arXiv subjects

Oscar Hernan Madrid Padilla

Publications and source records attributed to Oscar Hernan Madrid Padilla.

At least 37 records · Page 2Linked to original sources

arXiv2025

Statistical Guarantees for Lifelong Reinforcement Learning using PAC-Bayes Theory

Lifelong reinforcement learning (RL) has been developed as a paradigm for extending single-task RL to more realistic, dynamic settings. In lifelong RL, the "life" of an RL agent is modeled as a stream of tasks drawn from a task distribution. We propose EPIC (Empirical PAC-Bayes that Improves Continuously), a novel algorithm designed for lifelong RL using PAC-Bayes theory. EPIC learns a shared policy distribution, referred to as the world policy, which enables rapid adaptation to new tasks while retaining valuable knowledge from previous experiences. Our theoretical analysis establishes a relationship between the algorithm's generalization performance and the number of prior tasks preserved in memory. We also derive the sample complexity of EPIC in terms of RL regret. Extensive experiments on a variety of environments demonstrate that EPIC significantly outperforms existing methods in lifelong RL, offering both theoretical guarantees and practical efficacy through the use of the world policy.

cs.LG↗

arXiv2025

Change Point Detection in Dynamic Graphs with Decoder-only Latent Space Model

This manuscript studies the unsupervised change point detection problem in time series of graphs using a decoder-only latent space model. The proposed framework consists of learnable prior distributions for low-dimensional graph representations and of a decoder that bridges the observed graphs and latent representations. The prior distributions of the latent spaces are learned from the observed data as empirical Bayes to assist change point detection. Specifically, the model parameters are estimated via maximum approximate likelihood, with a Group Fused Lasso regularization imposed on the prior parameters. The augmented Lagrangian is solved via Alternating Direction Method of Multipliers, and Langevin Dynamics are recruited for posterior inference. Simulation studies show good performance of the latent space model in supporting change point detection and real data experiments yield change points that align with significant events.

stat.ME↗

arXiv2025

Bayesian variance change point detection with credible sets

This paper introduces a novel Bayesian approach to detect changes in the variance of a Gaussian sequence model, focusing on quantifying the uncertainty in the change point locations and providing a scalable algorithm for inference. Such a measure of uncertainty is necessary when change point methods are deployed in sensitive applications, for example, when one is interested in determining whether an organ is viable for transplant. The key of our proposal is framing the problem as a product of multiple single changes in the scale parameter. We fit the model through an iterative procedure similar to what is done for additive models. The novelty is that each iteration returns a probability distribution on time instances, which captures the uncertainty in the change point location. Leveraging a recent result in the literature, we can show that our proposal is a variational approximation of the exact model posterior distribution. We study the algorithm's convergence and the change point localization rate. Extensive experiments in simulation studies illustrate the performance of our method and the possibility of generalizing it to more complex data-generating mechanisms. We apply the new model to an experiment involving a novel technique to assess the viability of a liver and oceanographic data.

stat.ME↗

arXiv2025

Confidence Interval Construction and Conditional Variance Estimation with Dense ReLU Networks

This paper addresses the problems of conditional variance estimation and confidence interval construction in nonparametric regression using dense networks with the Rectified Linear Unit (ReLU) activation function. We present a residual-based framework for conditional variance estimation, deriving nonasymptotic bounds for variance estimation under both heteroscedastic and homoscedastic settings. We relax the sub-Gaussian noise assumption, allowing the proposed bounds to accommodate sub-Exponential noise and beyond. Building on this, for a ReLU neural network estimator, we derive non-asymptotic bounds for both its conditional mean and variance estimation, representing the first result for variance estimation using ReLU networks. Furthermore, we develop a ReLU network based robust bootstrap procedure (Efron, 1992) for constructing confidence intervals for the true mean that comes with a theoretical guarantee on the coverage, providing a significant advancement in uncertainty quantification and the construction of reliable confidence intervals in deep learning settings.

stat.ML↗

arXiv2024

Multilayer random dot product graphs: Estimation and online change point detection

We study the multilayer random dot product graph (MRDPG) model, an extension of the random dot product graph to multilayer networks. To estimate the edge probabilities, we deploy a tensor-based methodology and demonstrate its superiority over existing approaches. Moving to dynamic MRDPGs, we formulate and analyse an online change point detection framework. At every time point, we observe a realization from an MRDPG. Across layers, we assume fixed shared common node sets and latent positions but allow for different connectivity matrices. We propose efficient tensor algorithms under both fixed and random latent position cases to minimize the detection delay while controlling false alarms. Notably, in the random latent position case, we devise a novel nonparametric change point detection algorithm based on density kernel estimation that is applicable to a wide range of scenarios, including stochastic block models as special cases. Our theoretical findings are supported by extensive numerical experiments, with the code available online https://github.com/MountLee/MRDPG.

