arXiv subjects

Ilmun Kim

Publications and source records attributed to Ilmun Kim.

At least 19 recordsLinked to original sources

arXiv2026

A Sharper Hoeffding Bound for Weighted Sums of Exchangeable Random Variables

We prove a Hoeffding-type moment generating function bound for weighted sums of bounded exchangeable random variables centered by their finite-population average. The bound improves the finite-population inflation factor in a recent weighted exchangeable Hoeffding inequality from logarithmic order to the rate-optimal inverse-population-size order, with an explicit constant. The proof reduces the problem to Hamming slices, identifies two-level extremizers for the relevant symmetric variational problem, and applies a hypergeometric martingale bound. We also give a lower bound showing that an inverse-population-size inflation is unavoidable.

math.PR↗

arXiv2026

Sharp Minimax Rates for Smooth Two-Sample Testing under Central Differential Privacy

We establish sharp minimax limits for two-sample testing of H\"older-smooth densities under central differential privacy. Given two independent samples, the goal is to decide whether the underlying distributions are identical or separated in $L_1$ distance, while releasing only an $\varepsilon$-differentially private decision. We show that privacy changes the classical smooth-testing boundary through multiple regimes: the optimal separation radius is the maximum of four terms, consisting of the classical nonprivate rate and three distinct privacy-induced barriers. Which barrier is active depends on the privacy budget and the smoothness-to-dimension ratio, yielding a sharp phase diagram. Our upper bound discretizes the samples, applies a private discrete two-sample test to the resulting histograms, and chooses the bin resolution to balance approximation bias, sampling fluctuations, and privacy noise. The procedure also admits a permutation-calibrated implementation with finite-sample type~I error control. For the lower bounds, we combine smooth perturbation constructions with privacy-specific coupling and transport inequalities, showing that all four terms are unavoidable. Finally, when the smoothness is unknown, we develop a multiscale private test that attains the optimal adaptive rate and prove a matching lower bound. Adaptation costs exactly an iterated-logarithmic factor, and this cost appears only in the classical nonprivate term.

math.ST↗

arXiv2026

A Two-Sample Test for Random Persistence Diagrams Against Alternatives Characterized by Persistence Intensity Differences

Persistence intensity functions provide informative first-order summaries of random persistence diagram distributions. We study the two-sample comparison of persistence diagram distributions through their persistence intensity functions. In particular, for persistence diagram distributions $P$ and $Q$ with respective intensities $p$ and $q$, we consider the two-sample testing problem: $ \mathrm{H}_0: P=Q \ \text{versus}\ \mathrm{H}_1 :p\neq q. $ We propose a kernel-based permutation test for this problem. The test has finite-sample validity under the distributional null, and its power is characterized in terms of the $L^2$ discrepancy between the weighted persistence intensity functions. To accommodate persistence diagrams with possibly unbounded cardinality, we introduce regularity conditions that control the effect of cardinality variation and yield a sharp variance bound for the test statistic. We further show that every probability density on $\{(x,y)\in\mathbb{R}^2:y>x > 0\}$ can be realized as the intensity function of a random diagram. Using these results, we establish that the proposed test attains minimax-optimal separation rates over anisotropic Sobolev balls. Since the optimal bandwidth is not directly accessible in practice, we adopt a bandwidth aggregation framework. Simulations demonstrate validity under the distributional null and high empirical power against intensity-separated alternatives.

math.ST↗

arXiv2026

Aggregation of Statistical Evidence under Exchangeability

We study aggregation of statistical evidence under unknown and potentially complex dependence using group-invariance. Building on permutation-based constructions that treat transformed datasets as exchangeable units, we aggregate evidence across statistics for each transformed dataset and calibrate the resulting aggregates across transformations. We develop a finite-sample power and adaptivity theory for this framework, together with extensions to sequential and data-dependent aggregation that preserve validity. For single-batch aggregation, which uses one collection of transformed datasets for both standardization and calibration, we show that the critical values uniformly improve on deterministic calibrations valid under arbitrary dependence, including Bonferroni correction, while adapting to the unknown dependence structure. We also introduce a sequential alpha-spending version that permits early rejection when evidence is strong, and a two-batch extension that separates standardization from calibration to accommodate learned aggregation rules and reduce computation. Applications to adaptive nonparametric testing and conformal prediction illustrate how these results sharpen existing aggregation methods.

