Oscar Hernan Madrid Padilla

Oscar Hernan Madrid Padilla

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.

Haiyan Jiang, Shanshan Qin, Oscar Hernan Madrid Padilla

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}.

Oscar Hernan Madrid Padilla, Wesley Tansey, Yanzhen Chen

Quantile regression is the task of estimating a specified percentile response, such as the median, from a collection of known covariates. We study quantile regression with rectified linear unit (ReLU) neural networks as the chosen model class. We derive an upper bound on the expected mean squared error of a ReLU network used to estimate any quantile conditional on a set of covariates. This upper bound only depends on the best possible approximation error, the number of layers in the network, and the number of nodes per layer. We further show upper bounds that are tight for two large classes of functions: compositions of Hölder functions and members of a Besov space. These tight bounds imply ReLU networks with quantile regression achieve minimax rates for broad collections of function types. Unlike existing work, the theoretical results hold under minimal assumptions and apply to general error distributions, including heavy-tailed distributions. Empirical simulations on a suite of synthetic response functions demonstrate the theoretical results translate to practical implementations of ReLU networks. Overall, the theoretical and empirical results provide insight into the strong performance of ReLU neural networks for quantile regression across a broad range of function classes and error distributions. All code for this paper is publicly available at https://github.com/tansey/quantile-regression.

Wesley Tansey, Oscar Hernan Madrid Padilla, Arun Sai Suggala et al.

We present Vector-Space Markov Random Fields (VS-MRFs), a novel class of undirected graphical models where each variable can belong to an arbitrary vector space. VS-MRFs generalize a recent line of work on scalar-valued, uni-parameter exponential family and mixed graphical models, thereby greatly broadening the class of exponential families available (e.g., allowing multinomial and Dirichlet distributions). Specifically, VS-MRFs are the joint graphical model distributions where the node-conditional distributions belong to generic exponential families with general vector space domains. We also present a sparsistent $M$-estimator for learning our class of MRFs that recovers the correct set of edges with high probability. We validate our approach via a set of synthetic data experiments as well as a real-world case study of over four million foods from the popular diet tracking app MyFitnessPal. Our results demonstrate that our algorithm performs well empirically and that VS-MRFs are capable of capturing and highlighting interesting structure in complex, real-world data. All code for our algorithm is open source and publicly available.

Zhi Zhang, Chris Chow, Yasi Zhang et al.

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.

Marcos Matabuena, Juan C. Vidal, Oscar Hernan Madrid Padilla et al.

We introduce a novel \textit{k}-nearest neighbor (\textit{k}-NN) regression method for joint estimation of the conditional mean and variance. The proposed algorithm preserves the computational efficiency and manifold-learning capabilities of classical non-parametric \textit{k}-NN models, while integrating a data-driven variable selection step that improves empirical performance. By accurately estimating both conditional mean and variance regression functions, the method effectively reconstructs the conditional distribution and density functions for multiple families of scale-and-localization generative models. We show that our estimator can achieve fast convergence rates, and we derive practical rules for selecting the smoothing parameter~$k$ that enhance the precision of the algorithm in finite sample regimes. Extensive simulations for low, moderate and large-dimensional covariate spaces, together with a real-world biomedical application, demonstrate that the proposed method can consistently outperform the conventional \textit{k-NN} regression algorithm while being more interpretable in the model output.

Carlos Misael Madrid Padilla, Oscar Hernan Madrid Padilla, Yik Lun Kei et al.

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.

Shivam Kumar, Haotian Xu, Carlos Misael Madrid Padilla et al.