stat.ME↗

arXiv2024

Network two-sample test for block models

We consider the two-sample testing problem for networks, where the goal is to determine whether two sets of networks originated from the same stochastic model. Assuming no vertex correspondence and allowing for different numbers of nodes, we address a fundamental network testing problem that goes beyond simple adjacency matrix comparisons. We adopt the stochastic block model (SBM) for network distributions, due to their interpretability and the potential to approximate more general models. The lack of meaningful node labels and vertex correspondence translate to a graph matching challenge when developing a test for SBMs. We introduce an efficient algorithm to match estimated network parameters, allowing us to properly combine and contrast information within and across samples, leading to a powerful test. We show that the matching algorithm, and the overall test are consistent, under mild conditions on the sparsity of the networks and the sample sizes, and derive a chi-squared asymptotic null distribution for the test. Through a mixture of theoretical insights and empirical validations, including experiments with both synthetic and real-world data, this study advances robust statistical inference for complex network data.

math.ST↗

arXiv2024

Conformal uncertainty quantification using kernel depth measures in separable Hilbert spaces

Depth measures have gained popularity in the statistical literature for defining level sets in complex data structures like multivariate data, functional data, and graphs. Despite their versatility, integrating depth measures into regression modeling for establishing prediction regions remains underexplored. To address this gap, we propose a novel method utilizing a model-free uncertainty quantification algorithm based on conditional depth measures and conditional kernel mean embeddings. This enables the creation of tailored prediction and tolerance regions in regression models handling complex statistical responses and predictors in separable Hilbert spaces. Our focus in this paper is exclusively on examples where the response is a functional data object. To enhance practicality, we introduce a conformal prediction algorithm, providing non-asymptotic guarantees in the derived prediction region. Additionally, we establish both conditional and unconditional consistency results and fast convergence rates in some special homoscedastic cases. We evaluate the model finite sample performance in extensive simulation studies with different function objects as probability distributions and functional data. Finally, we apply the approach in a digital health application related to physical activity, aiming to offer personalized recommendations in the US. population based on individuals' characteristics.

stat.ME↗

arXiv2024

High confidence inference on the probability an individual benefits from treatment using experimental or observational data with known propensity scores

We seek to understand the probability an individual benefits from treatment (PIBT), an inestimable quantity that must be bounded in practice. Given the innate uncertainty in the population-level bounds on PIBT, we seek to better understand the margin of error for their estimation in order to discern whether the estimated bounds on PIBT are tight or wide due to random chance or not. Toward this goal, we present guarantees to the estimation of bounds on marginal PIBT, with any threshold of interest, for a randomized experiment setting or an observational setting where propensity scores are known. We also derive results that permit us to understand heterogeneity in PIBT across learnable sub-groups delineated by pre-treatment features. These results can be used to help with formal statistical power analyses and frequentist confidence statements for settings where we are interested in assumption-lean inference on PIBT through the target bounds with minimal computational complexity compared to a bootstrap approach. Through a real data example from a large randomized experiment, we also demonstrate how our results for PIBT can allow us to understand the practical implication and goodness of fit of an estimate for the conditional average treatment effect (CATE), a function of an individual's baseline covariates.

stat.ME↗

arXiv2024

Variance estimation in graphs with the fused lasso

We study the problem of variance estimation in general graph-structured problems. First, we develop a linear time estimator for the homoscedastic case that can consistently estimate the variance in general graphs. We show that our estimator attains minimax rates for the chain and 2D grid graphs when the mean signal has total variation with canonical scaling. Furthermore, we provide general upper bounds on the mean squared error performance of the fused lasso estimator in general graphs under a moment condition and a bound on the tail behavior of the errors. These upper bounds allow us to generalize for broader classes of distributions, such as sub-exponential, many existing results on the fused lasso that are only known to hold with the assumption that errors are sub-Gaussian random variables. Exploiting our upper bounds, we then study a simple total variation regularization estimator for estimating the signal of variances in the heteroscedastic case. We also provide lower bounds showing that our heteroscedastic variance estimator attains minimax rates for estimating signals of bounded variation in grid graphs, and $K$-nearest neighbor graphs, and the estimator is consistent for estimating the variances in any connected graph.