stat.ME↗

arXiv2026

High-Confidence Minimax Testing with Prescribed Errors

Classical minimax lower bounds for testing are typically derived for fixed error probabilities, while high-confidence results often impose a common failure probability. We study prescribed-error testing, in which the level and the target type II error may be small and of different orders. Standard prior-based reductions generally aggregate the two errors into a single quantity and therefore do not capture their distinct roles. We develop a general lower-bound technique based on a binary reduction that preserves the separate roles of the two error targets. The reduction yields two directed Kullback-Leibler information requirements, corresponding respectively to the level and the target type II error. When both directed mixture divergences can be controlled, they combine into a binary Jeffreys divergence, leading to the logarithmic dependence on the level and the target type II error. Applying the framework to Gaussian sequence testing, multinomial uniformity testing, and continuous uniformity testing over H\"older balls, we obtain lower bounds that match corresponding high-confidence upper bounds and hence establish prescribed-error minimax rates sharp up to constant factors.

math.ST↗

arXiv2026

When Do Generalized Permutation Tests Achieve Optimal Power? A Dispersion Characterization

We study generalized Monte Carlo permutation tests under a non-uniform distribution on permutations. Focusing on the difference-in-means statistic, we introduce two scalar dispersion measures that quantify departures from complete randomization at the individual and pairwise levels. We show that if both dispersions vanish asymptotically, then the conditional permutation distribution converges to its Gaussian benchmark, the critical value stabilizes, and the test attains optimal Pitman local power. Conversely, if these dispersions fail to vanish, the permutation distribution does not self-average, the critical value need not stabilize, and optimal local power cannot in general be guaranteed. We further show that beyond the standard Pitman local model, suitably chosen non-uniform permutation distributions can strictly dominate the uniform distribution by exploiting nuisance structure in the data.

stat.ME↗

arXiv2026

More Permutations Do Not Always Increase Power: Non-monotonicity in Monte Carlo Permutation Tests

Monte Carlo permutation tests are a cornerstone of valid, model-free statistical inference. A widely held practical intuition is that increasing the number of sampled permutations improves test performance, in particular that statistical power tends to increase with the Monte Carlo budget. In this paper, we show that these intuitions are false in general. Leveraging the saw-toothed structure of power arising from distributional discreteness, we provide a simple structural explanation for why power can decrease as the number of sampled permutations increases, and we prove that such decreases occur infinitely often as the Monte Carlo budget grows.

stat.CO↗

arXiv2026

A Semi-Supervised Kernel Two-Sample Test

We consider the problem of two-sample testing in a semi-supervised setting with abundant unlabeled covariate data. Standard two-sample tests neglect covariate information, which has the potential to significantly boost performance. However, incorporating covariates potentially breaks the exchangeability assumption under the null, which further complicates a calibration procedure. To address these issues, we propose a semi-supervised method that produces a test statistic with asymptotic normality, while effectively integrating additional information from covariates. Our test is straightforward to calibrate due to the asymptotic normality under the null and achieves asymptotic power that is often much higher than existing kernel tests without covariates. Furthermore, we formally show that the proposed method is consistent in power against fixed and local alternatives. Simulations confirm the practical and theoretical strengths of our approach.

stat.ML↗

arXiv2026

Multi-LLM Adaptive Conformal Inference for Reliable LLM Responses

Ensuring factuality is essential for the safe use of Large Language Models (LLMs) in high-stakes domains such as medicine and law. Conformal inference provides distribution-free guarantees, but existing approaches are either overly conservative, discarding many true-claims, or rely on adaptive error rates and simple linear models that fail to capture complex group structures. To address these challenges, we reformulate conformal inference in a multiplicative filtering setting, modeling factuality as a product of claim-level scores. Our method, Multi-LLM Adaptive Conformal Inference (MACI), leverages ensembles to produce more accurate factuality-scores, which in our experiments led to higher retention, while validity is preserved through group-conditional calibration. Experiments show that MACI consistently achieves user-specified coverage with substantially higher retention and lower time cost than baselines. Our repository is available at https://github.com/MLAI-Yonsei/MACI

cs.LG↗

arXiv2025

Transfer Learning for Benign Overfitting in High-Dimensional Linear Regression

Transfer learning is a key component of modern machine learning, enhancing the performance of target tasks by leveraging diverse data sources. Simultaneously, overparameterized models such as the minimum-$\ell_2$-norm interpolator (MNI) in high-dimensional linear regression have garnered significant attention for their remarkable generalization capabilities, a property known as benign overfitting. Despite their individual importance, the intersection of transfer learning and MNI remains largely unexplored. Our research bridges this gap by proposing a novel two-step Transfer MNI approach and analyzing its trade-offs. We characterize its non-asymptotic excess risk and identify conditions under which it outperforms the target-only MNI. Our analysis reveals free-lunch covariate shift regimes, where leveraging heterogeneous data yields the benefit of knowledge transfer at limited cost. To operationalize our findings, we develop a data-driven procedure to detect informative sources and introduce an ensemble method incorporating multiple informative Transfer MNIs. Finite-sample experiments demonstrate the robustness of our methods to model and data heterogeneity, confirming their advantage.