We study denoising of a third-order tensor when the ground-truth tensor is not necessarily Tucker low-rank. Specifically, we observe $$ Y=X^\ast+Z\in \mathbb{R}^{p_{1} \times p_{2} \times p_{3}}, $$ where $X^\ast$ is the ground-truth tensor, and $Z$ is the noise tensor. We propose a simple variant of the higher-order tensor SVD estimator $\widetilde{X}$. We show that uniformly over all user-specified Tucker ranks $(r_{1},r_{2},r_{3})$, $$ \| \widetilde{X} - X^* \|_{ \mathrm{F}}^2 = O \Big( κ^2 \Big\{ r_{1}r_{2}r_{3}+\sum_{k=1}^{3} p_{k} r_{k} \Big\} \; + \; ξ_{(r_{1},r_{2},r_{3})}^2\Big) \quad \text{ with high probability.} $$ Here, the bias term $ξ_{(r_1,r_2,r_3)}$ corresponds to the best achievable approximation error of $X^\ast$ over the class of tensors with Tucker ranks $(r_1,r_2,r_3)$; $κ^2$ quantifies the noise level; and the variance term $κ^2 \{r_{1}r_{2}r_{3}+\sum_{k=1}^{3} p_{k} r_{k}\}$ scales with the effective number of free parameters in the estimator $\widetilde{X}$. Our analysis achieves a clean rank-adaptive bias--variance tradeoff: as we increase the ranks of estimator $\widetilde{X}$, the bias $ξ(r_{1},r_{2},r_{3})$ decreases and the variance increases. As a byproduct we also obtain a convenient bias-variance decomposition for the vanilla low-rank SVD matrix estimators.

Carlos Misael Madrid Padilla, Oscar Hernan Madrid Padilla, Sabyasachi Chatterjee

This work examines risk bounds for nonparametric distributional regression estimators. For convex-constrained distributional regression, general upper bounds are established for the continuous ranked probability score (CRPS) and the worst-case mean squared error (MSE) across the domain. These theoretical results are applied to isotonic and trend filtering distributional regression, yielding convergence rates consistent with those for mean estimation. Furthermore, a general upper bound is derived for distributional regression under non-convex constraints, with a specific application to neural network-based estimators. Comprehensive experiments on both simulated and real data validate the theoretical contributions, demonstrating their practical effectiveness.

Carlos Misael Madrid Padilla, Zhi Zhang, Xiaokai Luo et al.

In this paper, we focus on fully connected deep neural networks utilizing the Rectified Linear Unit (ReLU) activation function for nonparametric estimation. We derive non-asymptotic bounds that lead to convergence rates, addressing both temporal and spatial dependence in the observed measurements. By accounting for dependencies across time and space, our models better reflect the complexities of real-world data, enhancing both predictive performance and theoretical robustness. We also tackle the curse of dimensionality by modeling the data on a manifold, exploring the intrinsic dimensionality of high-dimensional data. We broaden existing theoretical findings of temporal-spatial analysis by applying them to neural networks in more general contexts and demonstrate that our proof techniques are effective for models with short-range dependence. Our empirical simulations across various synthetic response functions underscore the superior performance of our method, outperforming established approaches in the existing literature. These findings provide valuable insights into the strong capabilities of dense neural networks (Dense NN) for temporal-spatial modeling across a broad range of function classes.

Marcos Matabuena, Rahul Ghosal, Pavlo Mozharovskyi et al.

Depth measures are powerful tools for defining level sets in emerging, non--standard, and complex random objects such as high-dimensional multivariate data, functional data, and random graphs. Despite their favorable theoretical properties, the integration of depth measures into regression modeling to provide prediction regions remains a largely underexplored area of research. To address this gap, we propose a novel, model-free uncertainty quantification algorithm based on conditional depth measures--specifically, conditional kernel mean embeddings and an integrated depth measure. These new algorithms can be used to define prediction and tolerance regions when predictors and responses are defined in separable Hilbert spaces. The use of kernel mean embeddings ensures faster convergence rates in prediction region estimation. To enhance the practical utility of the algorithms with finite samples, we also introduce a conformal prediction variant that provides marginal, non-asymptotic guarantees for the derived prediction regions. Additionally, we establish both conditional and unconditional consistency results, as well as fast convergence rates in certain homoscedastic settings. We evaluate the finite--sample performance of our model in extensive simulation studies involving various types of functional data and traditional Euclidean scenarios. Finally, we demonstrate the practical relevance of our approach through a digital health application related to physical activity, aiming to provide personalized recommendations

Gabriel Ruiz, Oscar Hernan Madrid Padilla, Qing Zhou

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.

Fan Wang, Oscar Hernan Madrid Padilla, Yi Yu et al.

