Matthew Reimherr

Zelin He, Haotian Lin, Boran Han et al.

Agentic reinforcement learning (RL) enables LLM agents to improve continuously from environment rewards, yet the resulting policies do not systematically accumulate reusable strategies that generalize across tasks. Modular skills can provide such reusable strategies, yet existing skill-augmented RL methods decouple skill creation from policy optimization, risking adopting skills that conflict with the evolving policy. Inspired by Anthropic's Skill Creator, we introduce ReSkill, an RL-in-the-loop skill creation framework that reconciles skill evolution with policy learning. ReSkill exploits the group-wise structure of GRPO to naturally embed three mechanisms with only marginal additional overhead: (1) an assertion-driven skill creator that diagnoses failures from past experience and proposes conditional, trigger-based skill revisions; (2) within-group rollout sampling that enables controlled comparison of skill versions, capturing which version best supports the policy's ongoing learning; and (3) Thompson Sampling with adaptive discounting to balance exploration and exploitation in skill version selection as the policy evolves. Across several domains, ReSkill consistently outperforms existing memory and skill-based RL methods, with the largest gains on unseen tasks. Analysis of the skill lifecycle shows skills being automatically created, tested, refined, and pruned as the policy improves, demonstrating reconciled skill-policy co-evolution.

Carlos Soto, Karthik Bharath, Matthew Reimherr et al.

It is common for data structures such as images and shapes of 2D objects to be represented as points on a manifold. The utility of a mechanism to produce sanitized differentially private estimates from such data is intimately linked to how compatible it is with the underlying structure and geometry of the space. In particular, as recently shown, utility of the Laplace mechanism on a positively curved manifold, such as Kendall's 2D shape space, is significantly influences by the curvature. Focusing on the problem of sanitizing the Fréchet mean of a sample of points on a manifold, we exploit the characterisation of the mean as the minimizer of an objective function comprised of the sum of squared distances and develop a K-norm gradient mechanism on Riemannian manifolds that favors values that produce gradients close to the the zero of the objective function. For the case of positively curved manifolds, we describe how using the gradient of the squared distance function offers better control over sensitivity than the Laplace mechanism, and demonstrate this numerically on a dataset of shapes of corpus callosa. Further illustrations of the mechanism's utility on a sphere and the manifold of symmetric positive definite matrices are also presented.

Haotian Lin, Matthew Reimherr

We study the transfer learning (TL) for the functional linear regression (FLR) under the Reproducing Kernel Hilbert Space (RKHS) framework, observing that the TL techniques in existing high-dimensional linear regression are not compatible with the truncation-based FLR methods, as functional data are intrinsically infinite-dimensional and generated by smooth underlying processes. We measure the similarity across tasks using RKHS distance, allowing the type of information being transferred to be tied to the properties of the imposed RKHS. Building on the hypothesis offset transfer learning paradigm, two algorithms are proposed: one conducts the transfer when positive sources are known, while the other leverages aggregation techniques to achieve robust transfer without prior information about the sources. We establish asymptotic lower bounds for this learning problem and show that the proposed algorithms enjoy a matching upper bound. These analyses provide statistical insights into factors that contribute to the dynamics of the transfer. We also extend the results to functional generalized linear models. The effectiveness of the proposed algorithms is demonstrated via extensive synthetic data as well as real-world data applications.

Carlos Soto, Matthew Reimherr, Aleksandra Slavkovic et al.

In this paper we consider the problem of releasing a Gaussian Differentially Private (GDP) 3D human face. The human face is a complex structure with many features and inherently tied to one's identity. Protecting this data, in a formally private way, is important yet challenging given the dimensionality of the problem. We extend approximate DP techniques for functional data to the GDP framework. We further propose a novel representation, face radial curves, of a 3D face as a set of functions and then utilize our proposed GDP functional data mechanism. To preserve the shape of the face while injecting noise we rely on tools from shape analysis for our novel representation of the face. We show that our method preserves the shape of the average face and injects less noise than traditional methods for the same privacy budget. Our mechanism consists of two primary components, the first is generally applicable to function value summaries (as are commonly found in nonparametric statistics or functional data analysis) while the second is general to disk-like surfaces and hence more applicable than just to human faces.