stat.ML↗

arXiv2025

Locally minimax optimal confidence sets for the best model

This paper tackles a fundamental inference problem: given $n$ observations from a distribution $P$ over $\mathbb{R}^d$ with unknown mean $\boldsymbol{\mu}$, we must form a confidence set for the index (or indices) corresponding to the smallest component of $\boldsymbol{\mu}$. By duality, we reduce this to testing, for each $r$ in $1,\ldots,d$, whether $\mu_r$ is the smallest. Based on the sample splitting and self-normalization approach of Kim and Ramdas (2024), we propose "dimension-agnostic" tests that maintain validity regardless of how $d$ scales with $n$, and regardless of arbitrary ties in $\boldsymbol{\mu}$. Notably, our validity holds under mild moment conditions, requiring little more than finiteness of a second moment, and permitting possibly strong dependence between coordinates. In addition, we establish the \emph{local} minimax separation rate for this problem, which adapts to the cardinality of a confusion set, and show that the proposed tests attain this rate. Furthermore, we develop robust variants that continue to achieve the same minimax rate under heavy-tailed distributions with only finite second moments. While these results highlight the theoretical strength of our method, a practical concern is that sample splitting can reduce finite-sample power. We show that this drawback can be substantially alleviated by the multi-split aggregation method of Guo and Shah (2025). Finally, empirical results on simulated and real data illustrate the strong performance of our approach in terms of type I error control and power compared to existing methods.

math.ST↗

arXiv2024

Minimax Optimal Two-Sample Testing under Local Differential Privacy

We explore the trade-off between privacy and statistical utility in private two-sample testing under local differential privacy (LDP) for both multinomial and continuous data. We begin by addressing the multinomial case, where we introduce private permutation tests using practical privacy mechanisms such as Laplace, discrete Laplace, and Google's RAPPOR. We then extend our multinomial approach to continuous data via binning and study its uniform separation rates under LDP over H\"older and Besov smoothness classes. The proposed tests for both discrete and continuous cases rigorously control the type I error for any finite sample size, strictly adhere to LDP constraints, and achieve minimax separation rates under LDP. The attained minimax rates reveal inherent privacy-utility trade-offs that are unavoidable in private testing. To address scenarios with unknown smoothness parameters in density testing, we propose an adaptive test based on a Bonferroni-type approach that ensures robust performance without prior knowledge of the smoothness parameters. We validate our theoretical findings with extensive numerical experiments and demonstrate the practical relevance and effectiveness of our proposed methods.

stat.ML↗

arXiv2024

General Frameworks for Conditional Two-Sample Testing

We study the problem of conditional two-sample testing, which aims to determine whether two populations have the same distribution after accounting for confounding factors. This problem commonly arises in various applications, such as domain adaptation and algorithmic fairness, where comparing two groups is essential while controlling for confounding variables. We begin by establishing a hardness result for conditional two-sample testing, demonstrating that no valid test can have significant power against any single alternative without proper assumptions. We then introduce two general frameworks that implicitly or explicitly target specific classes of distributions for their validity and power. Our first framework allows us to convert any conditional independence test into a conditional two-sample test in a black-box manner, while preserving the asymptotic properties of the original conditional independence test. The second framework transforms the problem into comparing marginal distributions with estimated density ratios, which allows us to leverage existing methods for marginal two-sample testing. We demonstrate this idea in a concrete manner with classification and kernel-based methods. Finally, simulation studies are conducted to illustrate the proposed frameworks in finite-sample scenarios.

stat.ML↗

arXiv2024

Computational-Statistical Trade-off in Kernel Two-Sample Testing with Random Fourier Features