We study the theoretical properties of the fused lasso procedure originally proposed by \cite{tibshirani2005sparsity} in the context of a linear regression model in which the regression coefficient are totally ordered and assumed to be sparse and piecewise constant. Despite its popularity, to the best of our knowledge, estimation error bounds in high-dimensional settings have only been obtained for the simple case in which the design matrix is the identity matrix. We formulate a novel restricted isometry condition on the design matrix that is tailored to the fused lasso estimator and derive estimation bounds for both the constrained version of the fused lasso assuming dense coefficients and for its penalised version. We observe that the estimation error can be dominated by either the lasso or the fused lasso rate, depending on whether the number of non-zero coefficient is larger than the number of piece-wise constant segments. Finally, we devise a post-processing procedure to recover the piecewise-constant pattern of the coefficients. Extensive numerical experiments support our theoretical findings.

Oscar Hernan Madrid Padilla, Yi Yu, Alessandro Rinaldo

We study piece-wise constant signals corrupted by additive Gaussian noise over a $d$-dimensional lattice. Data of this form naturally arise in a host of applications, and the tasks of signal detection or testing, de-noising and estimation have been studied extensively in the statistical and signal processing literature. In this paper we consider instead the problem of partition recovery, i.e.~of estimating the partition of the lattice induced by the constancy regions of the unknown signal, using the computationally-efficient dyadic classification and regression tree (DCART) methodology proposed by \citep{donoho1997cart}. We prove that, under appropriate regularity conditions on the shape of the partition elements, a DCART-based procedure consistently estimates the underlying partition at a rate of order $σ^2 k^* \log (N)/κ^2$, where $k^*$ is the minimal number of rectangular sub-graphs obtained using recursive dyadic partitions supporting the signal partition, $σ^2$ is the noise variance, $κ$ is the minimal magnitude of the signal difference among contiguous elements of the partition and $N$ is the size of the lattice. Furthermore, under stronger assumptions, our method attains a sharper estimation error of order $σ^2\log(N)/κ^2$, independent of $k^*$, which we show to be minimax rate optimal. Our theoretical guarantees further extend to the partition estimator based on the optimal regression tree estimator (ORT) of \cite{chatterjee2019adaptive} and to the one obtained through an NP-hard exhaustive search method. We corroborate our theoretical findings and the effectiveness of DCART for partition recovery in simulations.

Yi Yu, Oscar Hernan Madrid Padilla, Daren Wang et al.

We study the problem of online network change point detection. In this setting, a collection of independent Bernoulli networks is collected sequentially, and the underlying distributions change when a change point occurs. The goal is to detect the change point as quickly as possible, if it exists, subject to a constraint on the number or probability of false alarms. In this paper, on the detection delay, we establish a minimax lower bound and two upper bounds based on NP-hard algorithms and polynomial-time algorithms, i.e., \[ \mbox{detection delay} \begin{cases} \gtrsim \log(1/α) \frac{\max\{r^2/n, \, 1\}}{κ_0^2 n ρ},\\ \lesssim \log(Δ/α) \frac{\max\{r^2/n, \, \log(r)\}}{κ_0^2 n ρ}, & \mbox{with NP-hard algorithms},\\ \lesssim \log(Δ/α) \frac{r}{κ_0^2 n ρ}, & \mbox{with polynomial-time algorithms}, \end{cases} \] where $κ_0, n, ρ, r$ and $α$ are the normalised jump size, network size, entrywise sparsity, rank sparsity and the overall Type-I error upper bound. All the model parameters are allowed to vary as $Δ$, the location of the change point, diverges. The polynomial-time algorithms are novel procedures that we propose in this paper, designed for quick detection under two different forms of Type-I error control. The first is based on controlling the overall probability of a false alarm when there are no change points, and the second is based on specifying a lower bound on the expected time of the first false alarm. Extensive experiments show that, under different scenarios and the aforementioned forms of Type-I error control, our proposed approaches outperform state-of-the-art methods.

Alfonso Landeros, Oscar Hernan Madrid Padilla, Hua Zhou et al.

