Low-rank adaptation (LoRA) has become a prevalent method for adapting pre-trained large language models to downstream tasks. However, the simple low-rank decomposition form may constrain the optimization flexibility. To address this limitation, we introduce Location-aware Cosine Adaptation (LoCA), a novel frequency-domain parameter-efficient fine-tuning method based on inverse Discrete Cosine Transform (iDCT) with selective locations of learnable components. We begin with a comprehensive theoretical comparison between frequency-domain and low-rank decompositions for fine-tuning pre-trained large models. Our analysis reveals that frequency-domain decomposition with carefully selected frequency components can surpass the expressivity of traditional low-rank-based methods. Furthermore, we demonstrate that iDCT offers a more efficient implementation compared to inverse Discrete Fourier Transform (iDFT), allowing for better selection and tuning of frequency components while maintaining equivalent expressivity to the optimal iDFT-based adaptation. By employing finite-difference approximation to estimate gradients for discrete locations of learnable coefficients on the DCT spectrum, LoCA dynamically selects the most informative frequency components during training. Experiments on diverse language and vision fine-tuning tasks demonstrate that LoCA offers enhanced parameter efficiency while maintains computational feasibility comparable to low-rank-based methods.

Uncertainty quantification in time series prediction is challenging due to the temporal dependence and distribution shift on sequential data. Conformal prediction provides a pivotal and flexible instrument for assessing the uncertainty of machine learning models through prediction sets. Recently, a series of online conformal inference methods updated thresholds of prediction sets by performing online gradient descent on a sequence of quantile loss functions. A drawback of such methods is that they only use the information of revealed non-conformity scores via miscoverage indicators but ignore error quantification, namely the distance between the non-conformity score and the current threshold. To accurately leverage the dynamic of miscoverage error, we propose Error-quantified Conformal Inference (ECI) by smoothing the quantile loss function. ECI introduces a continuous and adaptive feedback scale with the miscoverage error, rather than simple binary feedback in existing methods. We establish a long-term coverage guarantee for ECI under arbitrary dependence and distribution shift. The extensive experimental results show that ECI can achieve valid miscoverage control and output tighter prediction sets than other baselines.

In this paper, we address conditional testing problems through the conformal inference framework. We define the localized conformal $p$-values by inverting prediction intervals and prove their theoretical properties. These defined $p$-values are then applied to several conditional testing problems to illustrate their practicality. Firstly, we propose a conditional outlier detection procedure to test for outliers in the conditional distribution with finite-sample false discovery rate (FDR) control. We also introduce a novel conditional label screening problem with the goal of screening multivariate response variables and propose a screening procedure to control the family-wise error rate (FWER). Finally, we consider the two-sample conditional distribution test and define a weighted U-statistic through the aggregation of localized $p$-values. Numerical simulations and real-data examples validate the superior performance of our proposed strategies.

Conformal prediction is a powerful tool for constructing prediction intervals for black-box models, providing a finite sample coverage guarantee for exchangeable data. However, this exchangeability is compromised when some entries of the test feature are contaminated, such as in the case of cellwise outliers. To address this issue, this paper introduces a novel framework called *detect-then-impute conformal prediction*. This framework first employs an outlier detection procedure on the test feature and then utilizes an imputation method to fill in those cells identified as outliers. To quantify the uncertainty in the processed test feature, we adaptively apply the detection and imputation procedures to the calibration set, thereby constructing exchangeable features for the conformal prediction interval of the test label. We develop two practical algorithms, $\texttt{PDI-CP}$ and $\texttt{JDI-CP}$, and provide a distribution-free coverage analysis under some commonly used detection and imputation procedures. Notably, $\texttt{JDI-CP}$ achieves a finite sample $1-2\alpha$ coverage guarantee. Numerical experiments on both synthetic and real datasets demonstrate that our proposed algorithms exhibit robust coverage properties and comparable efficiency to the oracle baseline.