math.ST↗

arXiv2023

2D score based estimation of heterogeneous treatment effects

Statisticians show growing interest in estimating and analyzing heterogeneity in causal effects in observational studies. However, there usually exists a trade-off between accuracy and interpretability for developing a desirable estimator for treatment effects, especially in the case when there are a large number of features in estimation. To make efforts to address the issue, we propose a score-based framework for estimating the Conditional Average Treatment Effect (CATE) function in this paper. The framework integrates two components: (i) leverage the joint use of propensity and prognostic scores in a matching algorithm to obtain a proxy of the heterogeneous treatment effects for each observation, (ii) utilize non-parametric regression trees to construct an estimator for the CATE function conditioning on the two scores. The method naturally stratifies treatment effects into subgroups over a 2d grid whose axis are the propensity and prognostic scores. We conduct benchmark experiments on multiple simulated data and demonstrate clear advantages of the proposed estimator over state of the art methods. We also evaluate empirical performance in real-life settings, using two observational data from a clinical trial and a complex social survey, and interpret policy implications following the numerical results.

stat.ME↗

arXiv2023

A Partially Separable Model for Dynamic Valued Networks

The Exponential-family Random Graph Model (ERGM) is a powerful model to fit networks with complex structures. However, for dynamic valued networks whose observations are matrices of counts that evolve over time, the development of the ERGM framework is still in its infancy. To facilitate the modeling of dyad value increment and decrement, a Partially Separable Temporal ERGM is proposed for dynamic valued networks. The parameter learning algorithms inherit state-of-the-art estimation techniques to approximate the maximum likelihood, by drawing Markov chain Monte Carlo (MCMC) samples conditioning on the valued network from the previous time step. The ability of the proposed model to interpret network dynamics and forecast temporal trends is demonstrated with real data.

stat.ME↗

arXiv2023

Change point detection and inference in multivariable nonparametric models under mixing conditions

This paper studies multivariate nonparametric change point localization and inference problems. The data consists of a multivariate time series with potentially short range dependence. The distribution of this data is assumed to be piecewise constant with densities in a Hölder class. The change points, or times at which the distribution changes, are unknown. We derive the limiting distributions of the change point estimators when the minimal jump size vanishes or remains constant, a first in the literature on change point settings. We are introducing two new features: a consistent estimator that can detect when a change is happening in data with short-term dependence, and a consistent block-type long-run variance estimator. Numerical evidence is provided to back up our theoretical results.

math.ST↗

arXiv2022

Change point localization in dependent dynamic nonparametric random dot product graphs

In this paper, we study the offline change point localization problem in a sequence of dependent nonparametric random dot product graphs. To be specific, assume that at every time point, a network is generated from a nonparametric random dot product graph model \citep[see e.g.][]{athreya2017statistical}, where the latent positions are generated from unknown underlying distributions. The underlying distributions are piecewise constant in time and change at unknown locations, called change points. Most importantly, we allow for dependence among networks generated between two consecutive change points. This setting incorporates edge-dependence within networks and temporal dependence between networks, which is the most flexible setting in the published literature. To accomplish the task of consistently localizing change points, we propose a novel change point detection algorithm, consisting of two steps. First, we estimate the latent positions of the random dot product model, our theoretical result being a refined version of the state-of-the-art results, allowing the dimension of the latent positions to diverge. Subsequently, we construct a nonparametric version of the CUSUM statistic \citep[e.g.][]{Page1954, padilla2019optimal} that allows for temporal dependence. Consistent localization is proved theoretically and supported by extensive numerical experiments, which illustrate state-of-the-art performance. We also provide in depth discussion of possible extensions to give more understanding and insights.

stat.ME↗

arXiv2022

Feature Grouping and Sparse Principal Component Analysis with Truncated Regularization

In this paper, we consider a new variant for principal component analysis (PCA), aiming to capture the grouping and/or sparse structures of factor loadings simultaneously. To achieve these goals, we employ a non-convex truncated regularization with naturally adjustable sparsity and grouping effects, and propose the Feature Grouping and Sparse Principal Component Analysis (FGSPCA). The proposed FGSPCA method encourages the factor loadings with similar values to collapse into disjoint homogeneous groups for feature grouping or into a special zero-valued group for feature selection, which in turn helps reducing model complexity and increasing model interpretation. Usually, existing structured PCA methods require prior knowledge to construct the regularization term. However, the proposed FGSPCA can simultaneously capture the grouping and/or sparse structures of factor loadings without any prior information. To solve the resulting non-convex optimization problem, we propose an alternating algorithm that incorporates the difference-of-convex programming, augmented Lagrange method and coordinate descent method. Experimental results demonstrate the promising performance and efficiency of the new method on both synthetic and real-world datasets. An R implementation of FGSPCA can be found on github {https://github.com/higeeks/FGSPCA}.