The current paper studies the problem of minimizing a loss $f(\boldsymbol{x})$ subject to constraints of the form $\boldsymbol{D}\boldsymbol{x} \in S$, where $S$ is a closed set, convex or not, and $\boldsymbol{D}$ is a matrix that fuses parameters. Fusion constraints can capture smoothness, sparsity, or more general constraint patterns. To tackle this generic class of problems, we combine the Beltrami-Courant penalty method with the proximal distance principle. The latter is driven by minimization of penalized objectives $f(\boldsymbol{x})+\fracρ{2}\text{dist}(\boldsymbol{D}\boldsymbol{x},S)^2$ involving large tuning constants $ρ$ and the squared Euclidean distance of $\boldsymbol{D}\boldsymbol{x}$ from $S$. The next iterate $\boldsymbol{x}_{n+1}$ of the corresponding proximal distance algorithm is constructed from the current iterate $\boldsymbol{x}_n$ by minimizing the majorizing surrogate function $f(\boldsymbol{x})+\fracρ{2}\|\boldsymbol{D}\boldsymbol{x}-\mathcal{P}_{S}(\boldsymbol{D}\boldsymbol{x}_n)\|^2$. For fixed $ρ$ and a subanalytic loss $f(\boldsymbol{x})$ and a subanalytic constraint set $S$, we prove convergence to a stationary point. Under stronger assumptions, we provide convergence rates and demonstrate linear local convergence. We also construct a steepest descent (SD) variant to avoid costly linear system solves. To benchmark our algorithms, we compare against the alternating direction method of multipliers (ADMM). Our extensive numerical tests include problems on metric projection, convex regression, convex clustering, total variation image denoising, and projection of a matrix to a good condition number. These experiments demonstrate the superior speed and acceptable accuracy of our steepest variant on high-dimensional problems.

Hangjian Li, Oscar Hernan Madrid Padilla, Qing Zhou

Structural learning of directed acyclic graphs (DAGs) or Bayesian networks has been studied extensively under the assumption that data are independent. We propose a new Gaussian DAG model for dependent data which assumes the observations are correlated according to an undirected network. Under this model, we develop a method to estimate the DAG structure given a topological ordering of the nodes. The proposed method jointly estimates the Bayesian network and the correlations among observations by optimizing a scoring function based on penalized likelihood. We show that under some mild conditions, the proposed method produces consistent estimators after one iteration. Extensive numerical experiments also demonstrate that by jointly estimating the DAG structure and the sample correlation, our method achieves much higher accuracy in structure learning. When the node ordering is unknown, through experiments on synthetic and real data, we show that our algorithm can be used to estimate the correlations between samples, with which we can de-correlate the dependent data to significantly improve the performance of classical DAG learning methods.

Oscar Hernan Madrid Padilla, James G. Scott

We present an approach for penalized tensor decomposition (PTD) that estimates smoothly varying latent factors in multi-way data. This generalizes existing work on sparse tensor decomposition and penalized matrix decompositions, in a manner parallel to the generalized lasso for regression and smoothing problems. Our approach presents many nontrivial challenges at the intersection of modeling and computation, which are studied in detail. An efficient coordinate-wise optimization algorithm for (PTD) is presented, and its convergence properties are characterized. The method is applied both to simulated data and real data on flu hospitalizations in Texas. These results show that our penalized tensor decomposition can offer major improvements on existing methods for analyzing multi-way data that exhibit smooth spatial or temporal features.

Mingyuan Zhou, Oscar Hernan Madrid Padilla, James G. Scott

We define a family of probability distributions for random count matrices with a potentially unbounded number of rows and columns. The three distributions we consider are derived from the gamma-Poisson, gamma-negative binomial, and beta-negative binomial processes. Because the models lead to closed-form Gibbs sampling update equations, they are natural candidates for nonparametric Bayesian priors over count matrices. A key aspect of our analysis is the recognition that, although the random count matrices within the family are defined by a row-wise construction, their columns can be shown to be i.i.d. This fact is used to derive explicit formulas for drawing all the columns at once. Moreover, by analyzing these matrices' combinatorial structure, we describe how to sequentially construct a column-i.i.d. random count matrix one row at a time, and derive the predictive distribution of a new row count vector with previously unseen features. We describe the similarities and differences between the three priors, and argue that the greater flexibility of the gamma- and beta- negative binomial processes, especially their ability to model over-dispersed, heavy-tailed count data, makes these well suited to a wide variety of real-world applications. As an example of our framework, we construct a naive-Bayes text classifier to categorize a count vector to one of several existing random count matrices of different categories. The classifier supports an unbounded number of features, and unlike most existing methods, it does not require a predefined finite vocabulary to be shared by all the categories, and needs neither feature selection nor parameter tuning. Both the gamma- and beta- negative binomial processes are shown to significantly outperform the gamma-Poisson process for document categorization, with comparable performance to other state-of-the-art supervised text classification algorithms.