Online multiple hypothesis testing has attracted a lot of attention in many applications, e.g., anomaly status detection and stock market price monitoring. The state-of-the-art generalized $\alpha$-investing (GAI) algorithms can control online false discovery rate (FDR) on p-values only under specific dependence structures, a situation that rarely occurs in practice. The e-LOND algorithm (Xu & Ramdas, 2024) utilizes e-values to achieve online FDR control under arbitrary dependence but suffers from a significant loss in power as testing levels are derived from pre-specified descent sequences. To address these limitations, we propose a novel framework on valid e-values named e-GAI. The proposed e-GAI can ensure provable online FDR control under more general dependency conditions while improving the power by dynamically allocating the testing levels. These testing levels are updated not only by relying on both the number of previous rejections and the prior costs, but also, differing from the GAI framework, by assigning less $\alpha$-wealth for each rejection from a risk aversion perspective. Within the e-GAI framework, we introduce two new online FDR procedures, e-LORD and e-SAFFRON, and provide strategies for the long-term performance to address the issue of $\alpha$-death, a common phenomenon within the GAI framework. Furthermore, we demonstrate that e-GAI can be generalized to conditionally super-uniform p-values. Both simulated and real data experiments demonstrate the advantages of both e-LORD and e-SAFFRON in FDR control and power.

Active learning (AL) reduces annotation costs by selecting the most informative samples based on both model sensitivity and predictive uncertainty. While sensitivity can be measured through parameter gradients in an unsupervised manner, predictive uncertainty can hardly be estimated without true labels especially for regression tasks, reducing the informativeness of actively selected samples. This paper proposes the concept of \textit{auxiliary data} to aid the uncertainty estimation for regression tasks. With detailed theoretical analysis, we reveal that auxiliary data, despite potential distribution shifts, can provide a promising uncertainty surrogate when properly weighted. Such finding inspires our design of AGBAL, a novel AL framework that recalibrates auxiliary data losses through density ratio weighting to obtain reliable uncertainty estimates for sample selection. Extensive experiments show that AGBAL consistently outperforms existing approaches without auxiliary data across diverse synthetic and real-world datasets.

Personalized federated learning addresses data heterogeneity across distributed agents but lacks uncertainty quantification that is both agent-specific and instance-specific, which is a critical requirement for risk-sensitive applications. We propose personalized federated conformal prediction (PFCP), a novel framework that combines personalized federated learning with conformal prediction to provide statistically valid agent-personalized prediction sets with instance-localization. By leveraging privacy-preserving knowledge transfer from other source agents, PFCP ensures marginal coverage guarantees for target agents while significantly improving conditional coverage performance on individual test instances, which has been validated by extensive experiments.

This paper studies the problem of post-selection predictive inference in an online fashion. To avoid devoting resources to unimportant units, a preliminary selection of the current individual before reporting its prediction interval is common and meaningful in online predictive tasks. As a result of the temporal multiplicity introduced by the online selection process, it becomes essential to control the real-time false coverage-statement rate (FCR) which measures the overall miscoverage level. We develop a general framework named CAP (Calibration after Adaptive Pick), which performs an adaptive pick rule on historical data to construct a calibration set if the current individual is selected. This is followed by the output of a conformal prediction interval for the unobserved label. We present a series of tractable procedures for the construction of the calibration set for various popular online selection rules. It has been demonstrated that CAP achieves an exact selection-conditional coverage guarantee in the finite-sample and distribution-free regimes. In order to address the issue of the distribution shift in online data, we also embed CAP into some recent dynamic conformal prediction algorithms and prove that the proposed method can deliver long-run FCR control. Numerical results on both synthetic and real data corroborate that CAP can effectively control FCR around the target level and yield narrower prediction intervals than existing baselines across various settings.