stat.ME↗

arXiv2022

High Dimensional Latent Panel Quantile Regression with an Application to Asset Pricing

We propose a generalization of the linear panel quantile regression model to accommodate both \textit{sparse} and \textit{dense} parts: sparse means while the number of covariates available is large, potentially only a much smaller number of them have a nonzero impact on each conditional quantile of the response variable; while the dense part is represent by a low-rank matrix that can be approximated by latent factors and their loadings. Such a structure poses problems for traditional sparse estimators, such as the $\ell_1$-penalised Quantile Regression, and for traditional latent factor estimator, such as PCA. We propose a new estimation procedure, based on the ADMM algorithm, consists of combining the quantile loss function with $\ell_1$ \textit{and} nuclear norm regularization. We show, under general conditions, that our estimator can consistently estimate both the nonzero coefficients of the covariates and the latent low-rank matrix. Our proposed model has a "Characteristics + Latent Factors" Asset Pricing Model interpretation: we apply our model and estimator with a large-dimensional panel of financial data and find that (i) characteristics have sparser predictive power once latent factors were controlled (ii) the factors and coefficients at upper and lower quantiles are different from the median.

econ.EM↗

arXiv2022

Kernel Biclustering algorithm in Hilbert Spaces

Biclustering algorithms partition data and covariates simultaneously, providing new insights in several domains, such as analyzing gene expression to discover new biological functions. This paper develops a new model-free biclustering algorithm in abstract spaces using the notions of energy distance (ED) and the maximum mean discrepancy (MMD) -- two distances between probability distributions capable of handling complex data such as curves or graphs. The proposed method can learn more general and complex cluster shapes than most existing literature approaches, which usually focus on detecting mean and variance differences. Although the biclustering configurations of our approach are constrained to create disjoint structures at the datum and covariate levels, the results are competitive. Our results are similar to state-of-the-art methods in their optimal scenarios, assuming a proper kernel choice, outperforming them when cluster differences are concentrated in higher-order moments. The model's performance has been tested in several situations that involve simulated and real-world datasets. Finally, new theoretical consistency results are established using some tools of the theory of optimal transport.

stat.ME↗

arXiv2022

Dynamic and heterogeneous treatment effects with abrupt changes

From personalised medicine to targeted advertising, it is an inherent task to provide a sequence of decisions with historical covariates and outcome data. This requires understanding of both the dynamics and heterogeneity of treatment effects. In this paper, we are concerned with detecting abrupt changes in the treatment effects in terms of the conditional average treatment effect (CATE) in a sequential fashion. To be more specific, at each time point, we consider a nonparametric model to allow for maximal flexibility and robustness. Along the time, we allow for temporal dependence on historical covariates and noise functions. We provide a kernel-based change point estimator, which is shown to be consistent in terms of its detection delay, under an average run length control. Numerical results are provided to support our theoretical findings.

stat.ME↗

arXiv2022

Sequentially learning the topological ordering of causal directed acyclic graphs with likelihood ratio scores

Causal discovery, the learning of causality in a data mining scenario, has been of strong scientific and theoretical interest as a starting point to identify "what causes what?" Contingent on assumptions and a proper learning algorithm, it is sometimes possible to identify and accurately estimate a causal directed acyclic graph (DAG), as opposed to a Markov equivalence class of graphs that gives ambiguity of causal directions. The focus of this paper is in highlighting the identifiability and estimation of DAGs with general error distributions through a general sequential sorting procedure that orders variables one at a time, starting at root nodes, followed by children of the root nodes, and so on until completion. We demonstrate a novel application of this general approach to estimate the topological ordering of a DAG. At each step of the procedure, only simple likelihood ratio scores are calculated on regression residuals to decide the next node to append to the current partial ordering. The computational complexity of our algorithm on a p-node problem is O(pd), where d is the maximum neighborhood size. Under mild assumptions, the population version of our procedure provably identifies a true ordering of the underlying DAG. We provide extensive numerical evidence to demonstrate that this sequential procedure scales to possibly thousands of nodes and works well for high-dimensional data. We accompany these numerical experiments with an application to a single-cell gene expression dataset.

stat.ME↗