Haotian Lin, Matthew Reimherr

Many existing mechanisms to achieve differential privacy (DP) on infinite-dimensional functional summaries often involve embedding these summaries into finite-dimensional subspaces and applying traditional DP techniques. Such mechanisms generally treat each dimension uniformly and struggle with complex, structured summaries. This work introduces a novel mechanism for DP functional summary release: the Independent Component Laplace Process (ICLP) mechanism. This mechanism treats the summaries of interest as truly infinite-dimensional objects, thereby addressing several limitations of existing mechanisms. We establish the feasibility of the proposed mechanism in multiple function spaces. Several statistical estimation problems are considered, and we demonstrate one can enhance the utility of sanitized summaries by oversmoothing their non-private counterpart. Numerical experiments on synthetic and real datasets demonstrate the efficacy of the proposed mechanism.

Zelin He, Boran Han, Xiyuan Zhang et al.

Time-series diagnostic reasoning is essential for many applications, yet existing solutions face a persistent gap: general reasoning large language models (GRLMs) possess strong reasoning skills but lack the domain-specific knowledge to understand complex time-series patterns. Conversely, fine-tuned time-series LLMs (TSLMs) understand these patterns but lack the capacity to generalize reasoning for more complicated questions. To bridge this gap, we propose a hybrid knowledge-injection framework that injects TSLM-generated insights directly into GRLM's reasoning trace, thereby achieving strong time-series reasoning with in-domain knowledge. As collecting data for knowledge injection fine-tuning is costly, we further leverage a reinforcement learning-based approach with verifiable rewards (RLVR) to elicit knowledge-rich traces without human supervision, then transfer such an in-domain thinking trace into GRLM for efficient knowledge injection. We further release SenTSR-Bench, a multivariate time-series-based diagnostic reasoning benchmark collected from real-world industrial operations. Across SenTSR-Bench and other public datasets, our method consistently surpasses TSLMs by 9.1%-26.1% and GRLMs by 7.9%-22.4%, delivering robust, context-aware time-series diagnostic insights.

Sarah Alnegheimish, Zelin He, Matthew Reimherr et al.

With the widespread availability of sensor data across industrial and operational systems, we frequently encounter heterogeneous time series from multiple systems. Anomaly detection is crucial for such systems to facilitate predictive maintenance. However, most existing anomaly detection methods are designed for either univariate or single-system multivariate data, making them insufficient for these complex scenarios. To address this, we introduce M$^2$AD, a framework for unsupervised anomaly detection in multivariate time series data from multiple systems. M$^2$AD employs deep models to capture expected behavior under normal conditions, using the residuals as indicators of potential anomalies. These residuals are then aggregated into a global anomaly score through a Gaussian Mixture Model and Gamma calibration. We theoretically demonstrate that this framework can effectively address heterogeneity and dependencies across sensors and systems. Empirically, M$^2$AD outperforms existing methods in extensive evaluations by 21% on average, and its effectiveness is demonstrated on a large-scale real-world case study on 130 assets in Amazon Fulfillment Centers. Our code and results are available at https://github.com/sarahmish/M2AD.

Haotian Lin, Matthew Reimherr

Many existing two-phase kernel-based hypothesis transfer learning algorithms employ the same kernel regularization across phases and rely on the known smoothness of functions to obtain optimality. Therefore, they fail to adapt to the varying and unknown smoothness between the target/source and their offset in practice. In this paper, we address these problems by proposing Smoothness Adaptive Transfer Learning (SATL), a two-phase kernel ridge regression(KRR)-based algorithm. We first prove that employing the misspecified fixed bandwidth Gaussian kernel in target-only KRR learning can achieve minimax optimality and derive an adaptive procedure to the unknown Sobolev smoothness. Leveraging these results, SATL employs Gaussian kernels in both phases so that the estimators can adapt to the unknown smoothness of the target/source and their offset function. We derive the minimax lower bound of the learning problem in excess risk and show that SATL enjoys a matching upper bound up to a logarithmic factor. The minimax convergence rate sheds light on the factors influencing transfer dynamics and demonstrates the superiority of SATL compared to non-transfer learning settings. While our main objective is a theoretical analysis, we also conduct several experiments to confirm our results.

