditions under which ASRs can and cannot be meaningfully compared. Using jailbreaking as a running example, we provide examples and extensive discussion of apples-to-oranges ASR comparisons and measurement validity challenges.

We study the problem of robustly estimating the edge density of Erdos Renyi random graphs $\mathbb{G}(n, d^\circ/n)$ when an adversary can arbitrarily add or remove edges incident to an $\eta$-fraction of the nodes. We develop the first polynomial-time algorithm for this problem that estimates $d^\circ$ up to an additive error $O\left({[\sqrt{\log(n) / n} + \eta\sqrt{\log(1/\eta)} ] \cdot \sqrt{d^\circ} + \eta \log(1/\eta)}\right)$. Our error guarantee matches information-theoretic lower bounds up to factors of $\log(1/\eta)$. Moreover, our estimator works for all $d^\circ \geq \Omega(1)$ and achieves optimal breakdown point $\eta = 1/2$. Previous algorithms [Acharya et al 2022, Chen et al 2024], including inefficient ones, incur significantly suboptimal errors. Furthermore, even admitting suboptimal error guarantees, only inefficient algorithms achieve optimal breakdown point. Our algorithm is based on the sum-of-squares (SoS) hierarchy. A key ingredient is to construct constant-degree SoS certificates for concentration of the number of edges incident to small sets in $\mathbb{G}(n, d^\circ/n)$. Crucially, we show that these certificates also exist in the sparse regime, when $d^\circ = o(\log n)$, a regime in which the performance of previous algorithms was significantly suboptimal.

We study the task of noiseless linear regression under Gaussian covariates in the presence of additive oblivious contamination. Specifically, we are given i.i.d.\ samples from a distribution $(x, y)$ on $\mathbb R^d \times \mathbb R$ with $x \sim \mathcal N(0,I_d)$ and $y = x^\top \beta + z$, where $z$ is drawn from an unknown distribution that is independent of $x$. Moreover, $z$ satisfies $\mathbb P[z = 0] = \alpha>0$. The goal is to accurately recover the regressor $\beta$ to small $\ell_2$-error. Ignoring computational considerations, this problem is known to be solvable using $O(d/\alpha)$ samples. On the other hand, the best known polynomial-time algorithms require $\Omega(d/\alpha^2)$ samples. Here we provide formal evidence that the quadratic dependence in $1/\alpha$ is inherent for efficient algorithms. Specifically, we show that any efficient Statistical Query algorithm for this task requires VSTAT complexity at least $\tilde{\Omega}(d^{1/2}/\alpha^2)$.

The m-out-of-n bootstrap—proposed by \cite{bickel1992resampling}—approximates the distribution of a statistic by repeatedly drawing $m$ subsamples ($m \ll n$) without replacement from an original sample of size n; it is now routinely used for robust inference with heavy-tailed data, bandwidth selection, and other large-sample applications. Despite this broad applicability across econometrics, biostatistics, and machine-learning workflows, rigorous parameter-free guarantees for the soundness of the m-out-of-n bootstrap when estimating sample quantiles have remained elusive. This paper establishes such guarantees by analysing the estimator of sample quantiles obtained from m-out-of-n resampling of a dataset of length n. We first prove a central limit theorem for a fully data-driven version of the estimator that holds under a mild moment condition and involves no unknown nuisance parameters. We then show that the moment assumption is essentially tight by constructing a counter-example in which the CLT fails. Strengthening the assumptions slightly, we derive an Edgeworth expansion that delivers exact convergence rates and, as a corollary, a Berry–Esséen bound on the bootstrap approximation error. Finally, we illustrate the scope of our results by obtaining parameter-free asymptotic distributions for practical statistics, including the quantiles for random walk MH, and rewards of ergodic MDP's, thereby demonstrating the usefulness of our theory in modern estimation and learning tasks.