Recent years have seen a surge in methods for two-sample testing, among which the Maximum Mean Discrepancy (MMD) test has emerged as an effective tool for handling complex and high-dimensional data. Despite its success and widespread adoption, the primary limitation of the MMD test has been its quadratic-time complexity, which poses challenges for large-scale analysis. While various approaches have been proposed to expedite the procedure, it has been unclear whether it is possible to attain the same power guarantee as the MMD test at sub-quadratic time cost. To fill this gap, we revisit the approximated MMD test using random Fourier features, and investigate its computational-statistical trade-off. We start by revealing that the approximated MMD test is pointwise consistent in power only when the number of random features approaches infinity. We then consider the uniform power of the test and study the time-power trade-off under the minimax testing framework. Our result shows that, by carefully choosing the number of random features, it is possible to attain the same minimax separation rates as the MMD test within sub-quadratic time. We demonstrate this point under different distributional assumptions such as densities in a Sobolev ball. Our theoretical findings are corroborated by simulation studies.

stat.ML↗

arXiv2024

Enhancing Sufficient Dimension Reduction via Hellinger Correlation

In this work, we develop a new theory and method for sufficient dimension reduction (SDR) in single-index models, where SDR is a sub-field of supervised dimension reduction based on conditional independence. Our work is primarily motivated by the recent introduction of the Hellinger correlation as a dependency measure. Utilizing this measure, we develop a method capable of effectively detecting the dimension reduction subspace, complete with theoretical justification. Through extensive numerical experiments, we demonstrate that our proposed method significantly enhances and outperforms existing SDR methods. This improvement is largely attributed to our proposed method's deeper understanding of data dependencies and the refinement of existing SDR techniques.

stat.ML↗

arXiv2024

Robust Kernel Hypothesis Testing under Data Corruption

We propose a general method for constructing robust permutation tests under data corruption. The proposed tests effectively control the non-asymptotic type I error under data corruption, and we prove their consistency in power under minimal conditions. This contributes to the practical deployment of hypothesis tests for real-world applications with potential adversarial attacks. For the two-sample and independence settings, we show that our kernel robust tests are minimax optimal, in the sense that they are guaranteed to be non-asymptotically powerful against alternatives uniformly separated from the null in the kernel MMD and HSIC metrics at some optimal rate (tight with matching lower bound). We point out that existing differentially private tests can be adapted to be robust to data corruption, and we demonstrate in experiments that our proposed tests achieve much higher power than these private tests. Finally, we provide publicly available implementations and empirically illustrate the practicality of our robust tests.

stat.ML↗

arXiv2024

Semi-Supervised U-statistics

Semi-supervised datasets are ubiquitous across diverse domains where obtaining fully labeled data is costly or time-consuming. The prevalence of such datasets has consistently driven the demand for new tools and methods that exploit the potential of unlabeled data. Responding to this demand, we introduce semi-supervised U-statistics enhanced by the abundance of unlabeled data, and investigate their statistical properties. We show that the proposed approach is asymptotically Normal and exhibits notable efficiency gains over classical U-statistics by effectively integrating various powerful prediction tools into the framework. To understand the fundamental difficulty of the problem, we derive minimax lower bounds in semi-supervised settings and showcase that our procedure is semi-parametrically efficient under regularity conditions. Moreover, tailored to bivariate kernels, we propose a refined approach that outperforms the classical U-statistic across all degeneracy regimes, and demonstrate its optimality properties. Simulation studies are conducted to corroborate our findings and to further demonstrate our framework.

math.ST↗

arXiv2023

Differentially Private Permutation Tests

Recent years have witnessed growing concerns about the privacy of sensitive data. In response to these concerns, differential privacy has emerged as a rigorous framework for privacy protection, gaining widespread recognition in both academic and industrial circles. While substantial progress has been made in private data analysis, existing methods often suffer from impracticality or a significant loss of statistical efficiency. This paper aims to alleviate these concerns in the context of hypothesis testing by introducing differentially private permutation tests. The proposed framework extends classical non-private permutation tests to private settings, maintaining both finite-sample validity and differential privacy in a rigorous manner. The power of the proposed test depends on the choice of a test statistic, and we establish general conditions for consistency and non-asymptotic uniform power. To demonstrate the utility and practicality of our framework, we focus on reproducing kernel-based test statistics and introduce differentially private kernel tests for two-sample and independence testing: dpMMD and dpHSIC. The proposed kernel tests are straightforward to implement, applicable to various types of data, and attain minimax optimal power across different privacy regimes. Our empirical evaluations further highlight their competitive power under various synthetic and real-world scenarios, emphasizing their practical value. The code is publicly available to facilitate the implementation of our framework.

math.ST↗