Changepoint detection typically relies on a grid-search strategy for optimal data segmentation. When model fitting itself is expensive, repeatedly fitting a model on every candidate segment dominates the computation. Existing approaches mitigate this by pruning the grid, thus reducing the number of segments (and model fits). We propose Reliever, which instead cuts the number of model fits directly and nests seamlessly within standard grid-search routines. Reliever fits a small, deterministic collection of proxy models and reuses them wherever they apply, making it compatible with a wide range of existing algorithms. For high-dimensional regression with changepoints, coupling Reliever with an optimal grid-search method yields changepoint and coefficient estimators that are rate-optimal up to a logarithmic factor. Extensive numerical experiments demonstrate that Reliever rapidly and accurately detects changepoints across a wide range of high-dimensional and nonparametric models.

Reducing the radiation dose in CT scanning is important to alleviate the damage to the human health in clinical scenes. A promising way is to replace the normal-dose CT (NDCT) imaging by low-dose CT (LDCT) imaging with lower tube voltage and tube current. This often brings severe noise to the LDCT images, which adversely affects the diagnosis accuracy. Most of existing LDCT image denoising networks are trained either with synthetic LDCT images or real-world LDCT and NDCT image pairs with huge spatial misalignment. However, the synthetic noise is very different from the complex noise in real-world LDCT images, while the huge spatial misalignment brings inaccurate predictions of tissue structures in the denoised LDCT images. To well utilize real-world LDCT and NDCT image pairs for LDCT image denoising, in this paper, we introduce a new Patch Similarity Purification (PSP) strategy to construct high-quality training dataset for network training. Specifically, our PSP strategy first perform binarization for each pair of image patches cropped from the corresponding LDCT and NDCT image pairs. For each pair of binary masks, it then computes their similarity ratio by common mask calculation, and the patch pair can be selected as a training sample if their mask similarity ratio is higher than a threshold. By using our PSP strategy, each training set of our Rabbit and Patient datasets contain hundreds of thousands of real-world LDCT and NDCT image patch pairs with negligible misalignment. Extensive experiments demonstrate the usefulness of our PSP strategy on purifying the training data and the effectiveness of training LDCT image denoising networks on our datasets. The code and dataset are provided at https://github.com/TuTusong/PSP.

Technological advances have led to the emergence of an increasing number of applications requiring the analysis of datastreams, that are characterized by an indefinitely long and time-evolving sequence, particularly in the healthcare domain. In such applications, the status of a stream can alternate, possibly many times, between a regular status and an irregular status. Consequently, it is necessary to develop statistical methodologies that constantly detect multiple changepoints in an online manner. While we may employ conventional methods of sequential change detection to trigger signals after the change occurs, no online procedure is available to quantify the uncertainty of the detected changes. In this work, we fill this gap by framing online multiple changepoint detection into an online multiple testing problem and proposing a new framework to test the null hypothesis that there is no change between neighboring signalled points. To obtain valid p-values for online multiple testing, we propose a data-fission-based procedure that is a simple yet effective way of dealing with the post-detection uncertainty quantification. It is shown that popular online false discovery rate control methods with those p-values can achieve finite-sample false discovery rate control. We evaluate the proposed method in simulation studies. The method is applied to health monitoring dataset, alleviating the false alarm issue in online data analysis.

Accurately detecting multiple change-points is critical for various applications, but determining the optimal number of change-points remains a challenge. Existing approaches based on information criteria attempt to balance goodness-of-fit and model complexity, but their performance varies depending on the model. Recently, data-driven selection criteria based on cross-validation has been proposed, but these methods can be prone to slight overfitting in finite samples. In this paper, we introduce a method that controls the probability of overestimation and provides uncertainty quantification for learning multiple change-points via cross-validation. We frame this problem as a sequence of model comparison problems and leverage high-dimensional inferential procedures. We demonstrate the effectiveness of our approach through experiments on finite-sample data, showing superior uncertainty quantification for overestimation compared to existing methods. Our approach has broad applicability and can be used in diverse change-point models.