Haotian Lin, Matthew Reimherr

When concept shifts and sample scarcity are present in the target domain of interest, nonparametric regression learners often struggle to generalize effectively. The technique of transfer learning remedies these issues by leveraging data or pre-trained models from similar source domains. While existing generalization analyses of kernel-based transfer learning typically rely on correctly specified models, we present a transfer learning procedure that is robust against model misspecification while adaptively attaining optimality. To facilitate our analysis and avoid the risk of saturation found in classical misspecified results, we establish a novel result in the misspecified single-task learning setting, showing that spectral algorithms with fixed bandwidth Gaussian kernels can attain minimax convergence rates given the true function is in a Sobolev space, which may be of independent interest. Building on this, we derive the adaptive convergence rates of the excess risk for specifying Gaussian kernels in a prevalent class of hypothesis transfer learning algorithms. Our results are minimax optimal up to logarithmic factors and elucidate the key determinants of transfer efficiency.

Matthew Reimherr, Karthik Bharath, Carlos Soto

In this work we consider the problem of releasing a differentially private statistical summary that resides on a Riemannian manifold. We present an extension of the Laplace or K-norm mechanism that utilizes intrinsic distances and volumes on the manifold. We also consider in detail the specific case where the summary is the Fréchet mean of data residing on a manifold. We demonstrate that our mechanism is rate optimal and depends only on the dimension of the manifold, not on the dimension of any ambient space, while also showing how ignoring the manifold structure can decrease the utility of the sanitized summary. We illustrate our framework in two examples of particular interest in statistics: the space of symmetric positive definite matrices, which is used for covariance matrices, and the sphere, which can be used as a space for modeling discrete distributions.

Aniruddha Rajendra Rao, Matthew Reimherr

We introduce a new class of non-linear function-on-function regression models for functional data using neural networks. We propose a framework using a hidden layer consisting of continuous neurons, called a continuous hidden layer, for functional response modeling and give two model fitting strategies, Functional Direct Neural Network (FDNN) and Functional Basis Neural Network (FBNN). Both are designed explicitly to exploit the structure inherent in functional data and capture the complex relations existing between the functional predictors and the functional response. We fit these models by deriving functional gradients and implement regularization techniques for more parsimonious results. We demonstrate the power and flexibility of our proposed method in handling complex functional models through extensive simulation studies as well as real data examples.

Aniruddha Rajendra Rao, Matthew Reimherr

We introduce a new class of non-linear models for functional data based on neural networks. Deep learning has been very successful in non-linear modeling, but there has been little work done in the functional data setting. We propose two variations of our framework: a functional neural network with continuous hidden layers, called the Functional Direct Neural Network (FDNN), and a second version that utilizes basis expansions and continuous hidden layers, called the Functional Basis Neural Network (FBNN). Both are designed explicitly to exploit the structure inherent in functional data. To fit these models we derive a functional gradient based optimization algorithm. The effectiveness of the proposed methods in handling complex functional models is demonstrated by comprehensive simulation studies and real data examples.