Models such as Word2Vec and GloVe construct word embeddings based on the co-occurrence probability $P(i,j)$ of words $i$ and $j$ in text corpora. The resulting vectors $W_i$ not only group semantically similar words but also exhibit a striking linear analogy structure---for example, $W_{\text{king}} - W_{\text{man}} + W_{\text{woman}} \approx W_{\text{queen}}$---whose theoretical origin remains unclear. Previous observations indicate that this analogy structure: (i) already emerges in the top eigenvectors of the matrix $M(i,j) = P(i,j)/P(i)P(j)$, (ii) strengthens and then saturates as more eigenvectors of $M (i, j)$, which controls the dimension of the embeddings, are included, (iii) is enhanced when using $\log M(i,j)$ rather than $M(i,j)$, and (iv) persists even when all word pairs involved in a specific analogy relation (e.g., king--queen, man--woman) are removed from the corpus. To explain these phenomena, we introduce a theoretical generative model in which words are defined by binary semantic attributes, and co-occurrence probabilities are derived from attribute-based interactions. This model analytically reproduces the emergence of linear analogy structure and naturally accounts for properties (i)--(iv). It can be viewed as giving fine-grained resolution into the role of each additional embedding dimension. It is robust to various forms of noise and agrees well with co-occurrence statistics measured on Wikipedia and the analogy benchmark introduced by Mikolov et al.

This paper addresses the problem of inverse covariance (also known as precision matrix) estimation in high-dimensional settings. Specifically, we focus on two classes of estimators: linear shrinkage estimators with a target proportional to the identity matrix, and estimators derived from data augmentation (DA). Here, DA refers to the common practice of enriching a dataset with artificial samples—typically generated via a generative model or through random transformations of the original data—prior to model fitting. For both classes of estimators, we derive estimators and provide concentration bounds for their quadratic error. This allows for both method comparison and hyperparameter tuning, such as selecting the optimal proportion of artificial samples. On the technical side, our analysis relies on tools from random matrix theory. We introduce a novel deterministic equivalent for generalized resolvent matrices, accommodating dependent samples with specific structure. We support our theoretical results with numerical experiments.

We introduce a novel framework for differentially private (DP) statistical estimation via data truncation, addressing a key challenge in DP estimation when the data support is unbounded. Traditional approaches rely on problem-specific sensitivity analysis, limiting their applicability. By leveraging techniques from truncated statistics, we develop computationally efficient DP estimators for exponential family distributions, including Gaussian mean and covariance estimation, achieving near-optimal sample complexity. Previous works on exponential families only consider bounded or one-dimensional families. Our approach mitigates sensitivity through truncation while carefully correcting for the introduced bias using maximum likelihood estimation and DP stochastic gradient descent. Along the way, we establish improved uniform convergence guarantees for the log-likelihood function of exponential families, which may be of independent interest. Our results provide a general blueprint for DP algorithm design via truncated statistics.

We study the fundamental problem of calibrating a linear binary classifier of the form \(\sigma(\hat{w}^\top x)\), where the feature vector \(x\) is Gaussian, \(\sigma\) is a link function, and \(\hat{w}\) is an estimator of the true linear weight $w^\star$. By interpolating with a noninformative \emph{chance classifier}, we construct a well-calibrated predictor whose interpolation weight depends on the angle \(\angle(\hat{w}, w_\star)\) between the estimator \(\hat{w}\) and the true linear weight \(w_\star\). We establish that this angular calibration approach is provably well-calibrated in a high-dimensional regime where the number of samples and features both diverge, at a comparable rate. The angle \(\angle(\hat{w}, w_\star)\) can be consistently estimated. Furthermore, the resulting predictor is uniquely \emph{Bregman-optimal}, minimizing the Bregman divergence to the true label distribution within a suitable class of calibrated predictors. Our work is the first to provide a calibration strategy that satisfies both calibration and optimality properties provably in high dimensions. Additionally, we identify conditions under which a classical Platt-scaling predictor converges to our Bregman-optimal calibrated solution. Thus, Platt-scaling also inherits these desirable properties provably in high dimensions.