Various robust estimation methods or algorithms have been proposed to hedge against Byzantine failures in distributed learning. However, there is a lack of systematic approaches to provide theoretical guarantees of significance in detecting those Byzantine machines. In this paper, we develop a general detection procedure, ByMI, via error rate control to address this issue, which is applicable to many robust learning problems. The key idea is to apply the sample-splitting strategy on each worker machine to construct a score statistic integrated with a general robust estimation and then to utilize the symmetry property of those scores to derive a data-driven threshold. The proposed method is dimension insensitive and p-value free with the help of the symmetry property and can achieve false discovery rate control under mild conditions. Numerical experiments on both synthetic and real data validate the theoretical results and demonstrate the effectiveness of our proposed method on Byzantine machine identification.

The widespread use of black box prediction methods has sparked an increasing interest in algorithm/model-agnostic approaches for quantifying goodness-of-fit, with direct ties to specification testing, model selection and variable importance assessment. A commonly used framework involves defining a predictiveness criterion, applying a cross-fitting procedure to estimate the predictiveness, and utilizing the difference in estimated predictiveness between two models as the test statistic. However, even after standardization, the test statistic typically fails to converge to a non-degenerate distribution under the null hypothesis of equal goodness, leading to what is known as the degeneracy issue. To addresses this degeneracy issue, we present a simple yet effective device, Zipper. It draws inspiration from the strategy of additional splitting of testing data, but encourages an overlap between two testing data splits in predictiveness evaluation. Zipper binds together the two overlapping splits using a slider parameter that controls the proportion of overlap. Our proposed test statistic follows an asymptotically normal distribution under the null hypothesis for any fixed slider value, guaranteeing valid size control while enhancing power by effective data reuse. Finite-sample experiments demonstrate that our procedure, with a simple choice of the slider, works well across a wide range of settings.

Real-time decision-making gets more attention in the big data era. Here, we consider the problem of sample selection in the online setting, where one encounters a possibly infinite sequence of individuals collected over time with covariate information available. The goal is to select samples of interest that are characterized by their unobserved responses until the user-specified stopping time. We derive a new decision rule that enables us to find more preferable samples that meet practical requirements by simultaneously controlling two types of general constraints: individual and interactive constraints, which include the widely utilized False Selection Rate (FSR), cost limitations, and diversity of selected samples. The key elements of our approach involve quantifying the uncertainty of response predictions via predictive inference and addressing individual and interactive constraints in a sequential manner. Theoretical and numerical results demonstrate the effectiveness of the proposed method in controlling both individual and interactive constraints.

The task of distinguishing individuals of interest from a vast pool of candidates using predictive models has garnered significant attention in recent years. This task can be framed as a *conformalized multiple testing* procedure, which aims at quantifying prediction uncertainty by controlling the false discovery rate (FDR) via conformal inference. In this paper, we tackle the challenge of conformalized multiple testing after data-dependent selection procedures. To guarantee the construction of valid test statistics that accurately capture the distorted distribution resulting from the selection process, we leverage a holdout labeled set to closely emulate the selective distribution. Our approach involves adaptively picking labeled data to create a calibration set based on the stability of the selection rule. This strategy ensures that the calibration data and the selected test unit are exchangeable, allowing us to develop valid conformal p-values. Implementing with the famous Benjamini-Hochberg (BH) procedure, it effectively controls the FDR over the selected subset. To handle the randomness of the selected subset and the dependence among the constructed p-values, we establish a unified theoretical framework. This framework extends the application of conformalized multiple testing to complex selective settings. Furthermore, we conduct numerical studies to showcase the effectiveness and validity of our procedures across various scenarios.

The high-dimensional single index model (SIM), which assumes that the response is independent of the predictors given a linear combination of predictors, has drawn attention due to its flexibility and interpretability, but its efficiency is adversely affected by outlying observations and heavy-tailed distributions. This paper introduces a robust procedure by recasting the SIM into a pseudo-linear model with transformed responses. It relaxes the distributional conditions on random errors from sub-Gaussian to more general distributions and thus it is robust with substantial efficiency gain for heavy-tailed random errors. Under this paradigm, we provide asymptotically honest group inference procedures based on the idea of orthogonalization, which enjoys the feature that it does not require the zero and nonzero coefficients to be well-separated. Asymptotic null distribution and bootstrap implementation are both established. Moreover, we develop a multiple testing procedure for determining if the individual coefficients are relevant simultaneously, and show that it is able to control the false discovery rate asymptotically. Numerical results indicate that the new procedures can be highly competitive among existing methods, especially for heavy-tailed errors.