Aniruddha Rajendra Rao, Matthew Reimherr

This work considers the problem of fitting functional models with sparsely and irregularly sampled functional data. It overcomes the limitations of the state-of-the-art methods, which face major challenges in the fitting of more complex non-linear models. Currently, many of these models cannot be consistently estimated unless the number of observed points per curve grows sufficiently quickly with the sample size, whereas, we show numerically that a modified approach with more modern multiple imputation methods can produce better estimates in general. We also propose a new imputation approach that combines the ideas of {\it MissForest} with {\it Local Linear Forest} and compare their performance with {\it PACE} and several other multivariate multiple imputation methods. This work is motivated by a longitudinal study on smoking cessation, in which the Electronic Health Records (EHR) from Penn State PaTH to Health allow for the collection of a great deal of data, with highly variable sampling. To illustrate our approach, we explore the relation between relapse and diastolic blood pressure. We also consider a variety of simulation schemes with varying levels of sparsity to validate our methods.

Tobia Boschi, Matthew Reimherr, Francesca Chiaromonte

Feature selection is an important and active research area in statistics and machine learning. The Elastic Net is often used to perform selection when the features present non-negligible collinearity or practitioners wish to incorporate additional known structure. In this article, we propose a new Semi-smooth Newton Augmented Lagrangian Method to efficiently solve the Elastic Net in ultra-high dimensional settings. Our new algorithm exploits both the sparsity induced by the Elastic Net penalty and the sparsity due to the second order information of the augmented Lagrangian. This greatly reduces the computational cost of the problem. Using simulations on both synthetic and real datasets, we demonstrate that our approach outperforms its best competitors by at least an order of magnitude in terms of CPU time. We also apply our approach to a Genome Wide Association Study on childhood obesity.

Matthew Reimherr, Jordan Awan

This paper presents a new mechanism for producing sanitized statistical summaries that achieve \emph{differential privacy}, called the \emph{K-Norm Gradient} Mechanism, or KNG. This new approach maintains the strong flexibility of the exponential mechanism, while achieving the powerful utility performance of objective perturbation. KNG starts with an inherent objective function (often an empirical risk), and promotes summaries that are close to minimizing the objective by weighting according to how far the gradient of the objective function is from zero. Working with the gradient instead of the original objective function allows for additional flexibility as one can penalize using different norms. We show that, unlike the exponential mechanism, the noise added by KNG is asymptotically negligible compared to the statistical error for many problems. In addition to theoretical guarantees on privacy and utility, we confirm the utility of KNG empirically in the settings of linear and quantile regression through simulations.

Matthew Reimherr, Jordan Awan

We study elliptical distributions in locally convex vector spaces, and determine conditions when they can or cannot be used to satisfy differential privacy (DP). A requisite condition for a sanitized statistical summary to satisfy DP is that the corresponding privacy mechanism must induce equivalent measures for all possible input databases. We show that elliptical distributions with the same dispersion operator, $C$, are equivalent if the difference of their means lies in the Cameron-Martin space of $C$. In the case of releasing finite-dimensional projections using elliptical perturbations, we show that the privacy parameter $\ep$ can be computed in terms of a one-dimensional maximization problem. We apply this result to consider multivariate Laplace, $t$, Gaussian, and $K$-norm noise. Surprisingly, we show that the multivariate Laplace noise does not achieve $\ep$-DP in any dimension greater than one. Finally, we show that when the dimension of the space is infinite, no elliptical distribution can be used to give $\ep$-DP; only $(ε,δ)$-DP is possible.

Jordan Awan, Ana Kenney, Matthew Reimherr et al.

The exponential mechanism is a fundamental tool of Differential Privacy (DP) due to its strong privacy guarantees and flexibility. We study its extension to settings with summaries based on infinite dimensional outputs such as with functional data analysis, shape analysis, and nonparametric statistics. We show that one can design the mechanism with respect to a specific base measure over the output space, such as a Guassian process. We provide a positive result that establishes a Central Limit Theorem for the exponential mechanism quite broadly. We also provide an apparent negative result, showing that the magnitude of the noise introduced for privacy is asymptotically non-negligible relative to the statistical estimation error. We develop an \ep-DP mechanism for functional principal component analysis, applicable in separable Hilbert spaces. We demonstrate its performance via simulations and applications to two datasets.