Most existing transfer learning algorithms for high-dimensional models employ a two-step regularization framework, whose success heavily hinges on the assumption that the pre-trained model closely resembles the target. To relax this assumption, we propose a co-regularization process to directly exploit beneficial knowledge from the source domain for high-dimensional generalized linear models. The proposed method learns the target parameter by constraining the source parameters to be close to the target one, thereby preventing fine-tuning failures caused by significantly deviated pre-trained parameters. Our theoretical analysis demonstrates that the proposed method accommodates a broader range of sources than existing two-step frameworks, thus being more robust to less similar sources. Its effectiveness is validated through extensive empirical studies.

We study the problem of robust estimation under heterogeneous corruption rates, where each sample may be independently corrupted with a known but non-identical probability. This setting arises naturally in distributed and federated learning, crowdsourcing, and sensor networks, yet existing robust estimators typically assume uniform or worst-case corruption, ignoring structural heterogeneity. For mean estimation for multivariate bounded distributions and univariate gaussian distributions, we give tight minimax rates for all heterogeneous corruption patterns. For multivariate gaussian mean estimation and linear regression, we establish the minimax rate for squared error up to a factor of $\sqrt{d}$, where $d$ is the dimension. Roughly, our findings suggest that samples beyond a certain corruption threshold may be discarded by the optimal estimators -- this threshold is determined by the empirical distribution of the corruption rates given.

The construction of confidence intervals for the mean of a bounded random variable is a classical problem in statistics with numerous applications in machine learning and virtually all scientific fields. In particular, obtaining the tightest possible confidence intervals is vital every time the sampling of the random variables is expensive. The current state-of-the-art method to construct confidence intervals is by using betting algorithms. This is a very successful approach for deriving optimal confidence sequences, even matching the rate of law of iterated logarithms. However, in the fixed horizon setting, these approaches are either sub-optimal or based on heuristic solutions with strong empirical performance but without a finite-time guarantee. Hence, no betting-based algorithm guaranteeing the optimal $\mathcal{O}(\sqrt{\frac{\sigma^2\log\frac1\delta}{n}})$ width of the confidence intervals are known. This work bridges this gap. We propose a betting-based algorithm to compute confidence intervals that empirically outperforms the competitors. Our betting strategy uses the optimal strategy in every step (in a certain sense), whereas the standard betting methods choose a constant strategy in advance. Leveraging this fact results in strict improvements even for classical concentration inequalities, such as the ones of Hoeffding or Bernstein. Moreover, we also prove that the width of our confidence intervals is optimal up to an $1+o(1)$ factor diminishing with $n$.

Quantifying differences between probability distributions is fundamental to statistics and machine learning, primarily for comparing statistical uncertainty. In contrast, epistemic uncertainty---due to incomplete knowledge---requires richer representations than those offered by classical probability. Imprecise probability (IP) theory offers such models, capturing ambiguity and partial belief. This has driven growing interest in imprecise probabilistic machine learning (IPML), where inference and decision-making rely on broader uncertainty models---highlighting the need for metrics beyond classical probability. This work introduces the Integral Imprecise Probability Metric (IIPM) framework, a Choquet integral-based generalisation of classical Integral Probability Metric to the setting of capacities---a broad class of IP models encompassing many existing ones, including lower probabilities, probability intervals, belief functions, and more. Theoretically, we establish conditions under which IIPM serves as a valid metric and metrises a form of weak convergence of capacities. Practically, IIPM not only enables comparison across different IP models but also supports the quantification of epistemic uncertainty~(EU) within a single IP model. In particular, by comparing an IP model with its conjugate, IIPM gives rise to a new class of epistemic uncertainty measures---Maximum Mean Imprecision (MMI) ---which satisfy key axiomatic properties proposed in the Uncertainty Quantification literature. We validate MMI through selective classification experiments, demonstrating strong empirical performance against established EU measures, and outperforming them when classical methods struggle to scale to a large number of classes. Our work advances both theory and practice in Imprecise Probabilistic Machine Learning, offering a principled framework for comparing and quantifying epistemic uncertainty under imprecision.