We consider a general trace regression model with multiple structural changes and propose a universal approach for simultaneous exact or near-low-rank matrix recovery and change-point detection. It incorporates nuclear norm penalized least-squares minimization into a grid search scheme that determines the potential structural break. Under a set of general conditions, we establish the non-asymptotic error bounds with a nearly-oracle rate for the matrix estimators as well as the super-consistency rate for the change-point localization. We use concrete random design instances to justify the appropriateness of the proposed conditions. Numerical results demonstrate the validity and effectiveness of the proposed scheme.

The identification of changes in linear models is a fundamental problem encountered in various applications. Traditional methods often encounter difficulties when attempting to identify changepoints in the presence of heavy-tailed distribution. This paper focuses on the study of high-dimensional linear models with multiple structural changes in the presence of heavy-tailed errors, especially for those errors without moment conditions. We first propose a robust method that simultaneously estimates regression coefficient and changepoint by incorporating$\ell _{1}$norm penalized Wilcoxon rank loss minimization for single changepoint estimation. Furthermore, we extend it to multiple changepoints estimation based on a novel two-step moving window mechanism that combines fast coarse grid screening and an efficient refinement. Our method exhibits robustness against heavy-tailed random errors while maintaining high efficiency for normal random errors. Theoretically, we establish non-asymptotic error bounds with a near-oracle rate for the estimates of both the coefficient and the changepoint under weak conditions on the random error distribution. Numerical results provide evidence for the validity and effectiveness of the proposed approach.

The matrix lasso, which minimizes a least-squared loss function with the nuclear-norm regularization, offers a generally applicable paradigm for high-dimensional low-rank matrix estimation, but its efficiency is adversely affected by heavy-tailed distributions. This paper introduces a robust procedure by incorporating a Wilcoxon-type rank-based loss function with the nuclear-norm penalty for a unified high-dimensional low-rank matrix estimation framework. It includes matrix regression, multivariate regression and matrix completion as special examples. This procedure enjoys several appealing features. First, it relaxes the distributional conditions on random errors from sub-exponential or sub-Gaussian to more general distributions and thus it is robust with substantial efficiency gain for heavy-tailed random errors. Second, as the gradient function of the rank-based loss function is completely pivotal, it overcomes the challenge of tuning parameter selection and substantially saves the computation time by using an easily simulated tuning parameter. Third, we theoretically establish non-asymptotic error bounds with a nearly-oracle rate for the new estimator. Numerical results indicate that the new estimator can be highly competitive among existing methods, especially for heavy-tailed or skewed errors.

Given an unsupervised novelty detection task on a new dataset, how can we automatically select a ''best'' detection model while simultaneously controlling the error rate of the best model? For novelty detection analysis, numerous detectors have been proposed to detect outliers on a new unseen dataset based on a score function trained on available clean data. However, due to the absence of labeled data for model evaluation and comparison, there is a lack of systematic approaches that are able to select a ''best'' model/detector (i.e., the algorithm as well as its hyperparameters) and achieve certain error rate control simultaneously. In this paper, we introduce a unified data-driven procedure to address this issue. The key idea is to maximize the number of detected outliers while controlling the false discovery rate (FDR) with the help of Jackknife prediction. We establish non-asymptotic bounds for the false discovery proportions and show that the proposed procedure yields valid FDR control under some mild conditions. Numerical experiments on both synthetic and real data validate the theoretical results and demonstrate the effectiveness of our proposed AutoMS method. The code is available at https://github.com/ZhangYifan1996/AutoMS.